University of Southern California Dissertations and Theses

Essays on commodity futures and volatility

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Essays on commodity futures and volatility

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ESSAYS ON COMMODITY FUTURES AND VOLATILITY

by

Georgi D. Vassilev

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
August 2010

Copyright 2010 Georgi D. Vassilev

ii

Dedication

To My Parents
iii

Acknowledgments

I would like to express my deepest gratitude to my advisor, Dr. Nake Kamrany,
for his endless support and encouragement without which this dissertation would not
have been possible. I would like to thank as well my dissertation committee members,
Dr. Zapatero and Dr. Dekle, for their helpful comments and advice. Last but not least, I
would like to thank my parents and my girlfriend, Desi, for their support, understanding
and belief in me.
iv

Table of Contents

Dedication ii

Acknowledgments iii

List of Tables vi

List of Figures vii

Abstract viii

Chapter 1: Price Volatility of Commodity Futures 1
1.1. Introduction 1
1.2. Methodology Review 7
1.2.1. ARCH Models for Volatility 7
1.2.2. Seasonality and Time to Maturity Effects 14
1.2.3. The Mean Equation 16
1.2.4. Volatility Forecasting 17
1.3. Data 18
1.4. Empirical Results 23
1.4.1. Unit Root Tests 23
1.4.2. Normality Test 24
1.4.3. Heteroskedasticity of Futures Returns 26
1.4.4. Long-memory of Futures Volatility 34
1.4.5. Estimated Models 40
1.4.6. Model Comparisons 58
1.5. Conclusion 79

Chapter 2: On the Relationship between Demand for Futures and Volatility 81
2.1. Introduction 81
2.2. Data 87
2.3. Empirical Analysis 97
2.3.1. Volatility Estimates 97
2.3.2. Base Model 98
2.3.3. Unit Root Tests 99
2.3.4. Ordinary Least Squares Estimation 102
2.3.5. Quantile Regression 105

v

2.3.6. Granger Causality 131
2.3.7. Causality in Quantiles 136
2.4. Conclusion 140

Bibliography 142
vi

List of Tables

Table 1.1: Commodity Futures Description 20

Table 1.2: Summary Statistics for Commodity Futures 22

Table 1.3: Unit Root Test of Futures Returns 24

Table 1.4: Heteroskedasticity Tests of Futures Returns 35

Table 1.5: Tests for Long-memory in Commodity Futures Volatility Process 41

Table 1.6: ARCH Model Estimates 51

Table 1.7: ARCH Model Comparisons 61

Table 1.8: In-Sample 1-day Ahead Value-at-Risk 66

Table 1.9: Conditional ARCH Model Comparisons 77

Table 1.10: ARCH Model Comparisons Summary 78

Table 2.1: Open Interest Descriptive Statistics 91

Table 2.2: Traders’ Positions as Percentage of Open Interest 92

Table 2.3: Unit Root Tests on Open Interest Series 100

Table 2.4: Contemporaneous Effect of Expected Volatility 103

Table 2.5: Tests of Equality of Volatility Effects in the Extreme Quantiles 132

Table 2.6: Tests of Symmetry of Volatility Effects 133

Table 2.7: Granger Causality in the Conditional Mean Test Results 135

Table 2.8: Causality in Quantiles Test Results 139
vii

List of Figures

Figure 1.1: Commodity Futures Price Characteristics 27

Figure 1.2: Autocorrelation in Returns, Squared Returns, and Absolute Returns 42

Figure 2.1: Open Interest By Traders’ Positions 93

Figure 2.2: Net Open Interest by Traders’ Positions 95

Figure 2.3: Quantile Regression Estimates for Volatility 110

viii

Abstract

This dissertation and the essays herein represent an effort to extend our
understanding of the time-series and distributional properties of commodity futures on
one hand, and to provide new evidence on the effects of volatility on demand for futures
on the other. The behavior of commodity futures volatility is studied via parametric and
non-parametric tests. It is found that commodity futures volatility exhibits long-memory
properties. Autoregressive conditionally heteroskedastic models appear to capture
satisfactorily the volatility dynamics of commodity futures in general. From four
competing autoregressive models through in-sample tests it is found that the FIAPARCH
model with Student’s t-errors better accounts for nonlinearities in commodity prices even
after accounting for seasonality in volatility. However, out-of-sample tests reveal that
simpler models such as GARCH perform equally well at least for short forecast horizons.
While the effect of trade demand for futures by different groups of market
participants on futures volatility has been frequently explored, the impact of volatility on
demand for futures in general, and by hedgers in particular, has not be thoroughly
studied. It is established that such an effect indeed exists through Granger and quantile
causality tests. It is also found that regardless of the position held by hedgers, increased
volatility is associated with decrease in demand for futures, a rather interesting and
unaccounted for phenomenon.
1

Chapter 1: Price Volatility of Commodity Futures
1.1. Introduction
Through the years a preeminent research topic in empirical economics and finance
has been the study of the stochastic properties of speculative prices. This interest is
understandable given the significant implications that the probabilistic structure of prices
has among other things on financial and macroeconomic models, asset pricing and market
risk management.
One of the very first contributions to the literature on price distribution is the
random walk model of Louis Bachelier (1900) for security and commodity markets. In its
simplest form, Bachelier's model assumes that successive future price differences are
normally distributed zero mean random variables, with variance proportional to the square
of the differencing period and independent of the price level. This random-walk view of
speculative prices has been further espoused in the 1930's by Working (1934), modified
and formalized assuming that price returns instead are independent and identically
distributed (normal) random variables with zero mean and constant variance.
Even at that early stage, however, empirical evidence has been pointing towards the
inadequateness of the normal distribution to capture the peakedness of the price change
distribution (Olivier, 1926; Mills, 1927). In order to account for the extraordinarily long
tails of the returns distribution Mandelbrot (1963) proposed the use of stable Paretian
distributions. Evaluating the price returns of wheat, Kendall (1953) noticed that instead of
constant, the variance of price returns was changing over time. Thus, the random walk
2

hypothesis while still the prevalent one had to be modified with successive price changes
(returns) still being independent but now not identically distributed random variables.
The independence of price changes (returns) has been typically tested by analyzing
the autocorrelation of the respective series. A proof that the series are uncorrelated was
treated as an evidence of independence as usually the tests were performed under the
assumption of normality. Not surprisingly, as most of the research has documented no
correlations, or small positive such, the independence paradigm of the random walk model
has been regarded as reasonable (Kendall, 1953; Working, 1934; Roberts, 1959). The
attractive feature of the random walk model with its independent but possibly not
identically distributed (price changes) returns was not only empirical - the notion of
unpredictability of the series was viewed as evidence in support of efficient markets. Fama
(1965) defined efficient markets as a market where, given the available information, actual
prices at every point in time represent very good estimates of intrinsic values, a definition
later refined to a market in which prices always fully reflect available information (Fama,
1970). In his empirical analysis Fama (1965) found that the random walk model well
describes stock prices.
In the mid 1960s, Samuelson (1965) developed a more formal economic based
model for futures in which he proved that properly anticipated futures prices fluctuate
randomly. In his analysis, however, the independence condition was not required anymore,
instead a martingale process was sufficient. The idea of price changes following a
martingale was further explored by Mandelbrot (1966, 1969) who notes the far less
restrictive nature of the concept than independence and the possibility that higher
3

conditional moments of prices may be dependent on past and present prices. Thus, the door
for exploring non-linear dependence in speculative prices driven by higher order moment
dependence was swung wide open.
Mandelbrot's (1963) has noted that large changes tend to be followed by large
changes -- of either sign -- and small changes tend to be followed by small changes
indicating towards the idea that although price changes could be uncorrelated they could be
still dependent. Since Mandelbrot's observation, numerous papers have established a now
well-known, stylized fact about financial time series -- return volatilities tend to be a highly
correlated and persistent process, suggesting that volatility can be decomposed into
predictable and unpredictable components. The volatility persistence and clustering, could
as well account to certain degree for another stylized fact -- the leptokurtic distribution of
price returns. It was not until the early 1980s that a martingale process for the returns with
higher order dynamic dependence was proposed. Engle's (1982) path-breaking paper
introduced an extremely influential family of non-linear autoregressive conditionally
heteroskedastic (ARCH) models that attempts to capture the observed dependence in the
second moment of returns.
Some thirty years later, the use of ARCH models to analyze volatility
1

1
Herein the term volatility is used in loose sense, referring to both the variance and the square root of the
variance. If confusion is possible then we use standard deviation or variance instead of volatility.
in financial
data has become ubiquitous, with the initial model being extended well beyond its original
specification to better reflect stylized features of high frequency data. The popularity of
ARCH models is not to a small degree due to the heightened nowadays emphasis placed on
4

efficient modeling and forecasting of financial market volatility as volatility, a quantity
itself unobservable, plays a prominent role in modern risk management, portfolio
allocation, option pricing, and economic theory tests.
While most of the research in the area of price volatility modeling has centered on
stock prices and exchange rates, the application of the ARCH framework to commodity
futures has been lagging behind. The purpose of this chapter is to study the time series
properties of commodity futures and evaluate the applicability to commodity futures and
the performance in-sample and out-of-sample of some recent developments in ARCH
modeling of volatility such as asymmetric power and fractionally integrated ARCH
models.
In a survey of volatility forecasting in financial markets by Poon and Granger
(2003), from 93 studies cited applying the GARCH framework only 6 have considered
commodity futures and even then typically only GARCH(1,1) or EGARCH (1,1) models
have been compared to other competing models. Some of the earliest research modeling
commodity price volatility as ARCH type processes are Taylor (1986), Aradhyula and Holt
(1989), Holt (1993), Jayne and Myers (1994), Yang and Brorsen (1992, 1993), Bailie and
Myers (1991). These studies invariably have been based on the GARCH (1, 1) framework
(with the notable exception of Taylor, 1986, who models the standard deviation instead of
the volatility as an autoregressive process). In the early and mid 1990s, a multitude of
enhanced ARCH models have been introduced trying to capture more stylized facts of
financial assets volatility such as long memory. The suitability of these second wave

5

ARCH models for commodity futures has not been systematically studied, nor readily
embraced in the various commodity futures applications.
It is only recently that the long memory characteristics of commodity futures higher
moments have been studied while such properties for stocks have been analyzed as early as
in Ding, Granger and Engle (1993), de Lima and Crato (1993), Bollerslev and Mikkelsen
(1996), Baillie, Bollerslev and Mikkelsen (1996) and Breidt, Crato, and de Lima (1998)
2

2
Kohzadi and Boyd (1995) and Barkoulas et al. (1997) analyzed long memory property of the levels of
commodity prices and not higher moments.
.
By long memory of a time series it is meant that current observations are dependent on
previous ones even if far distant, or more precisely, the autocorrelation function of the
series decays slowly at a hyperbolic rate. Crato and Ray (2000) were among the first to
explore the persistence in volatility for commodities. Utilizing three methods (the modified
R/S test of Lo, 1991; the spectral regression based estimator of Geweke and Porter-Hudak,
1983; and the nonparametric spectral test of Lobato and Robinson, 1998) they find that for
17 commodity, 5 currency, and an index futures while no long-term memory in returns is
present, significant persistence in volatility is established with commodity futures
volatility's persistence exceeding that for exchange rates. Jin and Frechette (2004)
analyzed the applicability of Baillie et al.'s (1996) fractionally integrated model to
commodity futures and found the model to provide a better fit as compared to the standard
GARCH (1,1) model for 13 of 14 commodity futures studied. Elder and Jin (2007)
provides further evidence of unambiguous long memory in volatility for 14 commodity
futures employing wavelet decomposition methodology. For 6 commodity futures Baillie
6

et al. (2007) also find that a fractionally integrated GARCH model provides a good fit to
the data. Utilizing Jin and Frechette's (2004) dataset, Sephton (2009) extends the analysis
by considering the He and Terasvitra's (1999) asymmetric fractionally integrated GARCH
model.
In this paper, the statistical properties of 21 commodity futures prices are analyzed
for the existence of conditional heteroskedasticity and long-term dependence in price
volatility. Unlike previous research on fractionally integrated GARCH models, we
explicitly model seasonality in volatility. Previously, Anderson (1985) found that one of
the main drivers of volatility of futures prices especially for grains is seasonality.
Furthermore, all of the above studies consider data only up to year 2000 while our dataset
encompasses a more exhaustive list of commodities and over the more volatile recent
periods. Also, while prior research on fractionally integrated ARCH models (with the
exception of Sephton, 2009) assumes that the conditional distribution of futures returns is
normal, i.e. the time varying volatility is able to account for the majority of the observed
kurtosis, here we explore the relevance of introducing a heavier-tail asymmetric
distribution. Lastly, we compare the relative performance and suitability of 4 models that
encompass to a great extent the majority of existing ARCH type of models. We perform
these comparisons explicitly both via in-sample and out-of-sample tests.

7

1.2. Methodology Review
1.2.1. ARCH Models for Volatility
Since Engle's (1982) seminal paper and Bollerslev's (1986) generalized version of
the Engle's initial model, a multitude of ARCH type of models have been introduced, each
trying to account for certain empirical regularities in financial and commodity prices.
Bollerslev et al. (1994) and Pagan (1995) note that return series across different asset
classes exhibit several common characteristics. Specifically:
• Stationarity of returns.
• Martingale or near martingale properties of the return series.
• Volatility clustering.
• Non-normality of the returns distribution (thick-tailed and skewed distributions)
• Asymmetric effect of positive and negative price changes on volatility (leverage effect)
Some of the most influential models through the years have been the GARCH
model of Bollerslev (1986) that represents a parsimonious representation of Engle's
original ARCH model, the asymmetric power ARCH model (APARCH) of Ding, Granger
and Engle (1993) that generalizes additional five popular models (the Taylor,1986,/
Schwert,1989, TS-GARCH model; the GJR-GARCH model of Glosten, Jaganathan, and
Runkle, 1993; the TARCH model of Zakoian, 1991; the NARCH model of Higgins and
Bera, 1990; and the log-ARCH model of Geweke, 1986, and Pentula, 1986), and the more
recent long-memory fractionally integrated GARCH model of Baillie et al. (1996) and the
fractionally integrated APARCH model of Tse (1998).
8

The literature on the application and estimation of ARCH models of volatility is
vast. There exist a number of surveys such as Bollerslev, Chou and Kroner (1992), Bera
and Higgins (1993), Bollerslev, Engle and Nelson (1994), Engle (2001), Li, Ling and
McAleer (2002), Poon and Granger (2003), Andersen et al. (2006) that provide a detailed
review of the GARCH literature.
Before Engle's path breaking paper, the majority of statistical research in financial
and commodity markets had centered on the estimation of the return series, that is, the
analysis of the mean process of the return series:
[ ]
t t t t
e F r E r + =
−1
|
where is an uncorrelated, zero mean process with unconditional variance of
2
σ , i.e.
[ ] 0 =
t
e E , [ ] 0 =
− τ t t
e e E for τ ≠ t , and [ ]
2 2
σ =
t
e E . Although the unconditional volatility
of the error is constant, the conditional variance [ ]
h t h t t
e E
+ +
= σ need not to be -- it could
instead be changing over time. To model the observed heteroskedasticity of returns, Engle
(1982) proposed that the error term in equation 1.1 above evolve according to the following
process:
t t t
u e σ =
where
t
u is an iid random variable with mean equal to zero and unit variance. It follows
then that ( )
2
1 t t t
e Var σ =
−
, i.e. the conditional variance could change over time. The
different ARCH models essentially differ in the fashion in which the conditional volatility
is assumed to evolve conditional on ,
1 − t
F the informational set at time 1 − t .
(1.1)
9

Specifically, for Bollerslev's GARCH (p,q) model we have:
2
1
2
1
2
) 1 , 0 (
j t j
p
j
i t i
q
i
t
t
t t t
e
N u
u e
−
=
−
=
∑ ∑
+ + =
∼
=
α σ β ω σ
σ

Restricting the number of lagged conditional variances to zero ( 0 = q ) we obtain the
original ARCH(p) model by Engle (1982). In the above specification the distribution of the
white noise random variable
t
u was assumed to be normal; this need not to be case
however. Bollerslev (1987) observed that only volatility heteroskedasticity may not be
sufficient to explain the high kurtosis and fat-tail distribution of asset returns and
consequently proposed the use of Student's t-distribution which has heavier tails than the
normal distribution. Nelson (1991) introduced a similar rationale for using the generalized
error distribution in his EGARCH model. Further, Fernandez and Steel (1998) and
Lambert and Laurent (2001) have suggested the use of a skewed Student's t-distribution in
order to better capture both the excess skewness and kurtosis of empirical data. Sometimes
it is useful to have a short-cut notation for the above model where the lag operator L
) (
k t t
k
y y L
−
= is used instead. Thus, equation (1.2) could be rewritten in a more generic
form as:
( ) ( )
2
1
2
1
2
) ; 1 , 0 (
− −
+ + =
∼
=
t t t
t
t t t
e L L
D u
u e
α σ β ω σ
θ
σ

where ) ; 1 , 0 ( θ D is a general distribution with mean zero, unit variance and θ as an
(1.2)
10

additional distribution specification parameter vector; ( ) L α and ( ) L β are simply
polynomials in L of p and q degrees respectively.
Zakoian (1991), Taylor (1986) and Schwert (1989) have shown that in modeling
the conditional volatility it may be more appropriate to specify the process in terms of the
conditional standard deviation instead of the conditional variance as far more persistence is
observed in absolute returns. Further, Ding et al. (1993) showed that there is no need of
restricting the power term in the volatility modeling equation to a particular value but
instead the appropriate power exponent could be endogenously determined by the
underlying data process. This power functional form could be viewed as a type of Box-Cox
transformation performed on the volatility process.
While the GARCH model assumes that negative and positive price changes have
symmetrical effect on volatility, for some time now it has been recognized that negative
shocks tend to increase volatility in asset prices disproportionately (relative to positive
shocks) as evidenced by Black (1976) and Christie (1982), i.e., the so-called leveraged
effect is observed whereby negative price changes are associated with heightened
volatility. To this end, Zakoian (1994), Nelson (1991), Engle and Ng (1993), Glosten et al.
(1993) have extended the ARCH models to incorporate the asymmetrical effect of negative
shocks.

11

As previously noted, Ding et al. (1993) proposed an extremely flexible asymmetric
power GARCH model (APARCH) that nests several widely used ARCH models. The
APARCH model is defined as follows:
( )
δ
δ δ
γ α σ β ω σ
θ
σ
j t i j t j
p
j
i t i
q
i
t
t
t t t
e e
D u
u e
− −
=
−
=
− + + =
∼
=
∑ ∑
1 1
) ; 1 , 0 (
where 0 > δ is the power term parameter and 1 <
i
γ is the coefficient determining the
asymmetric impact of price changes on conditional volatility. By restricting the power
transformation and asymmetry terms the following models are nested within the APARCH
model:
• Engle's ARCH model for 2 = δ , 0 =
i
γ , 0 =
j
β
• Bollerslev's GARCH model for 2 = δ , 0 =
i
γ
• Glosten et al. GJR-GARCH model for 2 = δ
• Taylor-Schwert GARCH model for 1 = δ , 0 =
i
γ
• Zakoian's TARCH model for 1 = δ
• Higgins-Bera NARCH model for 0 =
i
γ , 0 =
j
β
• Geweke-Pantula log-ARCH for 0 → δ
For further details on the nested models within APARCH, proofs and properties see Ding
et al. (1993), Hentschel (1995), and He and Teräsvirta (1999).

12

In motivating the use of the power transformation term in the APARCH model,
Ding et al (1993) noted that the correlation in the squared and absolute returns was
significant at lags even beyond 2000 days highlighting the persistent nature of the volatility
process. Furthermore, for popular ARCH specifications it has been noted that the sum of
the coefficient estimates for the volatility process is close to unity, implying that shocks to
volatility tend to die out very slowly. However, the correlation function of the squared
residuals in the stationary GARCH framework implies a much faster, exponential decay of
the volatility shocks then it is empirically observed. Building on the ideas of the
fractionally integrated ARMA modeling (ARFIMA), Baillie et al. (1996) proposed a
simple way of merging the long-term persistence in volatility with the short-term memory
of the standard GARCH framework by introducing an additional difference parameter , d
( ) 1 0 ≤ ≤ d , that would control the order of integration of the volatility process. Allowing
for fractional integration helps in modeling the long-term persistence of the volatility
shocks - now shocks to volatility decay hyperbolically. Specifically, the conditional
variance in FIGARCH(p,d,q) is modeled as follows:
( ) [ ] ( )
2 1 2
1
t t
e L L λ β ω σ + − =
−

where ( ) ( ) [ ] ( ) ( )
d
L L L L − − − =
−
1 1 1
1
φ β λ and ( )
d
L − 1 is given by
( )
( ) ( )
j
d j
d j
j
L
− Γ + Γ
− Γ ∞
=
∑
1
0
,
with ( ) . Γ being the gamma function, or alternatively ( ) ( )
j j
j
d
j
d
L L 1 1
0
−








∑ = −
∞
=
. The
magnitude of the fractional difference parameter indicates not only persistence of the
volatility process but also the stationarity of the process (the process is stationary and
13

short-memeory, i.e. I(0), if d=0; if 5 . 0 < d then the process is still stationary but exhibits
long-memory; for 5 . 0 ≥ d the process is non-stationary). In estimating the FIGARCH
and FIAPARCH models, however, a slightly different set of volatility processes is
estimated as recommended by Chung (1999), namely:
( ) ( )
2 2 2 2
σ λ σ σ − + =
t t
e L
where σ is the unconditional volatility of the return series.
Baillie et al.'s (1996) fractionally integrated GARCH model (FIGARCH) was
extended by Bollerslev et al. (1996) to the fractionally integrated exponential GARCH
model (FIEGARCH). Tse (1998) further extended Baillie et al. (1996) by incorporating the
ideas of Ding et al.'s (1993) APARCH model and thus creating the fractionally integrated
APARCH model given by following process for the volatility:
( ) [ ] ( ) ( )
δ
δ
γ λ β ω σ
t t t
e e L L − + − =
−1
1
where ( ) L λ is defined as before.
In the description of the volatility ARCH models investigated in this study we have
allowed for the latent error term to come from a generic distribution. However, most
of the applications of the GARCH, APGARCH and especially FIGARCH and FIAPARCH
models have assumed that } {
t
u are distributed as standard normal random variables.
Under such an assumption, the log-likelihood function is given by:
( ) ( ) [ ]
2 2
1
log 2 log
2
1
t t
T
t
N
u + + − =
∑
=
σ π L
Although the conditional heteroskedasticity of the volatilities does create a leptokurtic
14

return distribution, it is generally observed that higher observed kurtosis is still not
adequately accounted for (see for example Pagan,1996; Bollerslev et al. ,1992, and
references therein). To better account for the empirically observed skewness and higher
kurtosis of futures returns, in this study we utilize the skewed Student's t-distribution, for
which the log-likelihood function is given by:
( ) ( ) ( )
( ) ( )
( )
















−
+
+ + + −








+








+
+ − −














Γ −













 +
Γ =






− ≥
−
=
∑
a
m
t
u
I
t
t
T
t
skwT
m su
s T
2
2
2
1
1
2
1 log 1 log
2
1
log
2
log 2 log
2
1
2
log
2
1
log
ξ
ν
ν σ
ξ
ν π
ν ν
ξ
L

where the degrees of freedom parameter, ν , is greater than 2, ξ is the skewness
parameter ( 1 = ξ is equivalent to no skewness), and m and
2
s are defined as follows:
( ) ( )
( )
2
2
2 2
2
5 . 0
2
1
1
1
1 2
m s
m
−








− + =








−
Γ
− Γ
=
+
ξ
ξ
ξ
ξ
π
ν
ν
ν

The ARCH models in this study are estimated by maximum likelihood and utilizing the
Garch module within the Ox console application (Doornik,2007, and Laurent and Peters,
2002).

1.2.2. Seasonality and Time to Maturity Effects
While the the APARCH, FIGARCH and FIAPARCH models have been widely
applied to financial assets, these models have been only recently applied to the study of
15

volatility of commodity futures. Before directly applying such models developed for
financial assets to commodity futures, certain characteristics of commodity futures have to
be taken into consideration. Specifically, prior research (Anderson, 1985; Kenyon et
al.,1987; Yang and Brorsen, 1993) has documented that many commodity futures,
particularly agricultural futures, exhibit seasonality in their volatility structures (as it can
be observed in figure1.1 panels 2 and 3). In this paper, we explicitly allow for seasonality
effects in the volatility process by introducing a deterministic seasonal component of the
following form:












+ 





=
∑
=
12
2
cos
12
2
sin
, 2 , 1
1
t
i
t
i
k
i
t
im
b
im
b s
π π

where
t
m is the month of the year corresponding to observation, and 2 = k . This
representation of seasonality in ARCH models is similar to Yang and Brorsen's (1993) and
Goodwyn and Schnepf's (2000) and in essence is a second order Fourier approximation for
the unobserved seasonality function.
Another distinct characteristic of futures volatility, as opposed to stock price
volatility for example, is the so called Samuelson's effect. Samuelson (1965) hypothesized
that the volatility of commodity futures is expected to be lower the more distant the futures
expiration is arguing that with the approach of the futures expiration more information is
relevant for the price determination of the futures and as information arrival is random,
more volatility is expected the shorter the maturity of the contract. The empirical evidence
on the Samuelson hypothesis is somewhat mixed (see Akin, 2003). Although Yang and
Brorsen (1993) confirm the existence of the maturity effect, they estimate that it has less
16

than 1% effect on volatility. Since in this study we consider only prompt month contracts,
the maturity effect on estimated volatilities, if any, would be negligible as the time to
maturity never exceeds 3 months and in most cases it is less than a month. Furthermore,
Anderson (1985) and Kenyon et al. (1987) reason that seasonality in futures could be a
better explanation for the maturity effect.

1.2.3. The Mean Equation
While important in its own right to correctly specify the mean process for the
returns (eq. 1.1), the present study is concerned with improving our understanding of the
process governing the evolution of commodity futures volatility. Hence, we specify the
mean equation for the commodity futures returns as a naive random process, i.e.
t t
e r + = µ
Due to the short time to maturity of the futures contracts considered herein, the
linear pattern of returns due to the overall shape of the futures curve would have minimal
impact. Further, Nelson (1992) shows that the effect of any misspecification of the mean
process tends to disappear in estimating the conditional variances. Engle (1982) proves that
asymptotically the volatility equation parameters are consistent when the mean equation is
uncorrelated with volatility, i.e., the covariance matrix is block-diagonal. Thus, we assume
that the commodity futures returns follow a simple random noise process around their
unconditional mean.

17

Nevertheless, when significant autocorrelation in the returns is noted, then the
returns are modeled as a low order ARMA(r,s) process, that is,
t i t i
s
i
i t i
r
i
t
e e r r + + + =
−
=
−
=
∑ ∑
α β µ
1 1

where
i
α and
i
β are the moving average and autoregressive coefficients respectively.

1.2.4. Volatility Forecasting
Forecasting volatility using ARCH models follows a similar line of reasoning as
forecasting with standard ARMA structures. For example, in the standard GARCH (1,1)
model the h-step ahead conditional variance estimate
2
|t h t +
σ is recursively given by:
2
1
2
|
1
2
|
~
~
~
j h t j
p
j
t i h t i
q
i
t h t
e
− +
=
− +
=
+ ∑ ∑
+ + = α σ β ω σ
where
2
|
2
| t h t t h t
e
+ +
= σ for 0 > h . Notice that the variance estimates in the GARCH(1,1)
model are not dependent on the distributional assumptions of the latent error term
However, this is not the case for out-of-sample forecasting with asymmetric ARCH
models. To see this, let's look at the APARCH(1,1) model considered in this study.
[ ]
( )
δ
δ
γ α σ β ω
σ σ
1 1 1 1 | 1 1
2 2
|
− + − + − +
+ +
− + + =
=
h t h t t t h t
h t t t h t
e e E
E

A closed form solution for the expression of ( )
δ
γ z z E
1
− when z follows a normal
distribution is derived by Ding et al. (1993). Lambert and Laurent (2001) derived a similar
closed form solution when z follows a skewed Student's t-distribution. Let
18

( )
δ
γ z z E k
1
− = and ( ) ξ ν , ; 1 , 0 t z ∼ where ν are the degrees of freedom of the
Student's t-distribution and is the skewness parameter of the distribution. . Then,
Lambert and Laurent (2001) show that:
( )
( )
( )
( ) [ ]
( ) ( ) ( )
( ) ( ) ( )
2
1
2 2
1
1 1
2
2
1 1
2
1
ν
ξ
δ ν δ
δ δ δ δ
ν π ξ
ν
γ ζ γ ζ
δ
Γ − +
− Γ Γ
− + + =
+
− +
+ + −
k
where as before ( ) . Γ is the standard gamma function, i.e., ( ) dt e t a
t a − − ∞
∫
= Γ
1
0
. Thus, the
conditional volatility forecast is given by:
( )
δ δ
δ
δ
σ α σ β ω
γ α σ β ω σ
t h t t h t
h t h t t t h t t h t
k
e e E
| 1 1 | 1 1
1 1 1 1 | 1 1
2
|
− + − +
− + − + − + +
+ + =
− + + =

1.3. Data
In this section the commodity futures considered are identified, the relevant data
described, and data summary statistics analyzed.
Commodity futures are one of the key instruments used to trade and manage
commodity risk. In many instances the volume of the futures markets dwarfs that of the
spot (cash) market for the underlying commodity. Hence, understanding the evolution of
the daily volatility of commodity futures is of utmost importance to both commodity
producers and consumers in order that they properly manage their risk exposures. Futures
are highly standardized derivative contracts traded on exchanges either through an open
outcry system and/or through electronic trading allowing for virtually around-the-clock
trading. Trading outside the regular (pit) trading hours is characterized by significantly
19

lower volumes and liquidity, and consequently higher bid-ask spreads. At the end of each
regular trading session, the exchange committee determines the settlement price for each
contract for which there is open interest even if no volume for that day. Typically the
closing and settlement prices are the same but not always as closing prices could be
influenced by reduced liquidity at closing.
In this study, we analyze 21 commodity futures contracts: 6 grain and oil seeds
commodities (corn, soybeans, soybean oil, soybean meal, oats, and wheat), 3 energy
commodities (crude oil, natural gas, and heating oil), 3 metal commodities (gold, silver,
and copper), 5 soft commodities (cocoa, coffee, sugar, cotton, and orange juice), and 4
livestock commodities (live cattle, feeder cattle, frozen pork bellies, and lean hogs). The
contract details are summarized in Table 1.1 below. Daily data on the settlement futures
prices for the considered commodity futures were obtained from ProphetX. Since
commodity futures typically exhibit highest volumes and open interest in the closest to
expiry (nearby) futures contract, we construct the return series for each commodity from
the nearby futures contracts only as they tend to be the most active, and thus representative
of the then current market sentiment. The data spans almost 16 years of daily data from
March 3, 1993 through May 1, 2009. Returns are calculated as the continuously
compounded percentage rate of difference of log daily close-to-close prices, i.e.
( )
1
log log 100
−
− =
t t t
p p r
where
t
p is the futures price for the nearby month contract at time t . Unlike when
calculating return series for other financial time series, calculating future returns is not as
20

straightforward due to the expiration of the futures contracts and the selected rolling
procedures, i.e., how and when the switch from an expiring contract to the next nearby
contract is performed. In creating the return series for the commodity futures in this study,
the returns for each commodity futures are calculated from the individual contracts only,
that is, the returns are strictly attributable to the relevant futures contract only. This is

Table 1.1: Commodity Futures Description

Deliverable Commodity Futures Exchange Ticker Contract size

1 SOYBEAN OIL CHICAGO BOARD OF TRADE BO 60,000 POUNDS
2 CORN CHICAGO BOARD OF TRADE C 5,000 BUSHELS
3 COCOA
COFFEE, SUGAR & COCOA
EXCHANGE/NYBOT
CC 10 METRIC TONS
4 CRUDE OIL, LIGHT 'SWEET'
NEW YORK MERCANTILE
EXCHANGE
CL 1,000 BARRELS
5 COTTON NO. 2
NEW YORK COTTON
EXCHANGE/NYBOT
CT 50,000 POUNDS
6 FEEDER CATTLE CHICAGO MERCANTILE EXCHANGE FC 50,000 POUNDS
7 GOLD COMMODITY EXCHANGE INC. GC
100 TROY
OUNCES
8 COPPER-GRADE #1 COMMODITY EXCHANGE INC. HG 25,000 POUNDS
9 NO. 2 HEATING OIL
NEW YORK MERCANTILE
EXCHANGE
HO
42,000 U.S.
GALLONS
10
FRZN CONCENTRATED
ORANGE JUICE
CITRUS ASSOC. OF N Y COTTON
EXCH/NYBOT
JO 15,000 POUNDS
11 COFFEE C
COFFEE, SUGAR & COCOA
EXCHANGE/NYBOT
KC 37,500 POUNDS
12 LIVE CATTLE CHICAGO MERCANTILE EXCHANGE LC 40,000 POUNDS
13 LEAN HOGS CHICAGO MERCANTILE EXCHANGE LH 40,000 POUNDS
14 NATURAL GAS
NEW YORK MERCANTILE
EXCHANGE
NG 10,000 MMBTU'S
15 OATS CHICAGO BOARD OF TRADE O 5,000 BUSHELS
16 FROZEN PORK BELLIES CHICAGO MERCANTILE EXCHANGE PB 40,000 POUNDS
17 SOYBEANS CHICAGO BOARD OF TRADE S 5,000 BUSHELS
18 SUGAR NO. 11
COFFEE, SUGAR & COCOA
EXCHANGE/NYBOT
SB 112,000 POUNDS
19 SILVER COMMODITY EXCHANGE INC. SI
5,000 TROY
OUNCES
20 SOYBEAN MEAL CHICAGO BOARD OF TRADE SM 100 TONS
21 WHEAT CHICAGO BOARD OF TRADE W 5,000 BUSHELS
21

necessary as simple rolling from one futures contract to the next introduces additional price
jumps at the time of the switch from one futures to another that are not truly related to
market volatility but rather the shape of the futures curves whether it is in contango or
backwardation
3
Another adjustment in the calculation of the return series is that the transition from
the expiring futures contract to the next nearby one is performed not at the expiry date of
the then prompt month futures but three days prior to that. This is necessitated by the fact
most futures contracts do not end up in the physical delivery of the underlying commodity
but rather these contracts are offset before their expiration. Since large positions cannot be
unwind within a single day without significantly affecting market prices, most commodity
futures traders would start rolling their positions sooner than the expiry date of the prompt
month futures.
. It has to be kept in mind though, that kurtosis is greatly affected by even a
single outlier even in large samples as the ones considered herein. Thus, not accounting for
this effect may significantly exaggerate the kurtosis and and to a lesser extent the volatility
of the commodity futures returns.
Summary statistics for the daily futures returns are reported in Table 1.2. It is
observed that 15 of the 21 commodity futures considered exhibit a positive albeit small
average return for the sample period considered in this study. Further, the median return for
the majority of the commodities is close to zero although for crude oil, copper, soybeans,

3
A futures market is said to be in contango when the overall shape of the futures curve is upward sloping, i.e.,
more distant futures contracts have higher prices on average ceteris paribus. Backwardation refers to a futures
market for which more distant future prices are lower on average. Certain commodity markets are typically in
contango while others in backwardation. Whether a market is in contango or backwardation is determined
among other things by the ability to store the underlying commodity and the associated storage costs, interest
rates, convenience yields and overall market expectations for the particular commodity.
22

and silver it is positive, and negative for natural gas. Looking at the sample volatility, it is
worth noting that the standard deviation of commodity futures runs the entire gamut from a
low of 12.75% on an annualized basis
4

for feeder cattle to 52.69% for natural gas; no
specific trends depending on the type of commodity class is observed apart from the
relatively higher overall volatility associated with energy based commodities.
Table 1.2: Summary Statistics for Commodity Futures

Mean Median Max Min Std. Dev. Skewness Kurtosis Jarque-Bera

BO 0.006% 0.000% 8.039% -7.239% 1.499% 0.093 5.224 842.7**
C -0.021% 0.000% 7.397% -8.096% 1.563% -0.037 5.381 960.2**
CC 0.008% 0.000% 9.962% -10.006% 1.947% -0.058 5.582 1124.1**
CL 0.032% 0.084% 13.340% -16.545% 2.264% -0.233 6.297 1856.5**
CT -0.040% 0.000% 11.349% -8.515% 1.697% 0.066 5.740 1272.1**
FC 0.017% 0.000% 6.591% -6.010% 0.803% -0.016 7.335 3170.5**
GC 0.021% 0.000% 8.887% -7.581% 1.053% 0.175 10.835 10248.5**
HG 0.044% 0.044% 11.742% -11.693% 1.805% -0.212 7.597 3577.4**
HO 0.032% 0.000% 10.403% -14.899% 2.217% -0.100 4.833 570.2**
JO -0.015% 0.000% 22.720% -12.912% 2.023% 0.533 12.358 15116.2**
KC 0.000% 0.000% 23.773% -15.031% 2.552% 0.438 10.320 9135.8**
LC 0.010% 0.000% 5.880% -6.357% 0.943% -0.128 5.029 706**
LH -0.034% 0.000% 11.650% -11.283% 1.552% -0.181 5.722 1273.5**
NG -0.065% -0.046% 18.764% -18.418% 3.319% -0.057 5.297 886.4**
O 0.026% 0.000% 11.102% -11.937% 1.967% -0.078 5.727 1261.5**
PB 0.026% 0.000% 8.281% -8.241% 2.170% 0.060 3.213 10.1**
S 0.031% 0.049% 7.411% -7.612% 1.503% -0.210 5.690 1253.8**
SB 0.015% 0.000% 8.622% -11.327% 1.981% -0.231 5.019 719.3**
SI 0.031% 0.076% 15.949% -14.755% 1.852% -0.478 11.377 11916**
SM 0.045% 0.000% 9.264% -8.236% 1.618% -0.084 5.885 1412.6**
W -0.022% 0.000% 8.794% -9.973% 1.782% 0.012 5.352 935.7**

4
Annualized assuming 252 trading days in a calendar year
23

1.4. Empirical Results
1.4.1. Unit Root Tests
For commodity futures it is usually found that prices are mean-reverting, i.e., prices
tend to be pulled towards their long-term equilibrium determined by demand and supply.
Thus, it is typically assumed that commodity futures prices are stationary. Recently Postali
and Picchetti (2006) have found that if long time series are considered (more than 100
years), commodity prices appear to be stationary; however, after accounting for structural
breaks non-stationarity appears again. Not to dwell on the stationarity of the data, in this
study the return series are analyzed instead which are shown to be stationary using the
Augmented Dickey-Fuller (ADF) test.
The ADF test, similarly to the Philips-Perron tests controls for serial correlation by
including higher autoregressive terms in the regression, i.e. after including p differenced
lags of the dependent variable the error term should be white noise:
t i t i
p
i
t t
r r r ε β δ + + = ∆
−
=
− ∑
1
1

The null hypothesis is that of a unit root, that is H0: 1 = δ versus the alternative of no unit
root -- H1: 1 < δ . The asymptotic distribution of
δ
t is independent of the number of lags
included and has the distribution of the Dickey-Fuller t-test. The number of lags to be
included was determined based on the modified Bayesian Information Criterion.

24

Table 1.3: Unit Root Test of Futures Returns

Deliverable Commodity Ticker ADF test statistic

no constant, no trend constant

1 SOYBEAN OIL BO -15.129** -15.131**
2 CORN C -11.316** -11.336**
3 COCOA CC -25.373** -25.372**
4 CRUDE OIL, LIGHT 'SWEET' CL -9.21** -9.222**
5 COTTON NO. 2 CT -11.243** -11.318**
6 FEEDER CATTLE FC -8.804** -8.871**
7 GOLD GC -26.268** -26.291**
8 COPPER-GRADE #1 HG -9.238** -9.311**
9 NO. 2 HEATING OIL, N.Y. HARBOR HO -11.712** -11.762**
10 FRZN CONCENTRATED ORANGE JUICE JO -64.259** -64.255**
11 COFFEE C KC -63.595** -63.587**
12 LIVE CATTLE LC -28.855** -28.865**
13 LEAN HOGS LH -12.035** -12.075**
14 NATURAL GAS NG -11.384** -11.464**
15 OATS O -12.887** -12.898**
16 FROZEN PORK BELLIES PB -18.53** -18.542**
17 SOYBEANS S -12.806** -12.868**
18 SUGAR NO. 11 SB -27.341** -27.341**
19 SILVER SI -12.759** -12.794**
20 SOYBEAN MEAL SM -17.026** -17.104**
21 WHEAT W -17.453** -17.47**

Notes: ** indicates that the null hypothesis of unit root is rejected at the 5% confidence level.

As it could be seen from Table 1.3, the commodity futures return series appear to be
stationary as the null hypothesis of unit root is rejected at the 5% significance level for all
commodities considered herein.

1.4.2. Normality Test
Financial and commodity returns are usually viewed as a stochastic process that
comes from a normal distribution. However, the null hypothesis that the return data for the
25

commodity futures considered herein is normally distributed is rejected at the 1%
significance level using the Jarque-Berra test statistic (Table 1.2). The Jarque-Berra
statistic effectively compares a weighted average of the third and fourth empirical
moments around the mean of the futures returns distribution to that of the normal
distribution, i.e.













 −
+ =
2
2
2
3
6
Kurt
Skew
T
JB
where Skew is the skewness and Kurt -- the kurtosis of the data
5
Panel 3 in Figure 1.1. shows further evidence that the futures return data is not
normally distributed. Comparing the empirical quantile distribution of the commodity
returns to that of the normal distribution if the returns distribution is close to normal then
the quantiles should lie on a straight line. It is observed, however, that significant
deviations at the extreme quantiles of the distributions are observed -- the quantile graph is
rather S-shaped than a straight line, with departures from normality not as severe in the top
quantiles. Intepretation of quantile graphs is not straightforward. Based on Chambers
(1983) and Fowlkes (1987) interpretations of commonly encountered departures from
linearity observed in quantile graphs, it could be deduced that long tails at both ends of the
. For a normal
distribution, the skewness and kurtosis coefficients are 0 and 3 respectively. The
Jarque-Berra statistic is distributed as a chi-square random variable with two degrees of
freedom under the null hypothesis of normality.

5
Skewness is defined as and the kurtosis coefficient as
[ ]
3
3
r
r r E
σ

−
and the kurtosis coefficient as
[ ]
4
4
r
r r E
σ

−

26

data distribution contribute to the non-normality of the data, with negative skewness
diminishing the reported departure from normality in the extreme top quantiles
6
prevalent is the futures on frozen pork bellies, for which the Jarque-Berra test statistic is the
lowest among all commodities considered and the quantile function appears to be close to
normal.
. The only
commodity futures for which it appears that the departures from normality are not that

1.4.3. Heteroskedasticity of Futures Returns
Panel 1 in Figure 1.1 reveals that commodity futures prices experienced substantial
variation from 1993 to 2009, with periods of significant price changes with the most
notable period being from 2005 to 2009. In section 1.4.1 it was shown that similar to other
asset classes commodity futures returns appear to be non-normally distributed. Moreover,
observation of panel 2 in figure1.1 indicates that the returns series appear to experience
periods of relative tranquility followed by periods of heightened volatility, a volatility
clustering phenomenon noticed initially by Mandelbrot (1963). Panels 2 and 3 in figure1.2
appear to further confirm this hypothesis - the autocorrelation function for all the
commodity futures is significantly different from zero for the squared returns and even
more so for the absolute return series. This finding is in line with Kariya et al. (1990),
Taylor (1986) and Schwert (1989) and the latter's approach of modeling the conditional
volatility instead of the conditional variance in the ARCH framework. The correlation in

6
From the commodities considered, 14 exhibit negative skewness, i.e., a heavier left-tail distribution as
compared to the normal.
27

Figure 1.1: Commodity Futures Price Characteristics

Mar93Mar97 Apr01 Apr05May09
20
40
60
Price
BO (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
BO Daily Returns
-5 0 5
-0.1
-0.05
0
0.05
0.1
Standard Normal Quantiles
Quantiles of BO
Futures returns distribution of BO
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
200
400
600
Price
C (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
C Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of C
Futures returns distribution of C
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
1000
2000
3000
Price
CC (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
-0.05
0
0.05
CC Daily Returns
-5 0 5
-0.2
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of CC
Futures returns distribution of CC
vs Gaussian distribution
27
28

Figure 1.1, Continued
Mar93Mar97 Apr01 Apr05May09
20
40
60
80
100
120
140
Price
CL (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
CL Daily Returns
-5 0 5
-0.2
-0.1
0
0.1
0.2
Standard Normal Quantiles
Quantiles of CL
Futures returns distribution of CL
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
40
60
80
100
Price
CT (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
0.1
CT Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of CT
Futures returns distribution of CT
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
60
80
100
Price
FC (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
FC Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of FC
Futures returns distribution of FC
vs Gaussian distribution
28
29

Figure 1.1, Continued

Mar93Mar97 Apr01 Apr05May09
400
600
800
1000
Price
GC (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
GC Daily Returns
-5 0 5
-0.1
-0.05
0
0.05
0.1
Standard Normal Quantiles
Quantiles of GC
Futures returns distribution of GC
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
100
200
300
400
Price
HG (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
HG Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of HG
Futures returns distribution of HG
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
1
2
3
4
Price
HO (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
HO Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of HO
Futures returns distribution of HO
vs Gaussian distribution
29
30

Figure 1.1, Continued

Mar93Mar97 Apr01 Apr05May09
100
150
200
Price
JO (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
0.2
JO Daily Returns
-5 0 5
-0.2
0
0.2
0.4
0.6
Standard Normal Quantiles
Quantiles of JO
Futures returns distribution of JO
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
100
200
300
Price
KC (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
0.2
KC Daily Returns
-5 0 5
-0.5
0
0.5
Standard Normal Quantiles
Quantiles of KC
Futures returns distribution of KC
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
60
80
100
Price
LC (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
LC Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of LC
Futures returns distribution of LC
vs Gaussian distribution
30
31

Figure 1.1, Continued

Mar93Mar97 Apr01 Apr05May09
40
60
80
Price
LH (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
-0.05
0
0.05
0.1
LH Daily Returns
-5 0 5
-0.2
-0.1
0
0.1
0.2
Standard Normal Quantiles
Quantiles of LH
Futures returns distribution of LH
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
5
10
15
Price
NG (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
NG Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of NG
Futures returns distribution of NG
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
100
200
300
400
Price
O (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
O Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of O
Futures returns distribution of O
vs Gaussian distribution
31
32

Figure 1.1, Continued

Mar93Mar97 Apr01 Apr05May09
40
60
80
100
120
Price
PB (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
PB Daily Returns
-5 0 5
-0.1
-0.05
0
0.05
0.1
Standard Normal Quantiles
Quantiles of PB
Futures returns distribution of PB
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
500
1000
1500
Price
S (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
S Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of S
Futures returns distribution of S
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
5
10
15
Price
SB (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
-0.05
0
0.05
SB Daily Returns
-5 0 5
-0.2
0
0.2
Standard Normal Quantiles
Quantiles of SB
Futures returns distribution of SB
vs Gaussian distribution
32
33

Figures 1.1, Continued

Mar93Mar97 Apr01 Apr05May09
5
10
15
20
Price
SI (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.1
0
0.1
SI Daily Returns
-5 0 5
-0.2
-0.1
0
0.1
0.2
Standard Normal Quantiles
Quantiles of SI
Futures returns distribution of SI
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
200
300
400
Price
SM (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
SM Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of SM
Futures returns distribution of SM
vs Gaussian distribution
Mar93Mar97 Apr01 Apr05May09
400
600
800
1000
1200
Price
W (Prompt Month) Futures Prices
Mar93 Mar97 Apr01 Apr05 May09
-0.05
0
0.05
W Daily Returns
-5 0 5
-0.1
0
0.1
Standard Normal Quantiles
Quantiles of W
Futures returns distribution of W
vs Gaussian distribution
33
34

the returns itself, however, is far less evident that it is statistically different from zero.
Additionally, the magnitude of the autocorrelations in the raw returns is distinctly lower
than that for the squared and absolute returns. To test statistically for the existence of
conditional heteroskedasticity, the ARCH-LM and the Ljung-Box Q-tests are performed.
The results from the tests reported in Table 1.4 provide further evidence for the existence
of conditional heteroskedasticity in all of the commodity futures returns considered herein
with the test-statistics rejecting at the 5% confidence level the null hypothesis of
homoskedasticity and no autocorrelation respectively. Thus, it is reasonable to model the
changing commodity futures returns volatilities as ARCH processes.

1.4.4. Long-memory of Futures Volatility
From panels 2 and 3 in Figure 1.2 it also appears that some commodities seem to
exhibit strong seasonality patterns in their autocorrelation function for the squared and
absolute returns, arguing in favor of the inclusion of seasonality functional when modeling
the volatility process. Furthermore, similar to Ding et al. (1993), Crato and Ray (2000),
and Jin and Frechette (2004) among others, it is observed that the autocorrelations in the
squared and absolute returns for commodity futures persist for more than a few hundred
observations pointing towards the existence of long-term memory in the volatility process.
To test empirically for the presence of long-term memory in volatility, we employ

35

Table 1.4: Heteroskedasticity Tests of Futures Returns

Ticker Lags
LB Q-Statistic of
the Squared
Returns
LB Q-Statistic of
the Absolute
Returns
ARCH test

BO
1 71.09** 59.571** 71.18**
2 146.41** 119.02** 129.55**
5 541.81** 365.28** 355.07**
10 1035.1** 757.65** 451.79**
20 1609.2** 1238.2** 481.67**
50 2771** 2132.6** 549.5**

C
1 172.9** 173.39** 172.24**
2 352.72** 379.8** 291.74**
5 819.12** 900** 459.63**
10 1384.9** 1576.9** 543.1**
20 2196.9** 2653.8** 583.76**
50 4326.7** 4858.8** 659.89**

CC
1 18.554** 21.589** 18.519**
2 52.14** 49.649** 48.966**
5 98.076** 113.83** 79.659**
10 134.61** 204.51** 96.933**
20 227.47** 412.32** 134.24**
50 376.43** 737.72** 176.77**

CL
1 72.95** 65.44** 73.899**
2 174.87** 176.97** 154.5**
5 600.63** 583.65** 382.7**
10 1027.9** 1085.1** 443.12**
20 1897.8** 2009.1** 546.88**
50 4047.8** 3999.6** 678.74**

CT
1 94.186** 89.182** 93.748**
2 144.66** 130.8** 127.18**
5 261.66** 293.19** 178.77**
10 476.91** 520.58** 249.65**
20 652.41** 842.89** 265.98**
50 1116.6** 1543.4** 328.97**

Notes: ** indicates rejection of the null hypothesis of no autocorrelation at the 5%
significance level

36

Table 1.4, Continued

Ticker Lags
LB Q-Statistic of
the Squared
Returns
LB Q-Statistic of
the Absolute
Returns
ARCH test

FC
1 182.11** 97.026** 185.39**
2 260.71** 190.06** 223.55**
5 382.55** 488.61** 262.99**
10 450.62** 738.64** 270.82**
20 555.46** 1115.8** 289.76**
50 742.44** 1667.3** 325.47**

GC
1 172.93** 204.62** 172.31**
2 210.05** 363.1** 183.99**
5 418.2** 883.63** 279.87**
10 552.88** 1534.9** 304.46**
20 898.68** 2801.8** 369.43**
50 1746.7** 6007.4** 446.6**

HG
1 197.99** 137.98** 198.06**
2 369.46** 283.89** 303.12**
5 962.58** 804.82** 521.86**
10 1607.7** 1509.4** 586.9**
20 2579.3** 2558.9** 625.64**
50 3883.3** 4380.2** 692.15**

HO
1 19.638** 26.813** 20.061**
2 68.888** 99.575** 64.577**
5 276.12** 318.14** 209.4**
10 440.37** 561.03** 248.29**
20 710.59** 975.37** 291.16**
50 1326.4** 1881.8** 357.42**

JO
1 6.429* 41.189** 6.358*
2 9.701** 69.986** 9.287**
5 12.468* 111.61** 11.197*
10 16.402 170.26** 14.091
20 26.367 277.37** 21.856
50 61.049 510.04** 45.693

Notes: ** indicates rejection of the null hypothesis of no autocorrelation at the 5%
significance level

37

Table 1.4, Continued

Ticker Lags
LB Q-Statistic of
the Squared
Returns
LB Q-Statistic of
the Absolute
Returns
ARCH test

KC
1 98.31** 122.34** 98.215**
2 178.99** 190.83** 154.77**
5 292.55** 390.07** 200.16**
10 584.42** 623.59** 358.54**
20 654.35** 798.5** 405.38**
50 893.48** 1183.8** 470.02**

LC
1 333.09** 63.329** 334.35**
2 394.38** 105.31** 341.64**
5 516.15** 287.06** 376.72**
10 632.97** 496.96** 406.3**
20 833.96** 851.33** 429.42**
50 1211** 1448.2** 463.29**

LH
1 475.9** 120.5** 473.97**
2 572.35** 194.17** 480.51**
5 900.63** 455.12** 573.9**
10 1280.7** 841.08** 618.53**
20 2049.3** 1556.1** 695.01**
50 2803.2** 2655.8** 713.02**

NG
1 132.11** 72.884** 131.63**
2 244.38** 168.46** 208.07**
5 463.94** 431.79** 298.81**
10 776.47** 823.96** 380.57**
20 1120.2** 1320.2** 422.9**
50 1335.7** 1709.4** 447.44**

O
1 149.64** 127.56** 149.07**
2 209.92** 217.41** 179.55**
5 330.24** 437.06** 226.99**
10 447.62** 683.83** 250.88**
20 576.17** 968.76** 280.27**
50 724.23** 1358.4** 316.82**

Notes: ** indicates rejection of the null hypothesis of no autocorrelation at the 5%
significance level

38

Table 1.4, Continued

Ticker Lags
LB Q-Statistic of
the Squared
Returns
LB Q-Statistic of
the Absolute
Returns
ARCH test

PB
1 290.95** 174.15** 289.45**
2 463.19** 295.38** 366.24**
5 857.74** 602.24** 463.81**
10 1661.9** 1064.2** 621.14**
20 2685.7** 1799.8** 668.69**
50 4277.3** 3091.1** 721.88**

S
1 66.149** 91.643** 67.086**
2 165.59** 206.96** 147.68**
5 559.56** 581.15** 374.26**
10 1012.4** 1101.2** 458.95**
20 1578.6** 1880.3** 484.7**
50 2752.9** 3334.6** 546.87**

SB
1 70.156** 83.415** 70.27**
2 90.302** 107.18** 81.813**
5 127.9** 215.84** 101.35**
10 206.73** 361.22** 138.52**
20 353.97** 698.94** 183.08**
50 673.47** 1434.9** 242.74**

SI
1 68.446** 144.67** 68.471**
2 106.63** 241.12** 94.748**
5 224.65** 569.87** 158.33**
10 430.03** 1046.1** 236.23**
20 859.31** 2081** 327.71**
50 1621.5** 4163.2** 430.04**

SM
1 81.711** 110.08** 82.154**
2 196.89** 234.74** 173.5**
5 537.33** 556.68** 363.91**
10 801.05** 841.01** 410.25**
20 1122** 1274.6** 434.99**
50 1791.8** 2343.8** 504.3**
50 2848** 2815** 592.46**

Notes: ** indicates rejection of the null hypothesis of no autocorrelation at the 5%
significance level

39

Table 1.4, Continued
Ticker Lags
LB Q-Statistic of
the Squared
Returns
LB Q-Statistic of
the Absolute
Returns
ARCH test

W
1 174.4** 91.36** 175.97**
2 271.4** 166.56** 229.46**
5 659.52** 486.12** 392.6**
10 1094.5** 880.34** 461.12**
20 1741.5** 1505.1** 524.06**

Notes: ** indicates rejection of the null hypothesis of no autocorrelation at the 5%
significance level

Mandelbrot's (1972) rescaled range statistic, R/S, test as well as the modified R/S test of Lo
(1991). Mandelbrot's R/S statistic for series { }
T
t
y
1
is given by:
[ ]
( ) ( )








− − − =
∑ ∑
= =
∈
y y y y
S
Q
t
k
j
t
k
j
T k
1 1
, 1
min max
1

where
2
S is the sample variance. Lo (1991) noted that Mandelbrot's test is sensitive to
low order autoregressive processes and suggested modifying the R/S statistic by replacing
the sample variance with the following estimate:
( )
j
q
j
q
j
S q γ σ ˆ
1
1 2 ˆ
1
2 2








+
− + =
∑
=

where
j
γ ˆ is the sample autocovariance at lag j. Then, Lo (1991) shows that if there is no
long-term dependence in the y series then the
( )
T
q Q
ratio approaches asymptotically the
difference of the maximum and minimum of standard Brownian bridges for appropriate
number of lags From Table 1.5 it is evident that for a wide range of lags considered and
40

commodity futures we can reject the null hypothesis of no-long term dependence using
Lo's (1991) modified R/S test (three exceptions are noted: for feeder cattle, lean cattle and
natural gas at lag 50 we cannot reject the null on no long-term dependence). Similar
indication is provided by Mandelbrot's original R/S test which, and hence we conclude that
commodity futures do exhibit long-memory in volatility.

1.4.5. Estimated Models
In this paper, the volatility of 21 commodity futures is estimated using the
GARCH(1,1), APARCH(1,1), FIGARCH(1,d,1), and FIAPARCH(1,d,1) models. The
GARCH(1,1) model was selected as it the widely accepted benchmark model for
estimating conditional volatility. As noted previously, the attractiveness of the APARCH
model lies in the fact that it nests several other popular ARCH models, while allowing for
asymmetric effect of negative and positive prices changes on power transformations of the
volatility. The FIGARCH and FIAPARCH are clear analogs to the previous two models
trying to account for the long-term memory of volatility shocks.
As noted previously, Bollerslev (1987), Hsieh (1989), Baillie and Bollerslev
(1989), Bollerslev, Chou, and Kroner (1992), Pagan (1996), and Palm and Vlaar (1997)
among others showed that the Student's t-distribution better captures the observed kurtosis
in empirical log-return time series. Furthermore, when applying volatility models for
estimating Value-at-Risk and similar "tail" risk measures (e.g., expected shortfall), the
skewness of the empirical distribution should be considered as it could significantly
41

Table 1.5: Tests for Long-memory in Commodity Futures Volatility Process

Hurst-Mandelbrot
R/S Statistic
Lo R/S Statistic
Ticker\
Lags (q)

1 2 3 4 5 10 50

BO
5.804 5.4816 5.2078 4.9729 4.7537 4.5452 3.8122 2.2688
C
7.1562 6.5149 5.9934 5.6153 5.2814 5.0031 4.1248 2.3466
CC
5.1112 4.934 4.76 4.5881 4.4562 4.3423 3.8893 2.5009
CL
5.4472 5.1299 4.8128 4.5371 4.2919 4.0713 3.3581 1.8816
CT
5.513 5.1449 4.9014 4.6987 4.5045 4.3415 3.7543 2.322
FC
4.2324 3.9386 3.7016 3.5071 3.3351 3.1837 2.7136 1.7252*
GC
8.2434 7.4446 6.8835 6.4483 6.0718 5.7544 4.7371 2.6031
HG
7.0317 6.4595 6 5.586 5.2719 4.9972 4.1034 2.3463
HO
5.9326 5.7044 5.4193 5.1758 4.9467 4.7333 4.0344 2.4073
JO
4.868 4.6408 4.4631 4.3378 4.2293 4.129 3.7564 2.6131
KC
4.7187 4.3548 4.1087 3.9065 3.7538 3.6076 3.1169 2.0847
LC
3.5987 3.393 3.2391 3.1064 2.9741 2.8599 2.4699 1.524*
LH
6.3187 5.8357 5.5004 5.2338 4.9921 4.7765 4.0171 2.3482
NG
4.1366 3.8835 3.6532 3.4768 3.3174 3.1682 2.6681 1.6598*
O
4.7611 4.3881 4.12 3.9078 3.7382 3.5878 3.0912 2.0244
PB
7.0238 6.3929 5.9547 5.6143 5.327 5.0955 4.2924 2.5363
S
6.6948 6.2424 5.8436 5.5309 5.2342 4.9764 4.1294 2.3744
SB
5.8758 5.4938 5.2702 5.0748 4.9073 4.7511 4.1914 2.5364
SI
7.1675 6.5716 6.1553 5.8184 5.5376 5.2813 4.3959 2.4524
SM
5.9721 5.534 5.1663 4.8856 4.6318 4.4184 3.7657 2.2993
W
6.9999 6.5276 6.1616 5.8264 5.5309 5.2786 4.4266 2.596

Notes: * Indicates rejection of the null hypothesis of no long-term dependence at the 10% significance level. The critical
ranges are: 90% confidence interval, [0.861, 1.747]; 95% confidence interval, [0.809,1.862
.
42

Figure 1.2: Autocorrelation in Returns, Squared Returns, and Absolute Returns

0 500 1000
-0.05
0
0.05
Autocorrelation Function with Bounds
for Raw Return Series for BO
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for BO
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for BO
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for C
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for C
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Absolote Return Series for C
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for CC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for the Squared Return Series for CC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for the Absolote Return Series for CC
Lag
Sample Autocorrelation
42
43

Figure 1.2, Continued

0 500 1000
-0.05
0
0.05
0.1
0.15
Autocorrelation Function with Bounds
for Raw Return Series for CL
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for CL
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for CL
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for CT
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Squared Return Series for CT
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for CT
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for FC
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for FC
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for FC
Lag
Sample Autocorrelation
43
44

Figure 1.2, Continued

0 500 1000
-0.1
-0.05
0
0.05
0.1
Autocorrelation Function with Bounds
for Raw Return Series for GC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for GC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for GC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for HG
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for HG
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Absolote Return Series for HG
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for HO
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Squared Return Series for HO
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for HO
Lag
Sample Autocorrelation
44
45

Figure 1.2, Continued

0 500 1000
-0.05
0
0.05
Autocorrelation Function with Bounds
for Raw Return Series for JO
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for JO
Lag
Sample Autocorrelation
0 500 1000
-0.05
0
0.05
0.1
0.15
Autocorrelation Function with Bounds
for the Absolote Return Series for JO
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for KC
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for KC
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for KC
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for LC
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for LC
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for LC
Lag
Sample Autocorrelation
45
46

Figure 1.2, Continued

0 500 1000
-0.05
0
0.05
0.1
0.15
Autocorrelation Function with Bounds
for Raw Return Series for LH
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
0.4
0.6
Autocorrelation Function with Bounds
for the Squared Return Series for LH
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for LH
Lag
Sample Autocorrelation
0 500 1000
-0.05
0
0.05
Autocorrelation Function with Bounds
for Raw Return Series for NG
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Squared Return Series for NG
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for NG
Lag
Sample Autocorrelation
0 500 1000
-0.05
0
0.05
Autocorrelation Function with Bounds
for Raw Return Series for O
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Squared Return Series for O
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for O
Lag
Sample Autocorrelation
46
47

Figure 1.2, Continued

0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for Raw Return Series for PB
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for PB
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for PB
Lag
Sample Autocorrelation
0 500 1000
-0.05
0
0.05
Autocorrelation Function with Bounds
for Raw Return Series for S
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for S
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for S
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for SB
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Squared Return Series for SB
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for SB
Lag
Sample Autocorrelation
47
48

Figure 1.2, Continued

0 500 1000
-0.05
0
0.05
0.1
0.15
Autocorrelation Function with Bounds
for Raw Return Series for SI
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Squared Return Series for SI
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
0.2
0.3
Autocorrelation Function with Bounds
for the Absolote Return Series for SI
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for SM
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for SM
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for SM
Lag
Sample Autocorrelation
0 500 1000
-0.1
0
0.1
Autocorrelation Function with Bounds
for Raw Return Series for W
Lag
Sample Autocorrelation
0 500 1000
-0.5
0
0.5
Autocorrelation Function with Bounds
for the Squared Return Series for W
Lag
Sample Autocorrelation
0 500 1000
-0.2
0
0.2
Autocorrelation Function with Bounds
for the Absolote Return Series for W
Lag
Sample Autocorrelation
48
49

affect the quantile functions of the returns distribution. Accordingly, in this study we
explicitly allow for skewness in the volatility distribution by considering the standardized
skewed Student's t-distributions.
Thus, the four previously identified ARCH models (GARCH, APGARCH,
FIGARCH, and FIAPARCH) have been estimated allowing for seasonality in the volatility
and utilizing the skewed Student's t-distributed errors. For frozen pork bellies only, a
normal distribution was assumed instead due to numerical instability when applying the
Student's t-distribution (the degree of freedom of the estimated t-coefficient varied from
several hundred to thousands indicating that normal distribution might be more
appropriate. That is in line with the initial observation in section 1.3 that the returns
quantile function for frozen pork bellies is close to normal). If the four seasonality terms
have been found to be insignificantly different from zero for all four of the models, the
models were re-estimated without seasonality terms. Similarly, if the skewness parameter
or the estimated degrees of freedom were found to be insignificant, the models have been
re-estimated without them. The estimation results tests are reported in Table 1.6.
A quick look at the summary results reveals that for the majority of the
commodities considered in this study the Student's t-distribution is appropriate for
modeling the error term with the degrees of freedom ranging from a low of 3.372 ( Frozen
orange juice, FIGARCH model) to a high of 19.702 (Wheat, FIAPARCH model) with an
average of 7.65 df. Connolly (1992) provides an interesting interpretation of the estimated
degree of freedoms of the Student's t-distribution in ARCH modeling, namely if the
50

estimate is below 10 df both non-normality and heteroskedasticity account for the observed
high kurtosis, and if the estimated degree of freedoms are greater than 30, then
heteroskedasticity alone could explain the leptokurtic distribution of the returns. From the
20 commodity futures, the estimated degrees of freedom are below 10 for at least 15 series,
with estimates for additional 3 series being close to 10 df for at least for some of the
estimated models; all of the commodity futures (but frozen pork bellies) degree of freedom
estimates are well below 30 df. Thus, incorporating a fat-tailed distribution in the
estimation of ARCH models appears desirable. However, most of the research in
commodity futures has considered only the normal distribution (see Baillie et al., 2007, Jin
and Fechetti, 2004) or have determined that normal distribution is preferred to Student's
t-distribution (Sephton, 2009). While using the normal distribution when the true one is
fat-tailed could be justified on the grounds of quasi-maximum likelihood estimation and its
asymptotic behavior, estimation efficiency could be lost in the process. Further, extreme
tail measures such as VaR could be significantly underestimated.
Furthermore, for six of commodities futures, modeling the skewness in the error
distribution seems appropriate (the skewness parameter ξ log is significantly different
from zero
7

7
For better convergence instead of
). Interestingly the three commodities that exhibit positive skewness, i.e., their
density distribution is skewed to the right are grains or grain derivatives - soybean oil, corn,
and wheat. This could be explained by appealing to the information flow patterns and their
influence on commodity futures prices whereby sudden good news (weather) tend to have
ξ we estimate ξ log ; hence, positive skewness is indicated by an
estimate greater than zero, and negative - an estimate lower than zero.
51

Table 1.6: ARCH Model Estimates

Model
Constant
In
Variance
Seasonality Terms In Variance
Fractional
estimate
ARCH GARCH Gamma Delta Skewness
Student's
t (df)

sin1 sin2 cos1 cos2

BO
GARCH 0.0331 0.0069* -0.0018* -0.0083* 0.0074*

0.0548 0.9309

0.0814 8.3944
APARCH 0.0249 0.005* -0.0006* -0.0056* 0.0055*

0.0591 0.9361 -0.1202* 1.5592 0.0814 8.3819
FIGARCH 1.8357 -0.022* 0.1139* -0.2921 -0.004* 0.3565 0.3017 0.6214

0.074 9.4086
FIAPARCH 1.1602 -0.0354* 0.1373* -0.3487 0.0187* 0.3224 0.3143 0.5981 -0.0862* 2.1753 0.0759 9.0879

C
GARCH 0.0302 0.0129 -0.0092* -0.0221 0.0139

0.0731 0.9171

0.0453 8.1552
APARCH 0.0157 0.0074 -0.0047* -0.0077 0.0055

0.068 0.9375 -0.1861 1.1267 0.0477 8.3909
FIGARCH 1.5738 -0.2439 0.1376* -0.5928 0.1284* 0.354 0.2329 0.5183

0.0389* 9.4952
FIAPARCH 1.0414 -0.2942 0.1467* -0.7094 0.1542* 0.3188 0.2171 0.4748 -0.1308 2.1663 0.0401* 9.2249

CC
GARCH 0.0112

0.0215 0.976

5.9772
APARCH 0.0168

0.0286 0.968 -0.2542 1.783

6.0638
FIGARCH 3.4855

0.3362 0.4339 0.709

6.2556
FIAPARCH 2.5245

0.2993 0.4268 0.6728 -0.2298 2.0659

6.1067

CL
GARCH 0.0221

0.0402 0.9566

-0.045 9.0031
APARCH 0.0154

0.0468 0.9571 0.2015 1.0242 -0.0472 9.2245
FIGARCH 2.7139

0.3755 0.2805 0.6366

-0.0423 10.2089
FIAPARCH 1.2083

0.2776 0.3035 0.5607 0.0682* 2.3879 -0.0484 9.4325

CT
GARCH 0.0123 0.01 -0.0007* -0.0051* 0.004*

0.0326 0.9644

6.0042
APARCH 0.0101 0.0067 -0.0003* -0.003* 0.0019*

0.0382 0.9645 -0.0108* 1.5403

5.995
FIGARCH 2.8449 -0.1019* 0.0998* -0.3798 -0.033* 0.3898 0.42 0.728

6.5426
FIAPARCH 2.3929 -0.1052* 0.1196* -0.428 -0.0241* 0.3777 0.4284 0.7247 -0.0089* 2.1165

6.3229

Notes: * indicates estimate insignificantly different from zero at 5% confidence level.

51
52

Table 1.6, Continued

Model
Constant
In
Variance
Seasonality Terms In Variance
Fractional
estimate
ARCH GARCH Gamma Delta Skewness
Student's
t (df)

sin1 sin2 cos1 cos2

FC
GARCH 0.0101 0.0001* -0.0037 0.0017* 0.0045

0.0742 0.9151

5.0834
APARCH 0.0116 0* -0.004 -0.0004* 0.0039

0.0778 0.9243 0.2602 1.3134

5.2417
FIGARCH 0.4416 0.0388* 0.0316* 0.0288* 0.0735 0.3496 0.2198 0.511

5.7389
FIAPARCH 0.2947 0.0003* 0.033* 0.0188* 0.0855 0.3067 0.2075 0.4606 0.2811 2.0697

5.6018

GC
GARCH 0.0011* 0* 0.002* 0.0033 0.0026

0.0423 0.9617

4.0687
APARCH 0.0029 -0.0002* 0.0008* 0.0018 0.0022

0.0518 0.9606 -0.4061 1.0328

4.1948
FIGARCH 0.3473 0.1174 0.0457 0.0071* -0.0415 0.4243 0.3428 0.6992

4.9753
FIAPARCH 0.1508* 0.0953 0.0505 0.0144* -0.0343* 0.399 0.3382 0.6772 -0.2495 2.069

4.7325

HG
GARCH 0.034

0.0449 0.9451

5.3323
APARCH 0.0232

0.0528 0.9475 0.017* 1.459

5.3381
FIGARCH 2.82

0.321 0.3133 0.5931

5.7402
FIAPARCH 2.2976

0.3006 0.3161 0.5762 -0.0106* 2.1384

5.5678

HO
GARCH 0.0198 -0.0118 0.0106* 0.0206 0.013*

0.0314 0.9657

9.7333
APARCH 0.0106 -0.0057* 0.0023* 0.0068 0.0036*

0.0345 0.9689 -0.0675* 1.1447

9.5705
FIGARCH 3.1859 0.4123* 0.1621* 0.6145 -0.1862* 0.3211 0.2401 0.5561

11.5818
FIAPARCH 2.1909 0.5601* 0.4328* 0.9004 -0.2215* 0.2202 0.2067 0.4276 -0.0725* 2.4494

10.7414

JO
GARCH 0.019

0.0196 0.9775

3.4856
APARCH 0.0147*

0.041 0.9655 0.1947* 1.1248

3.4991
FIGARCH 9.0151

0.3392 0.5369 0.737

3.372
FIAPARCH 7.494

0.3492 0.468 0.7266 0.3143 1.5511

3.6001

Note: * indicates estimate insignificantly different from zero at 5% confidence level.

52
53

Table 1.6, Continued

Model
Constant
In
Variance
Seasonality Terms In Variance
Fractional
estimate
ARCH GARCH Gamma Delta Skewness
Student's
t (df)

sin1 sin2 cos1 cos2

KC
GARCH 0.1854

0.0584 0.9147

4.5075
APARCH 0.0812

0.066 0.9193 -0.6461 1.0169

4.7174
FIGARCH 7.6003

0.3059 0.4505 0.6457

4.4345
FIAPARCH 4.3064

0.2501 0.3891 0.5694 -0.8556 1.3345

4.8831

LC
GARCH 0.0113

0.0493 0.9385

9.8412
APARCH 0.0109

0.0521 0.9473 0.2616 1.1679

9.4401
FIGARCH 0.7198

0.397 0.3576 0.6938

10.6295
FIAPARCH 0.5481

0.3817 0.3552 0.6834 0.1632 2.0376

10.3325

LH
GARCH 0.0314 -0.0042* 0.0103* -0.0047* -0.0056*

0.048 0.9384

-0.0895 11.5037
APARCH 0.0205 -0.0026* 0.008* -0.0043* -0.0038*

0.029 0.9598 0.4843 1.6862 -0.0815 12.5352
FIGARCH 2.2648 -0.145* -0.0435* 0.0495* -0.2082 0.3938 0.3504 0.6876

-0.0871 11.553
FIAPARCH 1.9134 -0.1271* -0.0454* 0.0229* -0.1755 0.3158 0.4238 0.6866 0.43 1.6787 -0.0755 13.3779

NG
GARCH 0.278 -0.1205 -0.0172* 0.0789* 0.0753

0.0633 0.9117

8.8603
APARCH 0.0964 -0.0326 -0.0042* 0.0128* 0.0248*

0.0706 0.9206 -0.3185 1.2015

9.4769
FIGARCH 10.1455 -0.9723* 1.0703 3.2368 0.8096* 0.3693 0.2852 0.6093

8.9934
FIAPARCH 8.1198 -0.9806 0.8943 2.4731 0.6498* 0.3178 0.2846 0.551 -0.3028 1.8126

9.8573

O
GARCH 0.1015 0.025 -0.0219* -0.0587 0.0248*

0.0793 0.901

5.1494
APARCH 0.0484 0.0084 -0.0066* -0.0116 0.004*

0.0846 0.9127 -0.2031 0.9748

5.2397
FIGARCH 3.6436 -0.3556* 0.3176* -1.1755 0.2909* 0.2736 0.2303* 0.3811

5.812
FIAPARCH 3.0359 -0.3647* 0.3383* -1.4113 0.3956 0.2588 0.2412* 0.3782 -0.1042 2.142

5.6467

Note: * indicates estimate insignificantly different from zero at 5% confidence level.
53
54

Table 1.6, Continued

Model
Constant
In
Variance
Seasonality Terms In Variance
Fractional
estimate
ARCH GARCH Gamma Delta Skewness
Student's
t (df)

sin1 sin2 cos1 cos2

PB
GARCH 0.0423 0.0154 0.0089* -0.0198 0.0065*

0.0486 0.9419

APARCH 0.2603* 0.035* 0.1963* -0.2293* -0.0186*

0.0177* 0.9277 0.0873* 4.6853

FIGARCH 5.7491 -0.5359 0.2846* -0.6975 -0.1814* 0.4271 0.3642 0.6935

FIAPARCH 6.2968 -0.3656 0.2275* -0.5558 -0.1706* 0.4274 0.3752 0.7079 0.1409 1.8407

S
GARCH 0.0213 0.0045* -0.001* -0.0156 0.0132

0.054 0.9381

7.0335
APARCH 0.0078 0.0024 -0.0005* -0.0034 0.0028

0.0512 0.9549 -0.3538 0.6522

7.4419
FIGARCH 0.9168 -0.1968 0.1303* -0.3959 0.1055* 0.3654 0.26 0.5988

8.1273
FIAPARCH 0.5281 -0.2629 0.1513 -0.4423 0.0777* 0.3228 0.2379 0.55 -0.2419 2.1144

7.9306

SB
GARCH 0.0125 0.009* 0.0036* 0.0022* 0.0034*

0.0338 0.9645

5.7346
APARCH 0.0112 0.0073* 0.0026* 0.0018* 0.0028*

0.0375 0.9643 0.0065* 1.7295

5.7238
FIGARCH 5.2658 0.2145* 0.1621* -0.4406 -0.094* 0.5132 0.4001 0.816

6.0459
FIAPARCH 4.74 0.2295* 0.1689* -0.4518 -0.0958* 0.5009 0.4078 0.8103 0.0151* 2.036

5.9685

SI
GARCH 0.0105 0.0025* 0.002* 0.0077 0.004*

0.0328 0.966

-0.0876 4.3145
APARCH 0.0074 0.0005* 0.0008* 0.0026* 0.0015*

0.0424 0.9647 -0.3246 1.2666 -0.0804 4.4099
FIGARCH 4.287 0.5026 0.2081 -0.1142* -0.105* 0.4733 0.4025 0.7924

-0.0761 4.523
FIAPARCH 3.955 0.4216 0.1991* -0.0943* -0.0926* 0.5021 0.3749 0.8063 -0.1712 2.0109 -0.0727 4.5563

SM
GARCH 0.0231 0.0053* 0.0015* -0.0075* 0.0091*

0.0599 0.9351

5.6757
APARCH 0.0102* 0.0022* 0.0001* -0.0019* 0.0024*

0.0512 0.955 -0.3613 0.7898

5.7668
FIGARCH -0.0626* 0.1005* -0.2858 0.1495* 0.3112 0.3112 0.2631 0.5056

6.9588
FIAPARCH 0.2752* 0.0128* 0.1032* -0.4094 0.1659* 0.216 0.1728* 0.3346 -0.1764 2.4192

6.2399

Note: * indicates estimate insignificantly different from zero at 5% confidence level
.
54
55

Table 1.6, Continued

Model
Constant
In
Variance
Seasonality Terms In Variance
Fractional
estimate
ARCH GARCH Gamma Delta Skewness
Student's
t (df)

sin1 sin2 cos1 cos2

W
GARCH 0.0257 0.0115 -0.0019* -0.0033* -0.0055*

0.0398 0.9518

0.0694 16.6308
APARCH 0.007* 0.0044 0* 0* -0.0018*

0.0317 0.9708 -0.4991 1.0978 0.068 19.1365
FIGARCH 2.516 0.0342* 0.0497* -0.3733 -0.046* 0.314 0.283 0.5642

0.061 18.5559
FIAPARCH 0.1499 0.0594* -0.008* -0.0588* -0.012* 0.2838 0.3222 0.5654 -0.3529 1.8792 0.0583 19.7022

Note: * indicates estimate insignificantly different from zero at 5% confidence level
55
56

a gradual impact on prices while such for bad weather may cause prices to jump instantly.
The three commodities for which the skewness parameter is negative are crude oil, live
hogs and silver, a rather disparate collection of types of commodities. Thus, overall while
we find some support for the use of a skewed distribution, it appears that for the majority of
commodities studied a symmetric error distribution should suffice.
The estimation results from the APARCH and FIAPARCH model seem to suggest
that commodity futures returns tend to exhibit asymmetric response in volatility due to
negative and positive price shocks, the so-called leverage effect. The coefficient of
leverage, γ , is significantly different from zero for 13 commodities, with γ being
negative for 10 of these. A negative γ indicates that volatility tends to decrease ceteris
paribus given a negative price shock and vice versa. This is in contrast to financial asset
returns for which it is generally found a large and positive asymmetric impact of negative
shocks to prices. To certain extent the above described information flow hypothesis could
account for the observed estimates. It is worth noting, however, that three commodities
with positive leverage coefficient of asymmetry γ , are all livestock commodities - feeder
cattle, lean cattle, and lean hogs, with frozen pork bellies having significant γ only under
the FIAPARCH model.
As initially noted by Ding et al. (1993), the persistence of the volatility process
need not be in absolute returns or squared returns, and from Table 1.6 it is observed that the
Box-Cox power transform δ , is significantly different from zero for all commodities and
on average below 2, varying between a low of 0.65 to as high as 4.685 when estimated
57

within the APARCH framework (the high of 685 . 4 = δ is for frozen pork bellies and it is
an outlier; the trimmed range is only 0.65-1.783, and trimmed mean of 1.2). Interestingly
the estimated power transform coefficient is consistently higher and for most cases not
significantly different from 2 when it is estimated under the FIAPARCH model, indicating
that after all perhaps modeling the conditional variance instead of other moments has some
additional appeal
8
The fractional term,
.
d , in FIGARCH (1,d,1) and FIAPARCH (1,d,1) is
significantly different from zero but also from unity for all commodities considered herein,
implying that although the sum of the ARCH and GARCH coefficients may be close to
unity the volatility process is well-behaved but fractional. The fractional term estimates are
fairly consistent across different groups of commodities with a median of approximately
0.33, an observation slightly smaller but still consistent with other studies' estimates from
fractional integrated GARCH models (Baillie et al., 2007, Jin and Fechette, 2004, Sephton,
2009). Thus, the results in Table 1.6 seem to suggest that the commodity futures volatility
process is governed by a short-term (ARMA) memory process but also and by a long-term
memory process (the fractionally integrated part) and thus shocks to volatility tend to
persist for up to several years.

8
The δ estimate from the APARCH model estimation is in general insignificantly different from 1 for
approximately 10 of the 21 commodities, and insignificantly different from 2 for 7 of the remaining 11
commodities.
58

1.4.6. Model Comparisons
The suitability of the estimated models in this study is examined in several ways.
Firstly, the four identified models are compared in-sample by way of information criteria
(AIC and BIC). This method was elected not only for ease of comparison, but also to avoid
spurious likelihood ratio type of tests as although the models are nested, when estimated
the different model coefficients domains had to be exactly matching. For example, Greene
(1997) points out that while the t-distribution does converge to Normal distribution as the
degree of freedom approach infinity, when actual optimization is performed only a
sub-domain of the real line is explored. Before ranking the different models according to
information criteria, the ARCH LM test is performed on the standardized residuals from
each model to check for any remaining heteroskedasticity at lags 2, 5, 10, 20, and 50. If a
model does not remove heteroskedasticity from the residuals as effectively as its peers, it is
removed from the information criteria comparison.
Secondly, the suitability of the four models is investigated via in-sample
Value-at-Risk tests. Since the introduction of RiskMetrics in the mid 1990s, the popularity
of Value-at-Risk (VaR) as a risk measure has consistently being increasing especially after
the introduction of the Basel II regulations for commercial banks, allowing them to utilize
internally developed risk measures such as VaR . The allure of the VaR as a risk measure
stems predominantly from the simplicity of the concepts itself and the highly aggregated
amount of information that it conveys to risk managers. Naturally, representing the risk
exposure of an entire business entity as a single parameter is powerful but yet,
59

oversimplifying approach. VaR is typically defined as the (dollar) amount of money that
would not be expected to be exceeded over the next n-days at a particular significance level
% α . Thus, VaR is simply the % α -th quantile of the n-day distribution of the portfolio
returns. Hence, forecasting VaR correctly is tantamount to testing a volatility model of
replicating the returns distribution in the tails (VaR is commonly thought to apply only to
long positions, that is the concern is only in the left tail of the ditribution; however, as
importantly is to model correctly the right tail of the distribution when short positions are
present). It is worth noting also that VaR does not specify what the maximum or average
loss (expected shortfall) at the % α confidence level would be. Furthermore, it is well
known that VaR is not a coherent risk measure and hence problematic when being
optimized (Artzner et al., 1999). Typically VaR measures are for one to five days out, and
at 99% through 95% level of significance. In this study, four quantiles are examined - the
99.75%, 95%, 5%, and 0.0025% quantiles for one-day ahead VaR. The actual performance
of the models is assessed my looking at the failure rate, i.e., the number of times that the
models forecasts exceed the specified % α significance level. Two statistical tests are
performed to quantify any differences from the specified quantiles: the Kupiec LR test
(1995) which compares the model's failure rate to the VaR significance level, and Engle
and Manganelli's (1999) tests of dynamic correlation in consecutive VaR breaches.
Thirdly, the performance of the ARCH models is compared out-of-sample by fixing the
initial period to the first 3500 observations, which for the differences commodity futures
results in approximately two years of out-of-sample of five day-ahead forecasts, and
60

performing the Mincer and Zarnowitz (1969) , MZ henceforth, test and comparing the
goodness of fit ( )
2
R for each model. An issue with applying the MZ test is that an
estimate of the true, ex-post variance is required first. Volatility, however, is an inherently
latent variable, and consequently it has itself to be approximated. A commonly used proxy
for realized volatility is the square of the returns, i.e., { }
2
t
r , which is unbiased estimate of
the true variance (see Cumby, Figlewski and Hasbrouck 1993; Figlewski, 1997; and
Jorion, 1996). To see this, note that if the volatility model is correctly specified and the
error term and mean equation are given as before, i.e., [ ]
t t t t
e F r E r + =
−1
| and
t t t
u e σ = ,
then [ ] [ ]
2 2
1 1
|
t t t t t
F r E r E σ = −
− −
. Unfortunately such a proxy is bound to be subject to
significant idiosyncratic disturbances caused by
2
t
u and consequently it is to be expected
any forecasts to produce a low
2
R from the simple linear regression of the ex-post
variance on a constant and the model forecast. For example, Cumby, Figlewski and
Hasbrouck (1993) reports
2
R as low as 0.003. More recently, Andersen and Bollerslev
(1998) have suggested the use of intraday tick data to estimate the realized volatility,
reporting significantly higher
2
R . However, in this study due to the unavailability of
high-frequency tick data, the squared return proxy described above is utilized. As an
alternative to the
2
R from the MZ test, we report the values from the standard mean
squared error loss function (MSE henceforth).
The results from the model comparisons are reported in Tables 1.7 and 1.8 and summarized
in Tables 1.9 and 1.10. While it is evident that for the majority of models and commodity
futures the considered models effectively reduce the correlation in the second moments,
61

Table 1.7: ARCH Model Comparisons

Ticker Model
Log
Likelihood
AIC BIC MZ MSE ARCH 2 ARCH 5
ARCH
10
ARCH
20
ARCH
50
Q(5)

BO
GARCH -7055.73 3.4812 3.4982 0.13075 56.169 0.1885 0.1845 1.0235 0.9703 0.8142 0.9362
APARCH -7052.82 3.4807 3.5009 0.09144 59.022 0.1607 0.2266 1.0962 0.9813 0.8081 1.1545
FIGARCH -7053.16 3.4804 3.4990 0.12887 56.085 0.1867 0.1381 0.5645 0.6985 0.6397 0.702
FIAPARCH -7049.67 3.4796 3.5014 0.09897 58.006 0.0648 0.0984 0.562 0.6789 0.6078 0.4934

C
GARCH -7047.61 3.4771 3.4942 0.06015 80.301 0.406 1.3629 1.1466 1.4472 1.1983 6.8994
APARCH -7037.86 3.4733 3.4935 0.01431 86.062 1.7493 2.8803* 1.8152 1.6697* 1.1843 14.4573**
FIGARCH -7047.72 3.4777 3.4963 0.0572 80.381 0.2873 1.3532 1.1067 1.2746 1.1399 6.8964
FIAPARCH -7041.57 3.4756 3.4974 na na 0.1116 1.3624 1.1172 1.2705 1.1035 6.8771

CC
GARCH -8123.918 4.0262 4.0340 0.01858 116.63 7.1658** 3.7325** 2.0098* 1.1558 1.0223 19.6309**
APARCH -8118.972 4.0248 4.0357 0.0175 116.99 5.7156** 2.7218* 1.4941 0.9243 0.9202 13.9738**
FIGARCH -8132.048 4.0307 4.0401 0.01326 117.6 5.4119** 3.4681** 2.0052* 1.1612 1.0182 16.6478**
FIAPARCH -8124.738 4.0281 4.0406 0.01509 117.47 3.2161* 2.1396 1.3549 0.8474 0.9089 10.2097*

CL
GARCH -8591.066 4.2771 4.2865 0.17728 381.27 0.3343 2.0378 1.1191 0.9024 0.9257 10.3299*
APARCH -8582.748 4.2740 4.2865 0.1893 374.89 0.5784 2.7225* 1.4487 1.1328 1.0105 13.8261**
FIGARCH -8587.716 4.2760 4.2869 0.17274 392.46 0.917 1.0792 1.0241 1.0103 0.9864 5.6586
FIAPARCH -8579.797 4.2730 4.2871 0.17002 390.53 0.7063 0.8158 0.7716 0.8532 0.907 4.1716

CT
GARCH -7545.623 3.7257 3.7412 0.0564 71.222 7.0659** 3.9219** 2.2657* 1.3758 1.0857 19.467**
APARCH -7544.084 3.7259 3.7445 0.05377 71.41 8.0686** 4.4782** 2.5655** 1.5321 1.1878 22.1713**
FIGARCH -7542.811 3.7248 3.7419 0.08195 69.403 0.1205 1.0198 0.8221 0.6605 0.7796 5.0745
FIAPARCH -7542.118 3.7254 3.7456 0.08089 69.372 0.0728 0.9598 0.7993 0.6605 0.7653 4.7514

Notes: AIC is the Akaike Information Criterion; BIC –the Bayesian (Schwarz) Information Criterion; M-Z R
2
is the R
2
from the Mincer and Zarnowitz (1969)
out-of-sample test; ARCH (n) stands for the Engle’s ARCH LM test of heteroskedasticity up to lag n; Q(n) is the Box-Pierce test statistic for autocorrelation in the
squared residuals; **(*) denotes rejection of the null hypotheses of no heteroskedasticity (ARCH) and no autocorrelation (Box-Pierce) at the 1% (5%) level of
significance.

61
62

Table 1.7, Continued

Ticker Model
Log
Likelihood
AIC BIC MZ MSE ARCH 2 ARCH 5
ARCH
10
ARCH
20
ARCH
50
Q(5)

FC
GARCH -4400.318 2.1785 2.1940 0.0044 6.2543 0.3468 0.1984 0.5747 0.9195 1.9644** 0.9701
APARCH -4385.38 2.1721 2.1908 0.00612 6.1802 0.2186 0.1626 0.5776 0.9757 1.4804* 0.8194
FIGARCH -4396.606 2.1771 2.1943 0.00573 6.1796 0.3415 0.348 0.5796 0.9537 1.8481** 1.7104
FIAPARCH -4380.663 2.1702 2.1905 0.00839 6.1554 0.6411 0.5005 0.7166 0.9917 1.4897* 2.4571

GC
GARCH -5005.035 2.5081 2.5239 0.05274 35.612 33.058** 14.542** 7.6315** 4.0671** 2.348** 71.0343**
APARCH -4979.69 2.4965 2.5154 0.03526 36.487 61.631** 26.803** 13.827** 7.1956** 3.2936** 129.056**
FIGARCH -5004.206 2.5082 2.5255 0.04698 35.509 5.9901** 2.9803* 2.0251* 1.2894 1.2042 14.8632**
FIAPARCH -4991.34 2.5028 2.5233 0.03068 36.763 4.8605** 2.3214* 1.6804 1.1395 1.2583 11.5759**

HG
GARCH -7543.68 3.7477 3.7571 0.13506 194.17 1.2929 1.1739 0.6649 0.6944 0.6598 5.8586
APARCH -7541.263 3.7475 3.7600 0.11691 197.23 3.1619* 1.9585 1.0563 1.0095 0.8161 9.8895*
FIGARCH -7542.101 3.7474 3.7583 0.14201 194.29 1.515 1.2279 0.7832 0.8019 0.6609 6.1531
FIAPARCH -7541.402 3.7480 3.7621 0.11973 198.12 1.1641 1.0541 0.6865 0.732 0.6378 5.2621

HO
GARCH -8640.274 4.2993 4.3150 0.11938 160.77 0.5384 4.7237** 2.7971** 1.6209* 1.0046 23.0469**
APARCH -8633.897 4.2972 4.3160 0.11202 162.76 0.8072 6.1922** 3.5773** 1.984** 1.219 30.4213**
FIGARCH -8638.723 4.2991 4.3163 0.11416 163.88 0.2698 0.9894 1.4556 1.0754 0.7523 5.0474
FIAPARCH -8630.326 4.2959 4.3162 0.09566 166.84 0.304 1.2898 1.3453 0.9874 0.7245 6.5733

JO
GARCH -8178.02 4.0020 4.0112 0.00562 99.761 0.5414 0.2303 0.4544 0.4804 0.3484 1.2666
APARCH -8160.521 3.9944 4.0067 0.00825 100.33 0.5257 0.1832 0.4406 0.4985 0.6101 1.2852
FIGARCH -8172.029 3.9995 4.0103 0.00529 100.59 0.2902 0.1743 0.5324 0.5096 0.4764 0.8493
FIAPARCH -8160.951 3.9951 4.0090 0.00841 100.64 0.1157 0.0723 0.4371 0.4962 0.5836 0.373

Notes: AIC is the Akaike Information Criterion; BIC –the Bayesian (Schwarz) Information Criterion; M-Z R
2
is the R
2
from the Mincer and Zarnowitz (1969)
out-of-sample test; ARCH (n) stands for the Engle’s ARCH LM test of heteroskedasticity up to lag n; Q(n) is the Box-Pierce test statistic for autocorrelation in the
squared residuals; **(*) denotes rejection of the null hypotheses of no heteroskedasticity (ARCH) and no autocorrelation (Box-Pierce) at the 1% (5%) level of
significance.

62
63

Table 1.7, Continued

Ticker Model
Log
Likelihood
AIC BIC MZ MSE ARCH 2 ARCH 5
ARCH
10
ARCH
20
ARCH
50
Q(5)

KC
GARCH -9037.612 4.4837 4.4931 0.00228 74.377 4.9253** 2.7327* 2.3452** 1.5784* 1.5738** 13.433**
APARCH -9010.823 4.4714 4.4839 0.00662 73.098 7.2356** 3.8521** 3.1217** 1.8931** 1.603** 19.1197**
FIGARCH -9028.278 4.4796 4.4905 0.00222 72.14 0.3219 0.8116 2.2944* 1.5552 1.5761** 4.0183
FIAPARCH -9001.103 4.4671 4.4811 0.00311 71.683 1.6792 1.4021 2.0067* 1.388 1.2927 6.8992

LC
GARCH -5262.065 2.5996 2.6089 0.06092 2.6038 11.797** 5.0783** 3.3198** 1.762* 1.0756 24.4418**
APARCH -5254.312 2.5967 2.6092 0.06133 2.5902 34.443** 14.177** 7.9636** 4.1185** 1.994** 65.7984**
FIGARCH -5262.45 2.6003 2.6112 0.05928 2.6115 6.8** 3.0224** 2.372** 1.2679 0.8538 14.8712**
FIAPARCH -5257.088 2.5986 2.6126 0.06283 2.5941 12.567** 5.3871** 3.5994** 1.9039** 1.1177 26.159**

LH
GARCH -7207.55 3.5621 3.5792 0.00154 14.642 12.377** 5.8031** 3.4764** 2.2914** 1.1338 27.8568**
APARCH -7191.6 3.5552 3.5754 0.00652 14.403 19.284** 8.7386** 4.8771** 3.1532** 1.4719* 41.8106**
FIGARCH -7206.48 3.5620 3.5807 0.00187 14.644 9.3667** 4.5022** 2.699** 1.994** 1.0213 22.0922**
FIAPARCH -7190.4 3.5551 3.5769 0.00713 14.367 4.1198* 2.5781* 1.7406 1.6871* 0.9394 12.5124**

NG
GARCH -10183.937 5.0686 5.0827 0.01937 286.81 0.3527 1.8017 2.3253* 1.9067** 1.3666* 8.8906*
APARCH -10171.082 5.0632 5.0804 0.01536 289.29 0.0295 2.5818* 2.7585** 2.1251** 1.4529* 12.8152**
FIGARCH -10184.479 5.0694 5.0850 0.01798 286.44 3.6737* 2.3262* 2.8266** 2.314** 1.5157* 11.2633*
FIAPARCH -10174.985 5.0656 5.0844 0.01955 284.99 0.7309 1.6554 2.6837** 2.3115** 1.5105* 7.9594*

O
GARCH -8098.905 3.9965 4.0120 0.02017 81.936 6.8577** 3.0202* 1.8386* 2.0717** 1.3495 15.345**
APARCH -8085.122 3.9907 4.0094 0.00952 83.194 13.065** 5.4517** 2.9803** 2.4833** 1.5671** 27.5147**
FIGARCH -8090.412 3.9928 4.0099 0.01661 82.165 0.8526 0.7086 0.6625 1.1254 0.9637 3.6091
FIAPARCH -8087.029 3.9921 4.0123 0.00934 82.983 0.4451 0.5524 0.5167 1.0323 0.9056 2.7918

Notes: AIC is the Akaike Information Criterion; BIC –the Bayesian (Schwarz) Information Criterion; M-Z R
2
is the R
2
from the Mincer and Zarnowitz (1969)
out-of-sample test; ARCH (n) stands for the Engle’s ARCH LM test of heteroskedasticity up to lag n; Q(n) is the Box-Pierce test statistic for autocorrelation in the
squared residuals; **(*) denotes rejection of the null hypotheses of no heteroskedasticity (ARCH) and no autocorrelation (Box-Pierce) at the 1% (5%) level of
significance.

63
64

Table 1.7, Continued

Ticker Model
Log
Likelihood
AIC BIC MZ MSE ARCH 2 ARCH 5
ARCH
10
ARCH
20
ARCH
50
Q(5)

PB
GARCH -8616.56 4.2522 4.2662 0.06773 25.513 3.6625* 1.8947 1.3606 1.5361 1.2573 9.8773*
APARCH -8603.49 4.2467 4.2638 0.05582 25.931 3.0987* 1.6546 1.1451 1.3065 1.1621 8.3338*
FIGARCH -8612.99 4.2509 4.2665 0.0727 25.54 0.7336 1.0074 1.1117 1.3482 1.138 5.1558
FIAPARCH -8608.75 4.2498 4.2685 0.07752 25.361 1.072 1.1326 1.2 1.3843 1.1431 5.8107

S
GARCH -6940.603 3.4239 3.4395 0.07779 57.535 0.5761 0.4782 0.4283 0.6726 1.04 2.4676
APARCH -6913.207 3.4114 3.4301 0.03327 60.932 0.5464 0.84 0.5 0.8602 1.4434* 4.2475
FIGARCH -6939.618 3.4239 3.4410 0.07976 57.738 0.554 0.5135 0.5643 0.7084 1.0564 2.6469
FIAPARCH -6926.473 3.4185 3.4387 na na 0.2853 0.614 0.5232 0.7754 1.1969 3.1001

SB
GARCH -8144.854 4.0511 4.0667 0.03169 112.83 9.2136** 3.843** 2.0614* 1.3818 1.0924 18.4632**
APARCH -8144.371 4.0518 4.0706 0.03279 112.7 9.5878** 4.0347** 2.1027* 1.4003 1.1073 19.3371**
FIGARCH -8142.524 4.0504 4.0676 0.03047 112.87 0.2281 0.2387 0.5182 0.8372 0.8504 1.2091
FIAPARCH -8142.412 4.0514 4.0717 0.03013 112.92 0.1653 0.2147 0.4993 0.819 0.844 1.0818

SI
GARCH -7415.38 3.6915 3.7071 0.06853 312.53 3.8869* 2.1442 1.7399 1.2861 0.7866 10.474*
APARCH -7404.88 3.6872 3.7060 0.0381 325.06 4.3506* 2.5993* 2.0861* 1.5291 0.9181 12.6141**
FIGARCH -7420.17 3.6943 3.7116 0.07627 310.32 0.8405 0.7165 1.1565 1.0858 0.7631 3.5713
FIAPARCH -7415.98 3.6933 3.7136 0.06545 316.04 1.1631 0.9925 1.4107 1.1943 0.7404 4.9471

SM
GARCH -7286.646 3.5939 3.6079 0.02886 75.444 0.5983 0.6616 0.7002 1.134 1.2119 3.3778
APARCH -7266.775 3.5851 3.6022 0.01052 77.1 2.7283 3.645** 2.1202* 1.8953** 1.564** 19.3616**
FIGARCH -7277.52 3.5899 3.6055 0.03426 75.202 0.0071 0.2691 0.674 1.042 1.0695 1.3427
FIAPARCH -7256.361 3.5805 3.5991 na na 0.3765 0.5008 0.6323 1.0749 1.0369 2.5538

Notes: AIC is the Akaike Information Criterion; BIC –the Bayesian (Schwarz) Information Criterion; M-Z R
2
is the R
2
from the Mincer and Zarnowitz (1969)
out-of-sample test; ARCH (n) stands for the Engle’s ARCH LM test of heteroskedasticity up to lag n; Q(n) is the Box-Pierce test statistic for autocorrelation in the
squared residuals; **(*) denotes rejection of the null hypotheses of no heteroskedasticity (ARCH) and no autocorrelation (Box-Pierce) at the 1% (5%) level of
significance.

64
65

Table 1.7, Continued

Ticker Model
Log
Likelihood
AIC BIC MZ MSE ARCH 2 ARCH 5
ARCH
10
ARCH
20
ARCH
50
Q(5)

W
GARCH -7768.91 3.8320 3.8475 0.03095 156.05 0.2938 1.8245 1.4538 1.0253 1.0254 9.3583*
APARCH -7753.76 3.8255 3.8441 0.03064 157.28 0.9306 2.26* 1.7253 1.1381 1.0696 11.6332**
FIGARCH -7763.61 3.8299 3.8469 0.03392 154.37 0.2319 0.4484 0.7393 0.6185 0.8655 2.2712
FIAPARCH -7750.31 3.8243 3.8445 0.03062 156.43 0.2993 0.1794 0.6084 0.6045 0.8581 0.8942

Notes: AIC is the Akaike Information Criterion; BIC –the Bayesian (Schwarz) Information Criterion; M-Z R
2
is the R
2
from the Mincer and Zarnowitz (1969)
out-of-sample test; ARCH (n) stands for the Engle’s ARCH LM test of heteroskedasticity up to lag n; Q(n) is the Box-Pierce test statistic for autocorrelation in the
squared residuals; **(*) denotes rejection of the null hypotheses of no heteroskedasticity (ARCH) and no autocorrelation (Box-Pierce) at the 1% (5%) level of
significance.

65
66

Table 1.8: In-Sample 1-day Ahead Value-at-Risk

BO C
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9488 0.9503 0.9458 0.9478

0.9525 0.9547 0.9473 0.9520
Kupiec LR test 0.1286 0.0052 1.4605 0.4143

0.5268 1.93 0.617 0.3361
Kupiec LR test p-value 0.7199 0.9426 0.2269 0.5198

0.4680 0.1648 0.4322 0.5621
EM test 2.7855 3.8515 3.6349 2.6956

6.8401 7.3811 4.1608 4.1919
EM test p-value 0.8353 0.6968 0.7259 0.8460

0.3359 0.2870 0.6549 0.6507

99.75% Quantile
Fail/Success rate 0.9973 0.9973 0.9975 0.9980

0.9951 0.9956 0.9953 0.9961
Kupiec LR test 0.0695 0.0695 0.0022 0.4926

7.4543* 4.9396* 6.144* 2.8721
Kupiec LR test p-value 0.7921 0.7921 0.9623 0.4828

0.0063 0.0262 0.0132 0.0901
EM test 0.2176 0.2176 0.1274 0.6589

5.3838 3.8493 4.5999 2.4708
EM test p-value 0.9998 0.9998 1.0000 0.9953

0.4956 0.6971 0.5961 0.8717

5% Quantile
Fail/Success rate 0.0485 0.0502 0.0510 0.0500

0.0510 0.0510 0.0554 0.0527
Kupiec LR test 0.1884 0.0052 0.0825 0

0.0825 0.0825 2.4282 0.617
Kupiec LR test p-value 0.6642 0.9426 0.7740 1.0000

0.7740 0.7740 0.1192 0.4322
EM test 6.5195 9.7964 5.2067 10.886

6.237 4.3243 6.8287 7.6912
EM test p-value 0.3676 0.1335 0.5176 0.0920

0.3972 0.6329 0.3370 0.2616

0.25% Quantile
Fail/Success rate 0.0015 0.0022 0.0025 0.0022

0.0022 0.0025 0.0025 0.0022
Kupiec LR test 1.9957 0.1358 0.0022 0.1358

0.1358 0.0022 0.0022 0.1358
Kupiec LR test p-value 0.1578 0.7125 0.9623 0.7125

0.7125 0.9623 0.9623 0.7125
EM test 2.9195 0.2485 0.1274 0.2485

0.2485 0.1274 0.1274 0.2485
EM test p-value 0.8189 0.9997 1.0000 0.9997

0.9997 1.0000 1.0000 0.9997

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

66
67

Table 1.8, Continued

CC CL
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9490 0.9502 0.9463 0.9510

0.9508 0.9478 0.9435 0.9475
Kupiec LR test 0.0871 0.0042 1.1617 0.0798

0.0474 0.4183 3.4043 0.5157
Kupiec LR test p-value 0.7679 0.9482 0.2811 0.7776

0.8277 0.5178 0.0650 0.4727
EM test 21.973* 14.397* 14.136* 9.9098

3.1473 3.1594 8.8508 3.7473
EM test p-value 0.0012 0.0255 0.0282 0.1285

0.7901 0.7886 0.1821 0.7108

99.75% Quantile
Fail/Success rate 0.9978 0.9978 0.9973 0.9975

0.9985 0.9978 0.9975 0.9978
Kupiec LR test 0.1236 0.1236 0.079 0.0009

1.9143 0.114 0.0002 0.114
Kupiec LR test p-value 0.7252 0.7252 0.7786 0.9761

0.1665 0.7356 0.9874 0.7356
EM test 49.682* 49.682* 66.694* 39.627*

115.93* 99.253* 38.861* 48.623*
EM test p-value 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

5% Quantile
Fail/Success rate 0.0493 0.0483 0.0503 0.0493

0.0537 0.0525 0.0562 0.0540
Kupiec LR test 0.044 0.251 0.0063 0.044

1.1515 0.5157 3.152 1.3082
Kupiec LR test p-value 0.8338 0.6164 0.9368 0.8338

0.2832 0.4727 0.0758 0.2527
EM test 1.9149 5.2879 1.4676 6.5761

6.0438 4.8156 8.7236 7.9796
EM test p-value 0.9274 0.5075 0.9616 0.3618

0.4183 0.5677 0.1897 0.2396

0.25% Quantile
Fail/Success rate 0.0032 0.0035 0.0037 0.0032

0.0027 0.0025 0.0032 0.0027
Kupiec LR test 0.7677 1.3503 2.0763 0.7677

0.0873 0.0002 0.794 0.0873
Kupiec LR test p-value 0.3809 0.2452 0.1496 0.3809

0.7676 0.9874 0.3729 0.7676
EM test 22.989* 1.3412 17.99* 0.8649

0.2355 39.429* 23.166* 0.2355
EM test p-value 0.0008 0.9693 0.0063 0.9902

0.9998 0.0000 0.0007 0.9998

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

67
68

Table 1.8, Continued

CT FC
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9468 0.9470 0.9433 0.9458

0.9476 0.9472 0.9462 0.9459
Kupiec LR test 0.8864 0.7583 3.6876 1.4961

0.4673 0.6815 1.2279 1.3888
Kupiec LR test p-value 0.3465 0.3839 0.0548 0.2213

0.4942 0.4091 0.2678 0.2386
EM test 15.289* 13.803* 8.3588 8.1428

10.783 18.04* 9.815 21.074*
EM test p-value 0.0181 0.0319 0.2130 0.2278

0.0953 0.0061 0.1327 0.0018

99.75% Quantile
Fail/Success rate 0.9975 0.9973 0.9970 0.9970

0.9975 0.9978 0.9973 0.9978
Kupiec LR test 0.0019 0.0711 0.3229 0.3229

0.0015 0.1297 0.0742 0.1297
Kupiec LR test p-value 0.9648 0.7897 0.5699 0.5699

0.9692 0.7188 0.7854 0.7188
EM test 0.1273 0.2193 0.4701 0.4701

0.127 0.2419 0.2223 0.2419
EM test p-value 1.0000 0.9998 0.9982 0.9982

1.0000 0.9997 0.9998 0.9997

5% Quantile
Fail/Success rate 0.0540 0.0535 0.0579 0.0567

0.0499 0.0521 0.0538 0.0521
Kupiec LR test 1.3291 1.0242 5.1315* 3.6876

0.0011 0.3751 1.2279 0.3751
Kupiec LR test p-value 0.2490 0.3115 0.0235 0.0548

0.9741 0.5402 0.2678 0.5402
EM test 18.288* 19.298* 17.803* 13.117*

13.289* 9.7229 13.924* 4.8242
EM test p-value 0.0056 0.0037 0.0067 0.0412

0.0387 0.1368 0.0305 0.5666

0.25% Quantile
Fail/Success rate 0.0022 0.0020 0.0022 0.0020

0.0017 0.0012 0.0030 0.0017
Kupiec LR test 0.1336 0.4884 0.1336 0.4884

1.0835 3.1983 0.3294 1.0835
Kupiec LR test p-value 0.7148 0.4847 0.7148 0.4847

0.2979 0.0737 0.5660 0.2979
EM test 0.2461 0.6536 0.2461 0.6536

1.4566 5.2857 0.4759 1.4566
EM test p-value 0.9997 0.9954 0.9997 0.9954

0.9623 0.5077 0.9981 0.9623

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

68
69

Table 1.8, Continued

GC HG
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9465 0.9495 0.9385 0.9445

0.9489 0.9501 0.9481 0.9486
Kupiec LR test 1.0169 0.0221 10.434* 2.4751

0.1074 0.0011 0.2944 0.1596
Kupiec LR test p-value 0.3133 0.8820 0.0012 0.1157

0.7431 0.9740 0.5874 0.6896
EM test 3.8148 4.5437 12.986* 4.9208

2.4156 2.8304 3.0793 3.7537
EM test p-value 0.7017 0.6035 0.0433 0.5540

0.8778 0.8298 0.7988 0.7100

99.75% Quantile
Fail/Success rate 0.9980 0.9975 0.9970 0.9970

0.9985 0.9983 0.9980 0.9980
Kupiec LR test 0.4297 0 0.3777 0.3777

1.9325 1.0528 0.4602 0.4602
Kupiec LR test p-value 0.5121 0.9994 0.5388 0.5388

0.1645 0.3049 0.4975 0.4975
EM test 166.05* 105.36* 72.753* 72.753*

2.8136 1.4126 0.6185 0.6185
EM test p-value 0.0000 0.0000 0.0000 0.0000

0.8319 0.9651 0.9961 0.9961

5% Quantile
Fail/Success rate 0.0515 0.0535 0.0575 0.0548

0.0511 0.0506 0.0531 0.0521
Kupiec LR test 0.1909 1.0169 4.5438* 1.8556

0.1074 0.0338 0.8073 0.377
Kupiec LR test p-value 0.6622 0.3133 0.0330 0.1731

0.7431 0.8540 0.3689 0.5392
EM test 7.6262 3.8851 6.1514 8.0382

9.4582 8.9128 7.8423 5.921
EM test p-value 0.2668 0.6922 0.4065 0.2353

0.1494 0.1785 0.2499 0.4321

0.25% Quantile
Fail/Success rate 0.0025 0.0025 0.0038 0.0033

0.0027 0.0025 0.0037 0.0035
Kupiec LR test 0 0 2.1727 0.8252

0.0831 0.0005 2.0983 1.3678
Kupiec LR test p-value 0.9994 0.9994 0.1405 0.3637

0.7731 0.9817 0.1475 0.2422
EM test 0.1271 0.1271 1.9628 0.9115

0.2313 0.1267 1.9107 1.3544
EM test p-value 1.0000 1.0000 0.9231 0.9888

0.9998 1.0000 0.9277 0.9686

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

69
70

Table 1.8, Continued

HO JO
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9496 0.9493 0.9438 0.9486

0.9504 0.9479 0.9511 0.9489
Kupiec LR test 0.0169 0.0408 3.0997 0.1744

0.0116 0.3671 0.105 0.1035
Kupiec LR test p-value 0.8966 0.8399 0.0783 0.6762

0.9142 0.5446 0.7460 0.7477
EM test 2.7846 3.9232 4.3361 3.7907

3.7902 7.2659 1.6062 8.6442
EM test p-value 0.8354 0.6871 0.6313 0.7050

0.7050 0.2970 0.9521 0.1946

99.75% Quantile
Fail/Success rate 0.9980 0.9978 0.9975 0.9983

0.9980 0.9978 0.9976 0.9973
Kupiec LR test 0.4551 0.1161 0.0004 1.0451

0.5249 0.1534 0.005 0.0575
Kupiec LR test p-value 0.5000 0.7333 0.9849 0.3066

0.4688 0.6953 0.9436 0.8106
EM test 62.66* 48.675* 38.902* 83.452*

0.6994 0.2676 0.1293 0.2053
EM test p-value 0.0000 0.0000 0.0000 0.0000

0.9945 0.9996 1.0000 0.9998

5% Quantile
Fail/Success rate 0.0497 0.0512 0.0549 0.0524

0.0548 0.0521 0.0553 0.0531
Kupiec LR test 0.0075 0.1196 1.9903 0.4949

1.901 0.3671 2.3043 0.7892
Kupiec LR test p-value 0.9308 0.7294 0.1583 0.4817

0.1680 0.5446 0.1290 0.3743
EM test 4.6211 5.6219 5.0439 3.5672

11.137 8.8572 6.8457 5.6962
EM test p-value 0.5932 0.4669 0.5382 0.7350

0.0842 0.1818 0.3354 0.4581

0.25% Quantile
Fail/Success rate 0.0035 0.0040 0.0042 0.0042

0.0012 0.0010 0.0012 0.0015
Kupiec LR test 1.3776 2.9775 3.97* 3.97*

3.3027 4.9512* 3.3027 2.0575
Kupiec LR test p-value 0.2405 0.0844 0.0463 0.0463

0.0692 0.0261 0.0692 0.1515
EM test 20.081* 16.386* 15.244* 15.244*

5.4977 9.7169 5.4977 3.024
EM test p-value 0.0027 0.0118 0.0184 0.0184

0.4817 0.1371 0.4817 0.8058

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

70
71

Table 1.8, Continued

KC LC
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9430 0.9445 0.9437 0.9462

0.9442 0.9438 0.9420 0.9425
Kupiec LR test 4.0065* 2.5094 3.2159 1.1935

2.7346 3.2138 5.182* 4.5739*
Kupiec LR test p-value 0.0453 0.1132 0.0729 0.2746

0.0982 0.0730 0.0228 0.0325
EM test 40.607* 14.285* 29.856* 7.9662

6.4911 10.096 7.2553 8.8315
EM test p-value 0.0000 0.0266 0.0000 0.2406

0.3705 0.1207 0.2979 0.1833

99.75% Quantile
Fail/Success rate 0.9975 0.9975 0.9978 0.9973

0.9988 0.9985 0.9988 0.9988
Kupiec LR test 0.0007 0.0007 0.1215 0.0808

3.2084 1.9814 3.2084 3.2084
Kupiec LR test p-value 0.9786 0.9786 0.7275 0.7762

0.0733 0.1593 0.0733 0.0733
EM test 0.1267 0.1267 0.233 0.229

5.3062 2.8954 5.3062 5.3062
EM test p-value 1.0000 1.0000 0.9998 0.9998

0.5052 0.8219 0.5052 0.5052

5% Quantile
Fail/Success rate 0.0548 0.0553 0.0516 0.0550

0.0526 0.0511 0.0533 0.0530
Kupiec LR test 1.8879 2.2927 0.2051 2.0855

0.5477 0.0976 0.9071 0.7775
Kupiec LR test p-value 0.1694 0.1300 0.6506 0.1487

0.4593 0.7547 0.3409 0.3779
EM test 13.923* 12.531 13.121* 17.221*

9.9034 11.374 8.4804 8.5665
EM test p-value 0.0305 0.0511 0.0412 0.0085

0.1288 0.0775 0.2050 0.1995

0.25% Quantile
Fail/Success rate 0.0012 0.0007 0.0007 0.0010

0.0017 0.0027 0.0022 0.0015
Kupiec LR test 3.1603 6.9078* 6.9078* 4.7812*

1.0897 0.0724 0.1319 1.9814
Kupiec LR test p-value 0.0754 0.0086 0.0086 0.0288

0.2965 0.7878 0.7165 0.1593
EM test 5.2092 16.756* 16.756* 9.2861

86.155* 32.533* 49.867* 2.8954
EM test p-value 0.5173 0.0102 0.0102 0.1581

0.0000 0.0000 0.0000 0.8219

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

71
72

Table 1.8, Continued

LH NG
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9455 0.9447 0.9462 0.9430

0.9461 0.9495 0.9461 0.9475
Kupiec LR test 1.7012 2.2928 1.1957 4.0023*

1.2915 0.0188 1.2915 0.5052
Kupiec LR test p-value 0.1921 0.1300 0.2742 0.0454

0.2558 0.8908 0.2558 0.4772
EM test 2.7352 10.101 2.4304 12.478

9.2121 8.0061 13.938* 9.9113
EM test p-value 0.8413 0.1205 0.8762 0.0521

0.1620 0.2377 0.0303 0.1284

99.75% Quantile
Fail/Success rate 0.9980 0.9980 0.9980 0.9975

0.9983 0.9975 0.9980 0.9973
Kupiec LR test 0.4852 0.4852 0.4852 0.0017

1.0421 0.0003 0.453 0.0864
Kupiec LR test p-value 0.4861 0.4861 0.4861 0.9667

0.3073 0.9861 0.5009 0.7688
EM test 0.6496 0.6496 0.6496 0.1272

1.3973 0.1267 0.6096 0.2346
EM test p-value 0.9955 0.9955 0.9955 1.0000

0.9660 1.0000 0.9962 0.9998

5% Quantile
Fail/Success rate 0.0582 0.0570 0.0572 0.0575

0.0502 0.0537 0.0507 0.0535
Kupiec LR test 5.4996* 4.0023* 4.2836* 4.5739*

0.0042 1.1359 0.0438 0.99
Kupiec LR test p-value 0.0190 0.0454 0.0385 0.0325

0.9481 0.2865 0.8342 0.3197
EM test 31.824* 20.102* 24.216* 17.863*

3.5559 4.1695 4.2407 4.2152
EM test p-value 0.0000 0.0027 0.0005 0.0066

0.7365 0.6537 0.6441 0.6476

0.25% Quantile
Fail/Success rate 0.0007 0.0005 0.0010 0.0007

0.0020 0.0020 0.0017 0.0017
Kupiec LR test 6.9748* 9.7909* 4.8387* 6.9748*

0.453 0.453 1.0421 1.0421
Kupiec LR test p-value 0.0083 0.0018 0.0278 0.0083

0.5009 0.5009 0.3073 0.3073
EM test 16.981* 33.09* 9.4302 16.981*

0.6096 0.6096 1.3973 1.3973
EM test p-value 0.0094 0.0000 0.1508 0.0094

0.9962 0.9962 0.9660 0.9660

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

72
73

Table 1.8, Continued

O PB
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9438 0.9421 0.9382 0.9414

0.9468 0.9423 0.9510 0.9473
Kupiec LR test 3.1481 5.098* 11.198* 6.0699*

0.8795 4.8083* 0.0774 0.6342
Kupiec LR test p-value 0.0760 0.0240 0.0008 0.0138

0.3483 0.0283 0.7809 0.4258
EM test 7.9334 8.4307 11.379 9.2039

7.2091 10.145 4.5267 6.3725
EM test p-value 0.2430 0.2082 0.0773 0.1624

0.3019 0.1187 0.6058 0.3828

99.75% Quantile
Fail/Success rate 0.9980 0.9980 0.9973 0.9975

0.9995 1.0000 1.0000 1.0000
Kupiec LR test 0.4905 0.4905 0.0703 0.0021

9.807*
Kupiec LR test p-value 0.4837 0.4837 0.7909 0.9636

0.0017 0.0000 0.0000 0.0000
EM test 0.6562 0.6562 0.2184 0.1274

33.172* 4.E+35* 4.E+35* 4.E+35*
EM test p-value 0.9954 0.9954 0.9998 1.0000

0.0000 0.0000 0.0000 0.0000

5% Quantile
Fail/Success rate 0.0476 0.0503 0.0532 0.0518

0.0564 0.0567 0.0542 0.0530
Kupiec LR test 0.5165 0.0063 0.8727 0.2587

3.4127 3.6735 1.4872 0.752
Kupiec LR test p-value 0.4723 0.9369 0.3502 0.6110

0.0647 0.0553 0.2227 0.3859
EM test 2.5762 4.4644 1.9473 2.3493

21.722* 16.238* 9.0256 5.992
EM test p-value 0.8598 0.6141 0.9245 0.8849

0.0014 0.0125 0.1721 0.4241

0.25% Quantile
Fail/Success rate 0.0017 0.0012 0.0020 0.0017

0.0005 0.0000 0.0007 0.0002
Kupiec LR test 1.0975 3.2211 0.4905 1.0975

9.807*

6.9889* 13.672*
Kupiec LR test p-value 0.2948 0.0727 0.4837 0.2948

0.0017 0.0000 0.0082 0.0002
EM test 1.4766 5.3319 0.6562 1.4766

33.172* 4.E+35* 17.029* 83.607*
EM test p-value 0.9610 0.5020 0.9954 0.9610

0.0000 0.0000 0.0092 0.0000

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

73
74

Table 1.8, Continued

S SB
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9480 0.9505 0.9441 0.9466

0.9483 0.9478 0.9481 0.9483
Kupiec LR test 0.3278 0.0208 2.8814 0.9949

0.2323 0.3905 0.3064 0.2323
Kupiec LR test p-value 0.5670 0.8853 0.0896 0.3185

0.6298 0.5320 0.5799 0.6298
EM test 13.621* 4.8617 7.9982 4.0675

13.484* 13.839* 4.2015 4.3754
EM test p-value 0.0342 0.5617 0.2382 0.6675

0.0360 0.0315 0.6494 0.6260

99.75% Quantile
Fail/Success rate 0.9975 0.9970 0.9968 0.9968

0.9983 0.9983 0.9980 0.9978
Kupiec LR test 0.0022 0.3192 0.7364 0.7364

1.0482 1.0482 0.4571 0.1172
Kupiec LR test p-value 0.9623 0.5721 0.3908 0.3908

0.3059 0.3059 0.4990 0.7321
EM test 0.1274 0.4668 0.8393 0.8393

1.406 1.406 0.6147 49.523*
EM test p-value 1.0000 0.9982 0.9910 0.9910

0.9655 0.9655 0.9962 0.0000

5% Quantile
Fail/Success rate 0.0502 0.0488 0.0554 0.0530

0.0477 0.0482 0.0507 0.0507
Kupiec LR test 0.0052 0.1307 2.4282 0.7332

0.459 0.2819 0.038 0.038
Kupiec LR test p-value 0.9426 0.7178 0.1192 0.3919

0.4981 0.5955 0.8455 0.8455
EM test 3.913 4.2458 9.3923 10.74

6.4783 6.1592 6.0051 4.4867
EM test p-value 0.6884 0.6435 0.1527 0.0967

0.3718 0.4056 0.4226 0.6111

0.25% Quantile
Fail/Success rate 0.0020 0.0022 0.0032 0.0022

0.0040 0.0040 0.0035 0.0035
Kupiec LR test 0.4926 0.1358 0.7364 0.1358

2.9716 2.9716 1.3737 1.3737
Kupiec LR test p-value 0.4828 0.7125 0.3908 0.7125

0.0847 0.0847 0.2412 0.2412
EM test 0.6589 0.2485 0.8393 0.2485

16.541* 16.541* 1.3588 1.3588
EM test p-value 0.9953 0.9997 0.9910 0.9997

0.0111 0.0111 0.9683 0.9683

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

74
75

Table 1.8, Continued

SI SM
GARCH APARCH FIGARCH FIAPARCH GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9508 0.9508 0.9486 0.9471

0.9461 0.9490 0.9394 0.9473
Kupiec LR test 0.0522 0.0522 0.1775 0.7216

1.2957 0.0825 9.007* 0.617
Kupiec LR test p-value 0.8193 0.8193 0.6736 0.3956

0.2550 0.7740 0.0027 0.4322
EM test 5.8349 8.7449 10.796 21.457*

16.184* 9.5317 16.452* 3.0423
EM test p-value 0.4419 0.1884 0.0949 0.0015

0.0128 0.1458 0.0115 0.8035

99.75% Quantile
Fail/Success rate 0.9975 0.9973 0.9970 0.9973

0.9970 0.9968 0.9958 0.9973
Kupiec LR test 0.0003 0.0859 0.3541 0.0859

0.3192 0.7364 3.8467* 0.0695
Kupiec LR test p-value 0.9855 0.7695 0.5518 0.7695

0.5721 0.3908 0.0498 0.7921
EM test 0.1267 0.2341 0.4979 0.2341

0.4668 0.8393 3.1374 0.2176
EM test p-value 1.0000 0.9998 0.9979 0.9998

0.9982 0.9910 0.7914 0.9998

5% Quantile
Fail/Success rate 0.0507 0.0507 0.0522 0.0527

0.0483 0.0498 0.0547 0.0498
Kupiec LR test 0.0423 0.0423 0.4043 0.6058

0.2569 0.0052 1.8191 0.0052
Kupiec LR test p-value 0.8370 0.8370 0.5249 0.4364

0.6123 0.9426 0.1774 0.9426
EM test 15.147* 12.938* 10.296 6.7309

3.4352 0.8601 6.6822 5.9431
EM test p-value 0.0191 0.0440 0.1127 0.3465

0.7526 0.9904 0.3512 0.4296

0.25% Quantile
Fail/Success rate 0.0025 0.0022 0.0030 0.0030

0.0025 0.0020 0.0037 0.0032
Kupiec LR test 0.0003 0.1156 0.3541 0.3541

0.0022 0.4926 2.0231 0.7364
Kupiec LR test p-value 0.9855 0.7339 0.5518 0.5518

0.9623 0.4828 0.1549 0.3908
EM test 0.1267 0.2268 27.071* 27.071*

0.1274 0.6589 1.8578 0.8393
EM test p-value 1.0000 0.9998 0.0001 0.0001

1.0000 0.9953 0.9323 0.9910

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .

75
76

Table 1.8, Continued

W
GARCH APARCH FIGARCH FIAPARCH

95% Quantile
Fail/Success rate 0.9530 0.9535 0.9495 0.9532
Kupiec LR test 0.7611 1.0393 0.0207 0.8946
Kupiec LR test p-value 0.3830 0.3080 0.8857 0.3442
EM test 12.174 14.418* 7.2265 6.0698
EM test p-value 0.0582 0.0253 0.3004 0.4154

99.75% Quantile
Fail/Success rate 0.9975 0.9973 0.9968 0.9970
Kupiec LR test 0.0022 0.0695 0.7364 0.3192
Kupiec LR test p-value 0.9623 0.7921 0.3908 0.5721
EM test 0.1274 0.2176 23.095* 0.4668
EM test p-value 1.0000 0.9998 0.0008 0.9982

5% Quantile
Fail/Success rate 0.0429 0.0441 0.0458 0.0466
Kupiec LR test 4.5731* 3.1054 1.54 1.0393
Kupiec LR test p-value 0.0325 0.0780 0.2146 0.3080
EM test 8.5348 5.0073 2.327 1.6315
EM test p-value 0.2015 0.5429 0.8873 0.9503

0.25% Quantile
Fail/Success rate 0.0032 0.0034 0.0032 0.0034
Kupiec LR test 0.7364 1.308 0.7364 1.308
Kupiec LR test p-value 0.3908 0.2528 0.3908 0.2528
EM test 23.358* 40.738* 23.358* 20.426*
EM test p-value 0.0007 0.0000 0.0007 0.0023

Notes: Kupiec LR test is a test of the null hypothesis that model forecasted fail rate is equal to the respective quantile. EM is the Engle-Manganelli dynamic
quantile regression test of the joint hypothesis that i) no expected VaR exceedences, and ii) VaR exceedences are uncorrelated .
76
77

Table 1.9: Conditional ARCH Model Comparisons

Ticker
Out-Of-Sample In-Sample

MZ R-square MSE AIC BIC VaR

BO GARCH FIGARCH FIAPARCH GARCH 4 tied
C GARCH GARCH FIAPARCH(2) GARCH(2) FIAPARCH
CC GARCH GARCH FIAPARCH(3) FIAPARCH(4) FIAPARCH
CL APARCH APARCH FIAPARCH GARCH(2) 2 tied (GARCH and FIAPARCH)
CT FIGARCH FIAPARCH FIGARCH FIGARCH(3) 2 tied (FIGARCH and FIAPARCH)
FC FIAPARCH* FIAPARCH FIAPARCH FIAPARCH 4 tied
GC GARCH FIGARCH FIAPARCH(2) FIAPARCH(2) 3 tied (GARCH, APARCH, FIAPARCH)
HG FIGARCH GARCH FIGARCH GARCH 4 tied
HO GARCH GARCH FIAPARCH FIAPARCH(3) 4 tied
JO FIAPARCH* GARCH APARCH APARCH 3 tied (GARCH, FIGARCH, FIAPARCH)
KC APARCH* FIAPARCH FIAPARCH FIAPARCH 3 tied (GARCH, APARCH, FIAPARCH)
LC FIAPARCH APARCH FIGARCH(4) FIGARCH(4) 3 tied (GARCH, APARCH, FIAPARCH)
LH FIAPARCH* FIAPARCH FIAPARCH FIAPARCH(2) 3 tied (GARCH, APARCH, FIGARCH)
NG FIAPARCH* FIAPARCH FIAPARCH(2) FIAPARCH(3) 3 tied (GARCH, APARCH, FIAPARCH)
O GARCH* GARCH FIAPARCH(2) FIGARCH(2) 3 tied (APARCH, FIGARCH, FIAPARCH)
PB FIAPARCH FIAPARCH FIAPARCH(2) FIGARCH(3) 2 tied (FIGARCH and FIAPARCH)
S FIGARCH GARCH FIAPARCH(2) FIAPARCH(2) 3 tied (APARCH, FIGARCH, FIAPARCH)
SB APARCH* APARCH FIGARCH FIGARCH(2) FIGARCH
SI FIGARCH FIGARCH FIAPARCH(3) FIGARCH(3) 3 tied(GARCH, APARCH, FIGARCH)
SM FIGARCH FIGARCH FIAPARCH FIAPARCH 2 tied (APARCH, FIAPARCH)
W FIGARCH* FIGARCH FIAPARCH FIAPARCH (2) FIAPARCH

Notes: * indicates that the R2 from the MZ test are less than 5%; numbers in brackets after a model indicate the model's relative ranking if
the ARCH LM constraint were not imposed.
77
78

the fractionally integrated models (FIGARCH and FIAPARCH) perform better in
removing the conditional heteroskedasticity, a finding in line with Baillie et al. (2007). It is
not unexpected that based on the Akaike Information Criterion (AIC) more frequently
models with relatively more parameters are selected as compared to when based on the
Bayesian (Schwartz) Information Criterion (BIC) as the latter imposes higher penalty on
the inclusion of extra variables. Nonetheless, the FIAPARCH model appears to provide
best fit based on AIC and BIC in 16 and 10 cases respectively. Furthermore, FIAPARCH
seems to model conditional volatility reasonably well for 18 of the 21 commodities as
judged by the 1-day ahead in-sample VaR tests.

Table 1.10: ARCH Model Comparisons Summary

Model
Out-Of-Sample In-Sample

MZ R-square MSE AIC BIC VaR

GARCH 6 (4) 7 0 4 12
APARCH 3 (1) 3 1 1 13
FIGARCH 6 (4) 5 4 6 12
FIAPARCH 6 (2) 6 16 10 18

Notes: in brackets are MZ results when models are compared only when the R-square is greater than 5%

When interpreting the results from the out-sample MZ tests, it should be noted that
the
2
R from the regressions of the model produced 5-day ahead forecast on the realized
variance, as proxied and described previously by the square of the returns, are extremely
low for four commodities (less than 1%) and lower than 5% for another 6. Again, as
described previously, such results should not be interpreted as clear indication of the low
79

forecasting ability of the ARCH models but rather as a result from the use of the noisy
proxy for realized volatility (see Anderson and Bollerslev, 1998). Thus, in Tables 1.7 and
1.8 when comparing the models out-of-sample, if the
2
R is less than 5%, the model
selection is marked with an asterisk, as differences between such low magnitude measures
may not be stable enough for robust inference. The results from the out-of-sample MZ
tests, however, indicate that neither of the models has a clear advantage when it comes to
forecasting volatility 5-days ahead. There is, however, a preference towards selecting the
relatively simpler models when the tests with
2
R from the Mincer and Zarnowitz test of
less than 5% are excluded - then the simple GARCH(1,1) model and the FIGARCH model
are selected most often as the best models. The results from the MZ test are largely
consistent when an alternative method for ranking the model base on MSE is applied with
no clear dominating out-of-sample model although GARCH(1,1) and FIAPARCH again
are most frequently chosen.

1.5. Conclusion
In this paper we have analyzed the distributional characteristics of commodity
future prices and the different conditional variance specifications that could adequately
describe the dynamic characteristics found in commodity futures. Our main findings can be
summarized as follows: we find that commodity futures returns exhibit little or no
autocorrelation and appear to be a stationary process. However, we found strong evidence
of heteroskedasticity and autocorrelation in higher moments. Rescaled range statistic tests
80

confirm results from prior research that the commodity volatility process exhibits
long-memory property. These results are consistent with other financial asset returns where
persistence in the volatility process is commonly found. Unlike extant research we have
explicitly modeled the seasonality of the volatility process while allowing for the error
distribution to be asymmetric and fat-tailed. It is established that Student's t-distribution is
relevant in describing the futures returns volatility while the asymmetric error distribution
is applicable only for a subset of commodities considered. The conditional volatility for
approximately half of the 21 commodity futures considered herein appear to respond
asymmetrically to positive and negative price shocks, with volatility increasing with
positive shocks. A comparison between four ARCH models (GARCH, APARCH,
FIGARCH, and FIAPARCH) indicates that while in-sample FIAPARCH appears to
outperform the rest of the models in terms of AIC, BIC and 1-day-ahead VaR, in
out-of-sample tests no model clearly prevails.

81

Chapter 2: On the Relationship between Demand for Futures
and Volatility
2.1. Introduction
The relationship between trade demand and price variability has garnered a
considerable amount of attention in the financial literature through the years. This is partly
due to the implications that many economic and financial models on market structure and
information flow have either directly on this relation or utilize it as a proxy for testing
between different theories, but also due to the bearing that this relation has on the issue of
the empirical distribution of speculative prices. In the studies on futures markets, the price
variability - trade demand relationship features prominently in the debate of whether
speculation and hedging activities are a stabilizing or a destabilizing factor in these markets
and the importance of public versus private information in determining market participants'
demand for futures contracts.
Extensive evidence on the price volatility -- trade demand in financial markets
exists with one of the earliest empirical studies being Granger and Morgenstern (1963)
which found no relation between a composite stock index price and volume changes. In an
authoritative survey Karpoff (1987) reports a stylized fact that for both equity and futures
markets the correlation between volume and the absolute value of price returns is positive
but that this association weakens as the measurement interval shortens. Among the citied
studies on commodity futures are Clark (1973), Cornell (1981) and Ruthledge (1984).
Using daily data from 1945 to 1958 on cotton futures, Clark (1973) finds a positive relation
82

between the squared price change and total volume. Similarly, Cornell (1981) and
Rutledge (1984) find positive relation between daily price variability and volumes for 17
and 13 commodity futures respectively for the sample periods being 1968-1979 and
1973-1976.
In the finance literature many studies have examined the impact of futures trading
on the volatility of the underlying securities. Employing quarterly open interest data on
futures Brorsen and Irwin (1987) find no significant relationship between price volatility
and traders' positions. In a study of eight financial and commodity futures Bessembinder
and Seguin (1992), and in a related paper Bessembinder and Seguin (1993), find that both
expected and unexpected trading volume are positively associated with volatility, while
expected open interest is negatively related to volatility. Similarly, Chang, Pinegar, and
Schachter (1997) find a positive relation between a constructed speculative trading volume
variable and volatility in two financial (SP 500 and T-Bond) and three commodity futures
(corn, soybeans and gold). Examining daily returns on the DJIA, Hiemstra and Jones
(1994) find no evidence of linear causality while at the same time a highly significant
bidirectional nonlinear causality is established. Utilizing data on trading volume
categorized by market participants (market makers, floor traders, clearing members and the
general public) Daigler and Wiley (1999) show that for five financial futures that they have
considered the general public tends to increase futures price volatility while floor traders
tend to dampen it. Using a historical survey data Irwin and Holt (2004) are able to find a
positive albeit small relationship between trading volume and volatility. Looking at net
positions by type of trader, Wang (2002a) finds that in exchange rate futures markets
83

measures of speculative activity and volatility are positively related while hedging activity
and volatility are negatively related. In a related study, Wang (2002b) provides additional
evidence on the relationship between price volatility and trading demand by type of trader.
This time, however, the author considers the Standard & Poor's (S&P) 500-stock index
futures market and finds that although expected trading demand by type of trader is
uncorrelated with price volatility, shocks in trading demands of speculators and hedgers
are significantly associated with volatility, with the relationship being negative for
speculative and positive for hedging demand shocks respectively. Employing a Directed
Acyclic Graph method to determine causality, Haigh et al. (2005) study the relationship
between speculation and price levels and price variability in crude oil and natural gas
markets. The authors find that speculators change their positions less frequently than
hedgers do, and that the speculators position changes are in response to price shifts that
prompted the hedgers to change their positions initially. Yang, Nessler, and Fung (2004)
using a modified Bray's (1981) model show that there exists a linear equilibrium
relationship between open interest and futures prices for storable commodities but not for
nonstorable commodities. Analyzing the long-run relationship between open interest and
futures prices for eleven futures via cointegration and Granger causality tests, it is found
that depending on whether the underlying commodity is storable or not the open interest
conveys the same information as the futures prices. Using Granger causality tests and
generalized forecast error variance decompositions to examine the lead-lag relationship
between futures trading activity (volume and open interest) and cash price volatility Yang,
Balyeat and Leatham (2005) show that an unexpected increase in futures trading volume
84

unidirectionally causes an increase in cash price volatility for major agricultural
commodities, that is futures trading has a destabilizing effect on agricultural commodity
markets.
While most of the research considered above has focused exclusively on the impact
that futures trading has on volatility or returns, much less attention is paid to the question of
how volatility affects trading demand for hedging and speculation in general and in
commodity futures in particular. In the theoretical literature there are two main paradigms
behind the trading demand and volatility relationship - the hedging motive and the
information transmission mechanism. In general, in the literature on optimal hedging
common theoretical models suggest that as price volatility increases, risk-averse hedgers
would likely increase their coverage (positions), and thus a significant positive relationship
is expected. The information theory models could be traced to the mixture of distribution
hypothesis of Clark (1973), and Epps and Epps (1976) where volatility and trade demand
are driven by a common latent information process and thus exhibit positive correlation.
Further refinements to the sequence of information flow have been suggested leading to
new and interesting perspectives on the volatility-trade demand relationship such as the De
Long et al. (1990) noise trader theory where uninformed traders often trade irrationally on
noise and overreact to information and cause higher volatility. Under this theory
uninformed traders are trend followers while rational traders act against noise driven price
movements, thus reducing volatility. Similar response to increased volatility is also
predicted by Chen, Cuny, and Haugen's (1995) market model where as market volatility
increases, investors lower their equity exposure by selling stocks and futures and hence the
85

new equilibrium is characterized by higher open interest and a lower futures price. Another
proposed mechanism through which higher volatility prompts higher trading demand could
be found in the noisy rational expectation model of Shalen (1993) where there are two
types of traders -- informed and uniformed, with the informed traders having private
information regarding market values. Uninformed speculators observe futures price
changes and attempt to extract price signals from them but since price changes are noisy
due to the random liquidity demand from hedgers, the uninformed speculators can then
misinterpret this liquidity trading as being due to information arrival, causing them to
adjust their positions and hence resulting in increases in trading demand.
Chen et al. (1995) investigate empirically the effect of volatility on trade demand
and find a significant positive relation for the SP 500 futures contract. In another study,
Chang, Chou, and Nelling (2000) investigate again the contemporaneous effect of
volatility on open interest held by speculators, hedgers, and small traders in the S&P
500-stock index futures market and find that the relation between volatility and open
interest is positive for hedgers (both long and short positions), but negative for speculators
(again for both long and short positions). Pan, Liu and Roth (2002) also examine how
volatility affects trading demand for futures in the SP 500 index futures market. The
authors find that increases in volatility prompts both hedgers and speculators to increase
their positions. Fujihara and Mougoue (1997) find evidence of significant bidirectional
nonlinear dependence in the returns and volume for the three petroleum futures contracts
utilizing the Hiemstra and Jones (1994) nonlinear causality tests. Utilizing a VAR
framework Fung and Patterson (1999) have found evidence of unidirectional causality in
86

currency markets. Chen and Daigler (2008) analyzed 6 futures markets (two agricultural
and four financial) via VAR and VAR-GARCH methods but are able to find Granger
causality from volatility to volume for only a few of the series, and to establish a positive
relationship between unexpected volatility and volume.
The purpose of this paper is to examine not only the contemporaneous but also and
the dynamic effects of expected volatility on demand for commodity futures by different
types of market participants. As noted previously, the emphasis of most of the research to
date on trade demand has centered on the return-volume relationship. When the trade
demand-volatility is explored typically this analysis is performed for financial markets and
rarely the effect of volatility on trade demand is studied. Furthermore, the vast body of
literature on the subject has predominantly studied the relationship either via ordinary least
square regression (for contemporaneous models) or Granger causality tests in the
conditional mean or variance. Utilizing a novel approach to test for contemporaneous
effects we establish that volatility has an asymmetric effect on the futures demand
distribution. Moreover, in stark contrast to prior research in financial futures, we find that
for majority of commodity futures volatility has negative effect on hedging demand for
futures. Further, our results on Granger causality extend the results from causility in
conditional mean to causality in quantiles, showing that that expected volatility
Granger-causes futures demand in distribution.

87

2.2. Data
In this paper, we analyze the contemporaneous as well dynamic relationship
between hedging and speculative demand for 21 commodity futures and futures price
volatility for the period March 9, 1993 through April 28, 2009
9

9
For lean hogs (LH) the sample period begins on 04/02/1996, while for frozen pork bellies (PB) the sample
ends on 07/22/2008.
. Daily commodity futures
price data has been obtained from ProphetX for the nearby futures contract (see Chapter 1
for more detailed description of the data). From the U.S. Commodity Futures Trading
Commission (CFTC) data on the historical weekly Commitment of Traders Reports have
been collected. The Commitments of Traders (COT) reports provide a breakdown of each
Tuesday's open interest for markets in which 20 or more traders hold positions equal to or
above the reporting levels established by the CFTC. Further, CFTC COT reports provide a
breakdown of all the trader participation by categories (commercial, non-commercial, and
non-reportable) and positions taken (long, short, or spread). All of a trader's reported
futures positions in a commodity are classified by CFTC as commercial if the trader uses
futures contracts in that particular commodity for hedging, or alternatively, the entities are
classified as non-commercial provided that their positions are above the commission set
minimal reportable level. Hence, it is natural to consider the reported commercial and
non-commercial positions to represent the hedging and speculative demand for futures
respectively. Ederington and Lee (2002), however, caution that such identification is not
always entirely accurate as there could be an incentive for speculators to classify
themselves as hedgers in order to gain from less stringent open position limits.

88

Nevertheless, given that CFTC has had well-established procedures for classifying
positions as commercial, that is determining whether an entity has an exposure to the
underlying risk associated with the particular futures, and the fact that such exposures
would typically be more unequivocally established for commodity futures than financial
futures, in this paper we assume that commercial positions represent a reasonable proxy for
commodity hedgers demand for futures. Consequently, we will interchangeably refer to
non-commercial traders as speculators and commercial traders as hedgers and vice versa.
Open interest has been used in a number of studies as a proxy for hedgers' opinions
(Kamara, 1993), or hedging demand (Chen et al., 1995), market depth (Bessembinder and
Seguin, 1993), and the difference in traders' opinions (Bessembinder et al., 1996).
In futures markets, the aggregate of all long open interest is equal to the aggregate of all
short open interest and thus given the three categories of positions reported by CFTC the
total open interest in commodity futures could be represented by either of the following:
nrs sprd ncs cs
nrl sprd ncl cl oi
+ + + =
+ + + =

where cl (cs), ncl (ncs), nrl (nrs) are the commercial long (short), non-commercial long
(short), and non-reportable long (short) positions and sprd is additional the number of open
interest of non-commercial positions holding both short and long positions. Further, it
should be self-evident from the above representations of total open interest that if we
define the net demand by type of position as the difference between long and short
positions then the following is true:
netnr netnc netc + + = 0
89

where netc is the net commercial positions, netnc - the net non-commercial, and netnr - the
net non-reportable positions. It is worth mentioning that in general it needs not to be the
case that a hedger's short position should be offset exactly by a speculator's long position
since a hedger's counterparty could very well be another hedger instead but with a long
position in the contract.
Summary statistics for open interest positions are reported in Tables 2.1 and 2.2
below. As it could be seen in Figure 2.1 open interest is increasing with time for all
commodities considered but for oats, silver (no clear trend) and pork bellies (decreasing).
The number of traders, both speculators and hedgers, is also increasing with time further
indicating the increasing economic significance of futures markets. It is observed also that
since 2003 open interest has increased exponentially for almost all commodities with an
even more dramatic collapse in 2008. For most commodities considered, hedgers tend to be
on average short (in 19 of 21 commodities), while speculators tend to take the opposite
side, i.e., they are net long in 18 of 21 commodities. Figure 2.2 illustrates another
interesting phenomenon -- even though net positions for hedgers and speculators have
oscillated for much of the sample period, there exists a clear relative increase in the net
long positions of speculators and respectively increase in net short positions of hedgers
since 2000, a fact commonly attributed to the abundance of funds commodity pools and
exchange traded commodity tracking funds have been able to draw post the 2001 tech
bubble. Although non-reportable positions are on average comparable in percentage of
total open interest as to those of non-commercial traders, speculator's positions volatility is
far higher which is to be expected as small traders positions consists of those of both small
90

hedgers and speculators as they are lumped together in the non-reportable positions and
thus decreasing their overall volatility. It is noted, however, that for natural gas only, both
hedgers and speculators are net short while small traders (non-reportable positions) have
on average the opposite, long, position. A commonly utilized index for measuring hedging
pressure in commodity markets is constructed as the ratio of net short hedgers positions
and the total hedgers open interest, i.e.,
cl cs
cl cs
hp
+
−
=

This index is usually interpreted as a signal whether a market would be expected to
be in backwardation or contango as per Keynes' (1939) normal backwardation theory. The
proposed argument has been that higher relative ratios indicate that more speculators
would need to be taking opposite positions and since hedgers are willing to pay a premium
in order to reduce their underlying risk exposure, futures prices would tend to be lower
(higher) if the hedging pressure ratio is positive (negative). Table 2.1 shows that on
average we observe a relatively higher hedging pressure in only a few markets - gold,
orange juice, oats, and silver.

91

Table 2.1: Open Interest Descriptive Statistics

oi ncl ncs sprd cl cs nrl nrs netc netnc netnr hp

BO 152,030 29,852 16,288 21,460 70,751 94,341 29,962 19,936 -23,590 13,564 10,027 0.146
C 578,990 125,990 63,831 58,076 283,810 304,640 111,110 152,440 -20,833 62,160 -41,327 0.005
CC 103,430 21,848 15,211 4,593 64,233 76,925 12,759 6,705 -12,691 6,637 6,054 0.084
CL 651,230 81,747 67,274 91,627 406,530 419,230 71,322 73,098 -12,697 14,473 -1,776 0.014
CT 95,305 20,058 20,215 6,412 54,299 58,296 14,536 10,383 -3,997 -156 4,154 0.011
FC 19,426 5,932 3,396 1,736 5,918 4,360 5,840 9,934 1,558 2,536 -4,094 -0.138
GC 228,680 73,487 37,673 27,405 83,587 139,790 44,204 23,812 -56,206 35,814 20,392 0.201
HG 73,387 17,434 13,788 4,375 35,060 43,450 16,518 11,774 -8,390 3,647 4,743 0.113
HO 163,900 16,707 12,002 12,691 94,447 110,170 40,057 29,041 -15,721 4,705 11,016 0.083
JO 27,575 8,040 4,881 1,749 12,028 18,228 5,757 2,717 -6,200 3,160 3,040 0.189
KC 70,586 18,006 12,672 6,493 34,483 44,549 11,604 6,872 -10,066 5,334 4,732 0.134
LC 131,550 31,711 19,498 16,291 56,928 58,916 26,623 36,847 -1,989 12,213 -10,224 0.041
LH 82,749 17,545 13,997 14,503 36,889 34,723 13,812 19,526 2,166 3,547 -5,714 NaN
NG 398,060 38,079 51,969 87,024 221,750 230,310 51,200 28,750 -8,560 -13,890 22,450 0.032
O 12,573 2,626 911 502 4,437 9,000 5,009 2,160 -4,563 1,715 2,848 0.360
PB 5,043 1,492 1,499 347 886 973 2,318 2,225 -87 -6 94 -0.001
S 232,010 53,028 28,256 27,127 98,480 122,200 53,372 54,420 -23,724 24,772 -1,048 0.117
SB 289,940 62,919 33,423 16,347 158,230 204,480 52,446 35,695 -46,247 29,497 16,751 0.136
SI 100,050 36,706 10,935 10,148 22,971 67,869 30,226 11,099 -44,898 25,771 19,127 0.492
SM 133,690 23,761 12,043 14,259 61,303 85,632 34,362 21,750 -24,330 11,718 12,612 0.166
W 170,820 36,983 34,579 21,492 81,720 83,484 30,623 31,264 -1,764 2,405 -641 0.076

91
92

Table 2.2: Traders’ Positions as Percentage of Open Interest

ncl ncs sprd cl cs nrl nrs

BO 18.51% 11.00% 13.23% 44.68% 59.98% 23.58% 15.79%
C 19.20% 11.37% 8.15% 49.11% 49.83% 23.55% 30.66%
CC 19.48% 14.56% 4.21% 63.04% 74.12% 13.26% 7.11%
CL 10.88% 9.03% 10.70% 64.94% 66.64% 13.48% 13.63%
CT 18.86% 21.28% 5.69% 57.59% 59.62% 17.86% 13.41%
FC 28.30% 17.45% 7.97% 30.08% 22.14% 33.65% 52.44%
GC 27.64% 18.27% 10.48% 40.91% 59.19% 20.97% 12.06%
HG 22.94% 17.85% 5.15% 47.23% 59.31% 24.68% 17.69%
HO 9.72% 6.98% 7.19% 57.24% 67.26% 25.85% 18.57%
JO 27.75% 18.18% 6.30% 45.34% 65.36% 20.61% 10.16%
KC 22.66% 16.69% 7.53% 47.32% 61.85% 22.49% 13.93%
LC 22.93% 14.10% 10.50% 41.57% 44.58% 25.01% 30.82%
LH 22.60% 16.76% 13.63% 41.36% 39.26% 22.41% 30.35%
NG 8.33% 9.76% 14.64% 60.93% 66.01% 16.11% 9.60%
O 21.59% 7.31% 3.66% 34.31% 70.62% 40.44% 18.40%
PB 27.95% 27.28% 6.43% 21.78% 19.70% 43.84% 46.59%
S 21.11% 12.60% 10.94% 40.88% 51.21% 27.07% 25.26%
SB 19.46% 10.78% 3.73% 53.21% 69.41% 23.61% 16.08%
SI 37.09% 11.85% 9.40% 22.82% 67.38% 30.70% 11.37%
SM 16.73% 8.75% 10.13% 45.15% 63.10% 27.99% 18.02%
W 22.42% 19.12% 9.90% 42.49% 48.71% 25.20% 22.27%
92
93

Figure 2.1: Open Interest By Traders’ Positions

1990 2000 2010
0
1
2
3
4
x 10
5
Positions for BO
1990 2000 2010
0
0.5
1
1.5
2
x 10
6
Positions for C
1990 2000 2010
0
0.5
1
1.5
2
x 10
5
Positions for CC
1990 2000 2010
0
0.5
1
1.5
2
x 10
6
Positions for CL
1990 2000 2010
0
1
2
3
x 10
5
Positions for CT
1990 2000 2010
0
2
4
6
x 10
4
Positions for FC
1990 2000 2010
0
2
4
6
x 10
5
Positions for GC
1990 2000 2010
0
5
10
15
x 10
4
Positions for HG
1990 2000 2010
0
1
2
3
x 10
5
Positions for HO

Total
NonComm Long
NonComm Short
Comm Long
Comm Short
94

Figure 2.1, Continued

1990 2000 2010
0
0.5
1
1.5
2
x 10
5
Positions for SI
1990 2000 2010
0
1
2
3
x 10
5
Positions for SM
1990 2000 2010
0
2
4
6
x 10
5
Positions for W
1990 2000 2010
0
2
4
6
x 10
4
Positions for JO
1990 2000 2010
0
1
2
3
x 10
5
Positions for KC
1990 2000 2010
0
1
2
3
4
x 10
5
Positions for LC
1990 2000 2010
0
1
2
3
x 10
5
Positions for LH
1990 2000 2010
0
5
10
x 10
5
Positions for NG
1990 2000 2010
0
1
2
3
x 10
4
Positions for O
1990 2000 2010
0
5000
10000
15000
Positions for PB
1990 2000 2010
0
2
4
6
8
x 10
5
Positions for S
1990 2000 2010
0
5
10
15
x 10
5
Positions for SB

Total
NonComm Long
NonComm Short
Comm Long
Comm Short
95

Figure 2.2: Net Open Interest by Traders’ Positions

1990 2000 2010
-2
-1
0
1
x 10
5
Net positions - BO
1990 2000 2010
-4
-2
0
2
4
x 10
5
Net positions - C
1990 2000 2010
-1
-0.5
0
0.5
1
x 10
5
Net positions - CC
1990 2000 2010
-2
-1
0
1
2
x 10
5
Net positions - CL
1990 2000 2010
-1
-0.5
0
0.5
1
x 10
5
Net positions - CT
1990 2000 2010
-2
-1
0
1
2
x 10
4
Net positions - FC
1990 2000 2010
-4
-2
0
2
4
x 10
5
Net positions - GC
1990 2000 2010
-10
-5
0
5
x 10
4
Net positions - HG
1990 2000 2010
-10
-5
0
5
x 10
4
Net positions - HO

NetComm
Net NonComm
NetNonReportable
96

Figure 2.2, Continued

1990 2000 2010
-4
-2
0
2
4
x 10
4
Net positions - JO
1990 2000 2010
-1
-0.5
0
0.5
1
x 10
5
Net positions - KC
1990 2000 2010
-5
0
5
10
x 10
4
Net positions - LC
1990 2000 2010
-4
-2
0
2
4
x 10
4
Net positions - LH
1990 2000 2010
-2
-1
0
1
2
x 10
5
Net positions - NG
1990 2000 2010
-2
-1
0
1
2
x 10
4
Net positions - O
1990 2000 2010
-5000
0
5000
Net positions - PB
1990 2000 2010
-2
-1
0
1
2
x 10
5
Net positions - S
1990 2000 2010
-4
-2
0
2
4
x 10
5
Net positions - SB
1990 2000 2010
-1
-0.5
0
0.5
1
x 10
5
Net positions - SI
1990 2000 2010
-2
-1
0
1
x 10
5
Net positions - SM
1990 2000 2010
-1
-0.5
0
0.5
1
x 10
5
Net positions - W

NetComm
Net NonComm
NetNonReportable
97

2.3. Empirical Analysis
2.3.1. Volatility Estimates
As discussed in Chapter 1, expected volatility of futures prices is not a readily
observable variable. In this paper, expected volatility is constructed from fitting a GARCH
model to the daily future returns. Results from chapter 1 indicate that for different
commodities and depending on the measure used for comparison different GARCH
models are found to best describe the underlying data. To side step this issue, here instead
we have chosen to employ the simple AR(1)-GARCH(1,1) model with seasonality and
skewed Student's t-distribution conditional errors
10












+ 





=
+ + + =
∼
=
+ + =
∑
=
− −
−
12
2
cos
12
2
sin
) ; 1 , 0 (
, 2 , 1
1
2
1 1
2
1 1
2
1
t
i
t
i
k
i
t
t t t t
t
t t t
t t t
im im
s
s e
D u
u e
e r r
π
γ
π
γ
α σ β ω σ
ν
σ
φ µ
, the estimation for which we have
described and performed previously in Chapter 1. Furthermore, most of the literature on
the subject has preferred utilizing a simple GARCH (1,1) model (Antoniou and Foster,
1992; Gulen and Mayhew, 2000; Yang, Balyeat, and Leatham, 2005). For the sake of
concreteness, the following GARCH model has been estimated for each commodity:

where
t
r is the continuously compounded return of futures prices,
t
σ is the conditional
volatility,
t
s are seasonality dummies, and ) ; 1 , 0 ( ν D is either an (asymmetric) Student's

10
Results utilizing the AR(p) FIAPARCH model are consistent with the simpler GARCH model used hereby.

(2.1)
98

t-distribution or the normal distribution. If seasonal dummies or the skewness parameter of
the Student's t-distribution are found to be insignificant, the model was reestimated
omitting these variables.

2.3.2. Base Model
Similarly to Pan et al. (2002), Chang et al. (2000) and Chen et al. (1995), the
contemporaneous effect of expected volatility on demand for futures
11
t t t
n m
t
n m
t
Time Time TTM Y Y ε θ δ γ βσ φ α + + + + + + =
−
2 2 ,
1
,
is analyzed by
utilizing the following model:

where Y is open interest for one of the following series: m={hedgers, speculators, small
traders}, n={long, short, net};
t
TTM is the time to maturity of the prompt month futures
contract in years, and Time captures the time trend in open interest. We do not include
seasonal dummies as open interest series are reported for all traded futures contracts thus
negating to a great extent the seasonality effect through averaging across different maturity
months. Time to maturity is included as a control variable to account for jumps in open
interest when prompt month contract is switched. Since we are investigating the effect of
volatility on demand for futures, the sign and magnitude of the β coefficient is of
particular interest - a large and positive β in general is expected based on the
predominant view and empirical results that as price volatility increases, hedgers

11
When we refer to demand for futures this should be interpreted loosely as under this term we include
collectively the demand for long, short and net positions for hedgers, speculators and small traders.

(2.2)
99

regardless of whether they are short or long would like to further protect themselves from
this additional variability introduced by futures prices.

2.3.3. Unit Roots Tests
Before proceeding with estimation of the above model, it needs to be established
whether the open interest series are stationary as otherwise any inferences could be the
result of spurious regressions. To that effect, all of the open interest series are tested for
unit root using the Augmented Dickey Fuller (ADF) test. Since unit root tests based on
testing the null hypothesis of a unit root are notorious for being plagued by lower power
against local alternative hypotheses, the Kwiatkowski, Phillips, Schmidt, and Shin (1992)
KPSS test is utilized as well since it is based on the null hypothesis that the evaluated series
are stationary. Results from both tests are reported in Table 2.3 below. Evidently, for some
of the series and commodities considered the ADF indicates that the null hypothesis for
unit root cannot be rejected at usual confidence levels (i.e. 95% or higher); however, for all
of these instances without exception the KPSS test's null hypothesis of stationarity cannot
be rejected as well. In light of such conflicting evidence as to the stationarity of the
investigated data series, the stationarity assumption have been maintained on economic
basis as it is unreasonable to expect that any of the open interest series would deviate to
infinity in the long term.
100

Table 2.3: Unit Root Tests on Open Interest Series

Ticker Test oi ncl ncs cl cs nrl nrs netc netnc netnr

BO
ADF -3.3104* -4.4254** -7.0244** -3.4184** -3.7578** -6.5936** -5.8203** -5.0978** -4.6776** -6.052**
KPSS 0.2735 0.1215** 0.207* 0.3029 0.2599 0.1326** 0.227 0.5329 0.1565* 0.1376**

C
ADF -1.87 -2.5251 -6.6848** -1.6544 -2.0897 -3.5962** -2.2497 -3.7832** -3.2132** -2.3091
KPSS 0.5036 0.4391 0.0592*** 0.4993 0.4437 0.4034 0.5208 0.5916 0.3481 1.6347

CC
ADF -4.4506** -3.8203** -5.5294** -4.4126** -3.8463** -4.3589** -6.5938** -4.4767** -4.4376** -3.1217**
KPSS 0.2027* 0.559 0.1594* 0.3125 0.2441 0.1702* 0.2742 0.592 0.5953 0.7273

CL
ADF -1.6986 -4.5168** -4.4771** -2.2695 -2.366 -3.2052* -2.1656 -6.8581** -6.5239** -6.546**
KPSS 0.6619 0.6965 0.7488 0.4735 0.4373 0.3982 0.5201 0.1663* 0.1078*** 0.6633

CT
ADF -2.4102 -4.4891** -5.209** -2.7826 -3.403* -3.8768** -5.4145** -5.5802** -5.3634** -4.8217**
KPSS 0.4908 0.3473 0.1725* 0.4952 0.3786 0.283 0.4519 0.3657 0.1103*** 0.7931

FC
ADF -3.7056** -4.1313** -6.1562** -4.3332** -5.0608** -3.6009** -3.4314** -4.9984** -4.5492** -2.4683
KPSS 0.2337 0.1145*** 0.234 0.1268** 0.1879* 0.2536 0.1216** 1.3576 0.0612*** 1.7785

GC
ADF -3.5457** -4.8971** -6.3607** -4.7282** -4.1742** -4.5181** -3.0886 -3.4762** -3.3793** -4.2558**
KPSS 0.6305 0.62 0.2215 0.3788 0.6799 0.4153 0.4016 2.5164 0.6208 1.8026

HG
ADF -5.6614** -5.476** -5.812** -5.1384** -4.3511** -4.3142** -3.4132* -5.0143** -5.149** -4.5173**
KPSS 0.2158* 0.2642 0.0451*** 0.1986* 0.4094 0.1948* 0.4242 0.4809 0.2029* 1.2239

HO
ADF -3.2934* -6.4731** -5.4611** -3.3148* -3.5971** -3.8797** -2.6139 -6.9672** -6.5398** -5.3826**
KPSS 0.3167 0.3385 0.2226 0.1467* 0.1868* 0.2642 0.3027 0.1076*** 0.1026*** 1.7767

JO
ADF -4.8588** -5.0868** -6.7119** -5.8216** -4.8767** -2.2545 -3.703** -4.8087** -5.2824** -2.9439**
KPSS 0.1257** 0.1141*** 0.066*** 0.078*** 0.1073*** 0.1497* 0.2837 0.2392 0.1043*** 0.2539

Notes: ***(**) indicate rejection of the null hypothesis (unit root for the ADF and stationarity for KPSS) at the 1% (5%) level.
100
101

Table 2.3, Continued

Ticker Test oi ncl ncs cl cs nrl nrs netc netnc netnr
KC
ADF -3.3507* -4.3277** -6.9227** -3.7902** -3.7405** -4.6285** -8.846** -5.7415** -5.4665** -3.9085**
KPSS 0.5933 0.4142 0.1169*** 0.6253 0.5209 0.2486 0.318 0.597 0.2589 0.3401

LC
ADF -2.6318 -4.3353** -4.4584** -2.1107 -3.7223** -5.3272** -4.4103** -4.2645** -4.6695** -2.4656
KPSS 0.4964 0.2213 0.4887 0.482 0.4059 0.1539* 0.2754 0.4585 0.0862*** 2.3825

LH
ADF -1.5946 -3.4262** -2.9207 -1.3058 -2.3476 -4.9267** -2.9869 -4.691** -4.953** -2.9556**
KPSS 0.5415 0.4271 0.543 0.5305 0.4164 0.3499 0.282 0.5621 0.0927*** 1.4892

NG
ADF -2.6816 -3.6498** -2.3374 -1.9684 -1.9176 -3.2837* -3.7111** -3.5207** -2.362 -3.2315**
KPSS 0.2988 0.4338 0.5369 0.2257 0.2965 0.2331 0.2668 0.794 0.3326 2.7441

O
ADF -3.6464** -4.1406** -5.5937** -2.6785 -3.7517** -5.0234** -5.7583** -4.6912** -4.2634** -3.2825**
KPSS 0.2757 0.1999* 0.1077*** 0.3328 0.3075 0.0881*** 0.1353** 0.6509 0.1685* 1.7802

PB
ADF -5.2959** -6.8435** -4.5813** -5.1675** -5.1405** -4.4256** -4.5182** -5.0948** -5.3036** -4.2037**
KPSS 0.152* 0.1694* 0.1072*** 0.1166*** 0.1035*** 0.1908* 0.076*** 1.4699 0.0497*** 0.7616

S
ADF -2.7745 -3.1301* -5.1083** -2.8512 -2.7842 -4.8431** -3.0084 -3.4219** -3.2212** -2.1907
KPSS 0.5223 0.3727 0.1169*** 0.5193 0.3504 0.2531 0.6081 0.1214** 0.2295 2.1257

SB
ADF -2.1659 -3.1947* -4.1439** -2.3332 -2.5134 -5.3423** -5.2783** -4.3814** -3.8129** -5.661**
KPSS 0.6898 0.5566 0.2183 0.6379 0.6446 0.4814 0.5524 0.9468 0.2977 0.4212

SI
ADF -4.1366** -6.4776** -7.0152** -5.1231** -4.6332** -3.6734** -5.242** -5.6638** -6.1712** -3.9139**
KPSS 0.5837 0.1245** 0.2329 0.5378 0.521 0.4641 0.5246 0.5714 0.1825* 0.4177

SM
ADF -3.3717* -5.2013** -4.1698** -3.8932** -3.3622* -6.2404** -4.6931** -4.5011** -4.3701** -5.7591**
KPSS 0.0717*** 0.1451** 0.0962*** 0.0724*** 0.0562*** 0.1233** 0.1903* 0.5694 0.0916*** 0.1694*
W
ADF -2.0652 -4.5061** -4.4029** -1.4887 -2.6716 -4.3349** -2.5898 -4.9487** -5.9477** -2.0773
KPSS 0.4291 0.4862 0.3563 0.4424 0.3437 0.3117 0.5149 1.8351 0.089*** 2.0318
101
102

2.3.4. Ordinary Least Squares Estimation
Since the tests of stationarity indicate that the open interest series appear to be
stationary we estimate equation 2.2 in levels. The results for the volatility coefficient from
the OLS tests in Table 2.4 appear to indicate that for a considerable number of
commodities and open interest series considered herein the effect of volatility on futures
demand is significantly different from zero utilizing White’s (1982) heteroskedasticity
robust standard errors. It appears, however, that the effect of volatility on futures demand is
rather surprising - an increase in expected volatility tends to decrease hedging demand for
both short and long hedgers, that is, as commodity futures market become more volatile
hedgers tend to reduce their positions. This finding is in stark contrast to previous research
which typically predicts a positive relationship. For most commodities it appears that the
magnitudes of the decreases in hedgers short positions is greater than those for hedgers'
long positions, thus resulting in an actual net positive volatility effect on hedgers' net
positions. For speculators, it is observed that higher volatility reduces their demand for
long futures positions but not by as much as for short positions, and consequently on the
net higher volatility tends to reduce their net positions. One model that to certain extent
could explain our findings is that of Peck (1981) where an increase in market liquidity
provided by speculators would lead to a decrease in volatility. An alternative explanation
that we hypothesize but leave for future research is that the marked-to-market rules to
which futures are subject to could potentially result in large cash outflows for hedgers who,

103

Table 2.4: Contemporaneous Effect of Expected Volatility

oi cl cs ncl ncs

BO -672.7*** -230.3 -502.18*** -394.74*** -48.71
C -1641.16*** -373.41 -1123.89*** -926.27*** -362.86
CC -334.69*** -62.18 -220.6** -245.79*** -57.29
CL -449.76** -210.35 -226.49 -203.65** -92.79
CT -649.71*** -273.34*** -484.63*** -298.88*** -132.22
FC 103.59 62.12 76.37** -62.12 -83.25
GC -2241.67*** -45.58 -1486.07*** -1634.86*** -409.44
HG -139.18** -28.54 -44.4 -81.86** -6.63
HO -122.49 -15.55 -36.28 -196.98*** -30.65
JO -25.67 -16.54 15.29 6.97 -36.43
KC -88.82*** -41.65*** -35.63 -22.57 -49.45***
LC -760.66** -396.97** -262.88 -417.45** -544.5***
LH -4.25 -35.72 19 -4.51 -58.2
NG -256.7*** -65.18 -150.44*** -84.76*** -29.93
O -29.07*** -0.49 -12.3 -10.85*** -14.02***
PB -26.77*** -9.86*** -4.87** -4.5 -13.31***
S -1193.64*** -200.34 -865.46*** -674.68*** -57.33
SB -1094.03*** -449.89*** -815.66*** -494.87*** -177.14
SI -337.77*** -74.9** -214.56*** -160.19*** -58.41**
SM -432.99*** -63.22 -356.3*** -303.12*** -73.17
W -681.4*** -220.58** -367.76*** -200.72** -248.34***

Notes: ***,**, and *, indicate significance at the 99%, 95%, and 90% level respectively.
The reported coefficients are those for the expected volatility from the OLS regression.

104

Table 2.4, Continued

nrl nrs netc netnc netnr

BO -125.44** -17 366.46 -288.64 -96.62
C -366.31** -376.93*** 775.54** -858.29** 89.61
CC -68.23** -192.45*** 212.66** -211.83** -14.05
CL -8.69 -90.05 91.15 -155.07 78.64
CT -134.41** -86.13** 210.79 -154.4 -81.51***
FC 44.96 80.26 -4.93 27.13 -47.1
GC -537.42*** -91.74 1478.31*** -1182.64** -344.31**
HG 12.35 -21.63 20.88 -49.66 36.6**
HO -32.82 -49.71 83.35 -106.8 6.62
JO -13.63 -1.08 -29.9 32.24 -2.05
KC -29.62** -16.46** -7.12 18.18 -12.15
LC -88.06 -166.44 -270.9 140.49 103.31
LH -6.61 13.38 -32.6 52.14 -21.35
NG -48.48*** -16.54 93.4*** -54.9 -41.23***
O -18.75*** -4.35 13.14 0.65 -14.46**
PB -14.58*** -8.82** -5.44 12.26** -6.14
S -213.05** -176.46 666.98*** -638.84*** -19.47
SB -364.94*** -189.89** 417.68 -292.35 -160.35
SI -76.15*** -9.95 156.98*** -108.51** -58.14***
SM -63.42 0.73 283.62** -211.73** -66.21
W -73.48 -120.34** -178.21 40.57 69.97

Notes: ***,**, and *, indicate significance at the 99%, 95%, and 90% level respectively.
The reported coefficients are those for the expected volatility from the OLS regression.

105

facing such liquidity risks, may be forced to close positions that are effectively still
hedging their overall economic risk exposures.

2.3.5. Quantile Regression
While the results from estimating equation 2.2 via OLS are quite intriguing, it is
quite possible that various parts of the distribution of futures demand could exhibit
different response to changes in volatility. The OLS approach provides us with an estimate
of the effects explanatory variables have on the conditional mean of the dependent variable
only. Mosteller and Tukey (1977) note that "[...] just as the mean gives an incomplete
picture of a single distribution, so the regression curve gives a correspondingly incomplete
picture for a set of distributions". Further, Granger (2003) notes that the study of the time
series of quantiles is relevant as the predictive distribution can be expressed in terms of
quantiles. Quantile regression as introduced by Koenker and Bassett (1978) could be
viewed as a natural extension of the classical estimation of the conditional mean via OLS,
providing a more complete description of the conditional distribution and the relationship
between variables. In particular, via quantile regression we could explore the existence of
any heterogeneity in the response of futures demand to changes in volatility by looking at
slices of the conditional distribution without any assumptions on global distribution.
The τ -th quantile of the distribution of a random variable Y is defined as the real number
y such that ( ) ( ) { } τ τ ≥ = y F y Q : inf where ( ) . F is the cumulative density function of
Y . Therefore, the τ -th quantile is simply the inverse cumulative distribution function at
106

the specified quantile, i.e., ( ) ( ).
1
τ τ
−
= F Q Although natural, the above definition for
quantile function could instead be cast in an equivalent linear optimization form,
effectively replacing the problem of sorting with that of optimizing, and thus allowing for
straightforward generalization for estimating more complex quantile functions. The
general form of this optimization problem for the quantile function of Y is defined
explicitly as follows:
( ) ( )






− =
∑
=
∈
ξ ρ τ
τ
ξ
i
T
i
y Q
1
min arg
R

where for a given random variable s ( )
{ }
( )
0 <
− =
s
I s s τ ρ
τ
and
{ } A
I is the indicator
function of event A . For example, the median quantile ( ) 5 . 0 Q which plays prominent
role in empirical analysis could be shown to be the optimum estimator under mean absolute
deviation loss function, that is:
( )






− =
∑
=
∈
ξ
ξ
i
T
i
y Q
1
min arg 5 . 0
R

Quantile regression has the additional appeal of mirroring the classical regression analysis
set up but instead of minimizing the sum of squared deviations, here the asymmetrically
weighted absolute deviations are minimized.
Similarly to the classical ordinary least squares where we define the conditional
mean of the dependent variable y as a linear function of exogenous variables, e.g.
[ ] β X X y E = | , we could define the conditional distribution of Y to be dependent on a set
of explanatory variables, that is, the conditional quantile function of Y --
(2.3)
(2.4)
107

( ) ( ) τ β τ X X Q
Y
= | . Proceeding in parallel fashion to equation 2.3 we have:
( )
( )
( ) ( )






− =
∑
=
∈
τ β ρ τ
τ
τ β
i i
T
i
x y X Q
1
min arg |
R

Koenker and Bessett (1978) have explored in some detail the properties of the
regression quantile estimates, ( ) τ β , and have shown that if the following conditions are
satisfied:
1. The cumulative distribution and density functions of are absolutely continuous, and
the density is bounded away from zero and infinity.
2. There ∃ 0 > Q such that Q T X X
T
= ′
∞ →
/ lim
3. There ∃ 0 > D such that ( ) ( ) D T x x x F f
i i i Y i T
= ∑
′
−
∞ →
/ | lim
1
τ
then
( ) ( ) ( ) ( ) ( )
1 1
1 , 0 ?
ˆ
− −
− − QD D N T τ τ τ β τ β
If it is assumed that errors are independently and identically distributed then the variance
matrix simplifies to:
1 2 1 1 − − −
= Q QD D ω
where
( )
( ) ( ) τ
τ τ
ω
1
1
−
−
=
F f
and is typically referred to as the sparsity function. In this paper,
however, we do not restrict errors to be identically distributed, i.e., it is assumed that the
errors are d ni i . . .

Consequently, we estimate the matrix D via Huber sandwich and Powell's (1984)
(2.5)
(2.6)
(2.7)
108

kernel approach. Explicitly,
( )
( )
( )
′








=
∑
=
i i
t
T
t
x x
T h
e
k
T T h
D
τ ˆ 1
ˆ
1

where ( ) . k is the kernel and ( ) T h - the kernel's bandwidth, and ( ) τ
t
e ˆ are the residual
errors from estimating the τ -th quantile function.
Koenker and Bessett (1982) develop a series of tests of linear restrictions on estimated
quantile coefficients for a given quantile [ ] u u − ∈ 1 , τ for some 0 > u and 5 . 0 < . In this
paper we utilize the Wald test. For a set of q restrictions on the set of k quantile
covariates let R be the k q × matrix such that ( ) r R = τ β . Then the Wald statistic under
the null hypothesis is given by:
( ) ( ) ( ) ( ) ( ) ( ) [ ] u u q r R R D Q D R r R W − ∈ − ′ − =
′ −
− −
′
1 , , ~
ˆ ˆ ˆ ˆ ˆ
2
1
1 1
τ χ τ β τ β
Figure 2.3 presents the results from estimating the slope coefficient of expected volatility
via quantile regression, that is the quantile function is assumed to be of the following
functional form:
( )
2 2
1
|
t t t y
Time Time TTM y c Q βσ θ δ γ φ τ + + + + + =
−
X
Notice that now for each quantile we have separate estimates for the slope coefficients and
for conciseness we report only the slope coefficient of volatility, β . For vast majority of
commodities and open interest series we observe that the ( ) τ β is negative and nearly linear
as a function of the quantiles apart from at the extremes tails. However, it is worth noting
that for many commodities at the tail quantiles the response of the demand for futures
variable is different from that in the median quantiles, with the two most common
(2.8)
(2.9)
(2.10)
109

occurrences being that volatility in the left tail of the conditional quantiles has a positive
effect and negative in the right tail, or the impact of volatility is significantly more negative
at the tails than at the median. In order to find whether these observations are statistically
significant we conduct two tests: a test of equality of the slope coefficients for
representative extreme quantiles, and a test of symmetry. For the first test the null
hypothesis is that of equality, i.e., ( ) ( ) τ β τ β − = 1 :
0
H . For the second test, we utilize the
Newey and Powell (1987) test of symmetry:
( ) ( )
( ) 5 . 0
2
1
:
0
β
τ β τ β
=
− +
H
Both hypotheses are tested via constructing the appropriate Wald statistics. From Table 2.5
it is observed that the extreme quantile of the conditional distribution of the sets of futures
demand and supply variables as represented by the 10-th and 90-th quantiles appear to be
equally influenced by volatility for the majority of the commodities and open interest series
but for net demand for futures by both hedgers and speculators where for 15 commodities
the null hypothesis that the volatility coefficient is the same is rejected. From Figure 2.3 it
is evident that volatility has positive effect on net demand for both hedgers and speculators
in the left tail of the conditional distribution and negative in the right. Not surprisingly the
OLS estimate was not significant as the effects in both tails of the distribution cancel each
other in OLS estimation. Alternatively from Table 2.6 we see that for the majority of series
and commodities the response to volatility is asymmetrical as we are able to reject the null
hypothesis of symmetry in the Newey-Powell test.
110

Figure 2.3: Quantile Regression Estimates for Volatility

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-4000
-2000
0
2000
BO - oi
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
BO - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
BO - cs
quantiles
β
0 0.5 1
-4000
-2000
0
2000
BO - ncl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
BO - ncs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
BO - nrl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
BO - nrs
quantiles
β
0 0.5 1
-5000
0
5000
BO - netc
quantiles
β
0 0.5 1
-5000
0
5000
BO - netnc
quantiles
β
111

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-10000
-5000
0
5000
C - oi
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
C - cl
quantiles
β
0 0.5 1
-10000
-5000
0
5000
C - cs
quantiles
β
0 0.5 1
-1
-0.5
0
0.5
1
x 10
4
C - ncl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
C - ncs
quantiles
β
0 0.5 1
-4000
-2000
0
2000
C - nrl
quantiles
β
0 0.5 1
-5000
0
5000
C - nrs
quantiles
β
0 0.5 1
-5000
0
5000
10000
C - netc
quantiles
β
0 0.5 1
-10000
-5000
0
5000
C - netnc
quantiles
β
112

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-2000
-1000
0
1000
CC - oi
quantiles
β
0 0.5 1
-2000
-1000
0
1000
CC - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CC - cs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
CC - ncl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
CC - ncs
quantiles
β
0 0.5 1
-1000
-500
0
500
CC - nrl
quantiles
β
0 0.5 1
-1500
-1000
-500
0
500
CC - nrs
quantiles
β
0 0.5 1
-2000
0
2000
4000
CC - netc
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CC - netnc
quantiles
β
113

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-5000
0
5000
CL - oi
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
CL - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
CL - cs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
CL - ncl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CL - ncs
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CL - nrl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CL - nrs
quantiles
β
0 0.5 1
-5000
0
5000
CL - netc
quantiles
β
0 0.5 1
-1
-0.5
0
0.5
1
x 10
4
CL - netnc
quantiles
β
114

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-3000
-2000
-1000
0
1000
CT - oi
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
CT - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CT - cs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
CT - ncl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
CT - ncs
quantiles
β
0 0.5 1
-1000
-500
0
500
CT - nrl
quantiles
β
0 0.5 1
-1000
-500
0
500
CT - nrs
quantiles
β
0 0.5 1
-2000
0
2000
4000
CT - netc
quantiles
β
0 0.5 1
-4000
-2000
0
2000
CT - netnc
quantiles
β
115

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-2000
-1000
0
1000
2000
FC - oi
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
FC - cl
quantiles
β
0 0.5 1
-500
0
500
FC - cs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
FC - ncl
quantiles
β
0 0.5 1
-400
-200
0
200
FC - ncs
quantiles
β
0 0.5 1
-400
-200
0
200
400
FC - nrl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
FC - nrs
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
FC - netc
quantiles
β
0 0.5 1
-500
0
500
1000
FC - netnc
quantiles
β
116

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-10000
-5000
0
5000
GC - oi
quantiles
β
0 0.5 1
-5000
0
5000
GC - cl
quantiles
β
0 0.5 1
-10000
-5000
0
5000
GC - cs
quantiles
β
0 0.5 1
-10000
-5000
0
5000
GC - ncl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
GC - ncs
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
GC - nrl
quantiles
β
0 0.5 1
-5000
0
5000
GC - nrs
quantiles
β
0 0.5 1
-5000
0
5000
10000
15000
GC - netc
quantiles
β
0 0.5 1
-10000
-5000
0
5000
GC - netnc
quantiles
β
117

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-1000
-500
0
500
HG - oi
quantiles
β
0 0.5 1
-500
0
500
1000
HG - cl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
HG - cs
quantiles
β
0 0.5 1
-500
0
500
HG - ncl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
HG - ncs
quantiles
β
0 0.5 1
-200
0
200
400
HG - nrl
quantiles
β
0 0.5 1
-400
-200
0
200
400
HG - nrs
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
HG - netc
quantiles
β
0 0.5 1
-1000
-500
0
500
HG - netnc
quantiles
β
118

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-4000
-2000
0
2000
HO - oi
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
HO - cl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
HO - cs
quantiles
β
0 0.5 1
-1500
-1000
-500
0
500
HO - ncl
quantiles
β
0 0.5 1
-400
-200
0
200
400
HO - ncs
quantiles
β
0 0.5 1
-400
-200
0
200
400
HO - nrl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
HO - nrs
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
HO - netc
quantiles
β
0 0.5 1
-1000
-500
0
500
HO - netnc
quantiles
β
119

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-1000
-500
0
500
1000
JO - oi
quantiles
β
0 0.5 1
-400
-200
0
200
400
JO - cl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
JO - cs
quantiles
β
0 0.5 1
-500
0
500
JO - ncl
quantiles
β
0 0.5 1
-400
-200
0
200
JO - ncs
quantiles
β
0 0.5 1
-400
-200
0
200
400
JO - nrl
quantiles
β
0 0.5 1
-400
-200
0
200
400
JO - nrs
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
JO - netc
quantiles
β
0 0.5 1
-1000
-500
0
500
JO - netnc
quantiles
β
120

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-400
-200
0
200
KC - oi
quantiles
β
0 0.5 1
-200
-100
0
100
KC - cl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
KC - cs
quantiles
β
0 0.5 1
-500
0
500
KC - ncl
quantiles
β
0 0.5 1
-300
-200
-100
0
100
KC - ncs
quantiles
β
0 0.5 1
-400
-200
0
200
400
KC - nrl
quantiles
β
0 0.5 1
-1000
0
1000
2000
KC - nrs
quantiles
β
0 0.5 1
-400
-200
0
200
KC - netc
quantiles
β
0 0.5 1
-500
0
500
1000
KC - netnc
quantiles
β
121

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-6000
-4000
-2000
0
2000
LC - oi
quantiles
β
0 0.5 1
-10000
-5000
0
5000
LC - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
LC - cs
quantiles
β
0 0.5 1
-4000
-2000
0
2000
LC - ncl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
LC - ncs
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
LC - nrl
quantiles
β
0 0.5 1
-2000
0
2000
4000
6000
LC - nrs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
LC - netc
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
LC - netnc
quantiles
β
122

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-500
0
500
1000
LH - oi
quantiles
β
0 0.5 1
-400
-200
0
200
400
LH - cl
quantiles
β
0 0.5 1
-400
-200
0
200
400
LH - cs
quantiles
β
0 0.5 1
-400
-200
0
200
400
LH - ncl
quantiles
β
0 0.5 1
-400
-200
0
200
LH - ncs
quantiles
β
0 0.5 1
-400
-200
0
200
400
LH - nrl
quantiles
β
0 0.5 1
-400
-200
0
200
400
LH - nrs
quantiles
β
0 0.5 1
-500
0
500
LH - netc
quantiles
β
0 0.5 1
-200
0
200
400
LH - netnc
quantiles
β
123

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-1000
-500
0
500
1000
NG - oi
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
NG - cl
quantiles
β
0 0.5 1
-1500
-1000
-500
0
500
NG - cs
quantiles
β
0 0.5 1
-400
-200
0
200
NG - ncl
quantiles
β
0 0.5 1
-400
-200
0
200
NG - ncs
quantiles
β
0 0.5 1
-200
-100
0
100
200
NG - nrl
quantiles
β
0 0.5 1
-500
0
500
NG - nrs
quantiles
β
0 0.5 1
-500
0
500
1000
NG - netc
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
NG - netnc
quantiles
β
124

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-600
-400
-200
0
200
O - oi
quantiles
β
0 0.5 1
-300
-200
-100
0
100
O - cl
quantiles
β
0 0.5 1
-400
-200
0
200
O - cs
quantiles
β
0 0.5 1
-150
-100
-50
0
50
O - ncl
quantiles
β
0 0.5 1
-100
-50
0
50
O - ncs
quantiles
β
0 0.5 1
-200
-100
0
100
200
O - nrl
quantiles
β
0 0.5 1
-60
-40
-20
0
20
O - nrs
quantiles
β
0 0.5 1
-100
0
100
200
O - netc
quantiles
β
0 0.5 1
-100
0
100
200
O - netnc
quantiles
β
125

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-150
-100
-50
0
50
PB - oi
quantiles
β
0 0.5 1
-40
-20
0
20
PB - cl
quantiles
β
0 0.5 1
-100
-50
0
50
PB - cs
quantiles
β
0 0.5 1
-100
-50
0
50
PB - ncl
quantiles
β
0 0.5 1
-100
-50
0
50
PB - ncs
quantiles
β
0 0.5 1
-50
0
50
100
PB - nrl
quantiles
β
0 0.5 1
-100
-50
0
50
PB - nrs
quantiles
β
0 0.5 1
-40
-20
0
20
40
PB - netc
quantiles
β
0 0.5 1
-100
-50
0
50
100
PB - netnc
quantiles
β
126

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-6000
-4000
-2000
0
2000
S - oi
quantiles
β
0 0.5 1
-4000
-2000
0
2000
S - cl
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
S - cs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
S - ncl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
S - ncs
quantiles
β
0 0.5 1
-2000
0
2000
4000
S - nrl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
S - nrs
quantiles
β
0 0.5 1
-2000
0
2000
4000
S - netc
quantiles
β
0 0.5 1
-5000
0
5000
S - netnc
quantiles
β
127

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-4000
-2000
0
2000
4000
SB - oi
quantiles
β
0 0.5 1
-4000
-2000
0
2000
SB - cl
quantiles
β
0 0.5 1
-5000
0
5000
SB - cs
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
SB - ncl
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
SB - ncs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
SB - nrl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
SB - nrs
quantiles
β
0 0.5 1
-5000
0
5000
10000
SB - netc
quantiles
β
0 0.5 1
-4000
-2000
0
2000
4000
SB - netnc
quantiles
β
128

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).
0 0.5 1
-1500
-1000
-500
0
500
SI - oi
quantiles
β
0 0.5 1
-400
-200
0
200
SI - cl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
SI - cs
quantiles
β
0 0.5 1
-1000
-500
0
500
SI - ncl
quantiles
β
0 0.5 1
-1500
-1000
-500
0
500
SI - ncs
quantiles
β
0 0.5 1
-300
-200
-100
0
100
SI - nrl
quantiles
β
0 0.5 1
-400
-200
0
200
SI - nrs
quantiles
β
0 0.5 1
-500
0
500
1000
1500
SI - netc
quantiles
β
0 0.5 1
-2000
-1000
0
1000
SI - netnc
quantiles
β
129

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-4000
-2000
0
2000
SM - oi
quantiles
β
0 0.5 1
-1000
0
1000
2000
SM - cl
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
SM - cs
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
SM - ncl
quantiles
β
0 0.5 1
-1000
-500
0
500
SM - ncs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
SM - nrl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
SM - nrs
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
SM - netc
quantiles
β
0 0.5 1
-2000
-1000
0
1000
SM - netnc
quantiles
β
130

Figure 2.3, Continued

Notes: Quantile estimate for the volatility coefficient (solid line) and 95% confidence intervals (dashed lines).

0 0.5 1
-6000
-4000
-2000
0
2000
W - oi
quantiles
β
0 0.5 1
-2000
-1000
0
1000
2000
W - cl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
W - cs
quantiles
β
0 0.5 1
-3000
-2000
-1000
0
1000
W - ncl
quantiles
β
0 0.5 1
-4000
-2000
0
2000
W - ncs
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
W - nrl
quantiles
β
0 0.5 1
-1000
-500
0
500
1000
W - nrs
quantiles
β
0 0.5 1
-4000
-2000
0
2000
W - netc
quantiles
β
0 0.5 1
-1000
0
1000
2000
W - netnc
quantiles
β
131

2.3.6. Granger Causality
In order to better understand the dynamics behind the effect of volatility on the
futures demand by different market participants, a Granger (1969) causality test is
performed. A variable x is said to not Granger cause a variable y if the conditional
distribution of y is not dependent on past values of x , i.e.,
( ) ( )
1 1 1
| , |
− − −
=
t t t t t
y F y F Y Y X

where
1 − t
X and
1 − t
Y are the filtration created by past values of the two random variables
and ( ) . F is the conditional distribution function of y . If on the other hand it is found that
the conditional distribution is dependent on past values of x then it is said that x
Granger-causes variable . y It has to be noted, however, that term causality here does not
carry any implications as to the fact whether y causes x but rather it only conveys
information whether variable y precedes variable x . Although Granger causality is
defined through the independence of the conditional distribution of y , typically the
Granger causality tests are conducted by evaluating whether specific moments of the
conditional distribution are dependent on past data of x , most often that being the
conditional mean (Granger, 1969,1980).
Definition 1: A variable
t
x does not Granger-cause a variable
t
y in mean if and only if
[ ] [ ]
1
. .
1 1
| , |
− − −
=
t t
s a
t t t
y E y E Y Y X

132

Table 2.5: Tests of Equality of Volatility Effects in the Extreme Quantiles

oi cl cs ncl ncs nrl nrs netc netnc netnr
BO
0.1739 0.0793 0.088 2.1451 1.9335 0.7941 0.068 4.7895** 5.368** 0.7107
C
0.0493 0.1968 0.6727 0.9153 2.1579 5.873** 0.0019 0.0259 0.4699 8.3067**
CC
1.4157 3.0631* 4.5547** 0.7986 6.952** 0.3804 5.1943** 5.9732** 7.7574** 0.3457
CL
3.9903** 0.0182 3.9987** 1.4479 0.4405 0.1287 1.9185 3.02* 0.7719 1.8227
CT
4.5622** 6.2345** 3.1333* 2.6759 29.0215** 0.0055 2.3468 21.6274** 34.2625** 1.7126
FC
0.46 0.1809 0.9287 10.3233** 0.2649 2.908* 1.668 3.2349* 5.7079** 1.332
GC
3.4413* 3.9397** 7.659** 1.8209 0.376 0.2949 0.701 8.5252** 21.2531** 0.0093
HG
0.7608 0.0079 0.0928 0.7468 0.0095 0.0416 0.9119 4.5466** 4.7116** 0.0859
HO
0.0009 0.3576 0.9486 7.0719** 0.0881 0.3345 1.5062 7.9782** 13.1668** 0.1912
JO
0.1971 1.9711 4.6016** 7.6328** 0.3456 0.1507 2.488 3.6882* 1.4692 0.1152
KC
0.8227 1.5889 0.7362 0.2124 5.4131** 0.6517 0.216 4.4562** 11.6899** 2.1063
LC
0.2201 0.9123 1.1173 0.811 2.4021 0.0113 0.1225 1.2347 7.4408** 0.1767
LH
11.857** 0.0582 17.9198** 3.274* 1.9204 0.1128 0.058 38.9593** 8.8363** 0.2085
NG
0.0446 2.5198 0.0239 25.2749** 13.3812** 0.0452 1.4602 52.1923** 54.4542** 9.1413**
O
0.0202 6.4319** 0.2839 0.5032 0.0001 0.1832 1.7335 1.1266 0.407 3.4259*
PB
0.4225 0.0013 0.957 1.0692 2.4764 2.1145 7.5034** 0.4142 1.8882 1.8253
S
0.0691 5.7756** 0.1232 1.8064 0.9752 1.734 1.4219 0.3786 31.0048** 0.5641
SB
3.0132* 0.471 17.9068** 2.739* 0.0266 0.3132 1.4947 26.4335** 57.5979** 0.3249
SI
0.5211 0.0045 1.3132 0.0017 2.0677 0.9138 0.7586 0.17 0.8836 0.6133
SM
2.1089 0.1503 3.4295* 1.0874 1.2149 1.7049 3.912** 8.7432** 15.0667** 0.44
W
0.2053 0.3205 3.3621* 2.4265 0.9975 6.8451** 11.0763** 5.4532** 26.4129** 0.2594

Notes: ***,**, and *, indicate significance at the 1%, 5%, and 10% level respectively. The test statistic is the Wald test of the null hypothesis of
equality of the volatility slope coefficient for the 10-th and 90-th quantiles.

132
133

Table 2.6: Tests of Symmetry of Volatility Effects

oi cl cs ncl ncs nrl nrs netc netnc netnr
BO
0.1739 0.0793 0.088 2.1451 1.9335 0.7941 0.068 4.7895** 5.368** 0.7107
C
7.6988** 0.792 1.2652 3.7886* 1.0353 1.1878 4.0062** 2.6496 4.8442** 2.2687
CC
4.3602** 1.808 6.9438** 0.2774 7.1919** 1.7235 0.4213 5.0782** 8.6186** 0.0644
CL
7.9237** 0.1783 3.0065* 5.1217** 0.5546 0.268 0.2746 0.0169 4.8209** 0.015
CT
28.6115** 10.9452** 9.114** 0.0033 21.0301** 4.7143** 6.5505** 9.3857** 22.5412** 4.8626**
FC
0.1144 0.0014 6.1422** 12.7832** 0.1348 2.0027 0.1894 2.8342* 1.4491 0.0073
GC
13.4201** 1.736 17.9822** 8.1403** 0.4835 2.9027* 1.0392 0.1258 25.1615** 0.8366
HG
2.1974 0.8172 0.2737 1.0237 0.0826 0.0012 0.0095 3.5154* 3.3957* 0.5994
HO
0.237 0.0526 0.4651 11.2261** 0.0022 0.327 3.3444* 0.5897 8.1209** 0.188
JO
1.1495 0.6014 2.983* 4.4251** 0.7512 2.1674 0.0727 1.0448 0.8414 0.0775
KC
19.2843** 6.6295** 3.5688* 3.2822* 11.7615** 2.1353 1.1614 1.2447 6.4235** 0.0034
LC
0.249 0.2043 0.0487 2.6909 8.1139** 0.1957 1.7686 0.6667 2.7023 0.3852
LH
5.7285** 0.0187 3.0726* 2.4455 3.0916* 0.5511 0.0209 21.2498** 0.0923 0.0686
NG
3.3435* 2.9693* 2.9706* 18.5336** 10.3041** 4.2809** 0.0402 6.8759** 26.6133** 13.3436**
O
1.9338 2.9955* 0.9684 5.2005** 0.2856 3.4572* 1.5317 0.0761 0.3226 6.4179**
PB
1.553 5.4764** 0.1326 0.8981 0.1524 2.0422 1.4436 0.0074 4.6373** 0.0123
S
5.6299** 3.1169* 1.8859 6.9798** 0.4606 0.2094 0.0048 1.7289 33.0184** 0.0772
SB
8.3167** 0.0786 18.733** 5.8249** 0.0037 1.8714 4.2344** 7.3128** 36.4853** 1.0064
SI
2.0205 1.9239 10.8665** 2.9177* 2.8276* 16.18** 0.2057 1.2643 3.5936* 1.3673
SM
15.6845** 0.3442 11.5714** 6.4142** 0.5642 0.0028 0.1473 1.9227 12.5757** 3.158*
W
4.6438** 0.6283 9.5987** 4.1535** 2.4535 5.734** 11.33** 3.1352* 8.7622** 0.001

Notes: ***,**, and *, indicate significance at the 1%, 5%, and 10% level respectively. The test statistic is the Wald test of the null hypothesis of
symmetry of the volatility slope coefficient for the 10-th and 90-th quantiles around the median
133
134

Usually tests exploring whether a variable x Granger causes a variable y in
mean are performed by utilizing a regression of the following form:

t i t i
p
i
i t i
p
i
t
x b y a c y ε + + + =
−
=
−
=
∑ ∑
1 1

The null hypothesis is that x does not Granger cause y , i.e., 0 :
0
=
i
b H , for p i : 1 = .
Thus, to test this null hypothesis any of the usual tests for redundant variables could be
used, e.g., Likelihood Ratio or Wald tests. Table 2.7 summarizes the results from
investigating whether volatility Granger causes futures demand by types of traders based
on the Likelihood ratio (LR) test. The optimal value of the lags p has been determined by
minimizing the Bayesian Information Criterion (BIC) among models with lags from 2 to 4
Further, to be consistent with the model examining the contemporaneous relationship
between demand for futures and volatility, the time to maturity and time dummies are also
included, i.e., the regression is:
t i t i
p
i
i t i
p
i
t t
b y a Time Time TTM c y ε σ ω δ γ + + + + + + =
−
=
−
=
∑ ∑
2
2 2
2

The LR test statistic is distributed as a ( ) p
2
χ random variable. In line with the results
from the OLS results from model 2.3, we find evidence of causality in the conditional mean
for more than a third of the commodities and open interest series pairs with total open
interest, hedgers short positions and speculators long positions exhibiting most evidence of
volatility Granger-causing demand for futures, a result consistent with findings of Fung
and Patterson (1999) and Chen and Daigler (2008). Nevertheless for many of the
commodities and open interest series we are not able to establish Granger causality.
(2.11)
(2.12)
135

Table 2.7: Granger Causality in the Conditional Mean Test Results

oi cl cs ncl ncs nrl nrs netc netnc netnr

BO 14.21*** 1.89 20.13*** 16.03*** 1.73 3.44 6.98** 12.37*** 11.92*** 8.51**
C 12.24*** 4.01 13.7*** 12.3*** 0.6 2.23 4.8 7.63** 9.47*** 1.99
CC 10.66*** 2.34 11.18*** 9.26*** 0.64 7.65** 18.98*** 6.9** 8.37** 5.85
CL 1.57 2.57 0.4 6.03** 1.84 2.78 3.1 2.19 2.3 2.72
CT 19.7*** 11.86*** 11.54*** 11.16*** 5.09 15.88*** 8.79** 2.67 1.8 8.5**
FC 4.84 5.16 3.55 2.64 6.44** 1.4 5.73 3.56 0.17 3.59
GC 15.53*** 0.07 10.62*** 14.87*** 3.35 12.3*** 7.87** 6.65** 6.61** 5.13
HG 3.53 1.44 0.66 0.87 0.07 1.13 2.4 0.02 0.43 3.49
HO 1.48 1.81 0.03 11.55*** 0.49 2.57 1.65 1.81 4.35 0.16
JO 4.67 0.68 0.29 1.06 0.73 0.61 6.55** 0.18 0.46 1.33
KC 4.18 0.61 1.69 2.38 2.44 21.25*** 6.93** 0.73 0.22 4.71
LC 1.42 3.84 3.83 5.22 9.59*** 0.51 0.42 6.17** 5.9 2.37
LH 0.24 1.76 0.95 0.31 2.52 0.44 1.52 2.19 1.45 4.18
NG 10.98*** 2.74 9.9*** 4.16 0.88 10.38** 1.2 4.23 2.03 9.92***
O 11.81*** 3.45 10.79*** 4.44 3.17 4.2 0.47 5.24 1.28 4.7
PB 13.51*** 8.7** 2.82 7.02** 8.82** 7.43** 4.27 1.12 0.89 0.42
S 12.69*** 1.62 16.93*** 16.92*** 6.36** 4.58 6.88** 10.36*** 9.23*** 4.18
SB 5.87 3.36 5.31 6.2** 3.48 7.78** 3.85 2.6 3.08 0.88
SI 16.51*** 6.86** 7.87** 4.24 3.13 14.39*** 4.74 2.39 1.3 6.17**
SM 8.01** 2.26 11.22*** 10.89*** 0.85 1.83 5.69 3.01 2.57 2.62
W 17.28*** 5.35 21.41*** 9.8*** 7.33** 2.71 4.59 5.19 7.9** 1.2

Notes: ***,**, and *, indicate significance at the 99%, 95%, and 90% level respectively. The reported statistic is the Likelihood Ratio statistic from testing
for causality in the conditional mean.
135
136

2.3.7. Causality in Quantiles
It is worth nothing that we find only weak support for volatility Granger-causing
demand for long futures by hedgers albeit the prevalent evidence of significant
contemporaneous effect. This observation as well as the failure to reject Granger
non-causality in other open interest series and commodities, however, should not be
interpreted as lack of overall causality as such could potentially exist in higher conditional
moments. Rather than investigate causality in variance, as for example in Granger (1986),
or non-linear causality as in Hiemstra and Jones (1994) and Silvapulle and Choi (1999), a
different and more comprehensive approach is pursued instead. Since Granger causality is
defined in conditional distribution, evaluating directly the distribution functions would be
a natural approach to test for causality.
Granger (2006) notes that “ [...] for most of its history time series theory considered
conditional means, but later conditional variances. The next natural development would be
conditional quantiles, but this area is receiving less attention than I expected. The last
stages are initially conditional marginal distributions, and finally conditional multivariate
distributions. Some interesting theory is starting in these areas but there is enormous
amount to be done”. Quantile regression has emerged as a comprehensive approach to the
statistical analysis of dynamic models. Recent developments in quantile regression seem to
provide perfectly tailored tools for such an approach. Investigating causality via quantile
regression methods have been only recently employed. Chuang, Kuan and Lin (2009)
explored the causality in quantiles between stock returns and volume in the SP500, DJIA,
137

and FTSE 100; Chen, Gerlach and Wei (2009) applied the quantile regression appoach to
test for causality between US and Asian stock indexes; Lee and Yang (2006) revisit the
money-income Granger-causality and find that money Granger-causes income more
strongly than it is usually found by using causality in conditional quantiles.
Definition 2: A variable
t
x does not Granger-cause a variable
t
y in quantiles if and
only if [ ] [ ]
1
. .
1 1
| , |
− − −
=
t y
s a
t t y
Q Q Y Y X τ τ for ( ) 1 , 0 ∈ ∀ τ
Empirically we could easily construct this test by estimating the conditional
quantile function via quantile regression and testing the joint significance of the
i
b
coefficients, i.e.
( ) i t
i
p
i
i t i
p
i
t t t y
b y a Time Time TTM c Q −
=
−
=
− − ∑ ∑
+ + + + + =
2
2 2
2
1 1
, | σ ω δ γ τ Y X
However, since we need to test the significance of
i
b for all quantiles in the ( ) 1 , 0 interval
the appropriate test is for the significance of the entire process for ( ) τ
i
b and thus the usual
Wald statistic is no longer applicable. Koenker and Machado (1999) show that under the
null hypothesis 0 :
0
= b H for [ ] u u − = ∈ 1 , T τ the supremum of the Wald tests for
[ ] u u − = ∈ 1 , T τ converges in distribution to that of the supremum of the norm of scaled
Brownian bridges. To see this note by equation 2.6 we have:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
q
I N R QD RD R T
R QD RD N R T
τ τ τ β τ β
τ τ τ β τ β
− ′ −
′ − −
−
− −
− −
1 , 0 ~
ˆ
1 , 0 ~
ˆ
2
1
1 1
1 1

A 1-dimensional Brownian bridge on the closed interval [ ] 1 , 0 is defined as
(2.13)
(2.14)
(2.15)
138

( )
1
tW W t B
t
− = where W is a standard Brownian motion. It is straightforward to show
that a Brownian bridge is distributed as ( ) ( ) t t N − 1 , 0 (Oksendal, 1995). Thus, the Wald
statistic (eq. 2.9) is equivalent in distribution to a q -dimensional Brownian Bridge.
Explicitly,
( )
( )
( )
[ ] u u
B
W
q
− ∈
−
1 , for ,
1
~
2
τ
τ τ
τ
τ
Then under the null hypothesis of 0 = b for T ∈ τ Koenker and Machado (1999) prove
(see Theorem 2.1 therein) that the appropriate test is that of ( ). sup τ
τ
W
T ∈
Under some
mild regularity conditions (see Andrews, 1993) it could be shown that
( )
( )
( ) τ τ
τ
τ
τ τ − ∈ ∈ 1
sup ~ sup
2
q
B
W
T T

In order to operationalize the above test, we set 05 . 0 = u , estimate the Wald statistics for
each of the 91 quantiles between 0.05 and 0.95, and replace the supremum with the
maximum of the quantile Wald statistics on those quantiles. Critical values for
( )
( ) τ τ
τ
τ − ∈ 1
2
sup
q
B
T
have been tabled by DeLong (1981), and Andrews (1993) and Chuang et al.
(2009) by simulation.
Results from the causality in quantiles are reported in Table 2.8. While previously
we have found that only for a third of the investigated pairs of commodities and open
interest series volatility Granger-causes in conditional mean futures demand, it is evident
that now for substantially all commodities and open interest series pairs considered herein
it volatility does cause demand for futures.
(2.16)
(2.17)
139

Table 2.8: Causality in Quantiles Test Results

oi cl cs ncl ncs nrl nrs netc netnc netnr

BO 15.14** 11.12* 35.12*** 25.03*** 10.7 9.81 10.02 52.3*** 53*** 12.74*
C 35.57*** 7.57 27.73*** 18.41*** 11.25* 22.04*** 6.01 18.05*** 44.15*** 10.59
CC 26.63*** 8.23 24.18*** 20.07*** 92.23*** 16.09** 28.01*** 23.82*** 28.71*** 14.43**
CL 52.64*** 17.07*** 35.62*** 26.21*** 16.95*** 11.41 16.72** 26.56*** 17.72*** 37.56***
CT 30.64*** 80.6*** 43.52*** 16.06** 96.4*** 29.03*** 21.23*** 41.8*** 67.3*** 20.36***
FC 21.88*** 27.74*** 51.79*** 69.73*** 27.98*** 60.72*** 97.08*** 69.72*** 28.8*** 12.75*
GC 46.22*** 15.54** 83.09*** 55.5*** 16.99*** 35.92*** 26.89*** 45.34*** 39.11*** 27.33***
HG 27.89*** 4.19 12.59* 5.67 29.09*** 5.7 6.06 15.32** 15.88** 11.68*
HO 7.93 5.88 7.91 38.1*** 6.96 18.97** 4.95 51.9*** 65.78*** 8.3
JO 26.63*** 4.65 53.63*** 19.34*** 47.85*** 54.63*** 54.7*** 21.42*** 21.5*** 29.67***
KC 63.62*** 23.55*** 12.82** 67.07*** 39.2*** 131.56*** 34.63*** 45.09*** 37.47*** 94.67***
LC 16.39*** 7.1 19.94*** 51.26*** 24.29*** 63.49*** 21.43*** 47.84*** 119*** 98.95***
LH 17.43*** 16.77*** 27.46*** 9.82 67.09*** 70.49*** 17.59*** 41.72*** 10.51 59.12***
NG 20.65*** 14.97* 68.18*** 63.49*** 46.58*** 74.62*** 4.88 76.31*** 42.48*** 50.71***
O 44.45*** 24.72*** 16** 16.85*** 14.23** 13.11** 10.66 54.53*** 33.53*** 31.24***
PB 24.65*** 14.61** 10.76 12.06* 11.85* 14.4** 14.94** 11.04 14.69** 11.91*
S 17*** 5.39 12.47* 25.71*** 35.84*** 9.12 18.35*** 18.41*** 75.89*** 16.23**
SB 32.94*** 22.65*** 32.01*** 18.38*** 26.59*** 25.18*** 82.67*** 44.5*** 46.17*** 9.91
SI 24.14*** 15.34** 39.95*** 15.39** 18.07*** 71.67*** 8.65 23.59*** 24.97*** 15.83**
SM 14.12** 12.65* 75.43*** 21.75*** 12.75* 8.32 10.57 63.42*** 29.81*** 14.99**
W 76.04*** 19.86*** 39.97*** 25.15*** 22.3*** 66.99*** 13.09** 42.77*** 44.17*** 23.75***

Notes: ***,**, and *, indicate significance at the 99%, 95%, and 90% level respectively. The test statistic is the sup-Wald test of the null hypothesis of no
causality in the [0.05,0.95] quantiles.
139
140

2.4. Conclusion
In this paper, the relationship between futures price volatility and demand for
futures by market participants as proxied by open interest was explored for twenty one
commodity futures. We found evidence of a significant contemporaneous relationship
between expected price volatility and open interest for both hedgers and speculators.
Contrary to much of prior empirical results on the volatility-trade demand relationship we
have found that hedgers react rather counter-intuitively to positive shocks in volatility by
reducing their open positions, a phenomenon at present not well captured by extant models.
Utilizing quantile regression the existence of heterogeneity in the response of net demand
for futures to volatility changes was established via symmetry tests on the quantile
conditional distribution. Further, our results from the Granger causality tests in conditional
mean reveal that the volatility does Granger-cause demand for futures but only for a third
of the considered commodities and open interest positions. However, employing a
relatively new approach of causality in quantiles we have been able to reveal that volatility
does cause for demand for futures for the vast majority of futures and trade positions
considered. This is in contrast to other researchers who find no or only weak evidence of
causal relationship between cash volatility and open interest possibly due to the limited
power of their tests as they do not consider the entire distribution but rather only particular
moments of it.
The empirical findings in this paper could help in understanding better the market
structure of commodity futures, the influence of volatility and the respective roles of
141

hedgers and speculators. The asymmetric response in demand for futures to volatility
shocks as well as the negative relationship between hedging demand and volatility should
prove to be a fertile area for exploration of new theoretical model attempting to explain
these singularities of commodity futures.

142

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Abstract (if available)

Abstract This dissertation and the essays herein represent an effort to extend our understanding of the time-series and distributional properties of commodity futures on one hand, and to provide new evidence on the effects of volatility on demand for futures on the other. The behavior of commodity futures volatility is studied via parametric and non-parametric tests. It is found that commodity futures volatility exhibits long-memory properties. Autoregressive conditionally heteroskedastic models appear to capture satisfactorily the volatility dynamics of commodity futures in general. From four competing autoregressive models through in-sample tests it is found that the FIAPARCH model with Student’s t-errors better accounts for nonlinearities in commodity prices even after accounting for seasonality in volatility. However, out-of-sample tests reveal that simpler models such as GARCH perform equally well at least for short forecast horizons.

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Asset Metadata

Creator Vassilev, Georgi D. (author)

Core Title Essays on commodity futures and volatility

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 06/17/2010

Defense Date 06/02/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag commodity futures,GARCH,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kamrany, Nake (committee chair), Dekle, Robert (committee member), Zapatero, Fernando (committee member)

Creator Email gvassile@usc.edu,vassilevg@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3140

Unique identifier UC1489602

Identifier etd-Vassilev-3838 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-355086 (legacy record id),usctheses-m3140 (legacy record id)

Legacy Identifier etd-Vassilev-3838.pdf

Dmrecord 355086

Document Type Dissertation

Rights Vassilev, Georgi D.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu