Generalized Taylor effect for main financial markets

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Generalized Taylor Effect for Main Financial Markets

by

Mingpu Song

University of Southern California

ii

Table of Contents
List of Figures ................................................................................................................................... iv
List of Tables ...................................................................................................................................... v
Abstract ................................................................................................................................................ 1
Chapter 1. Introduction ................................................................................................................. 2
Chapter 2. The Data ......................................................................................................................... 7
Chapter 3. The analysis for Taylor effect and data property .......................................... 9
Chapter 4. The ARCH model and Autocorrelation Function ......................................... 20
4.1 ACF in EGARCH(1, 1) model ..................................................................................... 23
4.2 Parameters and Taylor Effect ................................................................................... 25
Chapter 5. Emperical results and Simulation...................................................................... 29
Chapter 6. Conclusion .................................................................................................................. 33
References ........................................................................................................................................ 34

iii

Appendix ........................................................................................................................................... 38

iv

List of Figures
Figure 1 .......................................................................................................................................... 10
Figure 2 .......................................................................................................................................... 11
Figure 3 .......................................................................................................................................... 13
Figure 4 .......................................................................................................................................... 15
Figure 5 .......................................................................................................................................... 18
Figure 6 .......................................................................................................................................... 19
Figure 7 .......................................................................................................................................... 19
Figure 8 .......................................................................................................................................... 27
Figure 9 .......................................................................................................................................... 28
Figure 10 ....................................................................................................................................... 32

v

List of Tables
Table 1 ............................................................................................................................................ 29
Table 2 ............................................................................................................................................ 30

1

Abstract
It has been tested that for financial time series, the autocorrelation does not exists
for the returns themselves, while exist for the power transformation of the
absoluted values. Also, lots of studies have shown that, known as Taylor Effect, the
absolute returns are more strongly autocorrelated than the squared values. Ding
(1996) further checked some other time series and found that for the exchange rate
returns, the power d maximizing the autocorrelation was around 1/4. So a
Generalized Taylor Effect that the power maximizes the autocorrelation varies for
different data is proposed and analyzed. To more widely concern, 18 stock indices,
12 exchange rate seires and 12 future contracts are further checked. The d is
analyzed with some other properties of the data-mean, standard error, kurtosis and
skewness. Inspired by the deduction of ACF in EGARCH models, He T (2002), this
paper take a deeper look at the relationship among those values. Tests and Mote
Carlo simulations are also carried out to see the goodness of fit of the EGARCH
model.

2

Chapter 1. Introduction
Hurst (1951) first found the ‘long-memory’ property when studying the time series
of the water flows of The River Nile. Like in the hydrology, time series are quite
important for the finance. In many Mathematical Finance theories, the Market
Efficiency hypothesis has usually been the foundation. However, it has been
examined that one of the hypothesis-the returns distribute normally-not to be true.
Taylor (1986), found that although the autocorrelation of the time series themselves
are small, the power transformation of the absolute values |

|

have strong
correlation. By examining several series of stock returns, he found there exists the
‘long-memory’ property. Also, it was found |

| has stronger autocorrelation than
the squared values. Similarly, Ding, Granger and Engle (1993) examined the stock
returns of S&P 500 from Jan 3, 1928 to Aug 30, 1991, and found the |

| have
strong autocorrelation after long lags. Furthermore, it is also found that, the power
form of the returns-|

|

, reaches its maximum value around d=1. It was defined as
Taylor effect in Ding and Granger (1995). To test this empirical property, Ding and
Granger (1996) then examined other 4 stock returns and 1 exchange return. An

3

interesting result is that the d that maximizes the autocorrelation, is always around
1 for the stock returns, and tend to be around 1/4 for the exchange return. Muller
and Dacorogna (1998) tested the exchange rates of USD/DEM and USD/FRF and
found the |

|

maximized around d=1/2.
The first thought about this phenomenon would be whether this d could be figured
out theoretically, when applying the appropriate model. So far, there are two main
kinds of models related to the non-linear time series. The first kind of models are
based on the ARCH structure, which was first introduced in Engle (1982). Bollerslev
(1986) then introduced the GARCH model to capture the long memory property by
improving the variance term. Engle, Lilien and Robins (1987) extended the
conditional variance and introduced ATCH-M model. To reduce some restrictions
and illustrate the persistence to ‘bad news’, Nelson (1991) proposed a new
approach named EGARCH. Zakoian (1994) deduced the TARCH model, Higgins and
Bera (1993) introduced NARCH model. And a conclusive study of those models is
introduced in Ding, Granger and Engel (1993). They proposed a generalized class of
models which could transform to other ARCH models in special cases. The second
kind of models-Stochastic Volatility model-were first introduced by Taylor (1982).
Breidt et al. (1998) and Harvey (1998) proposed the LMSV model, which considered
the volatility as a long memory ARFIMA process. The SV models have simpler form

4

and are easier to get the Taylor property. Both models could capture the properties
asuc as the volatility persistence, autocorrelation and the Taylor effect. For more
difference, it was reviewed, see Carnero (2004). Among all of the weak points of the
ARSV models, the most important one is that the maximum likelihood can not be
derived except by numerical calcution with computer. Various methods have been
proposed to estimate this. But it is still of great interest to check if the esitimation is
good enough, especially when both models are reasonable to explore the time sereis.
However, the SV models has a strong point when it is about to get the
autocorrelation function. The acf could be derived easily while it is particularly hard
in most of the ARCH type models. More specifically, considering about the Taylor
Effect, one can only base his results by simulation. Ding, Granger and Engel (1993)
found for the S&P 500 stock returns, the Taylor Effect holds when d=2. But later in
Ding (1996), the property holds for the DEM/USD exchange rates returns when d
equals to 1/4 instead of 1. So it is naturally to say the d is not a constant, or say, it
varies from sample to sample. Though it is extremely hard to deduce the analytical
forms of the acf of |

|

for free parameter, studies based on the ARCH type models
are considerable. He and T (1999a) derived the acf for GARCH (1,1) when d=1 and
d=2, and further proved the Taylor effect. Then based the on the asymmetric power
ARCH models, in He and T (1999c), the acf of the logarithmed observations was

5

given for the EGARCH model. The final result was introduced in He and T (2002),
which gave the general form of the acf using the EGARCH model. When it comes to
the ARSV model, on the other hand, Harvey (1998) derives the acf of |

|

for all
powers d in a consise form. In both models, it is still impossible to get a derivative,
or say, get an function of the parameters for the power maximizing the acf. However,
we could at least get the relationship between the parameters and the acf. Then to
some aspects, explore a more generalized Taylor effect.
The main purpose of this paper is to check the Generalized Taylor Effect in some
other financial time series. The results show that further studies are needed because
the d changes a lot for the different data samples. Then to explore this
phenomeneon, by the derived autocorrelation function in EGARCH model, we will
focus on the determinants that affect the power maximizing the autocorrelation. In
other words, a more generalized Taylor effect will be shown in both models. By
caltulation and plotting, the relationship between the maximum d and some data
properties is explicitly shown. Though both models can explain this property, due to
the efficiency of the estimation, emperical results are checked. Besides, Monte-Carlo
simulations on EGARCH model are carried to see if the model is well fitted to
capture this stylized fact.

6

The paper is organized as follows. Section 2 gives a brief introduction to the data we
use. Section 3 gives the analysis about the data. Especially, Taylor Effect and the
autocorrelation is concerntrated. Then the d maximizing the autocorrelation would
be compared with mean, standard error, skewness and kurtosis. Section 4
introduces the ARCH type model and we will analyze the acf in EGARCH models to
get a brief view of the generalized Taylor effect. In Section 5 the Monte-Carlo
simulations are conducted to the fitted models. Based on the results, Section 6
concludes.

7

Chapter 2. The Data
To widely check the Taylor property, various data are chosen in this paper. The
basic idea is to check the dominate time sereis in the worldwide market. The stock
market, exchange rate market and futures contract market are the three markets we
may concern.
For the stock market, the indices are S&P 500 Index, Dow Jones Industrial Average,
NASDAQ Composite Index, S&P/TSX Composite Index, EURO STOXX 50 Price EUR,
FTSE 100 Index, CAC 40 Index, Deutsche Boerse AG German Stock Index DAX, IBEX
35 Index, FTSE MIB Index, AEX-Index, OMX Stockholm 30 Index, Nikkei 225,
S&P/ASX 200, Hong Kong Hang Seng Index, Shanghai Stock Exchange Composite
Index, Mexican Stock Exchange Mexican Bolsa IPC Index and Ibovespa Brasil Sao
Paulo Stock Exchange Index.
For the exchange rate market, the currencies chosen are all based on the US dollars.
In other words, the value of currencies per US dollar. They are CAD, EUR, GBP, AUD,
JPY, BRL, CNY, HKD, INR, MXN, RUB, CHF.

8

For the future contract market, the commidies chosen are from 4 main kinds-Gold,
Silver, High Grade Copper and Platinum are from metal; Corn, Soybean, Wheat and
Cotton are from angricultural products; Crude Oil and Natural Gas are resource type;
US Treasury Bond and US 10-year note are financial futures.
Notice that for various data, the daily data available till now are quite different, so
the data range is mainly chosen from Jan, 1984 to Mar, 2014, in order to be
consistent. But differences of the number of the ovservations and the start date do
exist for various countries. The explicit ranges for different data are given in the
Appendix.
If

denote the daily price, then the return series are defined as

= log

− log

(1)

9

Chapter 3. The analysis for Taylor effect and data property
We may take a first look at the return series. Due to the large amount of samples, the
S&P 500 stock index, USDCAD exchange rate and Gold futures are chosen from the
three kinds of markets respectively. Figure 1 is the daily returns of the three typical
financial time series. As we can see, the DJI3d is more stable than the other 2.
Especially for the futures of Gold, the returns fluctuate a lot. Moreover, we can see a
trend that a large return is more likely to be followed by a small one (the absolute
value).

10

Figure 1

To better illustrate the data, we then plot the histograms and calculate some
descriptive values of the three samples. As shown in Figure 2, we can clearly see that
the return series are normally distributed and all the JB statistics are significant.
Also the skewness is quite small while the kurtosis is large. But both the skewness
and kurtosis varies a lot.
0 2000 4000 6000
-0.20 -0.10 0.00 0.10
S&P500
Return
0 2000 4000 6000 8000
-0.04 -0.01 0.01 0.03
USDCAD
Return
0 2000 4000 6000
-0.10 0.00 0.10
Gold Futures
Return

11

Figure 2

0
400
800
1,200
1,600
2,000
2,400
2,800
3,200
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10
Series: SP500R
Sample 1 7607
Observations 7605
Mean 0.000319
Median 0.000586
Maximum 0.109572
Minimum -0.228997
Std. Dev. 0.011582
Skewness -1.274191
Kurtosis 31.08254
Jarque-Bera 251954.8
Probability 0.000000
0
400
800
1,200
1,600
2,000
2,400
-0.0375 -0.0250 -0.0125 0.0000 0.0125 0.0250
Series: USDCADR
Sample 1 7846
Observations 7845
Mean -1.52e-05
Median -6.84e-05
Maximum 0.032986
Minimum -0.037665
Std. Dev. 0.004473
Skewness 0.159480
Kurtosis 8.748080
Jarque-Bera 10833.34
Probability 0.000000

12

Then we need to check the autocorrelation at the data. Figure 3 show that all of
these three time series have little correlation of themselves but strong correlation of
the |

|. Although the autocorrelation decays, it still exists after long lags because the
rate is quite slow. This, in other words, shows the long memory property. The
autocorrelations of

are all smaller than that of |

|. Especially in figures for the
two exchange rates, we can see that the autocorrelation of

almost does not
exist as well.
0
500
1,000
1,500
2,000
2,500
3,000
-0.10 -0.05 -0.00 0.05 0.10
Series: GOLDR
Sample 1 7578
Observations 7577
Mean 0.000168
Median 0.000124
Maximum 0.111278
Minimum -0.098206
Std. Dev. 0.010569
Skewness -0.126994
Kurtosis 11.89349
Jarque-Bera 24991.07
Probability 0.000000

13

Figure 3

In Ding, Granger and Engel (1993), it is tested that |

|

maximizes the
autocorrelation around d=1 for the S&500. However, in Ding and Granger (1996),
other four stock returns are examined and they found that the autocorrelation of
|

|

maximizes around d=1/4 for the exchange returns. Inspired by this, three
0 20 40 60 80 100
0.0 0.1 0.2 0.3
lags
Autocorrelation
r
|r|
r^2
0 20 40 60 80 100
0.0 0.1 0.2 0.3
lags
Autocorrelation
r
|r|
r^2
0 20 40 60 80 100
0.0 0.1 0.2 0.3
lags
Autocorrelation
r
|r|
r^2

14

typical series from the free markets are checked. Figure 4 shows the autocorrelation
of the power transformation |

|

, changes along with the power d. As we can see,
the maximum d is quite different for different time series. One may consider it
varies for different financial derivatives, e.g. d=1 for stock, d=1/4 for exchange rate,
d=5/4 for futures etc. However, for these 3 specific data, the d are 1.22, 1.37 and
1.48, which vary a lot. Especially, for the USDCAD data, it is extremely different from
the exchange rate tested in Ding Granger (1996).

15

Figure 4

Based on the previous analysis, it is necessary to explore more samlpes. Also, the
relationship between the d maxmizing the autocorrelation and the descriptive
variables of the data is of our interest. The detailed values are given in Apendix B.
Intuitively, we will see the relationship in the pairs plot, which is Figure 5. First
notice that even the outliers exist for all the plots, it is obvious that the d maxmizing
0 1 2 3 4 5
0.00 0.10 0.20
d
rou
0 1 2 3 4 5
0.05 0.15 0.25
d
rou
0 1 2 3 4 5
0.05 0.10 0.15
d
rou

16

the autocorrelation is strongly related to the kurtosis and skewness. On the other
hand, the d is randomly distributed with the mean and standard error. The
relationship between the d and kurtosis, skewness is then analyzed. For the kurtosis,
the outliers eliminated are FTSED, FTMIBD, USDEUR, USDBRL, USDCNY, USDHKD,
USDINR, USDRUB, silver, high grade copper, soybean. Figure 6 is the plot for the
data without outliers for d and kurtosis. It shows that there is a negative relation
between d and the kurtosis. Which is the larger the kurtosis is, the smaller value of
maximizing d is. Also, it is shown there is an approximately linear relation between
the d and the log values of the kurtosis. So a regression is applied to d and log values
of the kurtosis. It is surprisingly that the p-values are very small. In other words, the
d is significantly linear correlated with the log values of the kurtosis. The result is as
follows,

17

For the skewness, the outliers eliminated are USDCNY, USDRUB, silver, high grade
copper, cotton. The result is shown in Figure 6. It shows that there is no specific
relation between these two variables.
In this section, a widely test on various data from the fincancial markets are
conducted. Mainly, the analysis on the relationship between the values of d, mean,
variance, kurtosis and skewness shows an interesting result that, the d that
maximizes the autocorrelation is strongly related to the kurtosis. This result
matches a lot of studies. In He T (2002), by the analytical form fo the
autocorrelation function, the realtionship between kurtosis and d would be then
studied.

18

Figure 5

data
0 1 2 3 0.0000 0.0006 0.0012 0 2000 4000 6000
0 10 20 30 40
0 1 2 3
d
mean
0.000 0.002
0.0000 0.0006 0.0012
se
skewness
-60 -20 20
0 10 20 30 40
0 2000 4000 6000
0.000 0.002 -60 -20 0 20 40
kurtosis

19

Figure 6

Figure 7

0.8 1.0 1.2 1.4 1.6 1.8
0 50 100 150 200 250 300
data1$d
data1$kurtosis
0 1 2 3
-2 0 2 4 6
data2$d
data2$skewness

20

Chapter 4. The ARCH model and Autocorrelation Function
The ARCH was first introduced in Engle (1982) to the properties of the volatility. It
was then developed and derived to other specific models. In this section, we mainly
talk about the GARCH, EGARCH and PGARCH models. If

follows the ARCH(p)
model, then

=

,

~ (0, 1) (2)
Where

= ω + ∑

(3)
In Bollerslev (1986), the generalized ARCH (GARCH) model was defined by adding
moving average to the volatility term as follows
σ

= ω + ∑ β

σ

+ ∑ α

ε

(4)
Actually, the ARCH model is a special case of the above GARCH model. Also, if
ARMA(r, m) model is added to the

to get new series

, then we can have the
ARMA-GARCH model where

= C + ∑

+ ∑

+

(5)

21

By the previous examination of the original data, the

has no correlation of itself,
so here we only take the MA(1) model to the

as a filter. The GARCH (1, 1) model,
meanwhile, was mostly used and studied. For other values of (p, q), the conclusion
was similar but tedious. So in this paper we actually refer the GARCH model as

= +

+

(6)

=

,

~ (0, 1) (7)

= +

+

(8)
As we can see from the model, the autocorrelation does not exist for

itself but
exists for the squared terms. So the model could capture this property of the
financial time series. More generally, Ding (1993) introduced a new asymmetric
power ARCH (A-PARCH) model, which includes several other ARCH models as
special cases, where the term

was defined as,

= + ∑

(|

| −

)

+ ∑

(9)
Consider the special case of the PARCH model when → 0, the conditional variance
term can be derived as
log

= + ∑

log

+ ∑

−

+ ∑

(10)

22

More generally, the variance term in the EGARCH(1, 1) mode is defined as
log

= + (

) + log

(11)
Here the g is a well-defined function of

, if
(

) =

+ (|

| − |

|) (12)
Then we could get the previous EGARCH model introduced by Nelson (1991).
Among the three parameters, measures the persistence and to ensure the log

to be stationary, < 1; is the asymmetry parameter, and stands for the
magnitude of

or say ,the shock of the volatility.
The Taylor effect was first found in Taylor (1986) and defined in Granger and Ding
(1995). It shows that for the various financial time series, the autocorrelation of the
absolute-valued prices is larger than the squared terms,
(|

|, |

|) > (|

|

, |

|

) (13)
Also, by using the distribution introduced by Nagao and Kadoya (1971), the Taylor
property was briefly shown in Granger and Ding (1995). However, there is little
evidence to show the relation between distributions to the time series tested. Then
in Ding and Granger (1996), acf for various ARCH (1, 1) type models were derived.

23

But on one hand, the analytical forms of free power were not given and on the other
hand, a specific result of the exchange rate return found the maximum d was equal
to 1/4. So it will be interesting to test and study if the d varies for different samples.
Section 3 has shown this is true by testing 4 other data. Then it comes to the
question that if we can capture this stylized fact by theoretical methods in some
specific models.
4.1 ACF in EGARCH(1, 1) model
The specific form of the ACF in EGARCH model was introduced by He T (2002), by
deriving the acf of the log ovservations. The acf of the |

|

is

() = (

∗

−

∗

) (

∗

−

∗

) ⁄ (14)
Where

= |

|

() = exp

= () ∏ [ ( )]

∗ ∏ [ ( )]

= ∏ ([ ( )] )

24

= ∏ [ ( )]

.
Here k denotes the lags of the acf, and g is the well-defined function of

. Note that
the g we have mentioned in Section 3 for Nelson’s EGARCH model. Then the
unknown determinants of the acf would be the distribution of

, the parameters
and , defined in g. Instead of using the moments of the data, we will use the
kurtosis of the data which was explained in He T(2002) and Harvey (1998). The
kurtosis of

in this model is
= (

) ∏ [ ( )]

[ [ ( )]

]

⁄

(15)
If we assume that

has normal distribution, the kurtosis and the form of the acf
could be derived as following.
= 3 exp
( )

∏

( )
( )

( )

( )
( )

( )

(16)

() =

(∙ ) ∏ Φ

∏ Φ

− ∏ Φ

/

∏ Φ

− ∏ Φ

(17)
Where

=
()

exp

( )

( )

25

=

exp

( )

(∙ ) =
()
−

( + )+ exp −

()

()
−

( − )
Φ

= Φ

( + )+ exp −

( )
Φ

( − )
Φ

= Φ

(1 +

)( + )+ exp −

( )
Φ (

(1 +

)( − ))
Φ

= Φ

( + )+ exp −2

( )
Φ (

( − ))
And Φ (∙ ) is the cdf of the standard normal distribution,

[ ] is called parabolic
cylinder function as,

[ ]=

()
∫

exp −−

(18)
Compared to the original model (17), this function is easier to calculate and has a
better approximation when applying numerical method.
4.2 Parameters and Taylor Effect
Based on the autocorrelation function, when specific distribution and parameters
are applied to the model, the Taylor property could then be checked. Notice that the
relationship between

(1) and

(2) has already been proved in various models.

26

And it shows the Taylor effect exists with some specific parameters. But the focus of
this paper is on the more generalized form of Taylor effect. Or say, by the acf derived
in EGARCH and ARSV model, the relationship between the maximum d and the
parameters is our interest. Becasue in EGARCH model, the kurtosis are functions of
, and .
But the acf can not be expressed only with kurtosis, which means we need to fix 2 of
the 3 parameters and change the 1 lest. Then along with the change the last
parameter, the acf and kurtosis both could be calculated. and are first set as
0.99 and 0.3, which is approximately shown in many financial data. Then Figure 8
gives a visual relationship among acf, kurtosis and power d.
So under the assumption that

has normal distribution, and fix the persistence
=0.99, the asymmetry = −0.19, we change the shock from 0.05 to 0.5. The
trend shows in 2 aspects: the peak power is changing with different kurtosis; the
autocorrelation decreases along when the kurtosis increases. More specifically,
Figure 8 shows the acf with power d for different kurtosis. It was explicit that the d
maximizing the autocorrelation does change from about 0.5 to 1 for different
kurtosis. Notice that the kurtosis increases when ||, || increase, so this plot

27

shows that the larger the kurtosis, the smaller the maximum d is. Similarly, when
the changing parameter is , we can get the relationship which is in Figure 9.
Figure 8

28

Figure 9

29

Chapter 5. Emperical results and Simulation
In this section, three data samples are fitted to EGARCH model, and simulations by
the fitted model are carried out to see if they are reasonable. The Dow jones
industial index, USDCAD exchange rate and Gold futures are chosen respectively.
Then by plugging in the estimated parameters to the acf derived in Section 4, the
power maximizing the acf are compared to the power from the original data.
To test the existence of ARCH property of the data, the Lagrange multiplier (LM) test
is first taken. Table 1 shows the p values of the LM statistics. The results strongly
indicate that the heteroscedasticity property exists for all the data in small lags.
However, for the exchange rates data, the property vanishes rapidly. And this is the
same with the results shown in Section 3.
Table 1

Lag[1] Lag[5] Lag[10]
DJI 0.0005526 0.0023267 0.0122245
USDCAD 0.01029 0.09467 0.39655
GCC 6.726e-12 1.379e-10 1.791e-08

30

Table 2 displays the estimated results. The models are revised by different results.
For example, for the USDCAD exchange rate returns, when applying the
MA(1)-EGARCH(1,1), the p-value of the MA term is 0.200128, which shows we
could reject. So EGARCH(1,1) with no ARMA model is then applied.
Table 2

Ma1
DJI 0.000347 0.127837 -0.083869 -0.057999 0.991350 0.132777
USDCAD NA NA -0.075081 0.023384 0.992425 0.325317
GCC 0.000131 -0.075670 -0.047619 0.033353 0.994762 0.119654

After the parameters are estimated, simulations are carried out 1000 larger than the
size of the 4 samples, and the first 1000 simulations are discarded. The Figures
compare the simulated acf of the squared values compared to the actual data, while
Figure is the plot of the absolute values. Though the simulated acf of the absolute
values are typically smaller than the actual ones, but the shape is more similar,
which indicates better fitted. Along with the power changing, Figure shows the acf of
the simulated values and of original data at lag=1. Also by the simulated parameters,
the values calculated by the theoretical form of the acf derived in Chapter 4 is added
to the figures. As we can see, the simulated acf fits well with the theoretical ones for

31

the DJI and USDCAD. However for the GCC exchange rates, the shape of acf of the
returns is irregular compared to the other three samples, and the simulated results
are not satisfactory either. But to our interest, the power that maximizes the acf are
most significant, which implies the generalized Taylor effect.
To capture the property of the autocorrelation of the varying volatility, several time
series models are introduced. Based on the ARCH structure, Bollerslev’s GARCH
model, Taylor/Schwert’s model and Ding, Granger, and Engle’s Power-ARCH model
are mostly used. The main difference is that the three models focus on different
power transformation of the volatility. In section 3, we can see the results show the
autocorrelation are mostly stronger for |

| than

, so one may consider the
Taylor/Schwert’s model better fits the data. Notice that Taylor/Schwert’s model is a
special case of the PARCH model when δ = 1. So here the estimations are taken in
two ways. From the estimation results in Appendix B, we can see the estimated
parameters are always significant in the same for both models. For example, all
parameters are significant for the Dow Jones Industrial Average Index for both
models. And for the other 3 indices, the parameters in GARCH model are not
significant when the parameters for PARCH are not simultaneously.

32

To better illustrate the model efficiency, Monte Carlo simulations are conducted.
Figure 10 shows the autocorrelation of |

|

for r and two simulation series. It is
shown that both models could catch the long memory property, while GARCH model
is better fitted. As tested in Section 3, the d that maximizes the autocorrelation of
the |

|

is 1.13, 0.29, 0.81, 1.25. And for the PARCH model, the estimated δ is 1.6,
0.7, 1.4 and 1.7. It is obvious that the d and δ have the same trend, or say, δ = d +

.
This could only be carried out by the computer, but in some aspects, this shows the
PARCH model could carry out the d. Compared to the ARSV model, the d could be
explained in terms of

,

and

, which have strong relation with δ. So from the
empirical applications, PARCH and ARSV models could figure it out numerically.
Figure 10
0 20 40 60 80 100
0.0 0.1 0.2 0.3
lags
Autocorrelation
0 20 40 60 80 100
0.0 0.1 0.2 0.3
lags
Autocorrelation

33

Chapter 6. Conclusion
In this paper, 42 other financial time series were examined. The results show the
autocorrelation does exist for the power transformation of the returns. On the other
hand, by maximizing it, it is shown that the Taylor Effect holds for different d in
different samples. However, the power that maximizes the autocorrelation varies a
lot. To explain it to some aspects, we analyze the relationship between the power
and other properties of the data. It is shown that there is a strong linear relation
between d and the log values of the kurtosis. Meanwhile, no obvious evidence shows
that the power is realted to other variables. Then by the derived form of acf in He T
(2002), this paper explores this relationship. Though we could not derive a
analytical form fo the maximum d, we could figure out the relation between d and
kurtosis. The result is in accordance to the result shown in previous part. Then the
EGARCH model is fitted to one of the data and simulations are then conducted. It
shows the model is reasonable enough to capture the property.

34

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38

Appendix
Table A: The data range
Data Start date End date Observations
SPXD 01/03/1984 03/03/2014 7607
DJI3D 01/03/1984 03/03/2014 7606
OTC_D 01/03/1984 03/03/2014 7607
GSPTSED 01/03/1984 03/03/2014 7594
EUSX5ED 12/31/1986 03/03/2014 6988
FTSED 01/03/1984 03/03/2014 7618
FCHID 07/09/1987 03/03/2014 6740
GDAXD 01/02/1984 03/03/2014 7587
IBEXD 01/05/1987 03/03/2014 6822
FTMIBD 12/31/1997 03/03/2014 4153
AEXD 01/02/1984 03/03/2014 7662
OMXSPID 01/31/1986 03/03/2014 6963
N225D 01/04/1984 03/03/2014 7594
AXJOD 01/31/1992 03/03/2014 5574
HSID 01/03/1984 03/03/2014 7497
SSECD 12/19/1990 03/03/2014 5672
MXXD 01/02/1985 03/03/2014 7300
BVSPD 01/02/1984 02/28/2014 7535
USDCAD 01/03/1984 03/03/2014 7846
USDEUR 01/02/1984 03/03/2014 7713
USDGBP 01/03/1984 03/03/2014 7834
USDAUD 01/03/1984 03/03/2014 7826

39

USDJPY 01/03/1984 03/03/2014 7844
USDBRL 01/02/1984 03/03/2014 7726
USDCNY 01/03/1984 03/03/2014 7675
USDHKD 01/03/1984 03/03/2014 7737
USDINR 01/03/1984 03/03/2014 7735
USDMXN 01/03/1984 03/03/2014 7743
USDRUB 01/03/1984 03/03/2014 6344
USDCHF 01/03/1984 03/03/2014 7832
gold 01/03/1984 03/03/2014 7578
silver 01/03/1984 03/03/2014 7572
high grade copper 01/03/1984 03/03/2014 7576
platinum 01/03/1984 03/03/2014 7547
corn 01/03/1984 03/03/2014 7608
soybean 01/03/1984 03/03/2014 7606
wheat 01/03/1984 03/03/2014 7606
cotton 01/03/1984 03/03/2014 7580
crude oil 01/03/1984 03/03/2014 7545
natural gas 04/03/1990 03/03/2014 5999
us treasury bond 01/03/1984 03/03/2014 7599
us 10-year bond 01/03/1984 03/03/2014 7600

Table B: The values for different data
data d mean se skewness kurtosis
SPXD 1.22 0.000318238 0.000132798 -1.273687 28.0744
DJI3D 1.18 0.000336319 0.000129787 -1.682733 41.91773
OTC_D 1.17 0.000359557 0.000161124 -0.2369475 8.195057
GSPTSED 1.56 0.000226186 0.000112886 -0.9315663 14.74038
EUSX5ED 1.14 0.000174739 0.00015973 -0.1531232 5.935494

40

FTSED 3.56 0.000250212 0.0001277 -0.4802456 9.500651
FCHID 1.5 0.000157664 0.000172026 -0.141156 5.282739
GDAXD 1.2 0.000325992 0.000163667 -0.3140695 6.19985
IBEXD 1.13 0.000220125 0.000170617 -0.07540204 5.47444
FTMIBD 0.01 -5.08E-05 0.000242278 -0.08131647 4.130582
AEXD 1.34 0.000213602 0.000154681 -0.2608978 8.289928
OMXSPID 1.47 0.000391667 0.000162094 0.05476509 5.984059
N225D 1.63 5.13E-05 0.000165372 -0.320745 8.199229
AXJOD 1.74 0.000215216 0.000129205 -0.4313148 5.931758
HSID 1.18 0.000433777 0.000198132 -2.251437 54.17237
SSECD 0.73 0.000534802 0.000320059 5.559207 152.7058
MXXD 1.81 0.001256751 0.000209985 -0.452504 17.52736
BVSPD 0.83 0.003445939 0.000350285 -0.623561 28.34601
USDCAD 1.37 -1.52E-05 5.05E-05 0.1594495 5.74585
USDEUR 2.33 -6.61E-05 7.97E-05 0.06161741 6.055585
USDGBP 1.32 -1.97E-05 7.11E-05 0.09932239 3.439354
USDAUD 1.58 5.27E-07 8.61E-05 0.5478854 12.02732
USDJPY 1.24 -0.000105738 7.87E-05 -0.3899569 6.139682
USDBRL 0.12 0.002926584 0.000119697 1.1652 22.99414
USDCNY 0.01 0.000147215 6.73E-05 53.52434 3311.074
USDHKD 1.17 -6.23E-07 6.94E-06 6.63162 907.2021
USDINR 0.41 0.000228829 5.26E-05 5.352201 134.5234
USDMXN 0.95 0.000563038 0.00011436 3.778311 113.5117
USDRUB 0.01 0.00170244 0.00051007 52.0332 3471.44
USDCHF 1.34 -0.000117513 8.28E-05 0.1013234 4.96924
gold 1.48 0.000167595 0.000121422 -0.1269693 8.890355
silver 0.34 -0.000491567 0.001366564 -53.77612 4966.037
high grade
copper
0.39 -0.000398595 0.000643552 -74.48101 6150.575
platinum 1.23 0.000175078 0.000165262 -0.2884284 4.182063

41

corn 0.75 4.49E-05 0.000198534 -1.392238 23.0615
soybean 3.64 7.68E-05 0.000180505 -0.9516823 16.56061
wheat 1 7.51E-05 0.000212174 -1.023314 17.00474
cotton 0.67 2.03E-05 0.000229307 -8.559391 323.2503
crude oil 1.13 0.000168638 0.000277101 -0.8388064 15.73106
natural gas 0.91 0.000168499 0.000457579 0.2054964 7.448934
us treasury bond 0.8 8.72E-05 8.01E-05 -2.937623 80.26898
us 10-year bond 0.81 6.23E-05 5.21E-05 -2.389452 61.14194

Asset Metadata

Creator Song, Mingpu (author)

Core Title Generalized Taylor effect for main financial markets

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School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Statistics

Publication Date 06/17/2014

Defense Date 05/07/2014

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

Tag autocorrelation,financial time series,GARCH model,OAI-PMH Harvest,Taylor effect

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Language English

Advisor Zhang, Jianfeng (committee chair), Lototsky, Sergey V. (committee member), Mikulevičius, Remigijus (committee member), Piterbarg, Leonid (committee member)

Creator Email 306829355@qq.com,mingpuso@usc.edu

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Abstract (if available)

Abstract It has been tested that for financial time series, the autocorrelation does not exists for the returns themselves, while exist for the power transformation of the absoluted values. Also, lots of studies have shown that, known as Taylor Effect, the absolute returns are more strongly autocorrelated than the squared values. Ding (1996) further checked some other time series and found that for the exchange rate returns, the power d maximizing the autocorrelation was around 1/4. So a Generalized Taylor Effect that the power maximizes the autocorrelation varies for different data is proposed and analyzed. To more widely concern, 18 stock indices, 12 exchange rate seires and 12 future contracts are further checked. The d is analyzed with some other properties of the data‐mean, standard error, kurtosis and skewness. Inspired by the deduction of ACF in EGARCH models, He T (2002), this paper take a deeper look at the relationship among those values. Tests and Monte Carlo simulations are also carried out to see the goodness of fit of the EGARCH model.