modeling of market volatility with aparch model417608/fulltext01.pdf · the asymmetric power arch...

U.U.D.M. Project Report 2011:6

Examensarbete i matematik, 30 hpHandledare och examinator: Maciej Klimek

Maj 2011

Modeling of Market Volatility with APARCH Model

Ding Ding

Department of MathematicsUppsala University

i

Abstract

The purpose of this paper is to discuss the APARCH model and its ability to forecast

and capture common facts about conditional volatility, such as fat-tails, persistence

of volatility, asymmetry and leverage effect. We investigate the forecasting

performance of APARCH model with the various density functions: normal

distribution, student’s t-distribution, skewed student’s t-distribution. We test on

three major stock market indexes: Standard&Poor 500 stock market daily closing

price index and MSCI EUROPE INDEX.

i i

1. Literature Review ................................................................................................- 1 -

2. The main characteristic of finance asset volatility..............................................- 2 -

2.1 Fat tails and Excess kurtosis ..........................................................................- 2 -

2.2 Volatility Clustering .......................................................................................- 2 -

2.3 Long Memory ................................................................................................- 2 -

2.4 Leverage Effects.............................................................................................- 3 -

2.5 Spillover Effects .............................................................................................- 3 -

3. Stochastic Volatility Models ................................................................................- 4 -

3.1 ARCH Model ..................................................................................................- 4 -

3.2 GARCH model ................................................................................................- 4 -

4. APARCH Model ....................................................................................................- 5 -

4.1 Normal Distribution ......................................................................................- 7 - 4.2 Student t Distribution .................................................................................- 10 -

4.3 Skewed student-t Distribution ....................................................................- 14 -

4.4 Forecasting Methods...................................................................................- 15 -

5. Empirical Application (Standard&Poor 500 Daily Index) ..................................- 17 -

5.1 Data Analysis ...............................................................................................- 17 - 5.2 Autocorrelation Analysis .............................................................................- 19 -

5.3 Selection of ARMA (p,q) ..............................................................................- 20 -

5.4 Estimation Result ........................................................................................- 21 -

6. Empirical Application (MSCI Europe Daily Index) .............................................- 23 -

6.1 Data Analysis ...............................................................................................- 23 -

6.2 Autocorrelation Analysis .............................................................................- 25 -

6.3 Selection of ARMA (p,q) ..............................................................................- 26 -

6.4 Estimation Result ........................................................................................- 26 -

7. Conclusion .........................................................................................................- 28 -

References ................................................................................................................- 30 -

Appendix A ...............................................................................................................- 33 -

Appendix B ...............................................................................................................- 42 -

- 1 -

1. Literature Review

The volatility of assets return is a topic that has been for a long time a major concern

to financial economists. Asset Portfolio Theory attempts to use the variance or

covariance to describe the volatility of returns in order to find the optimal portfolio.

The capital asset pricing model (CAPM) is used to determine a theoretically

appropriate rate of return of an asset. Option pricing formula gives options and other

derivatives prices based on the potentially volatility of assets.

However, the traditional financial econometrics models of risk are vague and the

volatility characteristics are difficult to understand. They are generally regarded as

variance independent, identically distributed constants. In 1960’s, a large number of

empirical research on price behavior in financial markets confirmed that the variance

changes with time. Mandelbrot (1963), the father of Fractal Theory, first discovered

the volatility of financial asset returns exhibits the clustering phenomenon. That is,

wider fluctuations cluster during certain times, while minor fluctuations cluster some

other time. This phenomenon is a common characteristic of financial markets. At the

same time the marginal contribution of asset returns is at the peak level, which

means they have wider tail compared to the standard normal distribution. Bera and

Higgins (1992) used the weekly exchange rate of the USD and GBP, the U.S. federal

government three months short-term bond rates and the growth rate of New York

Stock Exchange monthly index to verify the Mandelbrot’s theory. It can be seen that

the traditional econometric models with the assumption that variance is

independent and constant are not suitable for financial market price changes. Many

econometricians began to try different models and methods to solve this problem.

One of them is Engle (1982), who proposed the ARCH Model (Autoregressive

Conditional Heteroskedasticity). The model best reflects the changes of variance and

is widely used in the economics of time series analysis. Bollerslev (1986), Engle, Lilien

and Robbins (1987) improved the ARCH model and proposed GARCH, ARCH-M and

other promotional models. These models constitute a relatively complete theory of

autoregressive conditional heteroskedasticity in the economic and financial fields .

The Asymmetric Power ARCH model (APARCH) of Ding et al. (1993) is one of the most

promising ARCH type models. First we review the literature related to the

phenomenon of volatility.

- 2 -

2. The main characteristic of finance asset volatility

2.1 Fat tails and Excess kurtosis

The tradition of financial theory hypothesizes that the Rate of return of financial

assets has normal distribution. Mandelbrot found that the Rate of return is more like

Levy distribution, which performs as fat tails and excess kurtosis, which is verified in

stock market( Alexander 1961) and other financial asset(Peter 1991).

With the development on financial data, there are two modern hypothesizes on the

Rate of return of stock. One is the rate of return is levy distribution support by

Mandelbrot. Another is using Mixture distribution instead of Normal distribution (e.g.

Bollerslev (1987) used t-distribution. Jorion (1988) used Normal mixture distribution

of a Poisson. Baillie&Bolleslev (1989) used power exponent distribution. Nelson

(1990) used Expansion of the exponential distribution). The Fat Tail feature exists

everywhere in timeline. The kurtosis will increase with the data frequency grow

according to Anderson&Bollerslev (1998).

2.2 Volatility Clustering

Volatility clustering refers to the observation, as noted by Mandelbrot (1963), that

"large changes tend to be followed by large changes, of either sign, and small

changes tend to be followed by small changes.” This phenomenon might be caused

by the continuous effect of the external shocks. The ARCH (Engle, 1982) and

GARCH (Bollerslev, 1986) models describe the phenomenon of volatility clustering

more accurately. ARCH model explained the regularity of the Return time series.

GARCH model explained the heteroscedasticity of the Return Sequence residuals.

2.3 Long Memory

Long memory in volatility occurs when the effects of volatility shocks decay slowly,

which is often detected by the autocorrelation of measures of volatility. The practical

explanation is that historical event has a long and lasting effect. Fama&French (1988)

and Poterba&Summers (1988) discovered positive correlation in short term and

negative correlation in long term of stock returns. The significance of the

phenomenon is that the existence of “long memory” enables to predict the returns.

http://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot

http://en.wikipedia.org/wiki/ARCH

http://en.wikipedia.org/wiki/Robert_F._Engle

http://en.wikipedia.org/wiki/GARCH

http://en.wikipedia.org/wiki/Tim_Bollerslev

- 3 -

This phenomenon fights against the market efficiency hypothesis. AR, MA, ARMA,

ARIMA models represent short memory features. In this circumstance, there models

are inadequate. Geweke&Porter-Hudak (1983) brought fractional differencing test

for long memory. Engle&Bollerslev (1986) use IGARCH model to simulate the long

memory, but temporal aggregation generated by the model reduced credibility.

FIGARCH model (Baillie, Bollerslev&Mikkelsen, 1996), FIEGARCH model

(Bollerslev&Mikkelsen, 1996), LM-ARCH (Zumbach, 2002) were developed to analyze

this characteristic.

2.4 Leverage Effects

Black (1976) discovered that the current return and future volatility have negative

correlation, which means bad news will cause violent fluctuations compare to good

ones. It is called Leverage Effects. In other words, positive and negative information

lead to different level of effect to volatility. EGARCH model (Nelson, 1991) analyzes

the effect on stock volatility from asymmetric conditional heteros kedasticity caused

by different information. Glosten, Jagannathan&Runkle (1993) use GJR-GARCH model

which adds seasonal terms to distinguish the positive and negative shocks. Ding,

Granger and Engle (1993) brought APARCH model (asymmetric power ARCH), which

increased two parameters based on the GARCH model. One of the parameter is used

to be measure Leverage Effect.

2.5 Spillover Effects

The phrase Spillover Effects refers to positive or negative effects of those who are not

directly involved in it. In the financial markets, not just one single market will be

affected by the historical fluctuation but also other financial markets. For Spillover

Effects, Ross (1989) pointed out that volatility is directly linked to the rate of

information flow between the markets. King&Wadhwani (1990) showed that even

the information is for one specific market, the information flow will cause

over-reaction in other markets. Engle, Ito&Lin (1990) have separated the world

market into four main regions: Japan Region, Pacific Region, New York Region and

Europe Region and have proven that the regions have fluctuation conductivity. Chart,

Chan&Karoyi (1991) used high dimensional ARCH to prove future market fluctuation

will aggravate the volatility of monetary market, and from monetary market to future

market also exists wave conduction. But the Spillover Effects are more visible in

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developed countries and markets, less visible in undeveloped countries and markets.

3. Stochastic Volatility Models

3.1 ARCH Model

ARCH (Auto-regressive Conditional Heteoskedastic Model) is the simplest model in

stochastic variance modeling which was developed by Engle (1982). The particularity

of this model is that restriction of the auto-regression residual has been changed

from constant (var(εt) = 𝜎2) to a random sequence which only depend on past

residuals (*ε1, … , εt−1+). And Bollerslev (1986) amended this model adding the

conditional heteroskedasticity moving average items.

The model can be expressed as follows:

yt = xtξ + εt t = 1,2… . . T,

σt2 = ω+∑αjεt−j

2

q

j=1

εt = σtzt, zt~N(0,1) .

To assure *σt2+ is asymptotically stationary random sequence, we can assume

that α1 +⋯+αq < 1.

In the ARCH model, the conditional variance of εt is an increasing function of Lag

errors. Autoregressive coefficient decides the influence on persistence for the

follow-up errors. Larger Q cause the longer time of volatility persistence.

3.2 GARCH model

The Generalized Auto-Regressive Conditional Heteoskedastic Model is based on an

infinite ARCH specification. It improves the ARCH model by reducing the number of

estimated parameters from infinity to two. Standard GARCH models assume that

positive and negative error terms have asymmetric effect on the volatility. Nelson

(1991) brought exponential GARCH model to work on the leverage effect.

The model can be expressed as follows:

yt = xtξ + εt t = 1,2… . . T,

- 5 -

σt2 = ω+∑βjσt−j

2

p

j=1

+∑αjεt−j2

q

j=1

εt = σtzt, zt~N(0,1) .

In the GARCH model, the impacts to conditional variance of positive and negative

side are symmetrical. So GARCH is unable to express the Leverage Effects. The

GARCH (p, q) model is the extension of ARCH models, the GARCH (p, q) also has the

ARCH (q) model features. However, the conditional variance of GARCH model is not

only a linear function of lagged squared residuals but also a linear function of lagged

conditional variance.

GARCH model has greater applicability for easy computation. But the GARCH model

has drawbacks in application for asset pricing. First, GARCH model cannot explain the

negative correlation between the fluctuations in stock returns. GARCH (p, q) model

assumes that the conditional variance is a function of lagged squared residuals . So

the symbol does not affect the residual volatility, that is positive and negative

changes are symmetric to conditional variance. However, empirical studies found

that negative information had more influence on the volatility than the positive

information. Second, the GARCH model assumes all coefficients are greater than zero,

which also makes the model hard to apply.

In order to measure the rate of return volatility asymmetry, Glosten, Jagannathan

and Runkel (1993) proposed a GJR model, adding the negative impact of leverage in

the conditional variance equation. Nelson (1991) proposed the EGARCH model.

GJR-GARCH model:

σt = ω +∑(αjεt−j2 + γi(max(0, εt−j))

2)

q

j=1

+∑βiσt−i

p

i=1

EGARCH model:

log (σt) = ω +∑βjlog (σt−j)

p

i=1

+∑(αiεt−j

√σt−j+ γi |

εt−j

√σt−j|)

q

j=1

4. APARCH Model

Ding, Granger and Engle (1993) brought APARCH (Asymmetric Power ARCH Model).

- 6 -

This model can well express the Fat tails, Excess kurtosis and Leverage Effects. The

general structure is as follows:

yt = xtξ + εt t = 1,2… . . T,

σtδ = ω+∑αj(|εt−j| − γjεt−j)

δ

q

j=1

+∑βi(σt−i)δ

p

i=1

εt = σtzt , zt~N(0,1)

𝑘(εt−j) = |εt−j| − γjεt−j .

The mean equation ( yt = xtξ + εt t = 1,2… . . T) could also be written as

yt = E(yt|ψ𝑡−1) + εt, where E(yt|ψ𝑡−1) is the conditional mean of yt given ψ𝑡−1.

ψ𝑡−1 the whole information at time t-1.

ψ𝑡 = {yt, yt−1, … , y1, y0,xt, xt−1, … ,x1,x0} ,

where ξ, ω , αj , γj , βi and δ are the parameters which are needed to be

estimated. γj, reflects the leverage effect. A positive γj means negative information

has stronger impact than the positive information on the price volatility. δ reflects

the leverage effect.

The APARCH equation (σt2 = ω +∑ αj(|εt−j| − γjεt−j)

δqj=1 +∑ βi(σt−i)

δpi=1 ) is

supposed to satisfy the following conditions.

1) ω > 0, αj ≥ 0, 𝑗 = 1,2, …𝑞, βi ≥ 0, 𝑖 = 1,2,… 𝑝, when αj = 0, 𝑗 = 1,2,… 𝑞,

βi = 0, 𝑖 = 1,2, … 𝑝, then σt2 = ω. Due to the variance is positive, so ω > 0.

2) 0 ≤ ∑ αjqj=1 + ∑ βi

pi=1 ≤ 1

The corresponding conditional expectation and conditional variance of the Mean

equation’s explanatory variables are:

E,yt|xt- = xtξ

Var,yt|xt- = σtδ .

For T → ∞, the unconditional variance of εt would be

σtδ =

ω

1− ∑ αj(1 − γj)δq

j=1−∑ βi

p

i=1

.

This model includes the ARCH and GARCH models, by changing the parameters we

can get different models.

- 7 -

When δ = 2, βi = 0(𝑖 = 1,… , 𝑝), γj = 0(𝑗 = 1,… , 𝑞), APARCH model is ARCH

model.

When δ = 2, γj = 0(𝑗 = 1,… , 𝑞), APARCH model is GARCH model.

When δ = 2, APRCH model is GJR-GARCH model.

When δ = 1, APRCH model is TARCH model.

When βi = 0(𝑖 = 1,… , 𝑝) , γj = 0(𝑗 = 1, … , 𝑞) , APARCH model is NARCH

model.

When δ = ∞, APRCH model is Log-ARCH model.

More detail can be found in Ding et al. (1993).

4.1 Normal Distribution

The Conditional density function of yt is

f (yt|xt, ψ𝑡−1) =1

√2𝜋σt2𝑒𝑥𝑝,

−(yt − xtξ)2

2σt2

- ,

where σtδ = ω +∑ αj(|εt−j| − γjεt−j)


δpi=1

= ω+∑αj(|yt − xtξ| − γj(yt − xtξ))δ

q

j=1

+∑βi(σt−i)δ

p

i=1

.

Use maximum log-likelihood method to estimate the parameters in the APARCH

model. First we define some vector parameters to simplify the formula. We define

the vector γ = (γ1, γ2, … , γq), which measures the leverage effect; the vector θ =

(ω, α1, α2 , … , αq , β1, β2, … , βp) and the vector η = (ξ, γ, θ, δ) , which is the vector set

of the unknown parameters.

From the density function of yt, we have the log-likelihood function as below:

Log L(η) =∑log f(yt|xt , A)

𝑇

𝑡=1

= −𝑇

2log(2𝜋) −

1

2∑log (

𝑇

𝑡=1

σt2) −

1

2∑

(yt − xtξ)2

σt2

𝑇

𝑡=1

.

We can use the log likelihood to calculate the parameters η’. So the function Log L (η’)

can get the largest value at η’. It is usual to assume that zt is normal distribution.

The differentiating functions with respect to vector η are as follows:

- 8 -

∂ Log L(η)

∂ η = −

1

2∑

∂log(σt2)

∂ η

𝑇

𝑡=1

−1

2∑*

1

σt2

∂(yt − xtξ)2

∂ η−(yt − xtξ)

2

σt2

∂σt2

∂ η+

𝑇

𝑡=1

=1

2∑*−

1

σt2

∂σt2

∂ η−

1

σt2

∂(yt − xtξ)2

∂ η+εt2

σt2

∂σt2

∂ η+

𝑇

𝑡=1

=1

2∑*

εt2 − σt

2

σt4

∂σt2

∂ η−1

σtδ

∂εt2

∂ η+

𝑇

𝑡=1

=1

2∑*

εt2 −σt

2

σt4

∂σt2

∂ η−2 εtσtδ

∂εt∂ η

+

𝑇

𝑡=1

.

The differentiating of the variance with the respect to the vector set η is as:

∂σtδ

∂η=∂ 0ω+∑ αj(|εt−j| − γjεt−j)


δpi=1 1

∂ η

=∂ω

∂ η+∑

∂αj(|εt−j| − γjεt−j)δ

∂ η

q

j=1

+∑∂βi(σt−i)

δ

∂ η

p

i=1

.

We can rewrite the σtδ to σt

2:

∂σt2

∂ η=2σt

2

δσtδ

∂σtδ

∂ η .

From the above we can tell that ∂εt

∂ ξ= −xt. To find a tractable solution of

∂σt2

∂ η, we

can separate calculate the different parameters.

The differentiating of σtδ with the respect to ξ:

∂σtδ

∂ ξ=∂ω

∂ ξ+∑


∂ ξ

q

j=1

+∑∂βi(σt−i)

δ

∂ ξ

p

i=1

=∑δαj(|εt−j| − γjεt−j)δ−1

∂(|εt−j| − γjεt−j)

∂ ξ

q

j=1

+∑δβi(σt−i)δ−1

∂(σt−i)

∂ ξ

p

i=1

.

If εt−j ≥ 0, then ∂(|εt−j|−γj εt−j)

∂ ξ= (γj − 1)xt−j .

If εt−j < 0, then ∂(|εt−j|−γj εt−j)

∂ ξ= (γj + 1)xt−j .

It is possible there exist some εt−j , which makes coefficient negative (t < j) .

- 9 -

According to Sebastien Laurent 2004, it is easy to do the recursion of the Equation

by setting unobserved components to the sample average.

Here we quote the formulas from Laurent (2004) to set unobserved components to

their sample average.

𝑘(εt−j) =1

𝑇∑ (|𝜀𝑆| − γiεs)

δ𝑇𝑠=1 , for 𝑡 ≤ 𝑗 σt

δ = (1

𝑇∑ 𝜀𝑠

2𝑇𝑠=1 )

𝛿

2 , for 𝑡 ≤ 0.

By bringing two new symbols, we can simplify the formula to computing easily. First we define:

𝐼𝜏 = {−1, 𝑖𝑓ε𝜏 ≥ 0

1, 𝑖𝑓ετ < 0

𝐹𝜏 = {1, 𝑖𝑓 𝜏 > 00, 𝑖𝑓 𝜏 ≤ 0

,

Then the formula above can be transformed as follows:

∂σtδ

∂ ξ= δ∑αj 0(|εt−j| − γjεt−j)

δ−1 (γj + 𝐼𝑡−𝑖)xt−j1

𝐹𝑡−𝑗

q

j=1

× [1

𝑇∑(|εt−j| − γjεt−j)

δ−1(γj + 𝐼𝑡−𝑖)xt−j

𝑇

𝑠=1

]

1−𝐹𝑡 −𝑗

+∑βi (∂(σt−i)

δ

∂ ξ)

𝐹𝑡 −𝑖p

i=1

[−δ

𝑇(1

𝑇∑εs

2

𝑇

𝑠=1

+

δ−22

∑εsxs

𝑇

𝑠=1

]

1−𝐹𝑡−𝑖

.

The differentiating of σtδ with the respect to γ:

∂σtδ

∂γ=∂ω

∂γ+∑


∂γ

q

j=1

+∑∂βi(σt−i)

δ

∂γ

p

i=1

=∑αj∂k(εt−j)

δ

∂γ

q

j=1

+∑βi∂(σt−i)

δ

∂γ

p

i=1

.

The differentiating will be different with t changes.

- 10 -

∂k(εt−j)δ

∂γ=

{

−δk(εt−j)

δ−1εt−j ,𝑓𝑜𝑟 𝑡 > 𝑗

−δ

𝑇∑(|𝜀𝑆| − γiεs)

δ−1εs ,𝑓𝑜𝑟 𝑡 ≤ 𝑗

𝑇

𝑠=1

,

and ∂(σt−i)

δ

∂γ= 0 for 𝑡 ≤ 0.

The differentiating of σtδ with respect to δ:

∂σtδ

∂δ=∂ω

∂δ+∑


∂δ

q

j=1

+∑∂βi(σt−i)

δ

∂δ

p

i=1

= δ∑αj 0(|εt−j| − γjεt−j)δ 𝐿𝑛(|εt−j | − γjεt−j)1

𝐹𝑡 −𝑗

q

j=1

× [1


δ𝐿𝑛(|εs| − γjεs)

𝑇

𝑠=1

]

1−𝐹𝑡−𝑗

+∑βi .(σt−i)δ𝐿𝑛(σt−i)/

𝐹𝑡−𝑖

p

i=1

[−1

𝑇(1

𝑇∑εs

2

𝑇

𝑠=1

+

δ2

𝐿𝑛(1

𝑇∑εs

2

𝑇

𝑠=1

)]

1−𝐹𝑡 −𝑖

.

The differentiating of σtδ with respect to θ:

∂σtδ

∂θ=∂ω

∂θ+∑


∂θ

q

j=1

+∑∂βi(σt−i)

δ

∂θ

p

i=1


δ

∂θ

q

j=1

+∑βi∂(σt−i)

δ

∂θ ,

p

i=1

and ∂(σt−i)

δ

∂θ= 0 for 𝑡 ≤ 0.

4.2 Student t Distribution

From the above study on the characteristic of the financial time series, it may be

more appropriate to use student t distribution to express the fat tail and excess

kurtosis than the normal distribution. The t-distribution was first discovered by

- 11 -

William S. Gosset in 1908. The t density curves are symmetric and bell-shaped like

the normal distribution and have their peak at 0. However, the spread is more than

that of the standard normal distribution. The degrees of freedom is larger, the

t-density is closer to normal density. If zt has the student t distribution with 𝑣

degree of freedom, the density functions of zt and εt are

f (zt|, 𝑣) =𝛤.𝑣 + 12 /

√(𝑣 − 2)𝜋𝛤 .𝑣2/

(1 +zt2

𝑣 − 2)−𝑣+12

f (εt|𝑣) =𝛤 .𝑣 + 12 /

√(𝑣 − 2)𝜋𝛤 .𝑣2/(1+

.εtσt/2

𝑣 − 2,

−𝑣+12

(−1

σt2* .

If 𝑣 is even,

𝛤 .𝑣 + 12 /

√𝑣𝜋𝛤 .𝑣2/=

(𝑣 − 1)(𝑣 − 3) ∙∙∙ 5 ∙ 3

√𝑣𝜋(𝑣 − 2)(𝑣 − 3) ∙∙∙ 4 ∙ 2 .

If 𝑣 is odd,

𝛤 .𝑣 + 12

/

√𝑣𝜋𝛤 .𝑣2/=

(𝑣 − 1)(𝑣 − 3) ∙∙∙ 4 ∙ 2

√𝑣𝜋(𝑣 − 2)(𝑣 − 3) ∙∙∙ 5 ∙ 4 .

We have the log-likelihood function as below:

Log L(η) = ∑log f(εt|η, 𝑣)

𝑇

𝑡=1

= 𝑇 {𝐿𝑛 𝛤(𝑣 + 1

2* − 𝐿𝑛𝛤 .

𝑣

2/ −

1

2𝐿𝑛,(𝑣 − 2)𝜋-} −

1

2∑𝐿𝑛(σt

2)

𝑇

𝑡=1

−∑(𝑣 + 1

2*

𝑇

𝑡=1

𝐿𝑛(1 +.εtσt/2

𝑣 − 2, .

When v →∞, student t distribution becomes the normal distribution.

We can use the log likelihood to calculate the parameters η’, for which function Log L

- 12 -

(η’) can get the largest value at η’. The differentiating function with respect to

vector η is as follows:

∂ Log L(η)

∂ η = −

1

2∑

∂ 𝐿𝑛(σt2)

∂ η

𝑇

𝑡=1

−∑(𝑣 + 1

2*

∂ 𝐿𝑛(1+.εtσt/2

𝑣 − 2,

∂ η

𝑇

𝑡=1

= −1

2∑

1

σt2

∂σt2

∂ η

𝑇

𝑡=1

− (𝑣 + 1

2*∑

∂ 𝐿𝑛(1 +.εtσt/2

𝑣 − 2,

∂ η

𝑇

𝑡=1

= −1

2∑

1

σt2

∂σt2

∂ η

𝑇

𝑡=1

− (𝑣 + 1

2(𝑣 − 2)*

1

(1 +zt2

𝑣 − 2*∑

∂ zt2

∂ η

𝑇

𝑡=1

∂ zt2

∂ η =

1

σt2

∂εt2

∂ η+ εt

2∂σt

−2

∂ η=2εtσt2

∂εt∂ η

−2εt

2

σt3

∂σt∂ η

.

From the above we can tell that ∂εt

∂ ξ= −xt and

∂σt

∂ η=

σt

δσtδ

∂σtδ

∂ η.

The differentiating of σtδ with the respect to δ, θ and γ will be the same as the

normal distribution as former.

The differentiating of σtδ with the respect to ξ:

∂σtδ

∂ ξ= δ∑αj 0(|εt−j| − γjεt−j)

δ−1 (γj + 𝐼𝑡−𝑖)xt−j1

𝐹𝑡−𝑗

q

j=1

× [1


δ−1(γj + 𝐼𝑡−𝑖)xt−j

𝑇

𝑠=1

]

1−𝐹𝑡 −𝑗

+∑βi (∂(σt−i)

δ

∂ ξ)

𝐹𝑡 −𝑖p

i=1

[−δ

𝑇(1

𝑇∑εs

2

𝑇

𝑠=1

+

δ−22

∑εsxs

𝑇

𝑠=1

]

1−𝐹𝑡−𝑖

,

where

𝐼𝜏 = {−1, 𝑖𝑓ε𝜏 ≥ 0

1, 𝑖𝑓ετ < 0

- 13 -

𝐹𝜏 = {1, 𝑖𝑓 𝜏 > 0

0, 𝑖𝑓 𝜏 ≤ 0 .

The differentiating of σtδ with respect to γ:

∂σtδ

∂γ=∂ω

∂γ+∑


∂γ

q

j=1

+∑∂βi(σt−i)

δ

∂γ

p

i=1


δ

∂γ

q

j=1

+∑βi∂(σt−i)

δ

∂γ

p

i=1

.

The differentiating will be different with t changes.

∂k(εt−j)δ

∂γ=

{

−δk(εt−j)

δ−1εt−j ,𝑓𝑜𝑟 𝑡 > 𝑗

−δ

𝑇∑(|𝜀𝑆| − γiεs)

δ−1εs ,𝑓𝑜𝑟 𝑡 ≤ 𝑗

𝑇

𝑠=1

,

and ∂(σt−i)

δ

∂γ= 0 for 𝑡 ≤ 0.

The differentiating of σtδ with respect to δ:

∂σtδ

∂δ=∂ω

∂δ+∑


∂δ

q

j=1

+∑∂βi(σt−i)

δ

∂δ

p

i=1

= δ∑αj 0(|εt−j| − γjεt−j)δ 𝐿𝑛(|εt−j | − γjεt−j)1

𝐹𝑡 −𝑗

q

j=1

× [1


δ𝐿𝑛(|εs| − γjεs)

𝑇

𝑠=1

]

1−𝐹𝑡−𝑗

+∑βi .(σt−i)δ𝐿𝑛(σt−i)/

𝐹𝑡−𝑖

p

i=1

[−1

𝑇(1

𝑇∑εs

2

𝑇

𝑠=1

+

δ2

𝐿𝑛(1

𝑇∑εs

2

𝑇

𝑠=1

)]

1−𝐹𝑡 −𝑖

.

The differentiating of σtδ with respect to θ:

- 14 -

∂σtδ

∂θ=∂ω

∂θ+∑


∂θ

q

j=1

+∑∂βi(σt−i)

δ

∂θ

p

i=1


δ

∂θ

q

j=1

+∑βi∂(σt−i)

δ

∂θ

p

i=1

,

And ∂(σt−i)

δ

∂θ= 0 for 𝑡 ≤ 0..

4.3 Skewed student-t Distribution

The Skewed student t distribution was first discovered by Fernandez and Steel (1998).

Skewness and kurtosis are important characteristics in financial time series. Skewed

student t distribution can describe these features appropriately. Lambert and Laurent

(2000, 2001) extended the Skewed Student density. The density function of the

standardized skewed generalized error distribution is

f (zt |, 𝑣) =𝑣

(2𝐴 ∙ 𝛤(1𝑣)exp (−

|zt − B|𝑣

,1 − 𝑠𝑖𝑔𝑛(zt − B)ρ-𝑣 ∙ 𝐴𝑣

)

𝐴 = 𝛤(1

𝑣*0.5

𝛤 (3

𝑣*−0.5

𝐶(ρ)−1

B = 2ρ ∙ D ∙ C(ρ)−1

𝐶(ρ) = √1+ 3ρ2 −4𝐷2ρ2

D = 𝛤 (2

𝑣* 𝛤 (

1

𝑣*0.5

𝛤 (3

𝑣*−0.5

,

where ρ is a shape parameter which is positive and describes the degree of

asymmetry of the time series.

The log-likelihood function is as below

- 15 -

Log L(η) = T [ Ln 𝛤(𝑣 + 1

2* − 𝐿𝑛 .

𝑣

2/ −

1

2𝐿𝑛(𝜋(𝑣 − 2)) + ln (

2

ρ +1ρ

, + ln (s)]

−1

2∑*ln(σt

2) + (1 + 𝑣)ln (1 +(𝑠zt +𝑚)

2

𝑣 − 2ρ−2𝐼𝑡 +

𝑇

𝑡=1

𝐼𝑡 = {1 𝑖𝑓 zt ≥ −

𝑚

𝑠

−1 𝑖𝑓 zt < −𝑚

𝑠

𝑚 =𝛤.𝑣 + 12 /√𝑣 − 2

√𝜋𝛤 .𝑣2/

(ρ −1

ρ*

𝑠 = √(ρ2 +1

ρ2−1* −𝑚2 .

(See Lambert and Laurent (2001) for more details.)

4.4 Forecasting Methods

Poon and Granger (2003) have discussed the forecasting ability of the ARCH/GARCH

models. According to their research, there are some popular evaluation measures

used in the former papers, including Mean Error (ME), Mean Squared Error (MSE),

Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute

Percent Error (MAPE). There are some measures that are less commonly used such

like Mean Logarithm of Absolute Errors (MLAE), Theil-U statstic and LINEX. Except for

Theil-U statistic and LINEX, others are self-explanatory.

We are going to use six common measures to evaluate the forecasting.

1) Mean Squared Error (MSE)

The mean squared error (MSE) is able to quantify the difference between values

implied by an estimator and the true values of the quantity being estimated.

MSE =1

𝑕 + 1∑(𝜎𝑡

2 −𝜎𝑡2)2

𝑆+ℎ

𝑡=𝑆

.

2) Mean Absolute Error (MAE)

- 16 -

The Mean Absolute Error (MAE) is the average of the absolute value of the residuals.

The MAE is very similar to the MSE but is less sensitive to large errors

MAE =1

𝑕 + 1∑|𝜎𝑡

2 − 𝜎𝑡2|

𝑆+ℎ

𝑡=𝑆

.

3) Adjusted Mean Absolute Percentage Error (AMAPE)

Adjusted Mean Absolute Percentage Error (AMAPE) is a measure based on

percentage (or relative) errors.

AMAPE =1

𝑕 + 1∑ |

𝜎𝑡2 − 𝜎𝑡

2

𝜎𝑡2

|

𝑆+ℎ

𝑡=𝑆

.

4) Theil’s Inequality Coefficient (TIC)

Thiel's inequality coefficient (TIC), also known as Thiel's U, provides a measure of

how well a time series of estimated values compares to a corresponding time series

of observed values.

TIC =√

1𝑕+ 1

∑ (�̂�𝑡2 −𝑌𝑡

2)2𝑆+ℎ

𝑡=𝑆

√ 1𝑕 + 1

∑ �̂�𝑡2𝑆+ℎ

𝑡=𝑆 −√1

𝑕 + 1∑ 𝑌𝑡

2𝑆+ℎ𝑡=𝑆

,

where 𝑕 is the number of head steps, 𝑆 is the sample size, 𝜎𝑡2 is the forecasted

variance, 𝜎𝑡2 is the actual variance.

5) Q-Statistic(Box-Pierce test)

Box-Pierce test is defined as weighted sum of squares of a sequence of

auto-correlations.

Q = n∑ 𝑟𝑘2

𝑚

𝑘=1

Where 𝑟𝑘 is the sample auto-correlation at the lag k, n is the sample size, m is the

number of lags tested.

6) Q-Statistic (Ljung–Box test)

Q = n(n + 2)∑𝑟𝑘2

𝑛 − 𝑘

𝑚

𝑘=1

Where 𝑟𝑘 is the sample auto-correlation at the lag k, n is the sample size, m is the

number of lags tested.

- 17 -

5. Empirical Application (Standard&Poor 500 Daily Index)

5.1 Data Analysis

Standard&Poor 500 Stock Market Daily Closing Price Index

The first data we use is Standard&Poor’s 500 index. The S&P 500 has been widely

regarded as the best single gauge of the large cap U.S. equities market since the

index was first published in 1957. The index has over US$ 4.83 trillion benchmark,

with index assets comprising approximately US$ 1.1 trillion of this total. The index

includes 500 leading companies in leading industries of the U.S. economy, capturing

75% coverage of U.S. equities. We choose S&P daily close index from 1950/1/3 to

2011/3/18. The data is from Yahoo Finance (http://finance.yahoo.com/).

Base on the empirical evidence it is common to assume that the logarithmic return

series rt = 100(log(𝑝𝑡) − log(𝑝𝑡−1)), where 𝑝𝑡 is the closing value of the index at

time t, is weakly stationary. The following figures 5.1, 5.2 and 5.3 give the plot of 𝑝𝑡,

rt and |rt|. In figure 5.1 we can see a movement of the Standard & Poor daily price

index from 1950 to 2011. The movement is an upward trend. In the figure 5.2 the

return of the index is quietly stable around the mean value. In figure 5.3 the absolute

values show the volatility clustering feature. According to Mandelbrot (1963) and

Fama (1965), the large absolute returns are more likely to follow another large

absolute returns rather than small absolute returns. Figure 5.3 shows a large

absolute cluster and small absolute cluster.

0

400

800

1200

1600

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

S&P 500 Index

Fig 5.1.1 Standard & Poor Daily index 03/01/1950-18/03/2011

http://finance.yahoo.com/

- 18 -

Table 5.4 gives the summary statistics of rt and |rt|. We can see from the table that

the kurtosis for the Standard & Poor daily return is 32.08357 which is much higher

than the value of the normal distribution (kurtosis=3). The kurtosis for the Standard

& Poor Daily absolute return is 82.64713. The value shows the financial time series

have the fat-tail characteristic. The Jarque-Bera for the Standard & Poor daily return

is 545625.8. The high Jarque-Bera statistics indicates the non-normality of the series.

Data Sample

Size Max Min Mean SD Skewness Kurtosis Jarque-Bera

rt 15400 4.758650 -9.945223 0.012242 0.422011 -1.057339 32.08357 545625.8

|rt| 15400 9.945223 0 0.284010 0.312373 5.002845 82.64713 4134758

Table 5.1.4 Summary of rt and |rt|

-12

-8

-4

0

4

8

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Return S&P 500 Index

Fig 5.1.2 Standard & Poor Daily Return

0

4

8

12

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Absolute Return of S&P 500

Fig 5.1.3 Standard & Poor Daily Absolute Return

- 19 -

5.2 Autocorrelation Analysis

According to Fama (1970) and Taylor (1986), the financial time series contain

autocorrelation. This is also known as the long memory characteristic. The GARCH

models are good at describing the autocorrelation. We examine the autocorrelation

of rt , |rt| and rt2from lag 1 to 100 for Standard & Poor 500 Daily Index. We can

test our result with the 95% confidence interval ±1.96/√T. In our case T=15400, so

the 95% confidence interval is ±0.015794 . The table 5.2.1 shows the

autocorrelation of rt , |rt| and rt2for lag 1 to 100. The first lag autocorrelation of

rt is 0.034, which is highly positive. The first lag autocorrelation of |rt| and rt2 are

0.247 and 0.142. This means that the efficient market is not strict for the financial

time series and Standard & Poor 500 Daily Index is not a realization of an i.i.d

process.

Data Lag 1 2 3 4 5 10 20 40 70 100

rt 0.034 -0.049 0.009 -0.006 0.005 0.008 0.011 -0.010 -0.012 0.004

|rt| 0.247 0.269 0.249 0.244 0.288 0.226 0.201 0.151 0.113 0.117

rt2 0.142 0.206 0.11 0.098 0.188 0.082 0.066 0.004 0.019 0.024

Table 5.2.1 Autocorrelation of rt , |rt| and rt2 (Standard & Poor 500 Daily Index)

Fig 5.2.2 shows the chart for autocorrelation of rt , |rt| and rt2. From the chart,

|rt| has the higher autocorrelation. The autocorrelation shows the Standard & Poor

500 Daily Index has features of the financial time series. The APARCH model can be

well fit for forecasting.

-.1

.0

.1

.2

.3

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fig 5.2.2 Autocorrelation of rt ,|rt| and rt2

(Standard & Poor 500 Daily Index)

- 20 -

5.3 Selection of ARMA (p,q)

To get the parameters p and q of AMRA to fit in the series, we use AIC (Akaike

Information Criterion) and BIC (Bayes information criterion) to evaluate ARMA

model.

AIC (Akaike Information Criterion):

AIC(p) = Ln(𝑆𝑆𝑅

𝑇* + (p + 1)

2

𝑇 .

BIC (Bayes information Criterion):

BIC(p) = Ln(𝑆𝑆𝑅

𝑇* + (p + 1)

𝐿𝑛 𝑇

𝑇 .

In both formulas SSR stands for the sum of squared residuals.

AIC BIC

ARMA(0,0) 1.770651 1.771147

ARMA(1,0) 1.537067 1.538059

ARMA(2,0) 1.765482 1.766475

ARMA(3,0) 1.769544 1.770536

ARMA(0,1) 1.114012 1.115004

ARMA(0,2) 1.765254 1.766247

ARMA(0,3) 1.769378 1.770370

ARMA(1,1) 1.113036 1.114525

ARMA(1,2) 1.114033 1.115522

ARMA(1,3) 1.537088 1.538577

ARMA(2,2) 1.765457 1.766946

Fig 5.3.1 Criteria for ARMA(p,q) Selection

- 21 -

The figure shows the AIC and BIC for ARMA(p,q) model. We choose ARMA(1,1) as the

mean equation considering the value of AIC and BIC. So the conditional mean

equation is as follows

yt = u + ξ yt−1+εt +θεt−1, t = 1,2… . . T.

5.4 Estimation Result

We use Software R to estimate ARMA(1,1)-APARCH(1,1) with normal distribution,

student t distribution and skew student t distribution. In table 5.4.1, we can see the

estimation of the model parameters for different distribution. The parameters

estimated for three distributions are all significant except for the coefficient of the

first term of the moving average process under normal distribution and student

distribution. The skew student t distribution has the better estimated parameters.

Normal Studen-t Skew Student-t

mu 0.011337

(0.000015)

0.016641

(0.000000)

0.013011

(0.000000)

Ar1 -0.076902

(0.196710)

-0.157163

(0.015857)

-0.177153

(0.006055)

Ma1 0.183169

(0.001871)

0.260204

(0.000044)

0.278119

(0.000010)

Omega 0.003356

(0.000000)

0.003361

(0.000000)

0.003390

(0.000000)

Alpha1 0.075356

(0.000000)

0.069686

(0.000000)

0.069503

(0.000000)

Gamma1 0.405938

(0.000000)

0.555057

(0.000000)

0.555698

(0.000000)

Delta 1.405458 1.164553 1.166390

- 22 -

(0.000000) (0.000000) (0.000000)

Beta1 0.923324

(0.000000)

0.933316

(0.000000)

0.933696

(0.000000)

Shape 7.301929

(0.000000)

7.421825

(0.000000)

Skew 0.944998

(0.000000)

Fig 5.4.1 Parameters estimation for ARMA(1,1)-APARCH(1,1)

Table 5.4.2 shows the criteria under the three distributions. Skew student t

distribution has larger log likelihood. Akaike Information Criterion and Bayes

information Criterion are smaller than the values under other distributions. This

means the model under skew student t distribution is better fitted.


LogLikelihood -5388.415 -4946.076 -4933.384

AIC 0.70083 0.64352 0.64200

BIC 0.70480 0.64798 0.64696

Q(20)

(Box-Pierce test) 21.74 21.43 21.51

Q(20)

(Ljung–Box test) 24.96 41.00 40.93

MSE 0.178146 0.178220 0.178506

MAE 0.283385 0.283405 0.282599

TIC 0.953938 0.946151 0.928713

Fig 5.4.2 Criteria for ARMA(1,1)-APARCH(1,1)

- 23 -

6. Empirical Application (MSCI Europe Daily Index)

6.1 Data Analysis

MSCI Europe Index

The MSCI Index was created by Morgan Stanley Capital International. Each MSCI

Index measures a different aspect of global stock market performance. The MSCI

Europe Index is a free float-adjusted market capitalization weighted index. The

purpose of the MSCI Europe Index is to measure the equity market performance of

the developed markets in Europe. Since June 2007, the MSCI Europe Index has

consisted of the following 16 developed market country indices: Austria, Belgium,

Denmark, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Norway,

Portugal, Spain, Sweden, Switzerland, and the United Kingdom. Here we choose

MSCI Europe daily Index from 2006/1/31 to 2011/3/18. The data is from Morgan

Stanley Capital International website (http://www.msci.com/).

Base on the empirical evidence it is common to assume that the logarithmic return

series rt = 100(log(𝑝𝑡) − log(𝑝𝑡−1)), where 𝑝𝑡 is the closing value of the index at

time t, is weakly stationary. The following figures 6.1.1, 6.1.2 and 6.1.3 give the plot

of 𝑝𝑡, rt and |rt|. In figure 6.1.1 we can see a movement of the MSCI Europe daily

price index from 2006 to 2011. The movement is smooth compared to the first set of

data, which may be caused by the fact it covers recent years. In the figure 6.1.2 the

return of the index is also stable around the mean value. In figure 6.1.3 the absolute

values show the volatility clustering feature. The large value always follows another

large value and the small values always come together.

400

800

1200

1600

2000

2400

2006 2007 2007 2008 2008 2009 2009 2010 2010 2011

MSCI Europe Index

Fig 6.1.1 MSCI Europe Daily index 31/01/2006-18/03/2011

http://www.msci.com/

- 24 -

Table 6.1.4 gives the summary of statistics of rt and |rt|. We can see from the table

that the kurtosis for the MSCI Europe daily return is 9.685301 which is much higher

than the value of the normal distribution (kurtosis=3). The kurtosis for the MSCI

Europe Daily absolute return is 16.09036. The value shows the financial time series

have the fat-tail characteristic. The Jarque-Bera for the MSCI Europe daily return is

2491.652. The high Jarque-Bera statistics indicates the non-normality of the series.

Data Sample

Size Max Min Mean SD Skewness Kurtosis Jaeque-Bera

rt 1338 4.646108 -4.420361 -0.001556 0.752475 0.003508 9.685301 2491.652

|rt| 1338 4.646108 0 0.507554 0.555353 2.963989 16.09036 11512.29

Table 6.1.4 Summary of rt and |rt|

-6

-4

-2

0

2

4

6

2006 2007 2007 2008 2008 2009 2009 2010 2010 2011

Return of MSCI Europe Index

Fig 6.1.2 MSCI Europe Daily Return

0

1

2

3

4

5

2006 2007 2007 2008 2008 2009 2009 2010 2010 2011

Absolute Return of MSCI Europe Index

Fig 6.1.3 MSCI Europe Daily Absolute Return

- 25 -

6.2 Autocorrelation Analysis

We examine the autocorrelation of rt ,|rt| and rt2 form lag 1 to 100 for MSCI

Europe daily index.. We can test our result with the 95% confidence interval ±1.96/

√𝑇. In our case T=1338, so the 95% confidence interval is ±0.05358. The table 6.2.1

shows the autocorrelation of rt , |rt| and rt2 for lag 1 to 100. The first lag

autocorrelation of rt is -0.022, which is outside the confidence interval. But the first

lag autocorrelation of |rt| and rt2 are 0.235 and 0.178 which are highly positive.

So the time series of MSCI Europe index is not i.i.d process.

Data Lag 1 2 3 4 5 10 20 40 70 100

rt -0.022 -0.046 -0.046 0.099 -0.068 -0.012 0.004 0.047 0.026 -0.016

|rt| 0.235 0.276 0.290 0.270 0.317 0.304 0.188 0.174 0.099 0.071

rt2 0.178 0.234 0.249 0.242 0.375 0.291 0.089 0.200 0.058 0.037

Table 6.2.1 Autocorrelation of rt ,|rt| and rt2(MSCI Europe Daily Index)

The Fig 6.2.2 show the chart for autocorrelation of rt , |rt| and rt2 . The

autocorrelation shows the MSCI Europe Daily Index has features of the financial time

series. The APARCH model can be well fit for forecasting.

-.1

.0

.1

.2

.3

.4

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fig 6.2.2 Autocorrelation of rt ,|rt| and rt2

(MSCI Europe Daily Index)

- 26 -

6.3 Selection of ARMA (p,q)

To get the parameters p and q of ARMA to fit in the series, we also use AIC (Akaike

Information Criterion) and BIC (Bayes information Criterion) to evaluate ARMA

model.

AIC BIC

ARMA(0,0) 2.984260 2.988148

ARMA(1,0) 2.715341 2.723122

ARMA(2,0) 2.986705 2.994490

ARMA(3,0) 2.982627 2.990417

ARMA(0,1) 2.273656 2.281432

ARMA(0,2) 2.985613 2.993389

ARMA(0,3) 2.980539 2.988315

ARMA(1,1) 2.275156 2.286827

ARMA(1,2) 2.275028 2.286698

ARMA(1,3) 2.716102 2.727773

ARMA(2,2) 2.977722 2.989400

Fig 6.3.1 Criteria for ARMA(p,q) Selection

The figure shows the AIC and BIC for ARMA(p,q) model. We choose ARMA(0,1) as the

mean equation considering the value of AIC and BIC. So the conditional man

equation is as follows

yt = u+ εt +θεt−1, t = 1,2… . . T.

6.4 Estimation Result

We use Software R to estimate ARMA(0,1)-APARCH(1,1) with normal distribution,

- 27 -

student t distribution and skew student t distribution. In table 6.4.1, we can see the

estimation of the model parameters for different distribution. The parameters

estimated are all significant under normal distribution, student t distribution and

skew student t distribution.


mu 0.011168

(3.7e-05)

0.016153

(0e+00)

0.012617

(0e+00)

Ma1 0.107136

(0.0e+00)

0.104689

(0e+00)

0.102623

(0e+00)

Omega 0.003381

(0.0e+00)

0.003389

(0e+00)

0.003435

(0e+00)

Alpha1 0.075501

(0.0e+00)

0.069825

(0e+00)

0.069730

(0e+00)

Gamma1 0.410184

(0.0e+00)

0.564772

(0e+00)

0.567494

(0e+00)

Delta 1.401537

(0.0e+00)

1.164147

(0e+00)

1.164126

(0e+00)

Beta1 0.923202

(0.0e+00)

0.933100

(0e+00)

0.933392

(0e+00)

Shape 7.338168

(0e+00)

7.454834

(0e+00)

Skew 0.946450

(0e+00)

Fig 6.4.1 Parameters estimation for ARMA(0,1)-APARCH(1,1)

Table 6.4.2 shows the criteria under the three distributions. Skew student t

- 28 -

distribution has larger log likelihood. Akaike Information Criterion and Bayes

information Criterion are smaller than the values under other distributions. This

means the model under skew student t distribution is better fitted.


LogLikelihood -5388.829 -4948.78 -4936.763

AIC 0.70076 0.64374 0.64231

BIC 0.70423 0.64771 0.64677

Q(20)

(Box-Pierce test) 24.68 26.55 26.48

Q(20)

(Ljung–Box test) 24.94 40.89 40.78

ARCH LM Tests

Lag[10] 21.70 38.44 38.38

MSE 0.574254 0.572946 0.571914

MAE 0.510888 0.510200 0.509651

TIC 0.913638 0.907747 0.901017

Fig 6.4.2 Criteria for ARMA(0,1)-APARCH(1,1)

7. Conclusion

To model the financial time series data, we first consider the five characteristics

which are Fat tails and Excess kurtosis, Volatility Clustering, Long Memory, Leverage

Effects and Spillover Effects from the former literature. After reviewing the

autoregressive conditional heteroscedasticity model (ARCH) and generalized

autoregressive conditional heteroscedasticity model (GARCH), we analyze the

derivates of the Asymmetric Power ARCH Model (APARCH) normal distribution,

student t distribution and skew student t distribution. By using the log likelihood

- 29 -

methods, we calculate the differential for each parameter. Bollerslev (1987)

proposed student t distribution which captures the excess kurtosis and fat tail

features. Skew student t distribution described the leverage effects precisely. We

use S&P 500 daily index and MSCI Europe Index for our simulation. Using AIC and BIC

as the criteria for AMRA model, we find AMRA (1, 1) is suitable for S&P 500 daily

index and AMRA (0, 1) is suitable for MSCI Europe Index. We used software R for

programing. We compared the forecasting performance of APARCH model under

normal distribution, student t distribution and skew student t distribution. We found

that the skew student t distribution is the most efficient. The parameters have better

significance. Based on the estimated model, we use LogLikelihood, AIC, BICA, mean

squared error (MSE), mean Absolute Error (MAE) and Theil’s Inequality Coefficient

(TIC) to evaluate the forecasting. Skew student t distribution has larger likelihood and

smaller errors compared to other distribution.

- 30 -

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- 33 -

Appendix A

setwd("c:/data")

#install rgarch

install.packages("rgarch", repos="http://R-Forge.R-project.org")

#Load rgarch

library(rgarch)

#Define SP500

sp500<-read.table("sp500.txt")

#Set model

variance.model=list(model="apARCH",garchOrder=c(1,1),submodel=NULL,external.regressors=N

ULL);

mean.model=list(armaOrder=c(1,1),include.mean=T,garchInMean=F,inMeanType=1,arfima=F,exte

ranl.regressors=NULL);

spec=ugarchspec(variance.model=variance.model,mean.mode=mean.model,distribution.model=

"norm");

#Fitting parameters

fit=ugarchfit(data=sp500,spec=spec,out.sample=0,solver="solnp")

#result

fit

#Set model


ULL);




"std");

#Fitting parameters


#result

fit

#Set model

- 34 -


ULL);




"sstd");

#Fitting parameters


#result

fit

Normal Distribution

*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------

Model : apARCH (1,1)

Exogenous Regressors in variance equation: none

Include Mean : TRUE

AR(FI)MA Model : (1,0,1)

Garch-in-Mean : FALSE

Exogenous Regressors in mean equation: none

Conditional Distribution: norm

Optimal Parameters

--------------------------

Estimate Std. Error t value Pr(>|t|)

mu 0.011337 0.002623 4.3215 0.000015

ar1 -0.076902 0.059569 -1.2910 0.196710

ma1 0.183169 0.058899 3.1099 0.001871

omega 0.003356 0.000476 7.0571 0.000000

alpha1 0.075356 0.004285 17.5862 0.000000

gamma1 0.405938 0.032991 12.3044 0.000000

delta 1.405458 0.072915 19.2752 0.000000

beta1 0.923324 0.004180 220.8987 0.000000

- 35 -

Robust Standard Errors:


mu 0.011337 0.002806 4.0398 0.000053

ar1 -0.076902 0.044275 -1.7369 0.082403

ma1 0.183169 0.045519 4.0240 0.000057

omega 0.003356 0.000883 3.8013 0.000144

alpha1 0.075356 0.010587 7.1179 0.000000

gamma1 0.405938 0.078120 5.1963 0.000000

delta 1.405458 0.146444 9.5973 0.000000

beta1 0.923324 0.011181 82.5773 0.000000

LogLikelihood : -5388.415

Information Criteria

--------------------------

Akaike 0.70083

Bayes 0.70480

Shibata 0.70083

Hannan-Quinn 0.70215

Q-Statistics on Standardized Residuals

--------------------------

statistic p-value

Lag10 12.72 0.2400

Lag15 16.71 0.3368

Lag20 21.74 0.3550

H0 : No serial correlation

Q-Statistics on Standardized Squared Residuals

--------------------------

statistic p-value

Lag10 21.55 0.01758

Lag15 23.12 0.08164

Lag20 24.96 0.20281

ARCH LM Tests

--------------------------

Statistic DoF P-Value

ARCH Lag[2] 17.56 2 0.0001537

ARCH Lag[5] 17.87 5 0.0031111

ARCH Lag[10] 21.72 10 0.0166101

- 36 -

Nyblom stability test

--------------------------

Joint Statistic: 17.497

Individual Statistics:

mu 0.7773

ar1 11.2098

ma1 11.3524

omega 0.8153

alpha1 0.9238

gamma1 0.7261

delta 0.9430

beta1 0.8940

Asymptotic Critical Values (10% 5% 1%)

Joint Statistic: 1.89 2.11 2.59

Individual Statistic: 0.353 0.47 0.748

Sign Bias Test

--------------------------

t-value prob sig

Sign Bias 0.3799 0.703991

Negative Sign Bias 2.3518 0.018694 **

Positive Sign Bias 1.1950 0.232103

Joint Effect 14.6159 0.002176 ***

Adjusted Pearson Goodness-of-Fit Test:

--------------------------

group statistic p-value(g-1)

1 20 247.1 1.188e-41

2 30 265.7 4.480e-40

3 40 285.5 2.991e-39

4 50 294.7 8.700e-37

Elapsed time : 23.792

Studen t distirbution

*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------

- 37 -



Include Mean : TRUE




Conditional Distribution: std

Optimal Parameters

--------------------------


mu 0.016641 0.002469 6.7386 0.000000

ar1 -0.157163 0.065154 -2.4122 0.015857

ma1 0.260204 0.063731 4.0828 0.000044

omega 0.003361 0.000531 6.3309 0.000000

alpha1 0.069686 0.004477 15.5642 0.000000

gamma1 0.555057 0.045491 12.2014 0.000000

delta 1.164553 0.072084 16.1556 0.000000

beta1 0.933316 0.004243 219.9681 0.000000

shape 7.301929 0.394216 18.5226 0.000000



mu 0.016641 0.002605 6.3880 0.000000

ar1 -0.157163 0.056075 -2.8027 0.005067

ma1 0.260204 0.055991 4.6472 0.000003

omega 0.003361 0.000596 5.6353 0.000000

alpha1 0.069686 0.005120 13.6105 0.000000

gamma1 0.555057 0.048942 11.3411 0.000000

delta 1.164553 0.077792 14.9702 0.000000

beta1 0.933316 0.005014 186.1519 0.000000

shape 7.301929 0.529233 13.7972 0.000000



--------------------------

Akaike 0.64352

Bayes 0.64798

Shibata 0.64352


- 38 -


--------------------------

statistic p-value

Lag10 12.39 0.2597

Lag15 16.15 0.3719

Lag20 21.43 0.3722



--------------------------

statistic p-value

Lag10 38.33 3.320e-05

Lag15 39.53 5.334e-04

Lag20 41.00 3.722e-03

ARCH LM Tests

--------------------------


ARCH Lag[2] 33.74 2 4.717e-08

ARCH Lag[5] 34.83 5 1.626e-06

ARCH Lag[10] 38.59 10 2.989e-05


--------------------------



mu 0.7005

ar1 15.6282

ma1 15.5121

omega 2.9589

alpha1 2.9668

gamma1 1.5548

delta 3.5792

beta1 3.2809

shape 1.3378




Sign Bias Test

--------------------------

- 39 -

t-value prob sig

Sign Bias 0.04106 0.96725



Joint Effect 10.42846 0.01525 **


--------------------------


1 20 34.69 0.0152017

2 30 51.11 0.0068452

3 40 74.80 0.0004902

4 50 82.23 0.0020617


Skew Student t Distribution

*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------



Include Mean : TRUE




Conditional Distribution: sstd

Optimal Parameters

--------------------------


mu 0.013011 0.002568 5.0668 0.000000

ar1 -0.177153 0.064542 -2.7448 0.006055

ma1 0.278119 0.063061 4.4103 0.000010

omega 0.003390 0.000531 6.3859 0.000000

alpha1 0.069503 0.004439 15.6559 0.000000

- 40 -

gamma1 0.555698 0.045308 12.2649 0.000000

delta 1.166390 0.071644 16.2804 0.000000

beta1 0.933696 0.004199 222.3667 0.000000

skew 0.944998 0.010616 89.0125 0.000000

shape 7.421825 0.407394 18.2178 0.000000



mu 0.013011 0.002719 4.7845 0.000002

ar1 -0.177153 0.055026 -3.2195 0.001284

ma1 0.278119 0.054898 5.0661 0.000000

omega 0.003390 0.000596 5.6928 0.000000

alpha1 0.069503 0.005039 13.7932 0.000000

gamma1 0.555698 0.048525 11.4518 0.000000

delta 1.166390 0.077227 15.1033 0.000000

beta1 0.933696 0.004922 189.7020 0.000000

skew 0.944998 0.010294 91.8013 0.000000

shape 7.421825 0.538871 13.7729 0.000000



--------------------------

Akaike 0.64200

Bayes 0.64696

Shibata 0.64200



--------------------------

statistic p-value

Lag10 12.47 0.2551

Lag15 16.29 0.3627

Lag20 21.51 0.3679



--------------------------

statistic p-value

Lag10 38.30 0.0000337

Lag15 39.46 0.0005474

Lag20 40.93 0.0038062

- 41 -

ARCH LM Tests

--------------------------


ARCH Lag[2] 33.69 2 4.833e-08

ARCH Lag[5] 34.79 5 1.660e-06

ARCH Lag[10] 38.56 10 3.027e-05


--------------------------



mu 0.6645

ar1 15.8886

ma1 15.6791

omega 2.9848

alpha1 2.9052

gamma1 1.5531

delta 3.5574

beta1 3.1889

skew 0.6969

shape 1.3687




Sign Bias Test

--------------------------

t-value prob sig

Sign Bias 0.05501 0.95613



Joint Effect 10.73777 0.01323 **


--------------------------


1 20 23.54 0.21449

2 30 36.09 0.17082

3 40 48.11 0.15039

4 50 68.31 0.03547

- 42 -


Appendix B

setwd("c:/data")

#install rgarch

install.packages("rgarch", repos="http://R-Forge.R-project.org")

#Load rgarch

library(rgarch)

#Define SP500

msci<-read.table("msci.txt")

#Set model-Normal Distribution


ULL);




"norm");

#Fitting parameters


#result

fit

#Set model-Student t distribution


ULL);




"std");

#Fitting parameters


- 43 -

#result

fit

#Set model-Skwe Student t distribution


ULL);




"sstd");

#Fitting parameters


#result

fit

*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------



Include Mean : TRUE




Conditional Distribution: norm

Optimal Parameters

--------------------------


mu 0.011168 0.002706 4.1270 3.7e-05

ma1 0.107136 0.008746 12.2492 0.0e+00

omega 0.003381 0.000484 6.9882 0.0e+00

alpha1 0.075501 0.004302 17.5511 0.0e+00

gamma1 0.410184 0.033333 12.3055 0.0e+00

delta 1.401537 0.073385 19.0985 0.0e+00

beta1 0.923202 0.004185 220.5976 0.0e+00

- 44 -



mu 0.011168 0.002943 3.7942 0.000148

ma1 0.107136 0.010790 9.9294 0.000000

omega 0.003381 0.000906 3.7312 0.000191

alpha1 0.075501 0.010619 7.1098 0.000000

gamma1 0.410184 0.079302 5.1724 0.000000

delta 1.401537 0.148051 9.4666 0.000000

beta1 0.923202 0.011179 82.5827 0.000000



--------------------------

Akaike 0.70076

Bayes 0.70423

Shibata 0.70076



--------------------------

statistic p-value

Lag10 15.69 0.1088

Lag15 19.64 0.1861

Lag20 24.68 0.2141



--------------------------

statistic p-value

Lag10 21.50 0.01785

Lag15 23.09 0.08216

Lag20 24.94 0.20355

ARCH LM Tests

--------------------------


ARCH Lag[2] 17.58 2 0.0001526

ARCH Lag[5] 17.87 5 0.0031190

ARCH Lag[10] 21.70 10 0.0166843

- 45 -


--------------------------



mu 0.7832

ma1 11.0360

omega 0.8164

alpha1 0.9245

gamma1 0.7262

delta 0.9447

beta1 0.8965




Sign Bias Test

--------------------------

t-value prob sig

Sign Bias 0.3247 0.745389



Joint Effect 14.3576 0.002457 ***


--------------------------


1 20 244.5 3.977e-41

2 30 264.7 6.917e-40

3 40 277.1 1.148e-37

4 50 287.2 1.996e-35


*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------


- 46 -


Include Mean : TRUE




Conditional Distribution: std

Optimal Parameters

--------------------------


mu 0.016153 0.002472 6.5339 0

ma1 0.104689 0.008275 12.6517 0

omega 0.003389 0.000532 6.3720 0

alpha1 0.069825 0.004486 15.5664 0

gamma1 0.564772 0.045799 12.3314 0

delta 1.164147 0.071607 16.2575 0

beta1 0.933100 0.004242 219.9696 0

shape 7.338168 0.396375 18.5132 0



mu 0.016153 0.002531 6.3829 0

ma1 0.104689 0.009376 11.1659 0

omega 0.003389 0.000597 5.6734 0

alpha1 0.069825 0.005124 13.6278 0

gamma1 0.564772 0.049066 11.5104 0

delta 1.164147 0.077482 15.0247 0

beta1 0.933100 0.005015 186.0619 0

shape 7.338168 0.530258 13.8389 0



--------------------------

Akaike 0.64374

Bayes 0.64771

Shibata 0.64374



--------------------------

- 47 -

statistic p-value

Lag10 17.55 0.06295

Lag15 21.23 0.12977

Lag20 26.55 0.14849



--------------------------

statistic p-value

Lag10 38.16 3.552e-05

Lag15 39.39 5.596e-04

Lag20 40.89 3.852e-03

ARCH LM Tests

--------------------------


ARCH Lag[2] 33.71 2 4.789e-08

ARCH Lag[5] 34.72 5 1.709e-06

ARCH Lag[10] 38.44 10 3.179e-05


--------------------------



mu 0.7116

ma1 15.0732

omega 2.9776

alpha1 2.9833

gamma1 1.5651

delta 3.5939

beta1 3.2980

shape 1.3415




Sign Bias Test

--------------------------

t-value prob sig

Sign Bias 0.1191 0.90517



- 48 -

Joint Effect 10.0522 0.01813 **


--------------------------


1 20 33.79 0.019428

2 30 52.17 0.005218

3 40 62.34 0.010202

4 50 80.44 0.003089


*---------------------------*

* GARCH Model Fit *

*---------------------------*

Spec

--------------------------



Include Mean : TRUE




Conditional Distribution: sstd

Optimal Parameters

--------------------------


mu 0.012617 0.002572 4.9057 1e-06

ma1 0.102623 0.008345 12.2980 0e+00

omega 0.003435 0.000539 6.3774 0e+00

alpha1 0.069730 0.004459 15.6397 0e+00

gamma1 0.567494 0.045874 12.3707 0e+00

delta 1.164126 0.071556 16.2687 0e+00

beta1 0.933392 0.004209 221.7603 0e+00

skew 0.946450 0.010629 89.0447 0e+00

shape 7.454834 0.410245 18.1717 0e+00

- 49 -



mu 0.012617 0.002660 4.7431 2e-06

ma1 0.102623 0.009603 10.6867 0e+00

omega 0.003435 0.000610 5.6341 0e+00

alpha1 0.069730 0.005067 13.7618 0e+00

gamma1 0.567494 0.049265 11.5191 0e+00

delta 1.164126 0.077888 14.9461 0e+00

beta1 0.933392 0.004946 188.7228 0e+00

skew 0.946450 0.010279 92.0799 0e+00

shape 7.454834 0.542792 13.7342 0e+00



--------------------------

Akaike 0.64231

Bayes 0.64677

Shibata 0.64231



--------------------------

statistic p-value

Lag10 17.49 0.06422

Lag15 21.20 0.13041

Lag20 26.48 0.15051



--------------------------

statistic p-value

Lag10 38.10 3.642e-05

Lag15 39.29 5.792e-04

Lag20 40.78 3.971e-03

ARCH LM Tests

--------------------------


ARCH Lag[2] 33.64 2 4.959e-08

ARCH Lag[5] 34.65 5 1.765e-06

- 50 -

ARCH Lag[10] 38.38 10 3.255e-05


--------------------------



mu 0.6826

ma1 15.2018

omega 3.0061

alpha1 2.9287

gamma1 1.5607

delta 3.5779

beta1 3.2143

skew 0.6786

shape 1.3739




Sign Bias Test

--------------------------

t-value prob sig

Sign Bias 0.1068 0.91492



Joint Effect 10.2329 0.01669 **


--------------------------


1 20 24.47 0.1787

2 30 37.17 0.1419

3 40 43.69 0.2790

4 50 56.82 0.2066


modeling of market volatility with aparch model417608/fulltext01.pdf · the asymmetric power arch...

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