Modeling of Market Volatility with APARCH Model

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

Examensarbete i matematik, 30 hp

Handledare och examinator: Maciej Klimek Maj 2011

Modeling of Market Volatility with APARCH Model

Ding Ding

Department of Mathematics Uppsala University

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.

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. 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. 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 g row 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.

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

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(ε_t) = 𝜎²) to a random sequence which only depend on past residuals (*ε₁, … , ε_t−1+). And Bollerslev (1986) amended this model adding the conditional heteroskedasticity moving average items.

The model can be expressed as follows:

y_t = x_tξ + ε_t t = 1,2 … . . T, σ_t² = ω + ∑ α_jε_t−j²

q

ε_t= σ_tz_t, z_tj=1~N(0,1) .

To assure *σ_t²+ is asymptotically stationary random sequence, we can assume that α₁+ ⋯ + α_q < 1.

In the ARCH model, the conditional variance of ε_t 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:

y_t = x_tξ + ε_t t = 1,2 … . . T,

σ_t² = ω + ∑ β_jσ_t−j²

p

j=1

+ ∑ α_jε_t−j²

q

ε_t= σ_tz_t, z_t~N(0,1) . j=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:

σ_t = ω + ∑(α_jε_t−j² + γ_i(max(0, ε_t−j))²)

q

j=1

+ ∑ β_iσ_t−i

p

i=1

EGARCH model:

log (σ_t) = ω + ∑ β_jlog (σ_t−j)

p

i=1

+ ∑(α_i ε_t−j

√σt−j

+ γ_i| ε_t−j

√σt−j

|)

q

j=1

4. APARCH Model

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

This model can well express the Fat tails, Excess kurtosis and Leverage Effects. The general structure is as follows:

y_t = x_tξ + ε_t t = 1,2 … . . T,

σ_t^δ= ω + ∑ α_j(|ε_t−j| − γ_jε_t−j)^δ

q

j=1

+ ∑ β_i(σ_t−i)^δ

p

i=1

ε_t = σ_tz_t, z_t~N(0,1) 𝑘(ε_t−j) = |ε_t−j| − γ_jε_t−j .

The mean equation ( y_t= x_tξ + ε_t t = 1,2 … . . T) could also be written as y_t = E(y_t|ψ_𝑡−1) + ε_t, where E(y_t|ψ_𝑡−1) is the conditional mean of y_t given ψ_𝑡−1. ψ_𝑡−1 the whole information at time t-1.

ψ_𝑡 = {y_t, y_t−1, … , y₁, y_0,x_t, x_t−1, … ,x_1,x₀} ,

where ξ, ω , α_j, γ_j, β_i and δ are the parameters which are needed to be estimated. γ_j, reflects the leverage effect. A positive γ_j means negative information has stronger impact than the positive information on the price volatility. δ reflects the leverage effect.

The APARCH equation ( σ_t² = ω + ∑^q_j=1α_j(|ε_t−j| − γ_jε_t−j)^δ+ ∑^p_i=1β_i(σ_t−i)^δ ) is supposed to satisfy the following conditions.

1) ω > 0, α_j ≥ 0, 𝑗 = 1,2, … 𝑞, β_i≥ 0, 𝑖 = 1,2,… 𝑝, when α_j= 0, 𝑗 = 1,2,… 𝑞, β_i= 0, 𝑖 = 1,2, … 𝑝, then σ_t²= ω. Due to the variance is positive, so ω > 0.

2) 0 ≤ ∑^q_j=1α_j+ ∑^p_i=1β_i ≤ 1

The corresponding conditional expectation and conditional variance of the Mean equation’s explanatory variables are:

E,y_t|x_t- = x_tξ Var,y_t|x_t- = σ_t^δ . For T → ∞, the unconditional variance of ε_t would be

σ_t^δ= ω

1 − ∑^q_j=1α_j(1 − γ_j)^δ− ∑^p_i=1β_i .

This model includes the ARCH and GARCH models, by changing the parameters we can get different models.

 When δ = 2, β_i= 0(𝑖 = 1,… , 𝑝), γ_j = 0(𝑗 = 1, … , 𝑞), APARCH model is ARCH model.

 When δ = 2, γ_j= 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 β_i= 0(𝑖 = 1,… , 𝑝) , γ_j = 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 y_t is f (y_t|x_t, ψ_𝑡−1) = 1

√2𝜋σ_t²𝑒𝑥𝑝 ,−(y_t− x_tξ)² 2σ_t² - ,

where σ_t^δ = ω + ∑^q_j=1α_j(|ε_t−j| − γ_jε_t−j)^δ+ ∑^p_i=1β_i(σ_t−i)^δ

= ω + ∑ α_j(|y_t− x_tξ| − γ_j(y_t− x_tξ))^δ

q

j=1

+ ∑ β_i(σ_t−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, … , γ_q), which measures the leverage effect; the vector θ = (ω, α₁, α₂, … , α_q, β₁, β₂, … , β_p) and the vector η = (ξ, γ, θ, δ) , which is the vector set of the unknown parameters.

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

Log L(η) = ∑ log f(y_t|x_t, A)

𝑇

𝑡=1

= −𝑇

2log(2𝜋) −1

2∑ log (

𝑇

𝑡=1

σ_t²) −1

2∑(y_t− x_tξ)² σ_t²

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 z_t is normal distribution.

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

∂ Log L(η)

∂ η = −1

2∑∂log(σ_t²)

∂ η

𝑇

𝑡=1

−1

2∑* 1 σ_t²

∂(y_t− x_tξ)²

∂ η −(y_t− x_tξ)² σ_t²

∂σ_t²

∂ η+

𝑇

𝑡=1

=1

2∑ *− 1 σ_t²

∂σ_t²

∂ η − 1 σ_t²

∂(y_t− x_tξ)²

∂ η + ε_t² σ_t²

∂σ_t²

∂ η+

𝑇

𝑡=1

=1

2∑ *ε_t²− σ_t² σ_t⁴

∂σ_t²

∂ η − 1 σ_t^δ

∂ε_t²

∂ η+

𝑇

𝑡=1

=1

2∑ *ε_t² − σ_t² σ_t⁴

∂σ_t²

∂ η −2 ε_t σ_t^δ

∂ε_t

∂ η+

𝑇

𝑡=1

.

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

∂σ_t^δ

∂η =∂ 0ω + ∑^q_j=1α_j(|ε_t−j| − γ_jε_t−j)^δ+ ∑^p_i=1β_i(σ_t−i)^δ1

∂ η

= ∂ω

∂ η+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂ η

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂ η

p

i=1

.

We can rewrite the σ_t^δ to σ_t²:

∂σ_t²

∂ η =2σ_t² δσ_t^δ

∂σ_t^δ

∂ η .

From the above we can tell that ^∂ε_{∂ ξ}^t= −x_t. To find a tractable solution of ^∂σ_{∂ η}^t2, we can separate calculate the different parameters.

 The differentiating of σ_t^δ with the respect to ξ:

∂σ_t^δ

∂ ξ = ∂ω

∂ ξ+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂ ξ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂ ξ

p

i=1

= ∑ δα_j(|ε_t−j| − γ_jε_t−j)^δ−1 ∂(|ε_t−j| − γ_jε_t−j)

∂ ξ

q

j=1

+ ∑δβ_i(σ_t−i)^δ−1 ∂(σ_t−i)

∂ ξ

p

i=1

.

If ε_t−j ≥ 0, then ^∂(|εt−j_{∂ ξ}^{|−γjεt−j)} = (γ_j− 1)x_t−j .

If ε_t−j < 0, then ^∂(|εt−j_{∂ ξ}^{|−γjεt−j)} = (γ_j+ 1)x_t−j .

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

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(ε_t−j) =¹_𝑇∑^𝑇_𝑠=1(|𝜀_𝑆| − γ_iε_s)^δ, for 𝑡 ≤ 𝑗 σ_t^δ= (¹_𝑇∑^𝑇_𝑠=1𝜀_𝑠²)^𝛿2 , for 𝑡 ≤ 0.

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

𝐼_𝜏 = {−1, 𝑖𝑓ε_𝜏 ≥ 0 1, 𝑖𝑓ε_τ < 0 𝐹_𝜏= {1, 𝑖𝑓 𝜏 > 0

0, 𝑖𝑓 𝜏 ≤ 0 , Then the formula above can be transformed as follows:

∂σ_t^δ

∂ ξ = δ ∑ α_j0(|ε_t−j| − γ_jε_t−j)^δ−1 (γ_j+ 𝐼_𝑡−𝑖)x_t−j1^{𝐹𝑡−𝑗}

q

j=1

× [1

𝑇∑(|ε_t−j| − γ_jε_t−j)^δ−1(γ_j+ 𝐼_𝑡−𝑖)x_t−j

𝑇

𝑠=1

]

1−𝐹_{𝑡 −𝑗}

+ ∑ β_i(∂(σ_t−i)^δ

∂ ξ )

𝐹_{𝑡 −𝑖} p

i =1

[−δ 𝑇(1

𝑇∑ ε_s²

𝑇

𝑠=1

+

δ−22

∑ ε_sx_s

𝑇

𝑠=1

]

1−𝐹_𝑡−𝑖

.

 The differentiating of σ_t^δ with the respect to γ:

∂σ_t^δ

∂γ =∂ω

∂γ + ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂γ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂γ

p

i=1

= ∑ α_j∂k(ε_t−j)^δ

∂γ

q

j=1

+ ∑ β_i∂(σ_t−i)^δ

∂γ

p

i=1

.

The differentiating will be different with t changes.

∂k(ε_t−j)^δ

∂γ =

{

−δk(ε_t−j)^δ−1ε_t−j ,𝑓𝑜𝑟 𝑡 > 𝑗

−δ

𝑇∑(|𝜀_𝑆| − γ_iε_s)^δ−1ε_s ,𝑓𝑜𝑟 𝑡 ≤ 𝑗

𝑇

𝑠=1

,

and ^∂(σ_∂γ^t−i)δ= 0 for 𝑡 ≤ 0.

 The differentiating of σ_t^δ with respect to δ:

∂σ_t^δ

∂δ =∂ω

∂δ+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂δ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂δ

p

i=1

= δ ∑ α_j0(|ε_t−j| − γ_jε_t−j)^δ 𝐿𝑛(|ε_t−j| − γ_jε_t−j)1^{𝐹𝑡 −𝑗}

q

j=1

× [1

𝑇∑(|ε_t−j| − γ_jε_t−j)^δ𝐿𝑛(|ε_s| − γ_jε_s)

𝑇

𝑠=1

]

1−𝐹_𝑡−𝑗

+ ∑ β_i.(σ_t−i)^δ𝐿𝑛(σ_t−i)/^{𝐹𝑡−𝑖}

p

i=1

[−1 𝑇(1

𝑇∑ ε_s²

𝑇

𝑠 =1

+

δ 2

𝐿𝑛(1

𝑇∑ ε_s²

𝑇

𝑠=1

)]

1−𝐹_{𝑡 −𝑖}

.

 The differentiating of σ_t^δ with respect to θ:

∂σ_t^δ

∂θ =∂ω

∂θ+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂θ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂θ

p

i=1

= ∑ α_j∂k(ε_t−j)^δ

∂θ

q

j=1

+ ∑ β_i∂(σ_t−i)^δ

∂θ ,

p

i=1

and ^∂(σ_∂θ^t−i)δ= 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

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 z_t has the student t distribution with 𝑣 degree of freedom, the density functions of z_t and ε_t are

f (z_t|, 𝑣) = 𝛤 .𝑣 + 12 /

√(𝑣 − 2)𝜋𝛤 .𝑣2/

(1 + z_t²

𝑣 − 2)^−𝑣+12

f (ε_t|𝑣) = 𝛤 .𝑣 + 12 /

√(𝑣 − 2)𝜋𝛤 .𝑣2/(1 +.εσ^t_t/² 𝑣 − 2,

−𝑣+1 2

(− 1 σ_t²* .

If 𝑣 is even,

𝛤 .𝑣 + 12 /

√𝑣𝜋𝛤 .𝑣 2/

= (𝑣 − 1)(𝑣 − 3) ∙∙∙ 5 ∙ 3

√𝑣𝜋(𝑣 − 2)(𝑣 − 3) ∙∙∙ 4 ∙ 2 .

If 𝑣 is odd,

𝛤 .𝑣 + 12 /

√𝑣𝜋𝛤 .𝑣 2/

= (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(ε_t|η, 𝑣)

𝑇

𝑡=1

= 𝑇 {𝐿𝑛 𝛤 (𝑣 + 1

2 * − 𝐿𝑛𝛤 .𝑣 2/ −1

2𝐿𝑛,(𝑣 − 2)𝜋-} −1

2∑ 𝐿𝑛(σ_t²)

𝑇

𝑡=1

− ∑ (𝑣 + 1 2 *

𝑇

𝑡=1

𝐿𝑛(1 +.εσ^t_t/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

(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(σ_t²)

∂ η

𝑇

𝑡=1

− ∑ (𝑣 + 1 2 *

∂ 𝐿𝑛 (1 +.εσ^t_t/² 𝑣 − 2,

∂ η

𝑇

𝑡=1

= −1

2∑ 1

σ_t²

∂σ_t²

∂ η

𝑇

𝑡=1

− (𝑣 + 1

2 * ∑

∂ 𝐿𝑛(1 +.εσ^t_t/² 𝑣 − 2,

∂ η

𝑇

𝑡=1

= −1

2∑ 1

σ_t²

∂σ_t²

∂ η

𝑇

𝑡=1

− ( 𝑣 + 1

2(𝑣 − 2)* 1 (1 + z𝑣 − 2*^t2

∑∂ z_t²

∂ η

𝑇

𝑡=1

∂ z_t²

∂ η = 1 σ_t²

∂ε_t²

∂ η + ε_t²∂σ_t⁻²

∂ η =2ε_t σ_t²

∂ε_t

∂ η−2ε_t² σ_t³

∂σ_t

∂ η .

From the above we can tell that ^∂ε_{∂ ξ}^t= −x_t and ^∂σ_{∂ η}^t =_δσ^σt

tδ

∂σ_t^δ

∂ η .

The differentiating of σ_t^δ with the respect to δ, θ and γ will be the same as the normal distribution as former.

 The differentiating of σ_t^δ with the respect to ξ:

∂σ_t^δ

∂ ξ = δ ∑ α_j0(|ε_t−j| − γ_jε_t−j)^δ−1 (γ_j+ 𝐼_𝑡−𝑖)x_t−j1^{𝐹𝑡−𝑗}

q

j=1

× [1

𝑇∑(|ε_t−j| − γ_jε_t−j)^δ−1(γ_j+ 𝐼_𝑡−𝑖)x_t−j

𝑇

𝑠=1

]

1−𝐹_{𝑡 −𝑗}

+ ∑ β_i(∂(σ_t−i)^δ

∂ ξ )

𝐹_{𝑡 −𝑖} p

i =1

[−δ 𝑇(1

𝑇∑ ε_s²

𝑇

𝑠=1

+

δ−22

∑ ε_sx_s

𝑇

𝑠=1

]

1−𝐹_𝑡−𝑖

,

where

𝐼_𝜏 = {−1, 𝑖𝑓ε_𝜏 ≥ 0 1, 𝑖𝑓ε_τ < 0

𝐹_𝜏= {1, 𝑖𝑓 𝜏 > 0 0, 𝑖𝑓 𝜏 ≤ 0 .

 The differentiating of σ_t^δ with respect to γ:

∂σ_t^δ

∂γ =∂ω

∂γ + ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂γ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂γ

p

i=1

= ∑ α_j∂k(ε_t−j)^δ

∂γ

q

j=1

+ ∑ β_i∂(σ_t−i)^δ

∂γ

p

i=1

.

The differentiating will be different with t changes.

∂k(ε_t−j)^δ

∂γ =

{

−δk(ε_t−j)^δ−1ε_t−j ,𝑓𝑜𝑟 𝑡 > 𝑗

−δ

𝑇∑(|𝜀_𝑆| − γ_iε_s)^δ−1ε_s ,𝑓𝑜𝑟 𝑡 ≤ 𝑗

𝑇

𝑠=1

,

and ^∂(σ_∂γ^t−i)δ= 0 for 𝑡 ≤ 0.

 The differentiating of σ_t^δ with respect to δ:

∂σ_t^δ

∂δ =∂ω

∂δ+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂δ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂δ

p

i=1

= δ ∑ α_j0(|ε_t−j| − γ_jε_t−j)^δ 𝐿𝑛(|ε_t−j| − γ_jε_t−j)1^{𝐹𝑡 −𝑗}

q

j=1

× [1

𝑇∑(|ε_t−j| − γ_jε_t−j)^δ𝐿𝑛(|ε_s| − γ_jε_s)

𝑇

𝑠=1

]

1−𝐹_𝑡−𝑗

+ ∑ β_i.(σ_t−i)^δ𝐿𝑛(σ_t−i)/^{𝐹𝑡−𝑖}

p

i=1

[−1 𝑇(1

𝑇∑ ε_s²

𝑇

𝑠 =1

+

δ2

𝐿𝑛(1

𝑇∑ ε_s²

𝑇

𝑠=1

)]

1−𝐹_{𝑡 −𝑖}

.

 The differentiating of σ_t^δ with respect to θ:

∂σ_t^δ

∂θ =∂ω

∂θ+ ∑∂α_j(|ε_t−j| − γ_jε_t−j)^δ

∂θ

q

j=1

+ ∑∂β_i(σ_t−i)^δ

∂θ

p

i=1

= ∑ α_j∂k(ε_t−j)^δ

∂θ

q

j=1

+ ∑ β_i∂(σ_t−i)^δ

∂θ

p

i=1

,

And ^∂(σ_∂θ^t−i)δ = 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 (z_t|, 𝑣) = 𝑣

(2𝐴 ∙ 𝛤(1𝑣)exp (− |z_t− B|^𝑣

,1 − 𝑠𝑖𝑔𝑛(z_t− B)ρ-^𝑣∙ 𝐴^𝑣)

𝐴 = 𝛤 (1

𝑣*^0.5𝛤 (3

𝑣*^−0.5𝐶(ρ)⁻¹ B = 2ρ ∙ D ∙ C(ρ)⁻¹ 𝐶(ρ) = √1 + 3ρ² − 4𝐷²ρ²

D = 𝛤 (2 𝑣* 𝛤 (1

𝑣*^0.5𝛤 (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

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(σ_t²) + (1 + 𝑣)ln (1 +(𝑠z_t+ 𝑚)² 𝑣 − 2 ρ^−2𝐼𝑡+

𝑇

𝑡=1

𝐼_𝑡 = {1 𝑖𝑓 z_t≥ −𝑚 𝑠

−1 𝑖𝑓 z_t< −𝑚 𝑠

𝑚 = 𝛤 .𝑣 + 12 / √𝑣 − 2

√𝜋𝛤 .𝑣 2/

(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∑(𝜎̂_𝑡²− 𝜎_𝑡²)²

𝑆+ℎ

𝑡=𝑆

.

2) Mean Absolute Error (MAE)

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∑|𝜎̂_𝑡²− 𝜎_𝑡²|

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∑ |𝜎̂_𝑡²− 𝜎_𝑡² 𝜎_𝑡² |

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 ∑^𝑆+ℎ_𝑡=𝑆(𝑌̂_𝑡²− 𝑌_𝑡²)²

√ 1𝑕 + 1 ∑^𝑆+ℎ_𝑡=𝑆𝑌̂_𝑡²− √ 1𝑕 + 1 ∑^𝑆+ℎ_𝑡=𝑆𝑌_𝑡² ,

where 𝑕 is the number of head steps, 𝑆 is the sample size, 𝜎̂_𝑡² is the forecasted variance, 𝜎_𝑡² 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 ∑ 𝑟_𝑘²

𝑚

𝑘=1

Where 𝑟_𝑘 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) ∑ 𝑟_𝑘² 𝑛 − 𝑘

𝑚

𝑘=1

Where 𝑟_𝑘 is the sample auto-correlation at the lag k, n is the sample size, m is the number of lags tested.