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ARCH and GARCH Estimation Dr. Chen, Jo-Hui

Arch and Garch

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Page 1: Arch and Garch

ARCH and GARCH Estimation

Dr. Chen, Jo-Hui

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The types of serial correlation discussed so far refer only to the error term ut. In AR(p) we postulated that ut depends linearly on the p past errors ut-1, ut-2, ……,ut-p. Another type of serial correlation is often encountered in time series data, especially when forecasts are generated. Some forecasters have observed that the variance of prediction errors is not a constant but differs from period to period. For instance, when the Federal Reserve Board switched to controlling money growth rather than the interest rate, as was done before, interest rates became quite volatile (that is, they began to vary a great deal around the mean). Forecast errors associated with interest rate predictions were thus heteroscedastic.

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A similar heteroscedasticity was observed when exchange rate policy switched from fixed exchange rates to flexible exchange rates. In the latter case, exchange rates fluctuated a great deal, making their forecast variances larger. In monetary theory and the theory of finance, financial asset portfolios are functions of the expected means and variances of the rates of returns. Increased volatility of security prices or rates of return are often indicators that the variances are not constant over time. Engle (1982) introduced a new approach to modeling heteroscedasticity in a time series context.

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The structure of this topic

A. The ARCH Specification B. Estimating ARCH models in EViews C. Asymmetric ARCH ModelsD. EViews Example

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Autoregressive Conditional Heteroskedasticity (ARCH) models are specifically designed to model and forecast conditional variances. The variance of the dependent variable is modeled as a function of past values of the dependent variable and independent, or exogenous variables. In developing an ARCH model, you will have to provide two distinct specifications—one for the conditional mean and one for the conditional variance.

A. The ARCH Specification

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1. The GARCH(1,1) Model

A. The ARCH Specification

= News about volatility from the previous period, measured as the

lag of the squared residual from the mean equation. (ARCH term)

= the conditional variance equation is a function of three termsw = the mean.

where

2t

21t

21t = last period’s forecast variance. (GARCH term)

varianceconditioal The(2).......

equation Mean(1)...... 2

12

12

'

ttt

ttt rxy

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1. The GARCH(1,1) Model

The (1,1) in GARCH(1,1) refers to the presence of a first-order GARCH term and a first-order ARCH term. An ordinary ARCH model is a special case of a GARCH specification in which there are no lagged forecast variances in the conditional variance equation.

For example, if the asset return was unexpectedly large in either the upward or the downward direction, then the trader will increase the estimate of the variance for the next period. This model is consistent with the volatility clustering often seen in financial returns data, where large changes in returns are likely to be followed by further large changes.

A. The ARCH Specification

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2. The ARCH-M model

The x’s in equation (1) represent exogenous or predetermined variables that are included in the mean equation. If we introduce the conditional variance into the mean equation, we get the ARCH-in-Mean (ARCH-M) model:

The ARCH-M model is often used in financial applications where the expected return on an asset is related to the expected asset risk. The estimated coefficient on the expected risk is a measure of the risk-return tradeoff.

A. The ARCH Specification

tttt rrxy ~2'

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3. The GARCH(p,q) model

Higher order GARCH models, denoted GARCH(p,q), can be estimated by choosing either p or q greater than 1. The representation of the GARCH(p,q) variance is

where p is the order of the GARCH terms and q is the order of the ARCH term.

The error variance at time t is assumed to depend on previous squared error terms. Also, the variance at time t is conditional on those in previous periods and hence the term conditional heteroscedasticity. The ARCH test is one the null hypothesis H0:α1=α2=…..= αq=0.

A. The ARCH Specification

p

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q

iitit

1

2

1

22

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a) Quick/Estimate Equation/ARCHb) Option: Heteroskedasticity Consistent Covariances: You should use this option if you

suspect that the residuals are not conditionally normally distributed.c) The Mean Equation: You can enter the specification in list form by listing the dependent variable

followed by the regressors. You should add the C to your specification if you wish to include a constant. If your specification includes an ARCH-M term, you should click on the appropriate radio button in the upper right-hand side of the dialog.

d) The variance Equation①Under the ARCH specification label, you should choose the number of ARCH and GARCH terms.

②In the edit box labeled Variance Regressors, you may optionally list variables you wish to include in the variance specification. Note that EViews will always include a constant as a variance regressor so that you do not need to add C to the list.

B. Estimating ARCH models in EViews

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Examples 1

GARCH(1,1) To estimate a standard GARCH(1,1) model with no regressors in the mean and

variance equations.

(a) File/Open/workfile/c:\program file\EView4\example file\data\gerus(b) Quick/Estimate Equation/ARCH(c) R c(d) Enter 1 for the number of ARCH terms, and 1 for the number of GARCH

terms, and select GARCH (symmetric) Select None for the ARCH-M(e) Leave blank the Variance Regressors edit box.

B. Estimating ARCH models in EViews

2

12

12

ttt

tt cR

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Mean Eq.

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Examples 2

To estimate the ARCH(4)-M model:

a) Quick/Estimate Equation/ARCH b) NYS c DUM c) Enter 4 for the ARCH term and 0 for the GARCH term, and select

GARCH (symmetric).d) Select std. Dev. for the ARCH-M terme) enter DUM in the Variance Regressors edit box.

B. Estimating ARCH models in EViews

tttttt

tttt

DUMr

rDUMrrNYS

32

442

332

222

112

210

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Click

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Examples 3

To estimate ARCH(3) model: Supposed that annual data for the U.S. for 1960-1995 on the Federal Reserve

discount rate (in percent), money supply (M2 in billions of dollars), and Federal deficit (D) in billions of current dollars. The model relating discount rates (r) to the money supply (M) and government budget deficits (D) lagged twice is as follows:

B. Estimating ARCH models in EViews

233

222

2110

2251423121

tttt

tttttt

uuu

uDDMMr

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Examples 3

Steps: The interest equation was first estimated, its estimated residuals squared, and

the auxiliary regression for the variance estimated. R2 for this regression was 0.126, but nR2=3.91 has p-value 0.27, which in not significant for x2 test. Thus, ARCH(3) is not supported. However, the ARCH(1) term was significant at the level 0.09, so an ARCH(1) specification was tested next. The p-value for this test was 0.048, which indicates significant at the 5 percent level. Because the deficit term D2 was insignificant, it was excluded from the model in order to improve the precision of the other variables and to reduce any multicollinearity that may be present. The final model is given below with p-values in parentheses:

B. Estimating ARCH models in EViews

464.R

0.01)( 0.01)( 0.01)( 0.01)( 0347.00262.00283.038.3ˆ

2

121

tttt DMMr

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1. ARCH Estimation Output

The output from ARCH estimation is divided into two sections:1) The upper part provides the standard output for the mean equation. 2) The lower part, labeled “Variance Equation” contains the coefficients, standard

errors, z-statistics and p-values for the coefficients of the variance equation. The ARCH parameters correspond toαand the GARCH parameters toβ.

3) Note that measures such as R2 may not be meaningful if there are no regressors in the mean equation. Here, for example, the R2 is negative.

4) The sum of the ARCH and GARCH coefficient (α+β) is very close to one, indicating that volatility shocks are quite persistent.

B. Estimating ARCH models in EViews

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2. Working with ARCH Model

Once your model has been estimated, EViews provides a variety of views and procedures for inference and diagnostic checking.

Views of ARCH Models Views/actual, fitted, residual /Conditional SD Graph /Covariance Matrix /Coefficient Tests /Residual Tests/Correlogram-Q-statistics /Correlogram Squared Residuals /Histogram-Normality Test /ARCH LM Test

B. Estimating ARCH models in EViews

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2. Working with ARCH Model The ARCH LM test statistic is computed from an auxiliary test

regression. To test the null hypothesis that there is no ARCH up to order q in the residuals, we run the regression

where e is the residual. This is a regression of the squared residuals on a constant and lagged squared residuals up to order q. EViews reports two test statistics for this test regression. The F-statistic is an omitted variable test for the joint significance of all lagged squared residuals. The Obs*R-squared statistic is Engle’s LM test statistic, computed as the number of observations times the R2 from the test regression. The exact finite sample distribution of the F-statistic under H0 is not known but the LM test statistic is asymptotically distributed x2(q) under quite general conditions.

B. Estimating ARCH models in EViews

,1

20

2t

q

sstst vee

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(1)Mean equation: By using ACF, PACF, and Q test, we can evaluate the form of mean equation via ARMA procedure.View/Residual Tests/Correlogram Q-statistics

This view can be used to test for remaining serial correlation in the mean equation and to check the specification of the mean equation. If the mean equation is correctly specified, all Q-statistics should not be significant.

(2) Variance equation:We can also apply Ljung-Box Q2 test: and use ACF and PACF to decide the number of lag for GARCH (p, q ).

View/Residual Tests/Correlogram Squared Residuals

This view can be used to test for remaining ARCH in the variance equation and to check the specification of the variance equation. If the variance equation is correctly specified, all Q-statistics should not be significant.

3. Working with GARCH Model

q

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1

2 )()2()(

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Q-statistics Q 2 -statistics

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For Equities, it is often observed that downward movements in the market are followed by higher volatilities than upward movements of the same magnitude. EViews estimates two models that allow for asymmetric shocks to volatility: TARCH and EGARCH.

C. Asymmetric ARCH Models

News

Volatility

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1. The TARCH Model: Threshold ARCH

In this model, good news (εt>0 ), and bad new (εt<0 ), have differential effects on the conditional variance—good news has an impact of α, while bad news has an impact of α+ r. If r>0 we say that the leverage effect exists. If r≠0, the news impact is asymmetric.

C. Asymmetric ARCH Models

211

21

21

2 ttttt dr

where dt=1 if εt<0 , and 0 otherwise.

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1. The TARCH Model: Threshold ARCH

For higher order specifications of the TARCH model, EViews estimates

1) Quick/Estimate Equation2) ARCH specification.3) R c4) TARCH

If the leverage effect term (r), represented by (RESID<0)*ARCH(1) in the output, is significantly positive but there appears to be no asymmetric effect.

C. Asymmetric ARCH Models

p

jjtjtt

q

iitit dr

1

21

21

1

22

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2. The EGARCH Model

The specification for the conditional variance is

Note that the left-hand side is the log of the conditional variance. This implies that the leverage effect is exponential, and that forecasts of the conditional variance are guaranteed to be nonnegative. The presence of leverage effects can be tested by the hypothesis that r > 0. The impact is asymmetric if r≠ 0.

C. Asymmetric ARCH Models

1

1

1

121

2 loglog

t

t

t

ttt r

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2. The EGARCH Model

The higher order specifications of EGARCH model:

To estimate an EGARCH model, simply select the EGARCH radio button under the ARCH specification settings.

The leverage effect term (r), denoted as RES/SQR[GARCH](1) in the output, is negative and statistically different from zero, indicating the existence of the leverage effect in future bonds returns R during the sample period.

C. Asymmetric ARCH Models

p

j

q

i it

iti

it

itijtjt r

1 1

22 loglog

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3. Plotting the Estimated News Impact Curve

Our goal is to plot the volatilityσ2, against z =ε/σthe impact, where

a) Procs/Make GARCH Variance Series

C. Asymmetric ARCH Models

112

12 loglog

tttt zrz

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Modeling the Volatility in the US Dollar/Deutschmark exchange rates from 1971 to 1993.

1. The dependent variable is the daily nominal return, where s is the spot rate. The return series clearly shows volatility

clustering, especially early in the sample.

2. GARCH(1,1)-MA(1) model

The mean equation includes the conditional variance and the errors are MA(1). The MON series in the conditional variance equation is a dummy variable for Monday, which is meant to capture the non-trading period effect during the weekend.

D. EViews Example

1log

t

tt s

sr

tttt

tttt

MONr

R

2

12

12

12

10

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3. Test a) Open/workfile/Example/data/gerusb) Quick/Estimate Equation/ARCHc) ARCH specification: R c ma(1)d) Variance Regressors: mon e) 1 1f) Select Variance for the ARCH-Mg) View/Residual Tests/ARCH LM Test/7

The top part of the output from testing up to an ARCH(7) is given by

D. EViews Example

F-statistic 0.1163 Probability 0.9970

Obs*R-squared 0.8836 Probability 0.9964

ARCH TEST:

Indicating that there does not appear to be any ARCH up to order 7.

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3. Test h) View/Residual Tests/Historgram-Normality test

The residuals are highly leptokurtic and the Jarque-Bera (JB) test decisively rejects the normal distribution.

JB test is a test statistic for testing whether the series is normally distributed. The test statistic measures the difference of the skewness and kurtosis of the series with those from the normal distribution.

Jarque-Bera=

Where S is the skewness, K is the kurtosis, and k represents the number of estimated coefficients used to create the series.

Under the null hypothesis of a normal distribution, the JB statistic is distributed as x2 with 2 degrees of freedom.

D. EViews Example

4

36

22 KSkN

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h) Procs/Make Residual Series/Standardized/View/Distribution Graphs/Quantile-Quantile/normal distribution If the residuals are normally distributed, the QQ-plots should lie

on a straight line. The plot shows that it is primarily a few large outliers that are driving the departure from normality.