Chapter 20. ARCH and GARCH Estimation

602—Chapter 20. ARCH and GARCH Estimation

mation, it is called the conditional variance. The conditional variance equation specified in (20.2) is a function of three terms:

•A constant term: ω .

•News about volatility from the previous period, measured as the lag of the squared

residual from the mean equation: 2t − 1 (the ARCH term).

• Last period’s forecast variance: σ2t − 1 (the GARCH term).

The (1, 1) in GARCH(1, 1) refers to the presence of a first-order autoregressive GARCH term (the first term in parentheses) and a first-order moving average ARCH term (the second term in parentheses). 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— i.e., a GARCH(0, 1).

This specification is often interpreted in a financial context, where an agent or trader predicts this period’s variance by forming a weighted average of a long term average (the constant), the forecasted variance from last period (the GARCH term), and information about volatility observed in the previous period (the ARCH term). 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 also 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.

There are two equivalent representations of the variance equation that may aid you in interpreting the model:

•If we recursively substitute for the lagged variance on the right-hand side of Equation (20.2), we can express the conditional variance as a weighted average of all of the lagged squared residuals:

We see that the GARCH(1,1) variance specification is analogous to the sample variance, but that it down-weights more distant lagged squared errors.

Thus, the squared errors follow a heteroskedastic ARMA(1,1) process. The autoregressive root which governs the persistence of volatility shocks is the sum of α plus

Basic ARCH Specifications—603

β . In many applied settings, this root is very close to unity so that shocks die out rather slowly.

The GARCH(q, p) Model

Higher order GARCH models, denoted GARCH(q, p ), can be estimated by choosing either q or p greater than 1 where q is the order of the autoregressive GARCH terms and p is the order of the moving average ARCH terms.

The representation of the GARCH(q, p ) variance is:

The GARCH-M Model

The Xt in equation (20.2) represent exogenous or predetermined variables that are included in the mean equation. If we introduce the conditional variance or standard deviation into the mean equation, we get the GARCH-in-Mean (GARCH-M) model (Engle, Lilien and Robins, 1987):

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.

Two variants of this ARCH-M specification use the conditional standard deviation or the log of the conditional variance in place of the variance in Equation (20.6).

Regressors in the Variance Equation

Equation (20.5) may be extended to allow for the inclusion of exogenous or predetermined regressors, z , in the variance equation:

Note that the forecasted variances from this model are not guaranteed to be positive. You may wish to introduce regressors in a form where they are always positive to minimize the possibility that a single, large negative value generates a negative forecasted value.

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Distributional Assumptions

To complete the basic ARCH specification, we require an assumption about the conditional distribution of the error term . There are three assumptions commonly employed when working with ARCH models: normal (Gaussian) distribution, Student’s t-distribution, and the Generalized Error Distribution (GED). Given a distributional assumption, ARCH models are typically estimated by the method of maximum likelihood.

For example, for the GARCH(1, 1) model with conditionally normal errors, the contribution

where σ2t is specified in one of the forms above.

For the Student’s t-distribution, the log-likelihood contributions are of the form:

By default, ARCH models in EViews are estimated by the method of maximum likelihood under the assumption that the errors are conditionally normally distributed.

Estimating ARCH Models in EViews

To estimate an ARCH or GARCH model, open the equation specification dialog by selecting

Quick/Estimate Equation… or by selecting Object/New Object.../Equation…. Select

ARCH from the method combo box at the bottom of the dialog.

The dialog will change to show you the ARCH specification dialog. You will need to specify both the mean and the variance specifications, the error distribution and the estimation sample.

Estimating ARCH Models in EViews—605

The Mean Equation

In the dependent variable edit box, you should enter the specification of 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 you have a more complex mean specification, you can enter your mean equation using an explicit expression.

If your specification

includes an ARCH-M term, you should select the appropriate item of the combo box in the upper right-hand side of the dialog.

The Variance Equation

Your next step is to specify your variance equation.

Class of models

To estimate one of the standard GARCH models as described above, select the GARCH/ TARCH entry from the Model combo box. The other entries (EGARCH, PARCH, and Component ARCH(1, 1)) correspond to more complicated variants of the GARCH specification. We discuss each of these models in “Additional ARCH Models” on page 612.

Under the Options label, you should choose the number of ARCH and GARCH terms. The default, which includes one ARCH and one GARCH term is by far the most popular specification.

If you wish to estimate an asymmetric model, you should enter the number of asymmetry terms in the Threshold order edit field. The default settings estimate a symmetric model.

Variance regressors

In the Variance Regressors edit box, 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.

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The distinction between the permanent and transitory regressors is discussed in “The Component GARCH (CGARCH) Model” on page 615.

The Error Distribution

To specify the form of the conditional distribution for your errors, you should select an entry from the combo box labeled Error Distribution. You may choose between the default Normal (Gaussian), the Student’s t, the Generalized Error (GED), the Student’s t with fixed d.f., or the GED with fixed parameter. In the latter two cases, you will be prompted to enter a value for the fixed parameter. See “Distributional Assumptions” on page 604 for details on the supported distributions.

Estimation Options

EViews provides you with access to a number of optional estimation settings. Simply click on the Options tab and fill out the dialog as desired.

Backcasting

By default, both the innovations used in initializing MA estimation and the initial variance required for the GARCH terms are computed using backcasting methods. Details on the MA backcasting procedure are provided in “Backcasting MA terms” on page 510.

When computing backcast initial variances for GARCH, EViews first uses the coefficient values to compute the residuals of the mean equa-

tion, and then computes an exponential smoothing estimator of the initial values,

t = 1

Estimating ARCH Models in EViews—607

and the smoothing parameter λ = 0.7 . Alternatively, you can choose to initialize the GARCH process using the unconditional variance:

If you turn off backcasting, EViews will set the presample values of the residual to zero to initialize an MA, if present, and will set the presample values of the variance to the unconditional variance using Equation (20.15).

Our experience has been that GARCH models initialized using backcast exponential smoothing often outperform models initialized using the unconditional variance.

Heteroskedasticity Consistent Covariances

Click on the check box labeled Heteroskedasticity Consistent Covariance to compute the quasi-maximum likelihood (QML) covariances and standard errors using the methods described by Bollerslev and Wooldridge (1992). This option is only available if you choose the conditional normal as the error distribution.

You should use this option if you suspect that the residuals are not conditionally normally distributed. When the assumption of conditional normality does not hold, the ARCH parameter estimates will still be consistent, provided the mean and variance functions are correctly specified. The estimates of the covariance matrix will not be consistent unless this option is specified, resulting in incorrect standard errors.

Note that the parameter estimates will be unchanged if you select this option; only the estimated covariance matrix will be altered.

Derivative Methods

EViews currently uses numeric derivatives in estimating ARCH models. You can control the method used in computing these derivatives to favor speed (fewer function evaluations) or to favor accuracy (more function evaluations).

Iterative Estimation Control

The likelihood functions of ARCH models are not always well-behaved so that convergence may not be achieved with the default estimation settings. You can use the options dialog to select the iterative algorithm (Marquardt, BHHH/Gauss-Newton), change starting values, increase the maximum number of iterations, or adjust the convergence criterion.

Starting Values

As with other iterative procedures, starting coefficient values are required. EViews will supply its own starting values for ARCH procedures using OLS regression for the mean equation. Using the Options dialog, you can also set starting values to various fractions of

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the OLS starting values, or you can specify the values yourself by choosing the User Specified option, and placing the desired coefficients in the default coefficient vector.

GARCH(1,1) examples

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

Rt = c + t

(20.16)

σ2t = ω + α 2t − 1 + βσ2t − 1

you should enter the various parts of your specification:

•Fill in the Mean Equation Specification edit box as r c

•Enter 1 for the number of ARCH terms, and 1 for the number of GARCH terms, and select GARCH/TARCH.

•Select None for the ARCH-M term.

•Leave blank the Variance Regressors edit box.

To estimate the ARCH(4)-M model:

Rt = γ0 + γ1DUMt + γ2σt + t

(20.17)

σ2t = ω + α1 2t − 1 + α2 2t − 2 + α3 2t − 3 + α4 2t − 4 + γ3DUMt

you should fill out the dialog in the following fashion:

•Enter the mean equation specification “R C DUM”.

•Enter “4” for the ARCH term and “0” for the GARCH term, and select GARCH (symmetric).

•Select Std. Dev. for the ARCH-M term.

•Enter DUM in the Variance Regressors edit box.

Once you have filled in the Equation Specification dialog, click OK to estimate the model. ARCH models are estimated by the method of maximum likelihood, under the assumption that the errors are conditionally normally distributed. Because the variance appears in a non-linear way in the likelihood function, the likelihood function must be estimated using iterative algorithms. In the status line, you can watch the value of the likelihood as it changes with each iteration. When estimates converge, the parameter estimates and conventional regression statistics are presented in the ARCH object window.