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5.5.4. Prediction

GARCH models are frequently used to forecast volatility of return. It is straightforward to forecast the conditional variance from an ARCH model. Assuming that the model parameters are known, the one-period ahead forecast is

Forecasting the conditional volatility for h periods ahead can be done by a recursion

The ^-period ahead variance forecast for a GARCH(1 1) model is

5.5.5. Example: GARCH Estimation

As an example of GARCH model estimation in EViews we consider a series of 2 minutes exchange rates between the Euro and the British Pound for 21 August 2007 between 7:00 and 16:00 GMT. The data is contained in the file EURGBP.wf1. Plot of the EUR/GBP returns is given in Figure?

2 minutes EUR/GBP returns on 21 August 2007

Figure 5.4: 2 minutes EUR/GBP returns on 21 August 2007

We can clearly see periods of high and low volatility of the returns, thus an ARCH type model can be appropriate to model volatility.

Let us first estimate an OLS regression of returns on a constant term. This will give us an opportunity to test for the presence of ARCH effect more formally. Having typed in the command line and pressed Enter, go to View/Residual Tests/ Heteroscedasticity Tests... in the equation object window and choose ARCH option there. The test result is given in Figure ??

ARCH test results

Figure 5.5: ARCH test results

The p-value of the test is very small which rejects the null hypothesis of ho-moscedasticity of residuals in favor of ARCH alternative. Thus, based on this result we decide to estimate regression with ARCH specification. Go to Quick/Estimate Equation option of the main workfile menu and specify the model as you were specifying it for the OLS regression. That is, type r_eurusd c in the Equation Specification box. Now, in the Method field, choose ARCH — Autoregressive Conditional Heteroscedasticity. This will open you more option for ARCH model specification. In the ARCH-M we can indicate whether we want to include ARCH-in-mean term in the equation and, if yes, whether variance or standard deviation should enter it. In the Variance and distribution specification part we can select between simple ARCH/GARCH/TARCH model, EGARCH, PGARCH or two component GARCH model. We will stay with the simplest first option. In the Order field we should write orders of ARCH and GARCH terms in the variance equation. Let us specify GARCH(3,3) model, so enter 3 and 3 respectively in each of the box. If we do not want to include GARCH terms, simply put 0 in front of the GARCH field. Variance regressors box will be useful if we want to include some exogenous variables in the variance equation. Errors distribution box allows us to choose between Gaussian and Student - distributions of the error term. The model estimation output is given in Figure ??.

GARCH estimation output

Figure 5.6: GARCH estimation output

The resid terms of the output correspond to a% coefficients (ARCH terms) and GARCH terms correspond to 3% coefficients in (5.3.2).

We can see that a3 and [33 coefficients are statistically significant and a2 is on a border line of significance.

In View/Representation section one can find the variance specification. Also EViews allow to plot both standardized and on-standardized residual plots (in Actual, Fitted, Residuals), test for parameter constancy, linear restriction, build correlogram of residuals and squared residuals in the same way as it is done for the OLS regression.

In order to estimate the above model using the command line one should type

where the term arch indicates that an ARCH estimation method should be used, order of the ARCH and GARCH terms follow in parentheses. Then you should specify the conditional mean equation as it is done on the least squares model case (the dependent variable should be in the first place).

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