Analysis of Financial Time Series, Third Edition

3.3 Model Building

Building a volatility model for an asset return series consists of four steps:

1. Specify a mean equation by testing for serial dependence in the data and, if necessary, building an econometric model (e.g., an ARMA model) for the return series to remove any linear dependence.

2. Use the residuals of the mean equation to test for ARCH effects.

3. Specify a volatility model if ARCH effects are statistically significant, and perform a joint estimation of the mean and volatility equations.

4. Check the fitted model carefully and refine it if necessary.

For most asset return series, the serial correlations are weak, if any. Thus, building a mean equation amounts to removing the sample mean from the data if the sample mean is significantly different from zero. For some daily return series, a simple AR model might be needed. In some cases, the mean equation may employ some explanatory variables such as an indicator variable for weekend or January effects.

In what follows, we use R (both with and without OX) and S-Plus in empirical illustrations. Other software packages (e.g., Eviews, SCA, and RATS) can also be used.

3.3.1 Testing for ARCH Effect

For ease in notation, let at = rt − μt be the residuals of the mean equation. The squared series is then used to check for conditional heteroscedasticity, which is also known as the ARCH effects. Two tests are available. The ...