GARCH – Modeling Conditional Variance & Useful Diagnostic Tests

1. Introduction

GARCH models provide a way of modelling conditional volatility. I.e. They are useful in situations where the volatility of a time series is a function of previous levels of volatility AKA volatility clustering.

A GARCH model is typically of the following form:

\sigma_t^2 = \omega+ \sum_{i=1}^{p} \alpha_i\epsilon_{t-i}^2+\sum_{i=1}^{q} \beta_i\sigma_{t-i}^2

which means that the variance ( \sigma_t^2) of the time series today is equal to a constant (\omega), plus some amount ( \alpha) of the previous residual (\epsilon_{t-1}), plus some amount ( \beta) of the previous variance (\sigma^2_{t-1}).

2. Fitting

Fitting GARCH models is usually trivial using modern software such as the rugarch package for R. However checking whether the fitted model is any good is less trivial since a range of diagnostic tests must be applied and most importantly understood to ensure that the model captures the intended behaviour.

3. GARCH Diagnostics

Ljung-Box Test

The Ljung-Box test provides a means of testing for auto-correlation within the GARCH model’s standardized residuals. If the GARCH model has done its job there should be NO auto-correlation within the residuals. The null-hypothesis of the Ljung-Box test is that the auto-correlation between the residuals for a set of lags k = 0. If at least one auto-correlation for a set of lags k > 0 then the test statistic indicates that the null-hypothesis may be rejected.
The rugarch package for R applies a weighted Ljung-Box Test on the standardized Residuals and the standardized Squared Residuals. Should the p-value be <= 0.05 (your significance level \alpha) then the null hypothesis should be rejected meaning that the GARCH model has not captured the auto-correlation.

ARCH LM Test

Similar to the Ljung-Box Test, the ARCH LM test provides a means of testing for serial dependence (auto-correlation) due to a conditional variance process by testing for auto-correlation within the squared residuals.The null hypothesis is that the auto-correlation between the residuals for a set of lags k = 0.

Nyblom Stability Test

The Nyblom stability test provides a means of testing for structural change within a time series. A structural change implies that the relationship between variables changes overtime e.g. for the regression y= \beta x beta changes over time. The null hypothesis is that the parameter values are constant i.e. zero variance, the alternative hypothesis is that their variance > 0.

Sign Bias Test

Engle & Ng sign bias tests provide a means of testing for mispecification of conditional volatility models. Specifically they examine whether the standardized squared residual is predictable using (dummy) variables indicative of certain information.

  • The sign bias test has a dummy variable S^- that is 1 when \epsilon_{t-1}<0. It tests for the impact of positive & negative shocks on volatility not predicted by the model.
  • The negative sign bias test uses a dummy variable S^- \epsilon_{t-1} – it focuses on the effect of large and small negative shocks.
  • The positive sign bias test uses a dummy variable S^+ \epsilon_{t-1} where S^+ = 1-S^- – it focuses on the effect of large and small positive shocks.
    The null hypothesis for these tests is that additional parameters corresponding to the additional (dummy) variables = 0. The alternative hypothesis is that the addition parameters are non-zero indicating mispecification of the model.

Adjusted Pearson Goodness-of-Fit Test

The adjusted pearson goodness-of-fit test compares the empirical distribution of the standardized residuals with the selected theoretical distribution. The null hypothesis is that the empirical and theoretical distribution is identical.

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