# 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.