## 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:

which means that the variance () of the time series today is equal to a constant (), plus some amount () of the previous residual (), plus some amount () of the previous variance ().

## 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 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 that is 1 when . 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 – it focuses on the effect of large and small negative shocks.
- The positive sign bias test uses a dummy variable where – 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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