ARCH Model (Autoregressive Conditional Heteroskedasticity)

Run the ARCH-LM (Lagrange Multiplier) test on OLS residuals and inspect the partial autocorrelation function of squared residuals. Choose q where significant autocorrelation disappears. Then compare AIC and BIC across candidate orders and prefer the model with the lowest information criterion. In practice GARCH(1,1) often outperforms ARCH(q) for q > 2.

The ARCH-LM test regresses squared OLS residuals on their own lags and tests whether the slope coefficients are jointly zero. A significant chi-squared statistic confirms ARCH effects in the residuals, justifying a conditional heteroskedasticity model.

GARCH(1,1) adds one lagged conditional variance term to the ARCH equation. This parsimonious specification mimics an ARCH model with infinitely many lags while estimating only three parameters, and typically fits financial data better than ARCH with many lags.

Standard ARCH treats positive and negative shocks of equal size identically because it uses squared errors. Asymmetric leverage effects — where negative shocks raise volatility more than positive ones — require EGARCH (exponential GARCH) or TGARCH (threshold GARCH).

Financial returns often exhibit fat tails, meaning the Gaussian assumption underestimates tail risk. Quasi-maximum likelihood (QML) under Gaussian assumption still gives consistent parameter estimates, but efficiency improves and VaR forecasts become more reliable when a Student-t or GED distribution is assumed.