Panel EGARCH — Exponential GARCH for Panel Data

Panel GARCH models the conditional variance directly, requiring non-negativity constraints on parameters. Panel EGARCH models the log of conditional variance, eliminating those constraints, and adds a signed shock term (gamma) to capture the leverage effect — the tendency for negative shocks to raise volatility more than positive ones.

A significant negative gamma indicates a leverage effect: negative innovations increase log-conditional variance by more than positive innovations of the same magnitude. This is typical in equity markets where bad news amplifies uncertainty more than good news.

You can impose full homogeneity for parsimony, but it is risky if units operate in different economic environments. A common compromise is to pool the persistence (beta) and leverage (gamma) parameters but allow unit-specific intercepts (omega_i), testing homogeneity restrictions with a Wald or likelihood-ratio test.

Compute standardised residuals z_it = epsilon_it / sqrt(h_it). They should be close to i.i.d. Apply Ljung-Box tests to both z_it and z_it-squared; significant autocorrelation in the squares indicates remaining ARCH effects. Also check whether the chosen innovation distribution (normal vs. Student-t) fits the tail behaviour.

The conditional variance recursion itself is unit-specific, but cross-sectional dependence in the innovations can inflate standard errors. Use cluster-robust standard errors or a DCC (Dynamic Conditional Correlation) model when contemporaneous correlation across units is a primary concern.