Likelihood Fit Is Not Directional Fit: A Time-arrow Diagnostic for GARCH Models

Abstract

Volatility models are normally selected by likelihood, information criteria, residual diagnostics, and forecast loss. These criteria need not reveal whether a model reproduces the temporal direction of risk. We compare Gaussian GARCH, Student-t GARCH, Student-t GJR-GARCH, and Student-t EGARCH using daily S&P 500, FTSE 100, Nikkei 225, and Hang Seng returns from 2000 through June 2026. Model adequacy is evaluated against a 1–22-day time-arrow profile formed from return–variance mixed moments. EGARCH wins AIC and BIC in every market, whereas GJR-GARCH has the lowest directional RMSE in the United States, United Kingdom, and Japan. Bootstrap intervals do not resolve the GJR– EGARCH gaps in those markets, so this is a criterion-dependent point-ranking reversal, not evidence of GJR dominance. Separately, the best asymmetric model reduces full-sample RMSE by 18.6%–74.6% relative to the best symmetric benchmark, and these asymmetric-versus-symmetric intervals exclude zero in all four markets. Raw Monte Carlo diagnostics reject both asymmetric models only for the Hang Seng, while Holm and false-discovery-rate corrections appropriately weaken several model-level rejection claims. In fixed-parameter 2019–2026 and reviewer-motivated 2021–2026 evaluations, asymmetric models remain directionally closer than symmetric GARCH but are not uniformly adequate. Likelihood fit and directional fit therefore answer different questions: a model can explain conditional scale and heavy tails while attenuating how negative returns propagate into future variance.

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