Wilks' Theorem: the Asymptotic χ² Distribution

Both worked examples in The Generalized Likelihood Ratio Test landed on an exact $\chi^2$ null distribution — but both leaned on a special gift of the normal family: $\bar X$ and $\sum_i(X_i-\bar X)^2$ have clean, tabulated exact distributions no matter the sample size. A Poisson jump-intensity model, a logistic default-probability model, a multi-regime GARCH specification — none of these come with an exact finite-sample distribution for $-2\log\Lambda$ at all, and deriving one by hand is generally hopeless. Wilks' theorem is the fix, and it is close to the single most-used result in applied hypothesis testing: under mild regularity conditions, $-2\log\Lambda(X)$ converges in distribution to a $\chi^2$ distribution as $n\to\infty$, for essentially any regular parametric model, with degrees of freedom given by a strikingly simple counting rule — the number of free parameters the null hypothesis pins down. This is what lets a risk desk test whether a proposed three-factor model needs its third factor, whether a regime-switching model needs a second regime, or whether a restricted GARCH specification loses anything against its unrestricted parent — all from the same asymptotic machinery, without ever deriving a model-specific exact distribution.

Learning objectives. By the end of this note you should be able to: