Kleibergen-van-Dijk (1993) Non-Stationarity in GARCH Models: A Bayesian Analysis

Summary

Kleibergen and Van Dijk reexamine GARCH stationarity by distinguishing weak stationarity ( $\alpha + \beta < 1$ ) from quasi-strict stationarity ( $E[\ln(\beta + \alpha z^2)] < 0$ , Nelson 1990a). Applied to US 3-month T-bill rates (Jan 1957–Apr 1989, $T=388$ ) with an AR(1)-GARCH(1,1)-t model, they show the weak criterion declares near-certain non-stationarity ( $\Pr \approx 0.99$ ) while the quasi-strict criterion assigns only 13% probability. They also find that fat-tailed (Student-t) error distributions substantially raise Bayesian posterior odds for a unit root in the mean, relative to Gaussian models, because t-likelihoods downweight large outliers that would otherwise push the posterior toward stationarity in mean.

Key Claims

Weak stationarity ( $\alpha + \beta < 1$ ) is strictly more restrictive than quasi-strict stationarity; the stationary region under the weak criterion is a proper subset of the quasi-strict stationary region.
The GARCH(1,1) conditional variance product $G_k = G_{k-1}(\beta + \alpha z_k^2)$ converges to 0 a.s. when $E[\ln(\beta + \alpha z^2)] < 0$ and diverges otherwise (Theorem 1); quasi-strict stationarity holds iff this log-moment condition holds (Theorem 2).
Under integrated GARCH (IGARCH) ( $\alpha + \beta = 1$ ) with normal errors, $\Pr[h_{t+1} < h_t] = 0.68$ — the conditional variance is still quasi-strict stationary despite $E[h_{t+k}] \to \infty$ .
Bayesian importance sampling (IS) estimates (T-bill data, GARCH(1,1)-t): $\hat\alpha = 0.18$ , $\hat\beta = 0.72$ ; weak criterion: $\Pr(\text{non-stat}) \approx 0.99$ , $K_{01} = 4.95$ ; quasi-strict criterion: $\Pr(\text{non-stat}) \approx 0.13$ , $K_{01} = 0.434$ .
Posterior odds for unit root in the AR mean coefficient $H_0\colon \rho=1$ : constant variance + normal gives $K_{01} = 0.17$ (rejects unit root); GARCH + normal gives $K_{01} = 0.04$ (strongly rejects); constant variance + t gives $K_{01} = 2.30$ (supports unit root); GARCH + t gives $K_{01} = 2.50$ (supports unit root). Error distribution matters far more than variance specification.
The $\rho$ – $\lambda$ negative posterior correlation is the mechanism: fat-tailed models attribute extreme observations to heavy tails rather than high persistence, freeing $\rho$ to concentrate near 1.
t-likelihood derivatives $-\partial\ln p/\partial\varepsilon^2 = (\lambda+1)/[2(\lambda+\varepsilon^2)]$ decrease for large $|\varepsilon|$ , downweighting outliers; normal likelihoods treat the same outliers as high-variance evidence, biasing the posterior toward stationarity.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The quasi-strict stationarity condition is less restrictive than the weak stationarity condition. So, in the case of stationarity, the quasi-strict stationary region is larger than the weak stationary region."

"The probability of non-stationarity of the conditional variance is about 0.13 [under quasi-strict criterion], while it is approximately 0.99 [under weak criterion]."

My Take

The paper makes two genuinely important contributions. The theoretical one — clarifying the hierarchy between weak and quasi-strict stationarity and proving convergence via the log law of large numbers (LLN) — is clean and consequential: it explains why near-IGARCH estimates need not be economically explosive. The empirical one — that fat tails and unit roots are entangled in the posterior through the $\rho$ – $\lambda$ correlation — is a warning against running standard unit root tests on financial series with heavy tails. The importance sampling approach (SISAM, Hop-Van Dijk 1992) was state-of-the-art for 1993 but now Markov chain Monte Carlo (MCMC) would be used. The small sample ( $T=388$ ) makes some Bayes factor conclusions delicate, but the directional result (t vs. normal matters enormously for unit root inference) is robust.