Asymptotic Normality of the Quasi‐Maximum Likelihood Estimator for p ‐Order Random Coefficient Autoregressive Models

ABSTRACT

We establish the asymptotic normality, at the general order , of the quasi‐maximum likelihood estimator of random coefficient autoregressive (RCA) models evaluated by the Kalman filter. Because the lagged states are observed, the filter is exact, and the criterion is the conditional Gaussian quasi‐likelihood; this closes the gap between the non‐constructive general‐order maximum‐likelihood theory and the first‐order quasi‐likelihood results, with an effectively computable estimator. The score is a square‐integrable martingale difference, so the limit theorem is the martingale central limit theorem. Because the score is self‐normalised by the conditional variance, no moment of  beyond the second—the second‐order stationarity already assumed—is required, and only finite fourth moments of the innovations enter; this is strictly weaker than the classical . The information matrices are in closed form and proved positive definite, and the covariance is of sandwich form; Gaussianity serves only to identify it with the inverse Fisher information, hence efficiency, and a distribution‐free version is our main robustness result. Each evaluation costs , and a Monte Carlo study confirms the rate, including a heavy‐tailed design with .