The Restricted Likelihood Ratio Test for Autoregressive Processes

15 Pages Posted: 7 Mar 2012

See all articles by Willa W. Chen

Willa W. Chen

Texas A&M University - Department of Statistics

Rohit Deo

Stern School of Business, New York University

Date Written: March 2012


The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (−1, 1] × (−1, 1)p−1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.

Keywords: Restricted maximum likelihood, likelihood ratio test, partial autocorrelation, reparametrization

Suggested Citation

Chen, Willa W. and Deo, Rohit, The Restricted Likelihood Ratio Test for Autoregressive Processes (March 2012). Journal of Time Series Analysis, Vol. 33, Issue 2, pp. 325-339, 2012. Available at SSRN: or

Willa W. Chen (Contact Author)

Texas A&M University - Department of Statistics ( email )

155 Ireland Street
447 Blocker
College Station, TX 77843
United States

Rohit Deo

Stern School of Business, New York University ( email )

44 West Fourth Street
New York, NY 10012
United States

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