Semiparametric Sieve-Type Generalized Least Squares Inference

Abstract

This article considers the problem of statistical inference in linear regression models with dependent errors. A sieve-type generalized least squares (GLS) procedure is proposed based on an autoregressive approximation to the generating mechanism of the errors. The asymptotic properties of the sieve-type GLS estimator are established under general conditions, including mixingale-type conditions as well as conditions which allow for long-range dependence in the stochastic regressors and/or the errors. A Monte Carlo study examines the finite-sample properties of the method for testing regression hypotheses.

Keywords:

• Autoregressive approximation
• Generalized least squares
• Linear regression
• Long-range dependence
• Short-range dependence

JEL Classification:

• C22

ACKNOWLEDGMENTS

The authors wish to thank an associate editor and a referee for their helpful comments and suggestions.

Notes

We note that Yule–Walker or Burg-type estimates may be used instead of the OLS estimates in the construction of without changing the first-order asymptotic properties of .

For a definition and more information on the ϕ-mixing and NED concepts, and the related size terminology, the reader is referred to Davidson (1994).

The estimator can be easily obtained from the OLS regression of on .

We are grateful to Štěpána Lazarová for providing us with the computer code used in the simulation study of Hidalgo and Robinson (2002).

Bickel and Bühlmann (1997) show that the closure (with respect to certain metrics) of the class of MA(∞) or AR(∞) processes is fairly large. Roughly speaking, for any stationary nonlinear process, there exists another process in the closure of linear processes having identical sample paths with probability exceeding 1/e ≈ 0.368.

The full set of simulation results is available from the authors upon request.

Note that the standard central limit theorem for martingale-difference sequences specifies that (20) and (21) hold for all T and not only in the limit, but a casual examination of the proof of Theorem 24.3 of Davidson (1994) shows that (20) and (21) are sufficient for the theorem to hold.

It is worth noting that Kuersteiner (2005) argues that the chi-square approximation in Lemma 5.1 of Ng and Perron (1995) is not valid. However, irrespectively of the validity of the result in question, it is straightforward to establish that, as long as the probabilities with which the null hypothesis of the test used in the sequential testing procedure of Definition 3.1 of Ng and Perron (1995) is accepted when true and rejected when false are bounded away from 1 and 0, respectively, Lemma 5.2 of Ng and Perron (1995) is valid, thereby establishing our result.