Estimation in dynamic regression with an integrated process

Abstract

This paper deals with a dynamic regression model y_t = αy_t−1 + βz_t + u_t, where z_t is an integrated process of order one abbreviated as z_t ∼ I(1). Generally speaking, nonstandard asymptotic theory is required to investigate asymptotic properties of statistics related to an integrated process and the asymptotic results are very different from standard ones. There are two distinctive properties in nonstandard asymptotics: the so-called ‘super-consistency’ or T-consistency (where T is a sample size) and the weak convergence to a functional of the Wiener process. In spite of z_t being involved in our model, however, it is shown that our asymptotic results are the same as in the standard asymptotics in classical dynamic regression models, or if the disturbance u_t is serially correlated the OLS estimators of α and β have √T-inconsistency. This is due to the cointegration between y_t−1 and z_t. Although this point was clarified by Park and Phillips (1989) in a general context, we examine this explicitly through our specific model and connect the standard asymptotic theory with the nonstandard one in our case. Furthermore we investigate the limiting properties of other statistics such as t-ratio, the Durbin-Watson test and h-test. We also propose a consistent estimator of α and β by making use of Durbin's 2-step method. Finally, we carry out simulation studies which support our theoretical results.