On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle

Abstract

Some asymptotic properties of the least-squares estimator of the parameters of an AR model of order p, p ≥ 1, are studied when the roots of the characteristic polynomial of the given AR model are on or near the unit circle. Specifically, the convergence in distribution is established and the corresponding limiting random variables are represented in terms of functionals of suitable Brownian motions.Further, the preceding convergence in distribution is strengthened to that of convergence uniformly over all Borel subsets. It is indicated that the method employed for this purpose has the potential of being applicable in the wider context of obtaining suitable asymptotic expansions of the distributions of leastsquares estimators.

Bootstrapping Autoregressive Processes with Possible Unit Roots

“…One extension involves generalizing the results of this paper to possibly integrated and/or cointegrated vector valued processes. This extension will require a generalization of the results in Jeganathan (1991) to vector valued processes. The other extension is a study of the conditions under which the bootstrap approximation described in this paper provides asymptotic re nements for the studentized estimator of the slope parameter (or smooth functions thereof).…”

“…A formal proof of this conjecture is likely to be conceptually straightforward, but tedious. This extension is beyond the scope of this paper, however, because it would require the generalization of the Jeganathan (1991) results to models with deterministic time trends.…”

“…The cases for which w e establish the validity of the bootstrap are cases for which the asymptotic normality of the OLS estimator in the local-to-unity model follows from Jeganathan (1991) or for which asymptotic normality m a y be established by similar arguments. Jeganathan's local-to-unity model implies that the limiting distribution of the OLS estimator of the autoregressive model without drift depends on nuisance parameters.…”

Asymptotic inference for nearly unstable multidimensional AR processes

“…see, for example, [18], [9], [7]. (The above model is called also near integrated and is applied often in economic theory; see [18].)…”

“…(The above model is called also near integrated and is applied often in economic theory; see [18].) Recently, Jeganathan [9] has considered nearly nonstationary one-dimensional AR(p) models, i.e., AR(p) models near to an unstable model. He proved that the appropriately normalized LSE of the coefficients converges in law and gave a very complicated represen tation for the limiting distribution in terms of multiple stochastic integrals with respect to Wiener processes.…”