Uniform interval estimation for an AR(1) process with AR errors

Abstract

An empirical likelihood method was proposed in Hill and Peng (2014) to construct a unified interval estimation for the coefficient in an AR(1) model, regardless of whether the sequence was stationary or near integrated. The error term, however, was assumed independent, and this method fails when the errors are dependent. Testing for a unit root in an AR(1) model has been studied in the literature for dependent errors, but existing methods cannot be used to test for a near unit root. In this paper, assuming the errors are governed by an AR(p) process, we exploit the efficient empirical likelihood method to give a unified interval for the coefficient by taking the structure of errors into account. Furthermore, a jackknife empirical likelihood method is proposed to reduce the computation of the empirical likelihood method when the order in the AR errors is not small. A simulation study is conducted to examine the finite sample behavior of the proposed methods.

Asymptotic Theory and Unified Confidence Region for an Autoregressive Model

“…This means that the method in Andrews and Guggenberger () cannot unify the cases of μ =0 and μ ≠0 in , which is not surprising as Phillips () showed that the asymptotic distribution of the least squares estimator for φ n is different for the cases of μ =0 and μ ≠0. Recently, by adding a pseudo sample and applying the empirical likelihood method to some weighted score equations, Hill et al () provided unified intervals for φ n under model without distinguishing the cases of zero intercept, nonzero constant intercept, stationary, unit root, near unit root, and explosive, but did not discuss the case of moderate deviation from a unit root. Although constructing a unified interval for φ n has been studied in the literature, as far as we are aware there is no existing method on constructing a unified confidence region for ( μ , φ n ) T under model regardless of zero intercept or nonzero constant intercept or stationary or explosive or near unit root or unit root or moderate deviation from a unit root.…”

Uniform Test for Predictive Regression With AR Errors

“…Our proofs are along the lines of Hill, Li and Peng (2015). Before proving Theorem 1, we need the following lemmas.…”

“…models; Hall and Yao (2003) for parametric regression models; Hill, Li and Peng (2015) for an AR process with a possible near unit root.…”

“…Instead of splitting the data, we employ the trick of adding pseudo samples proposed by Li, Chan and Peng (2014) and Hill, Li and Peng (2015) to achieve a uniform inference.…”

A Unified test for the Intercept of a Predictive Regression Model*

“…In view of this, it is important to develop some unified tests which are robust against (i) – (iii). For the AR(1) process X t 's, uniform inference procedures for ϕ have already been discussed by many authors; See So and Shin (1999), Mikusheva (2007), Chan, Li and Peng (2012) and Hill, Li and Peng (2016). Recently, Zhu, Cai and Peng (2014) investigated the unified predictability test for β in (1) by empirical likelihood method.…”