An asymptotic theory is given for autoregressive time series with a root of the form r n ¼ 1 þ c=k n , which represents moderate deviations from unity when ðk n Þ n2N is a deterministic sequence increasing to infinity at a rate slower than n, so that k n ¼ oðnÞ as n ! 1. For co0, the results provide a ffiffiffiffiffiffiffi nk n p rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the ffiffi ffi n p and n convergence rates for the stationary (k n ¼ 1) and conventional local to unity (k n ¼ n) cases. For c40; the serial correlation coefficient is shown to have a k n r n n convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when r n 41. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for k n ¼ 1, where the convergence rate of the serial correlation coefficient is ð1 þ cÞ n and no invariance principle applies. r 2005 Elsevier B.V. All rights reserved.JEL classification: C22

This study examines stock return predictability via lagged …nancial variables with unknown stochastic properties. We propose a novel testing procedure that (1) robusti…es inference to regressors' degree of persistence, (2) accommodates testing the joint predictive ability of …nancial variables in multiple regression, (3) is easy to implement as it is based on a linear estimation procedure, and (4) can be used for long-horizon predictability tests. We provide some evidence in favor of short-horizon predictability during the 1927-2012 period. Nevertheless, this evidence almost entirely disappears in the post-1952 period. Moreover, predictability becomes weaker, not stronger, as the predictive horizon increases.

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An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form ρni = 1 + ci/nα, involving moderate deviations from unity when α ∈ (0, 1) and ci ∈ ℝ are constant parameters. When the data are moderately integrated in the stationary direction (with ci < 0), it is shown that least squares regression is consistent and asymptotically normal but suffers from significant bias, related to simultaneous equations bias. In the moderately explosive case (where ci > 0) the limit theory is mixed normal with Cauchy-type tail behavior, and the rate of convergence is explosive, as in the case of a moderately explosive scalar autoregression (Phillips and Magdalinos, 2007, Journal of Econometrics 136, 115–130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.

An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form ρ n = 1+c/n α , involving moderate deviations from unity when α ∈ (0, 1) and c ∈ R are constant parameters. The limit theory combines a functional law to a diffusion on D[0, ∞) and a central limit theorem. For c > 0, the limit theory of the first order serial correlation coefficient is Cauchy and is invariant to both the distribution and the dependence structure of the innovations. To our knowledge, this is the first invariance principle of its kind for explosive processes. The rate of convergence is found to be n α ρ n n , which bridges asymptotic rate results for conventional local to unity cases (n) and explosive autoregressions ((1 + c) n). For c < 0, we provide results for α ∈ (0, 1) that give an n (1+α)/2 rate of convergence and lead to asymptotic normality for the first order serial correlation, bridging the √ n and n convergence rates for the stationary and conventional local to unity cases. Weakly dependent errors are shown to induce a bias in the limit distribution, analogous to that of the local to unity case. Linkages to the limit theory in the stationary and explosive cases are established.