Limit Theory for Moderate Deviations From a Unit Root Under Weak Dependence

Abstract

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.

Limit theory for moderate deviations from a unit root

“…When c40, the signal of the moderately explosive regressor y tÀ1 is strong enough to allow for serially correlated errors without affecting the Cauchy limit theory of Theorem 4.3. In subsequent work, Phillips and Magdalinos (2005) extend Theorem 4.3 for moderately explosive processes with linear process errors u t ¼ P 1 j¼0 c j e tÀj , where e t is a sequence of i.i.d. ð0; s 2 Þ random variables and c j is a sequence of constants satisfying P 1 j¼1 jjc j jo1.…”

“…Giraitis and Phillips (2006) prove a version of Theorem 3.2 for martingale difference errors with constant conditional variance E F ntÀ1 ðu 2 t Þ ¼ s 2 for all t. Serial correlation in the errors, however, induces an asymptotic bias forr n and contributes to the variance of the Gaussian limiting distribution. For linear process innovations u t ¼ P 1 j¼0 c j e tÀj as above with Ee 4 1 o1, Phillips and Magdalinos (2005) derive an expression for the asymptotic bias ofr n and provide formulae for the asymptotic variance of the normalized and centered serial correlation coefficient. …”

Mean and autocovariance function estimation near the boundary of stationarity

“…Thus, (1.2) is a set valued result which holds for all ρ in a region whose width ultimately depends on the sample size n. It follows from (1.2) that standard asymptotic estimation and inferential theory applies over the whole region of ρ for which (1.2) holds. Similarly, in more general autoregressions than (1.1) and linear regressions where moderate deviations from a unit root occur, asymptotic normality will prevail although the rate of convergence may increase or slow down depending on the value of ρ and bias effects may emerge because of endogeneity in the regressors (Phillips and Magdalinos, 2007b;Magdalinos and Phillips, 2008). The present paper seeks to explore generalizations of (1.2) for sample mean, autocovariance and autocorrelation functions near the boundary of stationarity and under a wider class of models that allow for linear process errors.…”

Estimation for nearly unit root processes with GARCH errors

“…A key question is to study the limiting behavior of φ n when φ n is near to one. In the recent years, some theories for the inference for nearly unit root processes have been established, see for example [2], [3], [13], [14] and the references therein. To the best of our knowledge, however, limit theory for a nearly unit root process with GARCH errors has not been discussed in the literature.…”