Boundary Limit Theory for Functional Local to Unity Regression

Abstract

This paper studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multi-directional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its t statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the paper.

Properties of the Power Envelope for Tests Against Both Stationary and Explosive Alternatives: The Effect of Trends

“…This measure could be adapted for any (simple) hypothesis test on a covariance matrix, when invariance with respect to the mean is required. It provides a simple measure of the sensitivity of power to the choice of deterministics, similar in spirit to the analysis of Leamer (). Bykhovskaya and Phillips () explore tests involving functional local alternatives where the local parameter depends on time. For example, H 0 : c t = 0 vs. so that only the initial value has a unit root.…”

“…Bykhovskaya and Phillips () explore tests involving functional local alternatives where the local parameter depends on time. For example, H 0 : c t = 0 vs. so that only the initial value has a unit root.…”

Nearly Efficient Likelihood Ratio Tests of a Unit Root in an Autoregressive Model of Arbitrary Order

“…where C(•) is some (possibly) nonconstant function (see also Bykhovskaya and Phillips, 2018). Letting C denote a set of functions containing the zero function, we assume that the maintained hypothesis is of the form C(•) ∈ C, in which case the unit-root testing problem is the problem of testing…”

Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion

“…Problems of joint and sequential limit theory have been addressed before in weak convergence theory (Billingsley, 1968, Theorem 4.2), in panel data asymptotics (Phillips and Moon, 1999) and in nearunit root limit theory (Phillips, 1987, Chan and Wei, 1987, Bykhovskaya and Phillips, 2018). The latter work, which bears some similarity to the present context in terms of taking limits to the boundary of the domain of de…nition of the parameters, deals with the …rst order autoregression (AR(1)) model fy t = ay t + u t ; t = 1; 2;…”

“…As suggested by the referee, in the sequential asymptotics, 0 in D( ) does not depend on n because n has already diverged to 1, so we maintain the notation 0 in the sequential asymptotics.4 The case where c > 0 and c ! 1 at the upper boundary of the domain of de…nition is also considered inPhillips (1987) Bykhovskaya and Phillips (2018). consider cases where c( ) = cg( ) is a function and the scale coe¢ cient c !…”