Point Optimal Testing with Roots That Are Functionally Local to Unity

Abstract

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases. It is shown that against FLUR alternatives, conventional constant point optimal tests can have extremely low power, particularly when the departure from unity occurs early in the sample period. Simulation results are reported and some implications for empirical practice are examined.

Boundary Limit Theory for Functional Local to Unity Regression

“…It is a particularly useful framework for studying the effects of departures from simple unit root and LUR models to more complex time series behavior. Bykhovskaya and Phillips (2017) show that on standardization the process X t satisfies the functional law given in the lemma below with a Gaussian limit process which extends the usual limit theory for LUR time series. This functional law embodies the impact in the limit process of the functional localizing coefficient that is present in (1).…”

“…Bykhovskaya and Phillips show that on standardization the process X t satisfies the functional law given in the lemma below with a Gaussian limit process which extends the usual limit theory for LUR time series. This functional law embodies the impact in the limit process of the functional localizing coefficient that is present in .…”

“…We may also consider the case where the standardized time series is measured in segments of length O ( k n ) rather than length O ( n ). The framework then matches the usual FLUR model of Bykhovskaya and Phillips but over a much wider (infinite) domain. More specifically, when we focus on the process X t with t =⌊ k n r ⌋ instead of t =⌊ nr ⌋, we have the following limit theory for the standardized process .…”

“…The present article studies the same FLUR model as Bykhovskaya and Phillips and examines behavior as the localizing function diverges, thereby determining the impact on the properties of 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 when the alternatives involve time varying functional forms of differing types.…”

“…Such models are called functional local unit root models 524 A. BYKHOVSKAYA AND P. C. B. PHILLIPS (FLURs). They were recently studied in Bykhovskaya and Phillips (2017) in the context of point optimal unit root tests, showing how different the power envelope can be when the departures from a unit root are time varying.…”