Efficiency Gains from Quasi-Differencing under Nonstationarity

A famous theorem on trend removal by OLS regression (usually attributed to Grenander and Rosenblatt, 1957) gave conditions for the asymptotic equivalence of GLS and OLS in deterministic trend extraction. When a time series has trend components that are stochastically nonstationary, this asymptotic equivalence no longer holds. We consider models with integrated and near-integrated error processes where this asymptotic equivalence breaks down. In such models, the advantages of GLS can be achieved through quasi-differencing and we give an asymptotic theory of the relative gains that occur in deterministic trend extraction in such cases. Some differences between models with and without intercepts are explore

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay, 1998). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate n/{1/6}, slower than /n, when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data

Efficiency Gains from Quasi-Differencing Under Nonstationarity

A famous theorem on trend removal by OLS regression (usually attributed to Grenander and Rosenblatt, 1957) gave conditions for the asymptotic equivalence of GLS and OLS in deterministic trend extraction. When a time series has trend components that are stochastically nonstationary, this asymptotic equivalence no longer holds. We consider models with integrated and near-integrated error processes where this asymptotic equivalence breaks down. In such models, the advantages of GLS can be achieved through quasi-differencing and we give an asymptotic theory of the relative gains that occur in deterministic trend extraction in such cases. Some differences between models with and without intercepts are explored.

Essays in unit root testing

This thesis is a collection of four essays with main focus on testing for a unit root under structural change, and on the behaviour of power-enhancing unit root tests that have recently emerged as a solution to the well-known power deficiency of traditional such tests. New tests and variants of commonly applied ones are introduced in response to the need for reliable statistical techniques in modelling economic series over time. The first essay explores the possibility that a time series may change structure from trend-stationarity to difference-stationarity, or vice versa as has been recognised by economists for several years. Taking difference-stationarity as the null hypothesis, tests are developed for this possibility, where neither the location nor direction of any possible change under the alternative hypothesis need be specified. Application of these tests to series on consumer price inflation in the G7 countries reveals evidence of a change from trend-stationarity to difference-stationarity in the majority of these countries. In the second essay we apply two elaboration principles of standard unit root tests in the more flexible setting of testing for a unit root against the alternative of stationarity around a smooth transition in linear trend. In comparison to the standard case, the modified tests within this context generate only moderate additional power, a phenomenon which appears to be related to the elaborate nature of the trend function under the alternative. An empirical application of the modified smooth transition tests to common macroeconomic time series in the US economy leads to stronger evidence in favour of the smooth transition alternative than do the unmodified tests. In the third essay we show that more powerful variants of commonly applied unit root tests to panel data, seeking mean or trend reversion, are readily available. Moreover, power gains persist when the modifications are applied to bootstrap procedures that may be employed when cross-correlation of a rather general sort among individual panel members is suspected. That such an approach can strongly influence inference is demonstrated through an application to a panel of real exchange rates against the US dollar. The final essay explores the behaviour of the power-enhancing unit root test most widely applied in the empirical literature. The principle issue is that such a test can have very low power for certain parameter configurations and sample sizes relative to conventional unit root tests. A theoretical attempt is made to identify these unsatisfactory cases relying on local to unity asymptotics, through investigation of the relative efficiencies in the case of an unknown mean. Extensive Monte Carlo results highlight the shortcomings of such a test under higher order autoregressive processes and indicate preference for its existing rivals