Estimation for Unit Root Testing

We revisit estimation and computation of the Dickey Fuller (DF) and DF-type tests. Firstly, we show that the usual one step approach, based on the "DF autoregression", is likely to be subject to misspecification. Secondly, we clarify a neglected two step approach for estimation of the DF test. (In fact, we introduce a new two step DF autoregression.) This method is always correctly specified and efficient under the circumstances. However, it is either neglected or misused in unit root testing literature. The commonly employed hybrid of the (correct) two step method is shown to be inefficient, even asymptotically. Finally, we further improve/robustify the proposed two step method by employing the missing initial observations. Our finally proposed method is to be used in unit root testing, since it is a new DF autoregression that retains the missing observations.Comment: 10 page

On Unit Root Testing with Smooth Transitions

Improved critical values are calculated for Dickey-Fuller-type t ratio unit root tests against trend stationarity about non-linear trend, which is based on one deterministic smooth transition function. Simulation employs finegrid-searchoverbothsmoothtransitionparameterstofind accurate staring values, as well as constrained optimization. In addition, two new parsimonious models are introduced. Finally, an application of the test to the log of Real per capita GNP of USA is provided

Generalized least squares transformation and estimation with autoregressive error

Approximations of the usual GLS transformation matrices are proposed for estimation with AR error that remove boundary discontinuities. The proposed method avoids constrained optimization or rules of thumb that unnecessarily enforce estimated parameters to be in the interior.Gaussian AR model GLS transformation NLS and QML estimation LR test

Unit root testing based on BLUS residuals

This paper introduces an efficient version of the Dickey-Fuller unit root test, which is based on BLUS residuals. Simulated critical values are provided, along with power simulation and an empirical example.

Remark on the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root autoregression

This paper considers the asymptotic distribution of the OLS estimator in a simple, Gaussian unit-root AR(1) with fixed, non-zero startup. All asymptotic possibilities are considered. The approach is new, relatively simple, and relies on observing and determining the asymptotic/limiting behavior of the underlying finite sample distribution. It does not rely on inversion of joint moment generating or characteristic functions to derive limiting distributions. The paper introduces small-sigma/parameter-based asymptotic theory and re-examines large-sample asymptotic theory. In addition, combinations of these asymptotic approaches are considered explicitly. The analysis provides a set of very interesting and sometimes surprising results.AR(1) OLS estimator Unit-root process Parameter-based asymptotic theory Small-sigma asymptotic theory Large-sample asymptotic theory

Application of the Dickey-Fuller test to the Nelson and Plosser (1982) Data

In this letter, Nelson and Plosser's (1982) (NP, hereafter) time series has been retested for unit roots using the Dickey and Fuller (1979) (DF, hereafter) test. There have been two potential misspecification sources identified that may have biased NP's inference towards non rejection of the unit root null hypothesis for all but one series in their data set. These are incorrect lag order choice and (conditional) heteroskedasticity. Furthermore, appropriate testing strategy has been undertaken to alleviate these problems and identify which NP time series have unit roots. In contrast to previous classical (non Bayesian) evidence, only a few time series of NP appear to have unit roots.

New exact ML estimation and inference for a Gaussian MA(1) process

For a Gaussian MA(1) process, a new exact ML estimator is proposed that avoids the pile-up phenomenon (boundary estimates). Finite sample comparison is undertaken, along with Wald-type inference for an MA unit root or over-differencing (stationarity).