Goodness of Fit Tests via Exponential Series Density Estimation

This paper explores the properties of a new nonparametric goodness of fit test, based on the likelihood ratio test of Portnoy (1988). It is applied via the consistent series density estimator of Crain (1974) and Barron and Sheu (1991). The asymptotic properties are established as trivial corollaries to the results of those papers as well as from similar results in Marsh (2000) and Claeskens and Hjort (2004). The paper focuses on the computational and numerical properties. Specifically it is found that the choice of approximating basis is not crucial and that the choice of model dimension, through consistent selection criteria, yields a feasible procedure. Extensive numerical experiments show that the usage of asymptotic critical values is feasible in moderate sample seizes. More importantly the new tests are shown to have significantly more power than established tests such as the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling. Indeed, for certain interesting alternatives the power of the proposed tests may be several times that of the established ones.

A Two-Sample Non-Parametric Likelihood Ratio Test

This paper proposes a test for the hypothesis that two samples have the same distribution. The likelihood ratio test of Portnoy (1988) is applied in the context of the consistent series density estimator of Crain (1974) and Barron and Sheu (1991). It is proven that the test, when suitably standardised, is asymptotically standard normal and consistent against any complementary alternative. In comparison with the established Kolmogorov-Smirnov and Cramer-von Mises procedures the proposed test enjoys broadly comparable finite sample size properties, but vastly superior power properties.

The Available Information for Invariant Tests of a Unit Root

This paper considers the information available to invariant unit root tests at and near the unit root. Since all invariant tests will be functions of the maximal invariant, the Fisher information in this statistic will be the available information. The main finding of the paper is that the available information for all tests invariant to a linear trend is zero at the unit root. This result applies for any sample size, over a variety of distributions and correlation structures and is robust to the inclusion of any other deterministic component. In addition, an explicit bound upon the power of all invariant unit root tests is shown to depend solely upon the information. This bound is illustrated via comparison with the local-to-unity power envelope and a brief simulation study illustrates the impact that the requirements of invariance have on power.

A Measure of Distance for the Unit Root Hypothesis

This paper proposes and analyses a measure of distance for the unit root hypothesis tested against stochastic stationarity. It applies over a family of distributions, for any sample size, for any specification of deterministic components and under additional autocorrelation, here parameterised by a finite order moving-average. The measure is shown to obey a set of inequalities involving the measures of distance of Gibbs and Su (2002) which are also extended to include power. It is also shown to be a convex function of both the degree of a time polynomial regressors and the moving average parameters. Thus it is minimisable with respect to either. Implicitly, therefore, we find that linear trends and innovations having a moving average negative unit root will necessarily make power small. In the context of the Nelson and Plosser (1982) data, the distance is used to measure the impact that specification of the deterministic trend has on our ability to make unit root inferences. For certain series it highlights how imposition of a linear trend can lead to estimated models indistinguishable from unit root processes while freely estimating the degree of the trend yields a model very different in character.

Saddlepoint Approximations for Optimal Unit Root Tests

This paper provides a (saddlepoint) tail probability approximation for the distribution of an optimal unit root test. Under restrictive assumptions, Gaussianity and known covariance structure, the order of error of the approximation is given. More generally, when innovations are a linear process in martingale differences, the estimated saddlepoint is proven to yield valid asymptotic inference. Numerical evidence demonstrates superiority over approximations for a directly comparable test based on simulation of its limiting stochastic representation. In addition, because the saddlepoint offers an explicit representation P-value sensitivity to model specification is easily analyzed, here in the context of the Nelson and Plosser data.

Constructing Optimal Tests on a Lagged Dependent Variable

Via the leading unit root case, the problem of testing on a lagged dependent variable is characterized by a nuisance parameter which is present only under the alternative (see Andrews and Ploberger (1994)). This has proven a barrier to the construction of optimal tests. Moreover, in their absence it is impossible to objectively assess the absolute power properties of existing tests. Indeed, feasible tests based upon the optimality criteria used here are found to have numerically superior power properties to both the original Dickey and Fuller (1981) statistics and the efficient detrended versions suggested by Elliott, Rothenberg and Stock (1996) and analysed in Burridge and Taylor (2000).Nuisance parameter, invariant test, unit root