Specification testing

Journal of EconometricsVolume 143, Issue 1,
March 2008, Pages 1-4 Specification testing
Miguel A. Delgado,
Department of Economics Universidad Carlos III de Madrid Getafe, Spain

A model is said to be statistically adequate, or correctly specified, when its underlying assumptions are supported by the observed data. The concept of statistical adequacy was first put forward by Fisher and by Koopmans (1937) referred to as Fisher's axiom of correct specification. Once a model is selected for performing statistical inferences, the next natural step is to test if the specification of the putative model is correct. If the model is misspecified, the resulting statistical inferences are usually invalid.

Specification testing, or goodness-of-fit testing, is a classical research topic in statistics since the early thirties. The pioneering work of Kolmogorov (1933), Smirnov (1936), Cramér (1928) and von Mises, 1931 R. von Mises, Wahrscheinlichkeitsrechnung, Deuticke, Vienna (1931).von Mises (1931) for testing simple hypotheses on distribution functions were extended in the late fifties by Kac et al. (1955) and Gikhman (1953) to test composite hypothesis, where parameters are estimated. The formalization of these results was provided in the seventies by Durbin (1973) and (Neuhaus, 1973) and (Neuhaus, 1977). Test statistics are suitable functionals of the standard empirical process resulting in omnibus tests, i.e. tests designed to detect general alternatives of nonparametric nature. The limiting null distribution of test statistics with estimated parameters is case dependent, but tests can be implemented with the assistance of a parametric bootstrap. Khmaladze (1981) proposed to use the martingale part of the empirical process with estimated parameters for constructing asymptotically distribution free tests. See D’Agostino and Stephens (1986) for an overview of classical specification tests.

As argued by Durbin and Knott (1972), see also Eubank and LaRicca (1992), classical omnibus specification tests are highly unlikely to detect many alternatives in practice. Recently, Janssen (2000) has shown that any of these tests has a preference for a finite dimensional space of alternatives. Apart from this set, the power function is almost flat on balls of alternatives. Furthermore, there exists no test which pays equal attention to an infinite number of orthogonal alternatives. As a compromise between omnibus and directional tests, smooth tests, introduced by Neyman (1937), are based on the Lagrange-Multiplier testing principle. They assume a flexible parametric model under the alternative hypothesis, usually belonging to the exponential family. If the number of parameters in the model under the alternative hypothesis increases with the sample size at a suitable rate, the smooth tests become omnibus. They are related to tests that compare the estimated parametric model under the null and a nonparametric fit using smoothers, as proposed by Rosenblatt (1975). See Rayner and Best (1989) and Hart (1997) for overviews of Neyman's smooth tests and tests using nonparametric smoothing.

The alternative testing methodologies were conceived to test the specification of distribution functions, but they have been extended to test the specification of other type of models. Specification testing of regression curves have been proposed by Härdle and Mammen (1993) using smoothers and by Stute (1997) using a type of CUSUM process. Many others have studied the properties of both procedures. The same alternative strategies have been applied for goodness-of-fit testing of survival curves, conditional variances, conditional distributions, spectral distributions and general conditional functionals.

There have also been important developments in the econometrics literature. Haavelmo (1944) discusses in his famous Econometrica monograph the crucial importance of a correct specification for performing valid inferences. Among the many recent contributions, it is worth mentioning Zheng (1996) on regression curves using smoothers, Fan and Li (1996) on semi-parametric models using smoothers, Andrews (1997) on conditional distributions using multi-parameter empirical processes, and Horowitz and Spokoiny (2001) on optimal smoothing based testing.
This Annals volume is edited on the occasion of an ‘Explanatory Workshop of the European Science Fundation’ on Specification Testing held at Santander (Spain) in December 2005. The workshop was interdisciplinary, aimed to bring together researchers working in specification testing and those with different applied interests, e.g. economics, finance, physics, medicine and engineering. Twenty-five invited participants from 13 countries presented articles in the conference. Among them, 17 have submitted their work and twelve articles have been accepted after the usual referee process in the Journal of Econometrics.

The papers in this volume provide an upto date perspective on the state of the art in specification testing. The majority of articles deals with omnibus specification testing under the two leading methodologies: empirical processes and nonparametric smoothing.