This is the first in a series of seminars held in honour of Jim Durbin (CeMMAP Fellow, Honorary Professor UCL, Emeritus Professor LSE). These seminars are organised by CeMMAP, UCL and LSE.
Abstract: According to a result by Bahadur and Savage (1956), any valid test procedure about the mean of an iid sequence of real random variables is everywhere biased when the distribution is not specified. We first recall Bahadur and Savage (1956), initial result and propose some extensions in more general settings. The distributions may be multivariate. The iid-ness assumption is not required. In the parametric case, the parameter of interest needs not coincide with the expectation and may be multidimensional. Next, we propose to put these results in perspective of “testability” problems. We show that no meaningful test procedure exists for the covariance or the linear correlation matrix parameters and we propose to refer to this property as non testability. Variance and support of a random variables suffers from a similar, albeit weaker, problem called partial non testability. Finally we show that when non testability (either partial or not) prevails, asymptotic consistent test procedures necessarily have level 1 in the limit, regardless of their claimed level. This property is true for any consistent procedure even when “corrections” (for instance bootstrapping) are made to reduce the discrepancies between the actual and the claimed level. Non testability problems are then investigated from an econometric viewpoint, i.e. when a distinction between exogenous and endogenous variables is made. We propose a valid and consistent test for the slope coefficients of a linear regression model when the error terms are iid. However, we also show that the slope parameters are typically non testable when the assumption of i.i.d. observations is relaxed (for instance if the error terms are assumed independent and homoskedastic). We show how to extend these results to non linear regression models. We also investigate the case of the non parametric regression model. In this case, neither the regression function nor the hypothesis of significance of a given regressor are testable. Finally, we address the more general question of testability in models based on moment conditions.