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Exact Tests and Confidence sets in Linear Regressions with Autocorrelated Errors

Jean-Marie Dufour

Abstract

This paper proposes a general method to build exact tests and confidence sets in linear regressions with first-order autoregressive Gaussian disturbances. Because of a nuisance parameter problem, we argue that generalized bounds tests and conservative confidence sets provide natural inference procedures in such a context. Given an exact confidence set for the autocorrelation coefficient, we describe how to obtain a similar simultaneous confidence set for the autocorrelation coefficient and any subvector of regression coefficients. Conservative confidence sets for the regression coefficients are then deduced by a projection method. For any hypothesis that specifies jointly the value of the autocorrelation coefficient and any set of linear restrictions on the regression coefficients, we get exact similar tests. For testing linear hypotheses about the regression coefficients only, we suggest bounds-type procedures. Exact confidence sets for the autocorrelation coefficient are built by "inverting" autocorrelation tests. The method is illustrated with two examples.