In this paper, we consider a linear regression model with Gaussian autoregressive errors of order p = 2, which may be nonstationary. Exact inference methods (tests and confidence region) are developed for the autoregressive parameters and the regression coefficients. We generalize the method proposed in Dufour (1990) for linear regression models with autoregressive errors of order p = 1: The proposed approach consists in three stages. First, we build an exact confidence set for the complete vector of the autoregressive coefficients (varphi). This region is obtained by inverting independence tests for model errors after the model has been transformed to get independent errors under the null hypothesis. The independence tests are based on combining tests for the presence of autocorrelation at lags one and two. Exploiting the duality between tests and confidence sets, an exact confidence set is then built by finding the set of autoregressive parameter values which are not rejected (test inversion). Second, using this confidence set for (varphi), simultaneous confidence sets for the autoregressive parameters and regression coefficients are obtained. Finally, marginal confidence intervals for the regression coefficients are derived using a projection approach. We also propose generalized bounds tests for the regression parameters. These methods are applied to time series models of the U.S. money stock (M2) and GNP deflator.

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