Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors (Q806872)

Consider a linear regression model with first-order autoregressive normal disturbances. Let \(\rho\) denote the autocorrelation coefficient of the process. The paper constructs optimal invariant tests of the null hypothesis \(H_ 0: \rho =\rho_ 0\) against the alternatives \(H_ a^+: \rho >\rho_ 0\) and \(H_ a^-: \rho <\rho_ 0\), where \(\rho_ 0\) is any admissible value of \(\rho\). Both locally best invariant and point-optimal invariant tests are derived for stationary as well as nonstationary processes. Numerical comparisons of the powers of alternative tests are provided. The point-optimal tests are found to be usually preferable to locally best tests, especially when values of \(\rho\) equal to or greater than one are tested.