Inference in a nearly integrated autoregressive model with nonnormal innovations

A comparison of minimum MSE and maximum power for the nearly integrated non-Gaussian model

“…Typical examples of such tests include the Studentized t statistic and the normalized autocorrelation. By contrast, Rothenberg and Stock (1997) show that under non-normality the same tests clearly display suboptimal power against a broad range of point alternatives, if the true likelihood is used to construct the tests. Therefore, there appears to be room for further improvement of the usual testing procedures when the true likelihood is non-Gaussian.…”

“…As explained in the introduction, we work in a local-to-unity framework. In particular, following Chan and Wei (1987) and Cox and Llatas (1991), we assume that c = n ( 1) is constant in the sample size n. T aking a locally quadratic approximation to the log-quasi-likelihood, see Rothenberg and Stock (1997), we obtain log Q(c; c 0 ; q) = n (ii) The density f( ) is twice continuously di erentiable and vanishes on the edge of its support. (iii) The log-quasi-density q( ) is twice continuously di erentiable and q 00 ( ) satis es a linear Lipschitz condition.…”

“…In particular, we will study the impact of di erent optimality criteria on the choice of q. Therefore, for simplicity w e follow Rothenberg and Stock (1997) and abstain from estimating the variance 2 " of the innovations. Note that consistent estimators of " are asymptotically ancillary for , so that the choice of any consistent method of estimating " does not a ect our subsequent derivations.…”

“…Using Assumption 1, the following theorem can be derived from Rothenberg and Stock (1997) for the limit of the quasi-likelihood ratio of testing c = c 0 against the xed alternative c = c 1 . …”

“…The upper bound for the power of any unit root test in this context, i.e., the power envelope, follows from the Neyman-Pearson lemma and is given by P[QLR(c 1 ; 0; c 1 ; f ) < ], where f is the unknown true density of the innovations " i , and P[QLR(0; 0; c 1 ; f ) < ] is equal to some prespeci ed signi cance level, see also Rothenberg and Stock (1997). Using the power envelope, we can evaluate the power performance of commonly employed unit root tests like the Studentized t statistic t q =^q 1 q ;…”