Likelihood Analysis of a First Order Autoregressive Model With Exponential Innovations

This paper derives the exact distribution of the maximum likelihood estimator of a first order linear autoregression with exponential innovations. We show that even if the process is stationary, the estimator is $T$-consistent, where $T$ is the sample size. In the unit root case the estimator is $T^{2}$-consistent, while in the explosive case the estimator is $\rho ^{T}$-consistent. Further, the likelihood ratio test statistic for a simple hypothesis on the autoregressive parameter is asymptotically uniform for all values of the parameter