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 an exponential disturbance term. We also 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 ρ 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.