In this paper, the pseudo-likelihood estimators and the limiting distribution of the score test statistic associated with several hypothesis tests such as unit root test for the linear regression models with stationary and nonstationary residuals are calculated. The limiting behavior of theses test statistics by using a simpler approach of the original presentation is derived. Also by using a Mont Carlo method, it is shown that the derived pseudo-likelihood estimators are appropriate. The quantiles of the limiting distribution of the test statistic for a unit root are also calculated and a new table is provided which can be used by researchers for the unit root test.

In this paper, a modified two-stage procedure in the Autoregressive model  AR(1) is considered, which investigates the point and the interval estimation of the mean based on the least-squares estimator. The modified two-stage procedure is as effective as the best fixed-sample size procedure. In this regard, the significant properties of the procedure, including asymptotic risk efficiency, first-order efficiency, consistent, and asymptotic distribution of the mean, are established. Then, a Monte Carlo simulation study is deduced to investigate the modified two-stage procedure. The performance of estimators and confidence intervals are evaluated utilizing a simulation study. Finally, real-time series data is considered to illustrate the applicability of the modified two-stage procedure.