Estimation for autoregressive processes

A univariate autoregressive process of order p with deterministic mean function and a root close to or equal to one in absolute value is studied. We assume that if the absolute value of the largest root is one then the root is a real root. The approximate bias of the sum of the autoregressive coefficients is expressed as a function of the test for a unit root. The expression is used to modify the ordinary least squares estimator of the coefficients of the autoregression to obtain an estimator with finite sample bias that is of smaller order than that of the ordinary least squares estimator. The performance of the proposed estimator is evaluated through Monte Carlo simulation. Monte Carlo results are given for autoregressive processes with constant mean and autoregressive processes with a linear trend.;The second problem studied is that of estimation and inference for the trend coefficient in a linear trend model with autoregressive errors. Different feasible generalized least squares estimators of the trend coefficient are compared via simulation. A test statistic for the hypothesis that the trend coefficient is zero is suggested. The limiting properties of the test statistic are derived and limiting results provide theoretical justifications for the proposed test statistic. The finite sample behavior of the suggested test statistic is compared to that of some existing test statistics via Monte Carlo simulation. The test statistic has finite sample properties superior to those of existing feasible generalized least squares test statistics.;The third problem investigated is maximum likelihood estimation in a Gaussian vector autoregression. The likelihood equations are nonlinear in parameters and have to be solved numerically. The numerical solution of the likelihood equations are complicated and numerically unstable, especially when the autoregressive process has roots close to or equal to one. A onestep approximation to the solution of the likelihood equations is suggested. The onestep estimator of the coefficient matrices of a vector autoregressive process is easy to compute and numerically stable. The limiting distribution of the onestep, estimator is derived for processes with some unit roots. The finite sample properties of the onestep estimator are compared with those of the ordinary least squares estimator of the coefficient matrices via Monte Carlo simulation.