A vector autoregressive process of order p and dimension k such that the associated autoregressive operator has r < k unit roots and has other roots larger than one in unit value is studied. We assume that there exists a linear transformation of rank k-r, such that k-r components of the transformed vector are stationary, in which case the process is said to be cointegrated of rank k-r. A test for cointegration similar to the likelihood ratio test but based on alternative estimators of the process parameters is considered. The asymptotic distribution of the test statistic is derived and the performance of the test evaluated via Monte Carlo studies. This test procedure provides a definite improvement in power relative to the likelihood ratio test for cointegration;The second problem studied is that of parameter estimation for long memory time series. Long memory time series are those whose covariance functions decay hyperbolically to zero. It is shown that if an observed series is the sum of a long memory Gaussian signal and noise, where the noise is an independent identically distributed zero mean sequence, then the parameter estimates obtained by maximising the Gaussian likelihood are asymptotically normal. In addition, regression models with long memory errors are studied. The ordinary least squares estimators of the regression model parameters with polynomial trends as regressors are shown to be asymptotically normal. A similar result is established for a weighted least squares estimator, which is known to be asymptotically efficient in the case of polynomial trend regressors. The asymptotic distribution of the periodogram of a long memory time series evaluated at a fixed Fourier frequency is also derived. An approximate maximum likelihood estimator is proposed for the parameters of a class of long memory time series and is proved to be asymptotically normally distributed.