The sample variance is commonly used to estimate the variance of stationary time series. When the second-order statistics of the process are known up to a scaling factor, this estimator is generally inefficient. In the case of an autoregressive (AR) process with unknown parameters, the sample variance is shown to be asymptotically efficient. However, the sample variance of a moving-average (MA) process with unknown parameters is generally an inefficient estimator. Closed-form expressions are derived for the Cramer-Rao hound associated with the variance estimation problem and for the variance of the sample-variance estimator, for both AR and MA processes.