We develop an approach to estimating variances for X-11 seasonal adjustments that recognizes the effects of sampling error and revisions (the latter result from errors in forecast extension). We assume that both the true underlying series and the sampling errors follow known time series models. In practice these models are estimated using the time series data and estimates of the variances and lagged covariances of the sampling errors. The model is used to extend the series with forecasts and backcasts, allowing use of the symmetric X-11 filter. In our approach seasonal adjustment error in the central values of a sufficiently long series results only from the effect of the X-11 filtering on the sampling errors (assuming an additive or log-additive decomposition and using a linear approximation to X-11). This agrees with an approach suggested by Wolter and Monsour (1981). However, towards either end of the series, our approach also recognizes the contribution to seasonal adjustment error from forecast and backcast errors. We extend the approach to produce variances of errors in X-11 trend estimates, and to recognize error in estimation of regression coefficients used to model, e.g., calendar effects. We present empirical results for several time series. In these series, the contribution of sampling error often dominated the seasonal adjustment variances. Trend estimate variances, however, showed large increases at the ends of series due to the effects of fore/backcast error. The relative contribution to the variances of error in estimating trading-day or holiday regression coefficients tended to be small, unless the series had no sampling error. Additive outlier and level shift effects were substantial but local. Nonstationarities in the sampling errors produced striking changes in the patterns of seasonal adjustment and trend estimate variances.
