Preface approaches proved a successful competitor to the X-ll variant, partly because revisions of seasonal factors tended to be larger.Tn 1965, we certainly did not expect that the X-ll variant of Method TT would still be playing an important role in seasonal adjustment 35 years later. It also seemed likely that an approach to measuring the standard errors in the seasonal factors in X-ll would be developed. The authors are to be congratulated for an up-to-date review of the X-ll Method. Perhaps this review will not only help further extend the life of X-ll, but will help in the development and widespread use of new methods that bear little resemblance to those of the past.

Hamilton (A standard error for the estimated state vector of a state-space model. J. Economet. 33 (1986), 387±97) and Ansley and Kohn (Prediction mean squared error for state space models with estimated parameters. Biometrika 73 (1986), 467±73) have both proposed corrections to the naive approximation (obtained via substitution of the maximum likelihood estimates for the unknown parameters) of the Bayesian prediction mean squared error (MSE) for state space models, when the model's parameters are estimated from the data. Our work extends theirs in that we propose enhancements by identifying missing terms of the same order as that in their corrections. Because the approximations to the MSE are often subject to a frequentist interpretation, we compare our proposed enhancements with their original versions and with the naive approximation through a simulation study. For simplicity, we use the random walk plus noise model to develop the theory and to get our empirical results in the main body of the text. We also illustrate the differences between the various approximations with the Purse Snatching in Chicago series. Our empirical results show that (i) as expected, the underestimation in the naive approximation decreases as the sample size increases; (ii) the improved Ansley±Kohn approximation is the best compromise considering theoretical exactness, bias, precision and computational requirements, though the original Ansley±Kohn method performs quite well; ®nally, (iii) both the original and the improved Hamilton methods marginally improve the naive approximation. These conclusions also hold true with the Purse Snatching series.

We have a monthly series of observations which are obtained from sample surveys and are therefore subject to survey errors. We also have a series of annual values, called benchmarks, which are either exact or are substantially more accurate than the survey observations; these can be either annual totals or accurate values of the underlying variable at a particular month. The benchmarking problem is the problem of adjusting the monthly series to be consistent with the annual values. We provide two solutions to this problem. The 6rst of these is a two-stage method in which we first fit a state space model to the monthly data alone and then combine the d t s obtained at this stage with the benchmark data. In the second solution we construct a single series from the monthly and annual values together and fit a state space model to this series in a single stage. The treatment is extended to series which behave multiplicatively. The methods are illustrted by applying them to Canadian retail sales series.

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