Measures of Variability for Model-Based Seasonal Adjustment Procedures

Abstract

An algorithm is derived that develops measures of variability for the estimates of the nonseasonal component computed from a model-based seasonal adjustment procedure. The measures of variability are developed from signal extraction theory. Properties of components of the variance are developed, and the behavior of the variance is investigated for one popular time series model. The results are illustrated by using real data.

Estimation error and the specification of unobserved component models

“…The third example consists of a class of models that are often found to approx imate reasonably well the stochastic properties of many series: the so-called Airline Model of Box and Jenkins (1970, chapter 9). This example extends the results in Hillmer (1985), and presents some stylized facts often found in actual time series.…”

“…In this paper, we analyse some of these properties, mostly in connection with the components estimation error. Burridge and Wallis (1985) within the Stsm ap proach, and Hillmer (1985) within the A bm approach, have provided algorithms for computing the variance of the components estimation error. In this paper, an alternative approach, close to the one in Watson (1987), is followed, which permits us to obtain relatively simple analytical expressions for the variances of the components estimation error for different admissible decompositions.…”

Computation of asymmetric signal extraction filters and mean squared error for ARIMA component models

“…Comparing with shows that η ( F ) can be obtained by computing the symmetric filter , multiplying it by ψ ( B ), and taking the terms involving powers of F . Pierce (1980, p. 99 and p. 104) and Hillmer (1985, p. 62) do just this and obtain expressions analogous to some of . Pierce obtains results on the MSE of revisions in signal extraction estimates for general difference stationary models, including non‐optimal signal extraction estimates.…”

“…An important consequence of the derivations of Section 5 is a direct derivation showing, in the nonstationary case, the optimality of the asymmetric signal extraction results computed using the filters given here. Section 6 provides an example illustrating the results of the previous sections, showing how to apply the algorithm to calculate the filter weights and MSE from a model for ARIMA model-based seasonal adjustment as in Burman (1980) andHillmer andTiao (1982).Some results of Sections 4 and 5 overlap with results of Pierce (1979, 1980), and these instances of overlap will be noted. Pierce obtained results for the general difference stationary case where 'differenced' S t and N t are assumed to be stationary time series although not necessarily following ARMA models.…”

“…An important consequence of the derivations of Section 5 is a direct derivation showing, in the nonstationary case, the optimality of the asymmetric signal extraction results computed using the filters given here. Section 6 provides an example illustrating the results of the previous sections, showing how to apply the algorithm to calculate the filter weights and MSE from a model for ARIMA model-based seasonal adjustment as in Burman (1980) andHillmer andTiao (1982).…”