Abstract
This article considers the familiar but very important problem of how to estimate the mean squared error (MSE) of seasonally adjusted and trend estimators produced by X-11-ARIMA or other decomposition methods. The MSE estimators are obtained by defining the unknown target components such as the trend and seasonal effects to be the hypothetical X-11 estimates of them that would be obtained if there were no sampling errors and the series were sufficiently long to allow the use of the symmetric filters embedded in the programme, which are time invariant. This definition of the component series conforms to the classical definition of the target parameters in design-based survey sampling theory, so that users should find it comfortable to adjust to this definition. The performance of the MSE estimators is assessed by a simulation study and by application to real series obtained from an establishment survey carried out by the Bureau of Labor Statistics in the U.S.A.
Keywords
References
1.
Bell, W.R. and M. Kramer. 1999. “Toward Variances for X-11 Seasonal Adjustments.” Survey Methodology 25: 13-29.
2.
Burck,L. andM. Sverchkov. 2001. “AGeneralMethod for Estimating theVariances of X-11- ARIMAEstimators.”FederalCommitteeonStatisticalMethodologyResearchConference 3: 1-11. November 14-16, 2001,Washington DC, U.S.A. Available at: http://fcsm.sites.usa.gov/files/2014/05/2001FCSM_Burck.pdf (accessed October 2014).
3.
Chen, Z.G., P. Wong, M. Morry, and H. Fung. 2003. Variance Estimation for X-11 Seasonal Adjustment Procedure: Spectrum Approach and Comparison. Statistics Canada (Report BSMD-2003-001E).
4.
Findley, D.F. and D.E.K. Martin. 2006. “Frequency Domain Diagnostics of SEATS and X-11/12-ARIMA Seasonal Adjustment Filters for Short and Moderate-Length Time Series.” Journal of Official Statistics 22: 1-34.
5.
Gómez, V. and A Maravall. 1996. Programs TRAMO and SEATS, Introduction for User (Beta Version). Banco de Espanã: Banco de Espanã Working Papers 9628.
6.
Harvey, A.C. 1989. Forecasting Structural Time Series With the Kalman Filter. Cambridge: Cambridge University Press.
7.
Hilmer, S.C. and G.S. Tiao. 1982. “An ARIMA-Model-Based Approach to Seasonal Adjustment.” Journal of the American Statistical Association 77: 63-70. DOI: http://dx.doi.org/10.1080/01621459.1982.10477767
8.
Pfeffermann, D. 1994. “A General Method for Estimating the Variances of X-11 Seasonally Adjusted Estimators.” Journal of Time Series Analysis 15: 85-116. DOI: http://dx.doi.org/10.1111/j.1467-9892.1994.tb00179.x
9.
Pfeffermann, D., M. Morry, and P. Wong. 1995. “Estimation of the Variances of X-11 ARIMA Seasonally Adjusted Estimators for a Multiplicative Decomposition and Heteroscedastic Variances.” International Journal of Forecasting 11: 271-283. DOI: http://dx.doi.org/10.1016/0169-2070(94)00573-U
10.
Pfeffermann, D. and S. Scott. 1997. “Variance Measures for X-11 Seasonally Adjusted Estimators: Some Developments with Application to Labor Force Series.” In Proceedings of the Section on Business & Economic Statistics: American Statistical Association, 211-216. August 10-14, 1997, Anaheim, California, U.S.A.
11.
Pfeffermann, D., M. Feder, and D. Signorelli. 1998. “Estimation of Auto-correlations of Survey Errors with Application to Trend Estimation in Small Areas.” Journal of Business and Economic Statistics 16: 339-348.
12.
Pfeffermann, D., S. Scott, and R. Tiller. 2000. “Comparison of Variance Measures for SeasonallyAdjusted andTrend Series.” InProceedings of the 2nd InternationalConference on Establishment Surveys, 755-764. June 17-21, 2000, Buffalo, New York, U.S.A.
13.
Pfeffermann, D. and R. Tiller. 2005. “Bootstrap Approximation to Prediction MSE for State-Space Models with Estimated Parameters.” Journal of Time Series Analysis 26: 893-916. DOI: http://dx.doi.org/10.1111/j.1467-9892.2005.00448.x
14.
Scott, S., M. Sverchkov, and D. Pfeffermann. 2012. “Estimating Variance in X-11 Seasonal Adjustment.” In Economic Time Series: Modeling and Seasonality, edited by William R. Bell, Scott H. Holan, and Tucker S. McElroy, 185-210. London: Chapman and Hall.
15.
Tiller, R.B. 1992. “Time Series Modeling of Sample Survey Data from the U.S. Current Population Survey.” Journal of Official Statistics 8: 149-166.
16.
Tiller, R.B. 2012. “Frequency Domain Analysis of Seasonal Adjustment Filters Applied to Periodic Labor Force Survey Series.” In Economic Time Series: Modeling and Seasonality, edited by William R. Bell, H. Holan Scott, and Tucker S. McElroy, 135-158. London: Chapman and Hall.
17.
Wecker, W.E. 1979. “A New Approach to Seasonal Adjustment.” In Proceedings of the Section on Business and Economic Statistics: American Statistical Association, 322-323. August 13-16, 1979, Washington DC, U.S.A.
18.
X-13A-S Reference Manual, Version 0.1 (Beta). Time Series Staff, Statistical Research Division, Room 3000-4, U.S. Census Bureau, Washington, DC 20233-9100. Available at: https://www.google.com/?gws_rd=ssl#q=X-13-A-S+Reference+Manual%2c+Version+0.1+(Beta).