An ARMA representation of unobserved component models under generalized random walk specifications: new algorithms and examples

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Instituto Complutense de Análisis Económico. Universidad Complutense de Madrid
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Among the alternative Unobserved Components formulations within the stochastic state space setting, the Dynamic Harmonic Regression (DHR) has proved particularly useful for adaptive seasonal adjustment signal extraction, forecasting and back-casting of time series. Here, we show first how to obtain ARMA representations for the Dynamic Harmonic Regression (DHR) components under several random walk specifications. Later, we uses these theoretical results to derive an alternative algorithm based on the frequency domain for the identification and estimation of DHR models. The main advantages of this algorithm are linearity, fast computing, avoidance of some numerical issues, and automatic identification of the DHR model. To compare it with other alternatives, empirical applications are provided.
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Akaike, H. (1980). Seasonal Adjustment by a Bayesian Modelling. Journal of Time Series Analysis, 1 , 1–13. Bell, W. (1984). Signal extraction for nonstationary time series. The Annals of Statistics, 12 , 646–664. Box, G., Hillmer, S., & Tiao, G. (1978). Analysis and Modelling of Seasonal Time Series. In A. Zellner (ed.), Seasonal Analysis of Economic Time Series, (pp. 309–334). U.S. Dept. of Commerce — Bureau of the Census. Box, G. E. P. & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control . San Francisco: Holden Day. Bujosa, A., Bujosa, M., & Garc´ıa-Ferrer, A. (2001). A Note on the Pseudospectrum and Pseudo-Covariance Generating Functions of ARMA Processes. Mimeo. Bujosa, M. (2000). Contribuciones al m´etodo de regresi´on arm´onica din´amica: Desarrollos te´oricos y nuevos algoritmos. Ph.D. thesis, Dpto. de An´alisis Econ´omico: Econom´ıa Cuantitativa, Universidad Autónoma de Madrid. Burman, J. (1980). Seasonal Adjustment by Signal Extraction. Journal of the Royal Statistical Society A, 143 , 321–337. Findley, D. F., Monsell, B. C., Bell, W. R., Otto, M. C., & Chen, B. C. (1996). New capabilities and methods of the X-12 ARIMA seasonal adjusment program. US Bureau of the Census. mimeo. García-Ferrer, A. & Bujosa-Brun, M. (2000). Forecasting OECD industrial turning points using unobserved components models with business survey data. International Journal of Forecasting, 16 , 207–227. García-Ferrer, A., del Hoyo, J., & Martín-Arroyo, A. S. (1997). Univariate forecasting comparisons: The case of the Spanish automobile industry. Journal of Forecasting, 16 , 1–17. García-Ferrer, A. & Queralt, R. (1998). Using long-, medium-, and short-term trends to forecast turning points in the business cycle: Some international evidence. Studies in Nonlinear Dynamics and Econometrics, 3 , 79–105. Gomez, V. & Maravall, A. (1996a). New Methods for Quantitative Analysis of Short-Term Economic Activity. In A. Prat (ed.), Proceedings in Computational Statistics, (pp. 65–76). Heidelberg: Physica-Verlag. Gomez, V. & Maravall, A. (1996b). Programs TRAMO and SEATS, instructions for the user (BETA Version: Sept. 1996). Working paper 9628, Bank of Spain, Madrid. Harvey, A. (1989). Forecasting Structural Time Series Models and the Kalman Filter . Cambridge: Cambridge University Press, first edition. Hillmer, S. & Tiao, G. (1982). An Arima-Model Based Approach to Seasonal Adjustment. Journal of the American Statistical Association, 77 , 63–70. Jakeman, A. & Young, P. C. (1984). Recursive filtering and the inversion of ill-posed causal problems. Utilitas Mathematica, 35 , 351–376. Koopmans, S. J., Harvey, A. C., Doornik, J. A., & Shephard, N. (1995). STAMP 5.0: Structural Time Series Analyser, Modeller and Predictor. London: Chapman & Hall. Maravall, A. (1993). Stochastic linear trends, models and estimators. Journal of Econometrics, 56 , 5–37. West, M. & Harrison, J. (1989). Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag. Young, P., Ng, C., & Armitage, P. (1988). A systems approach to recursive economic forecasting and seasonal adjustment. Computers Math. Applic., 18 , 481–501. Young, P. & Pedregal, D. (1999). Recursive and en-bloc approaches to signal extraction. Journal of Applied Statistics, 26 , 103–128. Young, P. C. (1984). Recursive Estimation and Time Series Analysis. Communications and control engieneering series. Berlin: Springer-Verlag, first edition. Young, P. C. (1991). Comments on likelihood and cost as path integrals. Journal of the Royal Statistical Society, Series B, 53 , 529–531. Young, P. C. (1994). Time variable parameters and trend estimation in nonstationary economic time series. Journal of Forecasting, 13 , 179–210. Young, P. C., Pedregal, D., & Tych, W. (1999). Dynamic Harmonic Regression. Journal of Forecasting, 18 , 369–394.