Missing observations in ARIMA models: Skipping approach versus additive outlier approach. (English) Zbl 1054.62621

Gómez, Víctor; Maravall, Agustín; Peña, Daniel

J. Econom. 88, No. 2, 341-363 (1999).

Summary: Optimal estimation of missing values in ARMA models is typically performed by using the Kalman filter for likelihood evaluation, ‘skipping’ in the computations the missing observations, obtaining the maximum likelihood (ML) estimators of the model parameters, and using some smoothing algorithm. The same type of procedure has been extended to nonstationary ARIMA models in Gómez and Maravall (1994). An alternative procedure suggests filling in the holes in the series with arbitrary values and then performing ML estimation of the ARIMA model with additive outliers (AO). When the model parameters are not known the two methods differ, since the AO likelihood is affected by the arbitrary values. We develop the proper likelihood for the AO approach in the general nonstationary case and show the equivalence of this and the skipping method. Finally, the two methods are compared through simulation, and their relative advantages assessed; the comparison also includes the AO method with the uncorrected likelihood.

MSC:

62P20

Applications of statistics to economics

Keywords:

times series; outliers; nonstationarity; likelihood; Kalman filter

Software:

AS 197; TRAMO

References:

[1]	Anderson, B.D.A., Moore, J.B., 1979. Optimal Filtering. Prentice-Hall, Englewood cliffs, NJ. · Zbl 0688.93058
[2]	Ansley, C. F.: An algorithm for the exact likelihood of a mixed autoregressive–moving average process. Biometrika 66, 59-65 (1979) · Zbl 0411.62059
[3]	Bell, W.: Signal extraction for nonstationary series. The annals of statistics 12, 646-664 (1984) · Zbl 0542.62081
[4]	Brockwell, P.; Davis, R.: Time series: theory and methods. (1987) · Zbl 0604.62083
[5]	Box, G.E.P., Jenkins, G.M., 1976. Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco. · Zbl 0363.62069
[6]	Brubacher, S. R.; Wilson, G. T.: Interpolating time series with application to the estimation of holiday effects on electricity demand. Applied statistics 25, No. 2, 107-116 (1976)
[7]	De Jong, Piet, 1991. The diffuse kalman filter. Annals of statistics 19, 1073–1083. · Zbl 0742.62093
[8]	EUROSTAT, 1996. Is the ARIMA Model Adapted to Economic Time Series? Informal Group on Seasonal Adjustment, EUROSTAT /D3/SA/04.
[9]	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 adjustment program. Journal of Business and Economic Statistics, forthcoming.
[10]	Gómez, V.; Maravall, A.: Estimation, prediction and interpolation for nonstationary series with the Kalman filter. Journal of the American statistical association 89, 611-624 (1994) · Zbl 0806.62076
[11]	Gómez, V., Maravall, A., 1996. Programs TRAMO and SEATS, Instructions for the User, Beta Version: September 1996. Working paper No. 9628, Bank of Spain.
[12]	Gómez, V., Maravall, A., Peña, D., 1997. Missing Observations in ARIMA Models. Working paper no. 9701, Bank of Spain.
[13]	Harvey, A. C.: Forecasting structural time series models and the Kalman filter. (1989)
[14]	Jones, R.: Maximum likelihood Fitting of ARMA models to time series with missing observations. Technometrics 22, 389-395 (1980) · Zbl 0451.62069
[15]	Kohn, R.; Ansley, C. F.: Efficient estimation and prediction in time series regression models. Biometrika 72, 694-697 (1985)
[16]	Kohn, R.; Ansley, C. F.: Estimation, prediction and interpolation for ARIMA models with missing data. Journal of the American statistical association 81, 751-761 (1986) · Zbl 0607.62106
[17]	Ljung, G. M.: A note on the estimation of missing values in time series, communications in statistics. Simulation and computation 18, No. 2, 459-465 (1989) · Zbl 0695.62213
[18]	Maravall, A., Peña, D., 1996. Missing observations and additive outliers in time series models. Working paper no. 9612, Bank of Spain. In: Mariano, R. (Ed.), Advances in Statistical Analysis and Statistical Computing, vol. 3, JAI Press, Greenwich, forthcoming.
[19]	Mélard, G.: A fast algorithm for the exact likelihood of autoregressive-moving average models. Applied statistics 35, 104-114 (1984) · Zbl 0548.65098
[20]	Morf, M., Sidhu, G.S., Kailath, T., 1974. Some new algorithms for recursive estimation on constant, linear, discrete-time systems. IEEE Transactions on Automatic Control AC-19, 315–323. · Zbl 0279.93043
[21]	Peña, D., 1987. Measuring the importance of outliers in ARIMA models. In: Puri et al. M.L. (Eds.), New Perspectives in Theoretical and Applied Statistics. Wiley, New York, pp. 109–118. · Zbl 0616.62124
[22]	Pearlman, J. G.: An algorithm for the exact likelihood of a high - order – autoregressive – moving average process. Biometrika 67, 232-233 (1980) · Zbl 0425.62078
[23]	Sargan, J. D.; Drettakis, E. G.: Missing data in an autoregressive model. International economic review 15, 39-58 (1974) · Zbl 0281.62100