Estimation, prediction, and interpolation for nonstationary series with the Kalman filter. (English) Zbl 0806.62076

Gómez, Víctor; Maravall, Agustín

J. Am. Stat. Assoc. 89, No. 426, 611-624 (1994).

Summary: We show how our definition of the likelihood of an autoregressive integrated moving average (ARIMA) model with missing observations, alternative to that of R. Kohn and C. F. Ansley [ibid. 81, 751-761 (1986; Zbl 0607.62106)] and based on the usual assumptions made in estimation of and forecasting with ARIMA models, permits a direct and standard state-space representation of the nonstationary (original) data, so that the ordinary Kalman filter and fixed point smoother can be efficiently used for estimation, forecasting, and interpolation. In this way, the problem of estimating missing values in nonstationary series is considerably simplified. The results are extended to regression models with ARIMA errors, and a computer program is available from the authors.

MSC:

62M20	Inference from stochastic processes and prediction
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

likelihood function; time series; likelihood of an autoregressive integrated moving average model; missing observations; estimation; forecasting; ARIMA models; state-space representation; Kalman filter; fixed point smoother; interpolation; nonstationary series; ARIMA errors

Software:

AS 197