Estimation, Prediction, and Interpolation for Nonstationary Series with the Kalman Filter

Abstract

The problem of estimating any sequence of missing observations in series with a nonstationary ARIMA model representation was solved by Kohn and Ansley (1986). In their approach, the likelihood is defined first by means of a transformation of the data; then, in order to obtain an efficient estimation procedure, a modified Kalman filter and a modified fixed point smoothing algorithm are used. In this paper we show how an alternative definition of the likelihood, 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. Our approach, like that of Kohn and Ansley (1986), can handle any arbitrary pattern of missing data and we show that the same results are obtained with both approaches. In this way, the problem of estimating missing values in nonstationary series is considerably simplified. When the available observations do not permit estimation of some of the missing values, the method indicates which are these values, and the forecasts that might be affected. Moreover, if linear combinations of the unestimable missing observations are estimable, the estimates are readily obtained. The method is illustrated using the same examples of Kohn and Ansley (1986), and an additional one for the case of unestimable missing values with estimable linear combinations thereof. It is shown that our likelihood is equal to that of Kohn and Ansley (1986); it also coincides with that of Harvey and Pierse (1984) when applicable, and to that of Box and Jenkins (1976) when no observation is missing. The results are extended to regression models with ARIMA errors, and a computer program, written in Fortran for MSDOS computers, is available from the authors.

Regression analysis with missing data and unknown colored noise: Application to the MICROSCOPE space mission

“…Following the notation of Gómez and Maravall [20], an AR process can be described by the state-space representation:…”

“…The second step is carried out via a Kalman filter algorithm based on the AR model that allows us to compute the weights, as shown by Jones [18]. In the third step we finally compute an approximation of the Generalized Least Squares (GLS) estimator of the regression parameters, in a way similar to maximum likelihood computation methods applied to regression models [19,20]. These steps can be reproduced to converge to the maximum likelihood estimator (MLE) of the parameters.…”

Monthly US business cycle indicators: a new multivariate approach based on a band-pass filter

“…As for p t in eq. (11), we select the multivariate version of the state space representation used by Gómez and Maravall (1994), which is an extension to the nonstationary case of the approach proposed by Akaike (1974). Thus, the state space representation of (11) is…”

Minimally conditioned likelihood for a nonstationary state space model

“…This requirement is not fulfilled, for example, when one wants to estimate structural time series models (Harvey, 1989). The AK approach has been further developed in many relevant works (Kohn and Ansley, 1986;Ansley and Kohn, 1990, in the univariate case; Bell and Hillmer, 1991;Gomez and Maravall 1994), but none of them addressed the two preciously mentioned issues.…”