Agustin Maravall

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

Abstract We show how our definition of the likelihood of an autoregressive integrated moving average (ARIMA) model with missing observations, alternative to that of Kohn and Ansley 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.

Stochastic linear trends : Models and estimators

Abstract The paper considers stochastic linear trends in series with a higher than annual frequency of observation. Using an approach based on ARIMA models, some of the trend models (or the model interpretation of trend estimation filters) most often found in statistics and econometrics are analysed and compared. The properties of the trend optimal estimator are derived, and the analysis is extended to seasonally adjusted and/or detrended series. It is seen that, under fairly general conditions, the estimator of the unobserved component is noninvertible, and will not accept a convergent autoregressive representation. This has implications concerning unit root testing and VAD model fitting.

Interpolation, outliers and inverse autocorrelations

The paper addresses the problem of estimating missing observations in linear, possibly nonstationary, stochastic processes when the model is known. The general case of any possible distribution of missing observations in the time series is considered, and analytical expressions for the optimal estimators and their associated mean squared errors are obtained. These expressions involve solely the elements of the inverse or dual autocorrelation function of the series. This optimal estimator -the conditional expectation of the missing observations given the available ones-is equal oto the estimator that results from filling the missing values in the series with arbitrary numbers, treating these numbers as additive outliers, and removing the outlier effects from the invented numbers using intervention analysis.

Encompassing Univariate Models in Multivariate Time-Series - A Case-Study

Abstract Through the encompassing principle, univariate ARIMA analysis could provide an important tool for diagnosis of VAR models: The univariate ARIMA models implied by the VAR should explain the results from univariate analysis. This comparison is seldom performed, possibly due to the paradox that, while the implied ARIMA models typically contain a very large number of parameters, univariate analysis yields highly parsimonious models. Using a VAR application to six French macro-economic variables, it is seen how the encompassing check is straight-forward to perform, and surprisingly accurate.

COMPUTING MISSING VALUES IN TIME SERIES

This work presents two algorithms to estimate missing values in time series. The first is the Kalman Filter, as developed by Kohn and Ansley (1986) and others. The second is the additive outlier approach, developed by Pefia, Ljung and Maravall. Both are exact and lead to the same results. However, the first is, in general, faster and the second more flexible.