U.S. Census Bureau

. The problem of initializing the Kalman filter for nonstationary time series models is considered. Ansley and Kohn have developed a ‘modified Kalman filter’ for use with nonstationary models to produce estimates from what they call a ‘transformation approach’. We show that the same results can be obtained with a suitable initialization of the ordinary Kalman filter. Assuming there are d starting values for the nonstationary series, we initialize the Kalman filter using data through time d with the transformation approach estimate of the state vector and its associated error covariance matrix at time d. We give details of the initialization for ARIMA models, ARIMA component models and dynamic linear models. We present an example to illustrate how the results may differ from results obtained under more naive initializations that have been suggested.

[1]  Robert Kohn,et al.  Signal extraction for finite nonstationary time series , 1987 .

[2]  Richard H. Jones,et al.  Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations , 1980 .

[3]  R. Kohn,et al.  Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions , 1985 .

[4]  H. Akaike Covariance Matrix Computation of the State Variable of a Stationary Gaussian Process , 1978 .

[5]  Robert Kohn,et al.  A structured state space approach to computing the likelihood of an ARIMA process and its derivatives , 1985 .

[6]  Ian McLeod Correction: Derivation of the Theoretical Autocovariance Function of Autoregressive-Moving Average Time Series , 1977 .

[7]  Andrew Harvey,et al.  FINITE SAMPLE PREDICTION AND OVERDIFFERENCING , 1981 .

[8]  Piet de Jong,et al.  The likelihood for a state space model , 1988 .

[9]  K. Wallis,et al.  CALCULATING THE VARIANCE OF SEASONALLY ADJUSTED SERIES , 1985 .

[10]  V. Klema LINPACK user's guide , 1980 .

[11]  Andrew Harvey,et al.  Maximum likelihood estimation of regression models with autoregressive-moving average disturbances , 1979 .

[12]  Andrew Harvey,et al.  Estimating Missing Observations in Economic Time Series , 1984 .

[13]  W. Allen Spivey,et al.  A Framework for Time Varying Parameter Regression Modeling , 1985 .

[14]  Genshiro Kitagawa,et al.  A NONSTATIONARY TIME SERIES MODEL AND ITS FITTING BY A RECURSIVE FILTER , 1981 .

[15]  W. Bell,et al.  Signal Extraction for Nonstationary Time Series , 1984 .

[16]  R. Kohn,et al.  Estimation, Prediction, and Interpolation for ARIMA Models with Missing Data , 1986 .