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Keywords:

  • State space models;
  • stationary;
  • non-stationary;
  • Kalman filter

Abstract. The concepts of time invariance, stationarity, non-stationarity and immemorial time are considered for state space models (SSMs). Necessary and sufficient conditions for stationarity are given and connected with standard conditions. Expressions are given for the unconditional mean and covariance matrix of the state of a time-immemorial SSM. Application of the results is made to a variety of theoretical and empirical models and the initialization of the Kalman filter in the non-stationary case and for the ARIMA (p, d, q) model.