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Minimally Conditioned Likelihood for a Nonstationary State Space Model

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Abstract

Computing the gaussian likelihood for a nonstationary state-space model is a difficult problem which has been tackled by the literature using two main strategies: data transformation and diffuse likelihood. The data transformation approach is cumbersome, as it requires nonstandard filtering. On the other hand, in some nontrivial cases the diffuse likelihood value depends on the scale of the diffuse states, so one can obtain different likelihood values corresponding to different observationally equivalent models. In this paper we discuss the properties of the minimally-conditioned likelihood function, as well as two efficient methods to compute its terms with computational advantages for specific models. Three convenient features of the minimally-conditioned likelihood are: (a) it can be computed with standard Kalman filters, (b) it is scale-free, and (c) its values are coherent with those resulting from differencing, being this the most popular approach to deal with nonstationary data.

Suggested Citation

  • José Casals & Sonia Sotoca & Miguel Jerez, 2012. "Minimally Conditioned Likelihood for a Nonstationary State Space Model," Documentos de Trabajo del ICAE 2012-04, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    • Handle: RePEc:ucm:doicae:1204
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    References listed on IDEAS

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    1. Craig F. Ansley & Robert Kohn, 1990. "Filtering And Smoothing In State Space Models With Partially Diffuse Initial Conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 11(4), pages 275-293, July.
    2. Marc K. Francke & Siem Jan Koopman & Aart F. De Vos, 2010. "Likelihood functions for state space models with diffuse initial conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(6), pages 407-414, November.
    3. William Bell & Steven Hillmer, 1991. "Initializing The Kalman Filter For Nonstationary Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 12(4), pages 283-300, July.
    4. Casals, Jose & Jerez, Miguel & Sotoca, Sonia, 2000. "Exact smoothing for stationary and non-stationary time series," International Journal of Forecasting, Elsevier, vol. 16(1), pages 59-69.
    5. Victor Gómez & Agustin Maravall & Daniel Peña, 1999. "Missing observations in ARIMA models: Skipping strategy versus outlier approach," Working Papers 9701, Banco de España.
    6. Mauricio, Jose Alberto, 2006. "Exact maximum likelihood estimation of partially nonstationary vector ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 50(12), pages 3644-3662, August.
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    More about this item

    Keywords

    State-space models; Conditional likelihood; Diffuse likelihood; Diffuse initial conditions; Kalman filter; Nonstationarity.;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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