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The State Space Model

  • Víctor Gómez
Chapter
Part of the Statistics and Computing book series (SCO)

Abstract

The state space model considered in SSMMATLAB is
$$\displaystyle \begin {array}{rcl} \alpha _{t+1} &=& W_t\beta + T_t\alpha _t + H_t\epsilon _t, \\ Y_t &=& X_t\beta + Z_t\alpha _t + G_t\epsilon _t, \qquad t=1,\ldots ,n, \end {array}$$
where {Yt} is a multivariate process with \(Y_{t}\in \mathbb {R}^{p}\), Wt, Tt, Ht, Xt, Zt, and Gt are time-varying deterministic matrices, \(\beta \in \mathbb {R}^{q}\) is a constant bias vector, \(\alpha _{t}\in \mathbb {R}^{r}\) is the state vector, and {𝜖t} is a sequence of uncorrelated stochastic vectors, \(\epsilon _{t}\in \mathbb {R}^{s}\), with zero mean and common covariance matrix σ2I.

References

  1. Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (revised ed.). San Francisco: Holden-Day.Google Scholar
  2. de Jong, P. (1991). The diffuse Kalman filter. The Annals of Statistics,19, 1073–1083.MathSciNetCrossRefGoogle Scholar
  3. Gómez, V. (2016). Multivariate time series models with linear state space structure. New York: Springer.zbMATHGoogle Scholar
  4. Harvey, A. C. (1989). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.Google Scholar
  5. Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. The Quarterly of Applied Mathematics,2, 164–168.MathSciNetCrossRefGoogle Scholar
  6. Marquardt, D. (1963). An algorithm for least–squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics,11, 431–441.MathSciNetCrossRefGoogle Scholar
  7. Reinsel, G. C.(1997). Elements of multivariate time series analysis. New York: Springer.CrossRefGoogle Scholar
  8. Tsay, R. S. (2014). Multivariate time series analysis with R and financial applications. New York: Wiley.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Víctor Gómez
    • 1
  1. 1.General Directorate of BudgetsMinistry of Finance and Public AdministrationsMadridSpain

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