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Analytic Derivatives for Estimation of Linear Dynamic Models


  • Peter A. Zadrozny


This paper develops two algorithms. Algorithm I computes the exact, Gaussian, log-likelihood function, its exact, gradient vector, and an asymptotic approximation of its Hessian matrix, for discrete-time, linear, dynamic models in state-space form. Algorithm 2, derived from algorithm I, computes the exact, sample, information matrix of this likelihood function. The computed quantities are analytic (not numerical approximations) and should, therefore, be useful for reliably, quickly, and accurately: (i) checking local identifiability of parameters by checking the rank of the information matrix; (ii) using the gradient vector and Hessian matrix to compute maximum likelihood estimates of parameters with Newton methods; and, (iii) computing asymptotic covariances (Cramer-Rao bounds) of the parameter estimates with the Hessian or the information matrix. The principal contribution of the paper is algorithm 2, which extends to multivariate models the univariate results of Porat and Friedlander (1986). By relying on the Kalman filter instead of the Levinson-Durbin filter used by Porat and Friedlander, algorithms 1 and 2 can automatically handle any pattern of missing or linearly aggregated data. Although algorithm 1 is well known, it is treated in detail in order to make the paper self contained.

Suggested Citation

  • Peter A. Zadrozny, 1988. "Analytic Derivatives for Estimation of Linear Dynamic Models," Working Papers 88-5, Center for Economic Studies, U.S. Census Bureau.
  • Handle: RePEc:cen:wpaper:88-5

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    References listed on IDEAS

    1. Sveikauskas, Leo A, 1983. "Science and Technology in United States Foreign Trade," Economic Journal, Royal Economic Society, vol. 93(371), pages 542-554, September.
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    Cited by:

    1. André Klein & Guy Melard & Toufik Zahaf, 1998. "Computation of the exact information matrix of Gaussian dynamic regression time series models," ULB Institutional Repository 2013/13738, ULB -- Universite Libre de Bruxelles.
    2. André Klein & Guy Melard & Abdessamad Saidi, 2008. "The asymptotic and exact Fisher information matrices," ULB Institutional Repository 2013/13766, ULB -- Universite Libre de Bruxelles.
    3. Peter A. Zadrozny, 1990. "Estimating A Multivariate Arma Model with Mixed-Frequency Data: An Application to Forecasting U.S. GNP at Monthly Intervals," Working Papers 90-5, Center for Economic Studies, U.S. Census Bureau.
    4. André Klein & Guy Melard, 2004. "An algorithm for computing the asymptotic Fisher information matrix for seasonal SISO models," ULB Institutional Repository 2013/13746, ULB -- Universite Libre de Bruxelles.
    5. Iskrev, Nikolay, 2008. "Evaluating the information matrix in linearized DSGE models," Economics Letters, Elsevier, vol. 99(3), pages 607-610, June.
    6. Nikolay Iskrev, 2013. "On the distribution of information in the moment structure of DSGE models," 2013 Meeting Papers 339, Society for Economic Dynamics.


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