Maximum-Likelihood State-Space Estimates

The estimation of the state-space parameters can be challenging given the complexities of the resulting equations. Consider the case where all the parameters, 9= Zq, a, B, C, T , Zg), are to be estimated. It will be assumed that the initial state xo is normally distributed with zero mean and a covariance matrix Zq, and the errors (a and e) are jointly normally distributed and uncorrelated. Assume that the time series contains m data points and that there are a total of n unknown parameters. 

In order to compute the maximum-likelihood estimates, compute the one-step-ahead prediction error, e, using Eq. (5.135) and the corresponding covariance matrix, Z, using Eq. (5.136). Ignoring constant terms, the log-likelihood function, 7(0), can be written as 

Solving this equation requires using various numerical methods. Irrespective of the approach taken, the procedure can be summarised as follows  

Using the basic Kalman filter as given in Theorem 5.5 and the initial parameter estimate, 6q, obtain the innovations and error covariances for t=l,..m. 

Perform one iteration of Newton s method with as the objective 

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See e.g. Hall and Nicholls (1980) for a direct maximum likelihood estimation or Metaxoglou and Smith (2007) for estimation based on a transformation into state-space models. 

Metaxoglou, K. and Smith, A. Maximum likelihood estimation of vanna models using a state-space em algorithm. Journal of Time Series Analysis, 28(5) 666- 5, 2007. 

State-space parameter estimates obtained using the maximum-likelihood approach are not asymptotically consistent. 

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