Summary and Directions for Further Research
The monograph accomplishes the following. First, for estimation, seven nonlinear filters are introduced and developed. Three out of the seven filters, the Monte-Carlo simulation filter (Section 3.2), the modified Kitagawa estimator (Section 4.3) and the simulation-based density estimator (Section 4.4), were new proposals. Second, I analyzed what has to be approximated when applying the nonlinear measurement and transition equations approximated by the Taylor series expansion to the conventional linear recursive Kaiman filter algorithm (Section 3.2). Third, it was shown in Section 3.3 that under a certain functional form of either the measurement equation or the transition equation there is no correlation between the error terms (or residuals). Recall that one of the approximations in expanding the measurement and transition equations is assuming that the correlated error terms (residuals) are uncorrelated. Fourth, a re-interpretation was given to the single-stage iteration filter (Section 3.4). Fifth, comparing seven nonlinear filters by the Monte-Carlo experiments, it was shown in Chapter 5 that the modified Kitagawa estimator (Section 4.3) and the simulation-based density estimator (Section 4.4) are better than the other estimators while the extended Kaiman filter (Section 3.2) and the Gaussian sum filter (Section 4.2) are clearly biased. Finally, an estimation of permanent and transitory consumption was taken as an application to the nonlinear filters (Chapter 6).
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