| The purpose of this dissertation is to develop nonlinear filters and demonstrate their applications. There are two approaches to nonlinear filters. One is approximating nonlinear measurement and transition equations by the Taylor series expansion. The approximated nonlinear functions are applied directly to the standard linear recursive Kalman filter algorithm. Another approach is approximating the underlying density functions of the state vector.; For the nonlinear filters based on the Taylor series expansion, first, it is shown that we need to impose some approximations on the disturbances. Next, I propose a nonlinear filter which combines the extended Kalman filter with Monte-Carlo stochastic simulation, where each expectation in the algorithm is evaluated by generating random numbers. Also, for the single-stage iteration filter, a reinterpretation is given, which is different from the conventional one.; It is, however, known that applying the linearized nonlinear measurement and transition equations to the conventional linear algorithm leads to biased filtering estimates. Therefore, it is essential to approximate the underlying conditional density functions rather than the nonlinear measurement and transition equations. A small extension is given to Kitagawa's density approximation by numerical integration, where the nodes are taken as random numbers. Furthermore, the simulation-based density estimator is proposed, where the importance sampling theory developed by Geweke is applied to the nonlinear filtering problem.; Monte-Carlo experiments are performed to examine the nonlinear filters. We find that the nonlinear filters based on the density approximation are better estimators than those based on the Taylor series expansion by the criteria of BIAS, RMSE and MAPE.; Finally, as applications of the nonlinear filters, two examples are taken in this thesis. One is predicting final data based on preliminary data, where we see that the precision of prediction might be improved using the nonlinear filtering techniques. Another example is estimating permanent consumption and transitory consumption separately, where the findings are as follows; (i) according to the likelihood ratio test, the hypothesis of no transitory consumption is rejected, and (ii) the transitory consumption is large just after the Second World War but very small in the last decade. |
