Cuốn sách này cung cấp một tổng hợp của các khái niệm và các vật liệu thông thường xuất hiện một cách riêng biệt trong chuỗi thời gian và văn học kinh tế, trình bày một đánh giá toàn diện của cả hai khái niệm lý thuyết và ứng dụng. Có lẽ tính năng mới nhất của cuốn sách là của Kalman lọc cùng với hàng loạt phương pháp kinh tế lượng và thời gian. | Interest Rate Model and Non-Linear Filtering 101 which are measures of the statistical properties of the noise process. The fact that these quantities satisfy the finite-dimensional stochastic dynamical system dZ0 t -I 2 t d t dZ t Zl_1 t -l z t gdt i 1 is the crucial observation which allows US to reduce the original non-Markovian system to Markovian form of the dimension indicated. For the purposes of estimation of HJM models we need to consider the stochastic differential equation for the bond price P t T . However we have considered instead the stochastic differential equation for the log of the bond price since the Markovian system developed in equation turns out to be linear in the state variables. This latter result will be most convenient from the point of view of implementing estimation procedures Bharand Chiarella 1995b . Since the volatility vector V t is independent of the state variables and is a function of time only the stochastic dynamic system is Gaussian. The quantities t for all i which summarize the history of the noise process as well as the instantaneous spot rate of interest r t are not readily observable. Certainly there exists a one-to-one mapping between the t and yields drawn from the term structure and it would be possible to take n 1 such points to tie downZ iệt . While this would in principle provide an alternative estimation procedure it has the practical drawback that it would require continuous observations of the term structure. Many empirical studies attempt to proxy r t by some short-term rate . 30-day treasury bill rates. Thus in developing estimation techniques we need also to consider the observation vector which in this case reduces to a scalar Y t cs t C 1 O . f. The system with the observation vector is now in a form to which we are able to apply Kalman filter estimation techniques to form the log-likelihood function and hence estimate the parameters specifying the volatility function. 102 .
