The Kalman Filter Sample Clauses

The Kalman Filter. ‌ The ST/LT model presents a unique challenge for estimation. The first problem is that the two state variables are unobserved. To estimate these state variables I can use the Kalman filter (Xxxxxx 1960; Xxxxxxxx 1994). The Kalman filter produces mean and variance of the state variables which facilitates the building of a likelihood function. This section discusses the derivation of the Kalman filter which will be used in maximum likelihood estimation in section 4.2. It is shown below that if observations of futures prices with different maturities are available, I can use the Kalman filter to produce estimates of the state variables. To compute the Kalman filter, a system of equations must first be cast into state- space representation. To derive the Kalman filter as in Xxxxxxxx (1994) I first cast the dynamic system of equations into state space representation and lay out the assumptions required for that system. Equations (4.1.1a) and (4.1.1b) show our system cast into state space representation. xt = c + Gxt−1 + ωt t =1, 2,3,...,t
Sample 1
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The Kalman Filter. ‌ The Kalman Filter (KF) was first introduced for linear problems [49], but has then been extended to non-linear problems with the so called Extended Kalman Filter [44] and Unscented Kalman Filter [48]. An exhaustive analysis of these methods is beyond the scope of this work, and we refer to the literature for further information p c [44,48,49]. Here, we limit to a brief overview of the idea behind the KF, applying it to the linear problem (2.3) with the dependency on µ dropped for the sake of notation. The KF can be considered as a two-step method. The first is a prediction step, whose solution we denote with xn, while the second is a correction step, whose solu- tion we denote with xn. In the prediction step, the system evolves according to the deterministic model, that is, without considering the presence of the model errors εn. The ideal step would be p xn Axn 1 bn. (2.6) However, since the “true” solution xn 1 is not available, the state evolves according to the best information available on xn 1, that is, xn Axn 1 bn. (2.7) p c In the correction step, the information coming from the data is used to update xn. More precisely, the KF performs a correction of the form xn Lnxn Kndn (2.8)
Sample 1

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