197 lines
7.2 KiB
TeX
197 lines
7.2 KiB
TeX
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\begin{document}
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\title{Kalman filtering and smoothing in Dynare}%
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\maketitle
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\section*{Introduction}
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\noindent \textbf{``Filtering and Smoothing of State Vector for
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Diffuse State Space Models''}, S.J. Koopman and J. Durbin (2003,
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in \textit{Journal of Time Series Analysis}, vol. 24(1), pp.
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85-98).\newline
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\noindent \textbf{``Fast Filtering and Smoothing for Multivariate
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State Space Models''}, S.J. Koopman and J. Durbin (2000, in
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\textit{Journal of Time Series Analysis}, vol. 21(3), pp.
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282-296).\newline
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\bigskip
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\noindent \textsc{The State-Space Model}\footnote{Note that in
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Dynare, matrices $T$, $Z$, $R$, $H$ and $Q$ are assumed to be time
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invariant.}:
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\[y_t = Z\alpha_t + \varepsilon_t\]
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\[\alpha_{t+1} = T \alpha_t + R\eta_t\]
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with:
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\[\alpha_1 = a + A\delta + R_0\eta_0\]
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\noindent $m\times q$ matrix $A$ and $m\times(m-q)$ matrix $R_0$
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are selection matrices (their columns constitue all the columns of
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the $m\times m$ identity matrix) so that $A'R_0 = 0$ and
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$A'\alpha_1 = \delta$. We assume that the vector $\delta$ is
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distributed as a $\mathcal{N}(0,\kappa I_q)$ for a given $\kappa
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> 0$. So that the expectation of $\alpha_1$ is a and its variance
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is $P$, with
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\[ P = \kappa P_{\infty} + P_{\star}\]
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\[ P_{\infty} = A A'\]
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\[ P_{\star} = R_0 Q_0 R_0'\]
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\noindent $P_{\infty}$ is a $m\times m$ diagonal matrix with $q$
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ones and $m-q$ zeros. and where: $y_t$ is a $pp\times 1$ vector,
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$\alpha_t$ is a $mm\times 1$ vector, $\varepsilon_t$ is a
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$pp\times 1$ multivariate random variable (iid
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$\mathcal{N}(0,H)$), $\eta_t$ is a $rr\times 1$ multivariate
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random variable (iid $\mathcal{N}(0,Q)$), $a_1$ is a $mm\times 1$
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vector, $Z_t$ is a $pp\times mm$ matrix, $T$ is a $mm \times mm$
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matrix, $H$ is a $pp\times pp$ matrix, $R$ is a $mm\times rr$
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matrix, $Q$ is a $rr\times rr$ matrix and $P_1$ is a $mm\times mm$
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matrix. \newline
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\section{Filtering}
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\noindent The filtering equations are given by:
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\begin{equation}
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\begin{split}
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v_t &= y_t - Z a_t\\
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F_t &= Z P_t Z' + H\\
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K_t &= P_t Z' F_t^{-1}\\
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a_{t+1} &= T (a_t + K_t v_t)\\
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P_{t+1} &= T (P_t - P_tZ'K') T' + R Q R'\\
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\end{split}
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\end{equation}
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$\{F_t\}$ and $\{v_t\}$ are used to evaluate the likelihood. A
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potentially faster algorithm (unfortunately not with matlab) is to
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consider a univariate approach to the multivariate Kalman filter
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(the covariance matrix associated to the measurement errors has to
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be diagonal: $H=diag(\sigma_1^2,\dots,\sigma_{pp}^2)$). Let $Z_i$
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be line $i$ of the selection matrix $Z$. The univariate algorithm
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is as follows :
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\begin{equation}
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\begin{split}
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v_{t,i} &= y_{t,i} - Z_i a_t^{(i)}\\
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F_t^{(i)} &= Z_i P_t^{(i)} Z_i' + \sigma_{i}^2\\
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K_t^{(i)} &= P_t^{(i)} Z_i'\\
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a_{t}^{(i+1)} &= a_t^{(i)} + K_t^{(i)} v_{t,i} / F_t^{(i)}\\
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P_{t}^{(i+1)} &= P_{t}^{(i)} - K_{t}^{(i)} \left.K_t^{(i)}\right.' / F_t^{(i)}\\
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a_{t+1}^{(1)} &= T a_t^{(pp)}\\
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P_{t+1}^{(1)} &= T P_t^{(pp)} T' + R Q R'\\
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\end{split}
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\end{equation}
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when $F_t^{(i)}$ is equal to zero we simply have
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$a_{t}^{(i+1)}=a_{t}^{(i)}$ and $P_{t}^{(i+1)}=P_{t}^{(i)}$. The
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log-likelihood is evaluated as follows:
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\begin{equation}
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\mathcal{L}_T = const -\frac{1}{2}\sum_{i=1}^{pp}\sum_{t=1}^n \log
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F_{t}^{(i)} + v_{t,i}^2 / F_{t}^{(i)}
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\end{equation}
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\bigskip
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\noindent The diffuse filtering equations are given by:
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\begin{equation}
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\begin{split}
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v_t &= y_t - Z a_t\\
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F_{\infty,t} &= Z P_{\infty,t} Z' + H\\
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K_{\infty,t} &= P_{\infty,t} Z' F_{\infty,t}^{-1}\\
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F_{\ast,t} &= Z P_{\ast,t} Z' + H\\
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K_{\ast,t} &= \left(P_{\ast,t} Z' - K_{\infty,t}F_{\ast,t}\right)F_{\infty,t}^{-1}\\
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P_{\ast,t+1} &= T (P_{\ast,t}-P_{\ast,t}Z'K_{\infty,t}' - P_{\infty,t}Z'K_{\ast,t}') T' + R Q R'\\
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P_{\infty,t+1} &= T(P_{\infty,t}-P_{\infty,t}Z'K_{\infty,t}')T'\\
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a_{t+1} &= T (a_t + K_{\infty,t} v_t)\\
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\end{split}
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\end{equation}
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When the condition $rank(P_{\infty,t+1})=0$ is satisfied we set
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$d=t$ and go back to the standard Kalman filtering equations. Here
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$F_{\infty,t}$ is assumed to be a full rank matrix. If this is not
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the case we switch to another algorithm. If $F_{\infty,t}=0$:
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\begin{equation}
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\begin{split}
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v_t &= y_t - Z a_t\\
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F_{\ast,t} &= Z P_{\ast,t} Z' + H\\
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K_{\ast,t} &= P_{\ast,t} Z'F_{\infty,t}^{-1}\\
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P_{\ast,t+1} &= T (P_{\ast,t}-P_{\ast,t}Z'K_{\ast,t}') T' + R Q R'\\
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P_{\infty,t+1} &= TP_{\infty,t}T'\\
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a_{t+1} &= T (a_t + K_{\ast,t} v_t)\\
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L_t &= T(I - K_t Z)\\
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\end{split}
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\end{equation}
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otherwise, we consider a diffuse version of the univariate
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approach described above.
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\section{Smoothing}
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\noindent The smoothing equations are given by:
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\begin{equation}
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\begin{split}
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r_{t-1} &= Z' F_t^{-1}v_t + L_t' r_t\\
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\widehat{\alpha}_t &= a_t + P_t r_{t-1} \\
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\widehat{\eta}_t &= Q R r_t\\
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\widehat{\varepsilon}_t &= H \left(F_t^{-1}v_t-K_t'r_t\right)\\
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\end{split}
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\end{equation}
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initializing with $r_n = 0$ and with $L_t = T-K_tZ$. The diffuse
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smoothing equations are given by :
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\begin{equation}
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\begin{split}
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r_{t-1}^{(0)} &= L_{\infty,t}r_{t}^{(0)}\\
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r_{t-1}^{(1)} &= Z' F_{\infty,t}^{-1}v_t - K_{\ast,t}'r_{t}^{(0)}+ L_{\infty,t}' r_t^{(1)}\\
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\widehat{\alpha}_t &= a_t + P_{\ast,t} r_{t-1}^{(0)} + P_{\infty,t}r_{t-1}^{(1)}\\
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\widehat{\eta}_t &= Q R r_t^{(0)}\\
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\widehat{\varepsilon}_t &= -H K_{\infty,t}'r_t^{(0)}\\
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\end{split}
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\end{equation}
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for $t = d,d-1,...,1$, where $d$ is such that $P_{\infty,d+1}=0$.
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This backward recurrence is initialized with $r_{d}^{(0)}=r_{d}$,
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obtained from the non diffuse Kalman smoother, and
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$r_{d}^{(1)}=0$. $L_{\infty,t} = T-K_{\infty,t}Z$.\newline
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\noindent A univariate smoothing algorithm has to be coded... In
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the smoothing part the matrix $F_t$ (or $F_{\infty,t}$) is assumed
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to be full rank...
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% ----------------------------------------------------------------
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%\bibliographystyle{amsplain}
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%\bibliography{}
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\end{document}
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