Some notes on Kalman Filtering

State Space form

Measurement Equation

\displaystyle \boxed{\mathbf{\underbrace{y_{t}}_{N \times 1}=\underbrace{Z_{t}}_{N \times m}\underbrace{a_{t}}_{m \times 1}+d_{t}+\varepsilon_{t}}}

\displaystyle Var(\varepsilon_{t})= \mathbf{H_{t}}

Transition Equation

\displaystyle \boxed{\mathbf{\underbrace{a_{t}}_{m \times 1} =\underbrace{T_{t}}_{m \times m} a_{t-1}+c_{t}+\underbrace{R_{t}}_{m \times g} \underbrace{\eta_{t}}_{g \times 1}}}

\displaystyle Var(\eta_{t})=\mathbf{Q}_{t}

\displaystyle E(a_{0})= \mathbf{a_{0} \; \; \; \; Var(a_{0})=P_{0}} \; \; \; \; E(\varepsilon_{t}a_{0}^{\top}) \; \; \; E(\eta_{t}a_{0}^{\top})

Future form

\displaystyle \mathbf{a_{t+1}=T_{t}a_{t}+c_{t}+R_{t}\eta_{t}}

Continue reading “Some notes on Kalman Filtering”

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LAN for Linear Processes

Consider a m-vector linear process

\displaystyle \mathbf{X}(t) = \sum\limits_{j=0}^{\infty} A_{\theta}(j)\mathbf{U}(t-j), \qquad t \in \mathbb{Z}

where {\mathbf{U}(t)} are i.i.d. m-vector random variables with p.d.f. {p(\mathbf{u})>0} on {\mathbf{R}^m}, {A_{\theta} (j)} are {m \times m} matrices depending on a parameter vector { \mathbf{\theta} = (\theta_1,...,\theta_q) \in \Theta \subset \mathbf{R}^q}.

Set

\displaystyle A_{\theta}(z) = \sum\limits_{j=0}^{\infty} A_{\theta}(j)z^j, \qquad |z| \leq 1.

Assume the following conditions are satisfied

A1 i) For some {D} {(0<D<1/2)}

\displaystyle \pmb{|} A_{\theta}(j) \pmb{|} = O(j^{-1+D}), \qquad j \in \mathbb{N},

where { \pmb{|} A_{\theta}(j) \pmb{|}} denotes the sum of the absolute values of the entries of { A_{\theta}(j)}.

ii) Every { A_{\theta}(j)} is continuously two times differentiable with respect to {\theta}, and the derivatives satisfy

\displaystyle |\partial_{i_1} \partial_{i_2}... \partial_{i_k} A_{\theta, ab}(j)| = O \{j^{-1+D}(logj)^k\}, \qquad k=0,1,2

for {a,b=1,...,m,} where {\partial_i = \partial/ \partial\theta_i}.

iii) {det A_{\theta}(z) \neq 0} for {|z| \leq 1} and {A_{\theta}(z)^{-1}} can be expanded as follows:

\displaystyle A_{\theta}(z)^{-1} = I_m + B_{\theta}(1)z + B_{\theta}(2)z^2 + ...,

where { B_{\theta}(j)}, {j=1,2,...,} satisfy

\displaystyle \pmb{|} B_{\theta}(j) \pmb{|} = O(j^{-1-D}).

iv) Every { B_{\theta}(j)} is continuously two times differentiable with respect to {\theta}, and the derivatives satisfy

\displaystyle |\partial_{i_1} \partial_{i_2}... \partial_{i_k} B_{\theta, ab}(j)| = O \{j^{-1+D}(logj)^k\}, \qquad k=0,1,2

for {a,b=1,...,m.}

A2 {p(.)} satisfies

\displaystyle \lim\limits_{\| \mathbf{u} \| \rightarrow \infty} p(\mathbf{u})=0, \qquad \int \mathbf{u} p(\mathbf{u}) d \mathbf{u} =0, \qquad \text{and} \qquad \int \mathbf{uu'}p(\mathbf{u}) d \mathbf{u}=I_m

A3 The continuous derivative {Dp} of {p(.)} exists on {\mathbf{R}^m}.

A4

\displaystyle \int \pmb{|} \phi(\mathbf{u}) \pmb{|}^4 p (\mathbf{u}) d \mathbf{u} < \infty,
where {\phi(\mathbf{u}) = p^{-1}Dp}.

From A1 the linear process can be expressed as

\displaystyle \sum\limits_{j=0}^{\infty} B_{\theta}(j) \mathbf{X}(t-j) = \mathbf{U}(t), \qquad B_{\theta} (0) = I_m
and hence

\displaystyle \mathbf{U}(t) = \sum\limits_{j=0}^{t-1}B_{\theta}(j)\mathbf{X}(t-j)+\sum\limits_{r=0}^{\infty}C_{\theta}(r,t)\mathbf{U}(-r),

where

\displaystyle C_{\theta}(r,t)= \sum\limits_{r'=0}^{r}B_{\theta}(r'+t)A_{\theta}(r-r').

Continue reading “LAN for Linear Processes”