# 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

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

# 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

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

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

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