A.1.1 Basic Setting

A.1.1 Basic Setting

The basic building block is what is called a model in state-space form. This assumes that the model can be written as a first-order vector autoregression in states, along with an equation relating the observed data (y) to the states (ξ ). Recall from section 7.2.1 that the basic model comprises a state equation,

$${\xi }_t=F{\xi }_{t-1}+v_t,\text{\hspace{1em}(A.1)}$$

and a measurement equation,

$$y_t=H{\xi }_t+w_t,\text{\hspace{1em}(A.2)}$$

where the error terms are both serially uncorrelated and mutually uncorrelated:

$$E\left[\left(\begin{array}{l}{v}_t\\ w_t\end{array}\right)\left(\begin{array}{ll}{v}_t^{\prime }& w_t^{\prime }\end{array}\right)\right]=\left(\begin{array}{ll}Q& 0\\ 0& R\end{array}\right).\text{\hspace{1em}(A.3)}$$

Here ${\gamma }_t$ denotes observed values, while the states, ${\xi }_t,$ can be either observed or unobserved. The basic setup can be generalized to let the parameters of the model F , H, Q, and R vary with time.

The state-space form is a flexible way to write many models. For example, the AR(1) model,

$$y_t=\rho y_{t-1}+{\epsilon }_t,\text{\hspace{1em}(A.4)}$$

can be written in state-space form as

$$\begin{array}{l}{\xi }_t=\rho {\xi }_{t-1}+{\epsilon }_t,\\ {\gamma }_t={\xi }_t,\end{array}$$

with $F=\rho ,H=1,Q={\sigma }_{\epsilon }^2$ , and $R=0$ . Similarly, the $\mathrm{AR}(2)$ model,

$$y_t={\rho }_1{y}_{t-1}+{\rho }_2{y}_{t-2}+{\epsilon }_t\text{\hspace{1em}(A.5)}$$

can be written in state-space form as

$$\begin{array}{rlr}{\left({\xi }_{1t}\right)}_L=\left(\begin{array}{cc}{\rho }_1& {\rho }_2\\ 1& 0\end{array}\right)\left(\begin{array}{c}{\xi }_{1t-1}\\ {\xi }_{2t-1}\end{array}\right)+\left(\begin{array}{c}1\\ 0\end{array}\right){\epsilon }_t,& & \\ y_t={\xi }_{1t},& & \end{array}$$

with the obvious choice of parameters. This works out since ${\xi }_{2t}={\xi }_{1t-1}=y_{t-1}$ , so ${\xi }_{2t-1}=y_{t-2}$

As a third example, the MA(1) model,

$$y_t={\epsilon }_t+\theta {\epsilon }_{t-1},\text{\hspace{1em}(A.6)}$$

can be written in state-space form

$$\begin{array}{rl}\left(\begin{array}{l}{\xi }_{1t}\\ {\xi }_{2t}\end{array}\right)=\left(\begin{array}{ll}0& 0\\ 1& 0\end{array}\right)\left(\begin{array}{l}{\xi }_{1t-1}\\ {\xi }_{2t-1}\end{array}\right)+\left(\begin{array}{l}1\\ 0\end{array}\right){\epsilon }_t,& \\ y_t=\left(\begin{array}{ll}1& \theta \end{array}\right)\left(\begin{array}{l}{\xi }_{1t}\\ {\xi }_{2t}\end{array}\right).& \end{array}$$

In this case the state $\left({\epsilon }_t\right)$ is typically unobserved. Some models lend themselves to multiple forms of representations which, however, are identical up to a normalization.

As an example of a multivariate model, consider the VAR(2),

$$y_t=A_1{y}_{t-1}+A_2{y}_{t-2}+{\epsilon }_t,\text{\hspace{1em}(A.7)}$$

where ${\gamma }_t$ and ${\epsilon }_t$ are $(n\times 1)$ vectors and $A_i$ is an $(n\times n)$ matrix for $i=1,2$ . Using the companion or state-space form, we can rewrite this as a first-order matrix autoregressive model:

$$\begin{array}{rl}& \left(\begin{array}{c}{\xi }_{1t}\\ {\xi }_{2t}\end{array}\right)=\left(\begin{array}{cc}{A}_1& A_2\\ I_n& 0\end{array}\right)\left(\begin{array}{c}{\xi }_{1t-1}\\ {\xi }_{2t-1}\end{array}\right)+\left(\begin{array}{c}{I}_n\\ 0\end{array}\right){\epsilon }_t,\\ & {\gamma }_t={\xi }_{1t}.\end{array}$$

As a final example, the unobserved components model,

$${\xi }_t={\xi }_{t-1}+v_t,\text{\hspace{1em}(A.8)}$$

$$y_t={\xi }_t+w_t,$$

is already in state-space form. In this example, the state is again unobserved.

Two reasons explain the popularity and usefulness of the Kalman filter. First, the state equation (A.1) is in AR(1) form which is easy to iterate forward. Specifically, the h-step-ahead forecast of the state is given by

$$E_t[{\xi }_{t+h}]=F^h{\xi }_t.\text{\hspace{1em}(A.9)}$$

From a computational point of view, this greatly simplifies matters.

Second, updating the Kalman filter through newly arrived information is easy and relies on only the first two moments, essentially reducing to least squares (projection) methods. To establish the analogy, consider the bivariate normal distribution,

$$\left(x_1\right)\sim \mathrm{N}\left[\left(\begin{array}{l}{\mu }_1\\ {\mu }_2\end{array}\right),\left(\begin{array}{ll}{\Sigma }_{11}& {\Sigma }_{12}\\ {\Sigma }_{21}& {\Sigma }_{22}\end{array}\right)\right].\text{\hspace{1em}(A.10)}$$

Rotating the variables, we get the equivalent representation,

$$\left(\genfrac{}{}{0ex}{}{x_1-{\mu }_1}{x_2-{\mu }_2}\right)\sim \mathrm{N}\left[\left(\begin{array}{c}0\\ {\Sigma }_{12}^{\prime }{\Sigma }_{11}^{-1}\left(x_1-{\mu }_1\right)\end{array}\right),\left(\begin{array}{cc}{\Sigma }_{11}& 0\\ 0& {\Sigma }_{22}-{\Sigma }_{21}{\Sigma }_{11}^{-1}{\Sigma }_{21}\end{array}\right)\right].\text{\hspace{1em}(A.11)}$$

Using this representation, it is easily seen that the optimal prediction for the second variable is

$$\begin{array}{r}E\left[(x_2-{\mu }_2)|x_1-{\mu }_1\right]={\Sigma }_{12}^{\prime }{\Sigma }_{11}^{-1}\left(x_1-{\mu }_1\right),\end{array}\text{\hspace{1em}(A.12)}$$

which is both the conditional mean and the formula for OLS regressions. Dropping the normality assumption, (A.12) becomes the optimal linear predictor.