A.2 KALMAN FILTER EQUATIONS

A.2 KALMAN FILTER EQUATIONS

To establish the prediction and updating equations for the Kalman filter for some random variable, x, let $x_{t\mid t-1}$ denote the best prediction of xt given t − 1 information while $x_{t\mid t}$ is the best “prediction” (or nowcast) of $x_t$ given time t information. Moreover, let the matrices P and G contain the MSE values associated with the forecasts of ${\xi }_t$ and ${\gamma }_t$ , respectively, i.e., $P_{t|t-1}=E[({\xi }_t-{\xi }_{t|t-1})({\xi }_t-{\xi }_{t|t-1}{)}^{\prime }]$ and $P_{t|t}=E\left[({\xi }_t-{\xi }_{t|t})({\xi }_t-{\xi }_{t|t}{)}^{\prime }\right]$ , while $G_{t|t-1}=E\left[(y_t-y_{t|t-1})(y_t-y_{t|t-1}{)}^{\prime }\right]$

Using the state (A.1), measurement (A.2), and error covariance (A.3) equations, we get the following set of prediction equations:

$$\begin{array}{r}{y}_{t|t-1}=H{\xi }_{t|t-1},\end{array}\text{\hspace{1em}(A.13)}$$

$$G_{t|t-1}=H{P}_{t|t-1}{H}^{\prime }+R,\text{\hspace{1em}(A.14)}$$

$${\xi }_{t|t-1}=F{\xi }_{t-1|t-1},\text{\hspace{1em}(A.15)}$$

$$P_{t\mid t-1}=F{P}_{t-1\mid t-1}{F}^{\prime }+Q.\text{\hspace{1em}(A.16)}$$

Similarly, we have a pair of updating equations:

$${\xi }_{t|t}={\xi }_{t|t-1}+P_{t|t-1}{H}^{\prime }{G}_{t|t-1}^{-1}(y_t-y_{t|t-1}),\text{\hspace{1em}(A.17)}$$

$$\begin{array}{r}{P}_{t\mid t}=P_{t\mid t-1}-P_{t\mid t-1}{H}^{\prime }{G}_{t\mid t-1}^{-1}H{P}_{t\mid t-1}.\end{array}\text{\hspace{1em}(A.18)}$$

Recall that the states may be unobserved, so current-time predictions, ${\xi }_{t\mid t},$ make sense. On the other hand, there is no ${\gamma }_{t\mid t}$ or $G_{t|t}$ since ${\gamma }_t$ is observed.

To see how the method works, start at $t=0$ . At this time we have not observed any data, so we must make our best guesses of ${\xi }_{1|0}$ and $P_{1|0}$ without data, which essentially means picking a pair of initial conditions. Using these along with the model parameters, the prediction equations (A.13) and (A.14) give us ${\mathbf{\gamma }}_{1|0}$ and $G_{1|0}$ Now all first period forecasts are in place.

At time $t=1$ we observe ${\gamma }_1$ . The updating equations (A.17) and (A.18) provide us with ${\xi }_{1|1}$ and $P_{1|1}$ . The prediction equations (A.13)–(A.16) then give us each of the forecasts for the second period.

At time $t=2$ we observe $y_2,$ and the cycle continues. The end result is a set of sequences of predictions of the states, $\{{\xi }_{t\mid t}\}$ and $\{{\xi }_{t\mid t-1}\}$