A.2.1 Derivation of the Kalman Filter

A.2.1 Derivation of the Kalman Filter

Having described how the Kalman filter works, we briefly derive the prediction equations. Using (A.1) and (A.2), we have

$$\begin{array}{rl}& {\xi }_{t\mid t-1}=E_{t-1}{\xi }_t=E_{t-1}\left[F{\xi }_{t-1}+v_t\right]=F{\xi }_{t-1\mid t-1},\\ & y_{t\mid t-1}=E_{t-1}{y}_t=E_{t-1}\left[H{\xi }_t+w_t\right]=H{\xi }_{t-1\mid t-1},\end{array}$$

which is (A.15) and (A.13), respectively. The MSE values follow from this. Notice that

$$\begin{array}{r}{\xi }_t-{\xi }_{t|t-1}=F{\xi }_{t-1}+v_t-F{\xi }_{t-1|t-1}\\ =F({\xi }_{t-1}-{\xi }_{t-1|t-1})+v_t,\end{array}$$

and so, using (A.3), we get (A.16):

$$\begin{array}{rl}& P_{t|t-1}\equiv E\left[({\xi }_t-{\xi }_{t|t-1})({\xi }_t-{\xi }_{t|t-1}{)}^{\prime }\right]\\ & =FE\left[({\xi }_{t-1}-{\xi }_{t-1|t-1})({\xi }_{t-1}-{\xi }_{t-1|t-1}{)}^{\prime }\right]F^{\prime }+Q\\ & =F{P}_{t-1|t-1}{F}^{\prime }+Q,\end{array}$$

where the covariance is 0 because $v_t$ is white noise.

Turning to $y_t.$

$$\begin{array}{rl}& G_{t\mid t-1}\equiv E\left[(y_t-y_{t\mid t-1})(y_t-y_{t\mid t-1}{)}^{\prime }\right]\\ & =E\left[H({\xi }_t-{\xi }_{t\mid t-1}+w_t)({\xi }_t-{\xi }_{t\mid t-1}+w_t{)}^{\prime }{H}^{\prime }\right]\\ & =H{P}_{t\mid t-1}{H}^{\prime }+R,\end{array}\text{\hspace{1em}(A.19)}$$

which yields (A.14).

For the updating equations, we need to work out the optimal way to update our forecast of the state given the observed ${\gamma }_t$ . The forecast errors for the state and the actual values are expected to be correlated, so

$$\left(\begin{array}{l}{\gamma }_t-y_{t\mid t-1}\\ {\xi }_t-{\xi }_{t\mid t-1}\end{array}\right)$$

has a nondiagonal variance–covariance matrix. This is where the linear projections matter. Once we know the variance–covariance matrix, we can use the observed forecast error in ${\gamma }_t$ to update ${\xi }_t$ . The covariance is

$$\begin{array}{rl}& E[({\xi }_t-{\xi }_{t\mid t-1})(y_t-y_{t\mid t-1}{)}^{\prime }]=E[({\xi }_t-{\xi }_{t\mid t-1})({\xi }_t-{\xi }_{t\mid t-1}+w_t{)}^{\prime }{H}^{\prime }]\\ & =E[({\xi }_t-{\xi }_{t\mid t-1})({\xi }_t-{\xi }_{t\mid t-1}{)}^{\prime }]H^{\prime }\\ & =P_{t\mid t-1}{H}^{\prime }.\end{array}$$

Hence, we have

$$\left(\begin{array}{c}{y}_t-y_{t|t-1}\\ {\xi }_t-{\xi }_{t|t-1}\end{array}\right)\sim \left[\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{G}_{t|t-1}& H{P}_{t|t-1}\\ P_{t|t-1}{H}^{\prime }& P_{t|t-1}\end{array}\right)\right]$$

and the linear predictor is

$${\xi }_t-{\xi }_{t|t-1}=P_{t|t-1}{H}^{\prime }{G}_{t|t-1}^{-1}(y_t-y_{t|t-1}).$$

This gives the updating equation for ${\xi }_{t\mid t}$ in (A.17).

By analogy with (A.11), the variance $\mathrm{i}{s}^1$

$$\begin{array}{r}{P}_{t\mid t}=P_{t\mid t-1}-P_{t\mid t-1}{H}^{\prime }{G}_{t\mid t-1}^{-1}H{P}_{t\mid t-1},\end{array}\text{\hspace{1em}(A.20)}$$

which is equation (A.18) and so completes the derivation of the prediction and updating equations.

To initialize the system, we need to choose a set of priors, i.e., starting values before we get to observe any data. We can choose from the unconditional distribution if the model is stationary. Provided that the roots of F fall outside the unit circle, this distribution is centered on 0 and hence we could just set ${\xi }_{1|0}=0$ and use

$$E\left[{\xi }_1{\xi }_1^{\prime }\right]=FE\left[{\xi }_0{\xi }_0^{\prime }\right]F^{\prime }+E\left[v_1{v}_1^{\prime }\right],$$

or, equivalently,

$${\Sigma }_{\xi }=F{\Sigma }_{\xi }{F}^{\prime }+Q,$$

so that

$$\mathrm{vec}({\Sigma }_{\xi })=[I_r-(F\otimes F^{\prime }){]}^{-1}\mathrm{vec}(Q).$$

We could alternatively draw ${\xi }_{1|0}$ from a distribution with moments $(0,{\Sigma }_{\xi })$ . For nonstationary models the choice of initial values is not obvious. One approach is to leave the initial values as parameters to be estimated.

A.2.2 Examples

Any time we can write a model in state-space form, we are on our way to use the above methods. The only additional problems will be whether the model is identified and whether convergence can be achieved independent of the initialization.

A.2.2.1 Constructing Coincident Indicators

Common economic activity measures such as GDP are measured after the event, and so are really lagging indicators of the state of the economy. Stock and Watson (2002a) use the Kalman filter to construct an estimate of an unobserved component tracking the state of the economy, denoted $C_t$ . Suppose we observe a set of macroeconomic variables, $Y_t=(Y_{1t},\dots ,Y_{nt}{)}^{\prime }$ which have a common component, $C_t,$ and idiosyncratic errors, ${\chi }_{it},$ both of which follow AR(1) processes:

$$\begin{array}{rcl}& & Y_{it}={\gamma }_i{C}_t+{\chi }_{it},\\ & & \\ & & C_t={\varphi }_c{C}_{t-1}+v_{ct},\\ & & \\ & & {\chi }_{it}={\varphi }_i{\chi }_{it-1}+v_{it}.\end{array}$$

Assuming that both $C_t$ and ${\chi }_{it},i=1,\dots ,n$ are unobserved, the state equation can be written

$${\xi }_t=\left[\begin{array}{c}{C}_t\\ {\chi }_{1t}\\ ⋮\\ {\chi }_{nt}\end{array}\right]=\left(\begin{array}{cccc}{\varphi }_c& 0& \cdots & 0\\ 0& {\varphi }_1& 0& 0\\ ⋮& 0& \ddots & 0\\ 0& 0& 0& {\varphi }_n\end{array}\right){\xi }_{t-1}+\left(\begin{array}{c}{v}_{ct}\\ v_{1t}\\ ⋮\\ v_{nt}\end{array}\right),$$

while the measurement equation for the observed $Y_t$ becomes

$$y_t=\left[\begin{array}{l}{y}_{1t}\\ y_{2t}\\ ⋮\\ y_{nt}\end{array}\right]=\left(\begin{array}{llll}{\gamma }_1& 1& 0& 0\\ ⋮& 0& \ddots & 0\\ {\gamma }_n& 0& 0& 1\end{array}\right){\xi }_t.$$