A.2.4 Kalman Smoothing

A.2.4 Kalman Smoothing

So far we constructed estimates of the state variable ${\xi }_{t\mid t}$ using current and past data. In “real time,” information dated at time t is of course the most up-to-date information available for computing estimates of period-t variables. In some situations, we might instead be interested in computing the best estimate of the unobserved state variable at time t given all past, current, and future data, ${\xi }_{t|T}:[{\xi }_{t|T}=E[{\xi }_t|y_1,\dots ,y_T],0\le$ $t\le T]$

The Kalman smoother accomplishes this using the recursive equations,

$${\xi }_{t|T}={\xi }_{t|t}+P_{t|t}{F}^{\prime }{P}_{t+1|t}^{-1}({\xi }_{t+1|T}-{\xi }_{t+1|t}),\text{\hspace{1em}(A.25)}$$

$$P_{t|T}=P_{t|t}+P_{t|t}{F}^{\prime }{P}_{t+1|t}^{-1}(P_{t+1|T}-P_{t+1|t})P_{t+1|t}^{-1}F{P}_{t|t}.\text{\hspace{1em}(A.26)}$$

Constructing the sequences $\{{\xi }_{t\mid T}\}$ and $\{P_{t\mid T}\}$ is trivial and proceeds by backwards recursion, starting from the end $(t=T)$ . The last terms of the Kalman filter equations give the starting points for the Kalman smoother, ${\xi }_{T\mid T}$ and $P_{T|T}$ . From these we can compute ${\xi }_{T-1|T}$ and $P_{T-1|T}$ using (A.25) and (A.26). Continuing in this fashion from the end to the start of the data, $t=T,T-1,\dots ,1$ , gives the smoothed sequences.

The Kalman smoother equations can be derived in much the same way as the updating equations. Consider the objects ${\xi }_T-{\xi }_{T|T-1}$ and ${\xi }_{T-1}-{\xi }_{T-1|T-1}$ . Given information in $y_1,\dots ,y_{T-1}$ , the expectations of these are given by $E_{t-1}[({\xi }_T-$ ${\xi }_{T|T-1})({\xi }_T-{\xi }_{T|T-1}{)}^{\prime }]=P_{T|T-1}$ and $E_{t-1}[({\xi }_{T-1}-{\xi }_{T-1|T-1})({\xi }_{T-1}-{\xi }_{T-1|T-1}{)}^{\prime }]=$ $P_{T-1|T-1}$ . Moreover, the expectation of the covariance is

$$\begin{array}{rl}& E_{t-1}[({\xi }_T-{\xi }_{T|T-1})({\xi }_{T-1}-{\xi }_{T-1|T-1}{)}^{\prime }]\\ & =E_{t-1}[(F({\xi }_{T-1}-{\xi }_{T-1|T-1})+v_T)({\xi }_{T-1}-{\xi }_{T-1|T-1}{)}^{\prime }]\\ & =F{P}_{T-1|T-1}.\end{array}$$

The best linear projection of ${\xi }_{T-1|T}-{\xi }_{T-1|T-1}$ given $y_1,\dots ,y_{T-1},{\xi }_T$ is

$${\xi }_{T-1|T}-{\xi }_{T-1|T-1}=P_{T|T}{F}^{\prime }{P}_{T|T-1}^{-1}({\xi }_T-{\xi }_{T|T-1}).\text{\hspace{1em}(A.27)}$$

Replacing the unobserved state, ${\xi }_{T;\xi }$ , with our best linear projection ${\xi }_{T\mid T}$ given $y_1,\dots ,y_T$ results in (A.27). Similar arguments lead to the recursive equation for the MSE matrix. Such smoothed estimates therefore use all the information in the data up to time T. Notice a key difference: the Kalman filter is a one-sided, time-varying linear filter where the time variation arises through the updates to the variances. In contrast, the Kalman smoother is a two-sided time-varying filter.

A.3 ORDERS OF PROBABILITY

The orders of probability concept is a direct extension of the order of a nonrandom infinite sequence which we therefore use to motivate the former.

Definition. For sequences $\{x_n{\}}_{n=1}^{\infty }$ and $\{a_n{\}}_{n=1}^{\infty },$ we say that $x_n=o(a_n)$ if $\begin{array}{r}{\lim }_{n\to \infty }(x_n/a_n)=0\end{array}$ . We say that $x_{n}isO(a_n)$ if there exists a constant K such that $|x_n/a_n|<K$ for all n that are sufficiently large.