7.5.3 Unobserved Components

7.5.3 Unobserved Components

The unobserved components model consists of two equations,

$$\begin{array}{l}{y}_t={\xi }_t+{\epsilon }_t,\\ {\xi }_t={\xi }_{t-1}+v_t,\end{array}\text{\hspace{1em}(7.47)}$$

where the process is initialized by ${\xi }_0$ . The idea is that the observed process, $y_t.$ , is a noisy version of the process of interest, ${\xi }_t$ . In this sense the unobserved sequence {ξt} can be seen as a smoothed version of the data ${\gamma }_t$

From a forecasting perspective we are interested in predicting ${\xi }_{T+1}$ given the observed data $\{{\mathbf{y}}_t{\}}_{t=1}^T$ . It is typically assumed that the innovations are serially uncorrelated and

$$\left(\genfrac{}{}{0ex}{}{{\epsilon }_t}{v_t}\right)\sim \mathrm{N}\left[\left(\genfrac{}{}{0ex}{}{0}{0}\right),\left(\genfrac{}{}{0ex}{}{{\sigma }^2}{0}\right){\tau }^2\right),$$

where the lack of contemporaneous correlation between the innovations is imposed for identification. The normality assumption in (7.48) makes it easy to write down the likelihood function used in estimation of the model.

Typically estimates of the model in (7.47)–(7.48) employ the Kalman filter (Kalman, 1960). The appendix explains the Kalman filter equations and operation. The Kalman filter was introduced for engineering applications and has subsequently found great use in many situations. For normally distributed variables it can be used to write down a likelihood function, or more generally a pseudo likelihood. Many models can be rewritten in the state-space form required by this method, but the approach is particularly convenient for estimating ARMA models with a nontrivial MA component.

The unobserved components model is in state-space form (see the appendix for the equations of the filter). Define ${\xi }_{t\mid t-1}$ and ${\xi }_{t\mid t}$ as the best prediction of ${\xi }_t$ given information at time t − 1 and $t,$ respectively; define ${\gamma }_{t\mid t-1}$ and ${\gamma }_{t\mid t}$ correspondingly, and let $P_{t|t-1}=E[({\xi }_t-{\xi }_{t|t-1})({\xi }_t-{\xi }_{t|t-1}{)}^{\prime }],G_{t|t-1}=E[(y_t-y_{t|t-1})(y_t-y_{t|t-1}{)}^{\prime }]$ be MSE matrices. Then we can write the updating equation for the unobserved $\cdots {\mathrm{mean}}^{\prime \prime }{\xi }_t$ as

$$\begin{array}{rl}& {\xi }_{t\mid t}={\xi }_{t\mid t-1}+P_{t\mid t-1}{G}_{t\mid t-1}^{-1}(y_t-y_{t\mid t-1})\\ & ={\xi }_{t-1\mid t-1}+P_{t\mid t-1}{G}_{t\mid t-1}^{-1}(y_t-{\xi }_{t-1\mid t-1}).\end{array}$$

This again simplifies to (7.44) if $P_{t|t-1}{G}_{t|t-1}^{-1}=1-\delta$ . Moreover, for the unobserved components model we have

$$P_{t|t-1}{G}_{t|t-1}^{-1}=\frac{P_{t-1|t-1}+{\tau }^2}{P_{t-1|t-1}+{\tau }^2+{\sigma }^2}.$$

This is simply $\delta ={\sigma }^2/({\tau }^2+{\sigma }^2)$ when the variances are not time varying for a filter with $P_{t-1|t-1}$ set to 0. This gives another insight into the choice of (or method for estimating) the discount parameter δ. When there is little variation in the mean relative to the variation in the observed data (so ${\tau }^2$ is small relative to ${\sigma }^2)$ , δ will be large and it becomes optimal to put more weight on “old” data that remains relevant for estimating the mean. When variation in the innovations to the unobserved mean is large relative to ${\sigma }^2$ , δ will be closer to 0.