A.2.2.2 Hodrick–Prescott Filter

A.2.2.2 Hodrick–Prescott Filter

The Hodrick–Prescott (HP) filter can be written in the form of a pair of equations,

$$\begin{array}{r}{\Delta }^2{g}_t=u_t,\end{array}$$

$$y_t=g_t+w_t.$$

This is almost in state-space form, except that the unobserved state follows an $\mathrm{AR}(2)$ rather than an AR(1) process. This can be handled by writing the model in companion form as follows:

$$\left(\genfrac{}{}{0ex}{}{{\xi }_{1t}}{{\xi }_{2t}}\right)=\left(\begin{array}{cc}2& -1\\ 1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{{\xi }_{1t-1}}{{\xi }_{2t-1}}\right)+\left(\begin{array}{c}1\\ 0\end{array}\right)u_t,$$

and then notice that

$$\begin{array}{r}{\xi }_{1t}=2{\xi }_{1t-1}-1{\xi }_{2t-1}+u_t\\ =2{\xi }_{1t-1}-1{\xi }_{1t-2}+u_t,\end{array}$$

or, rearranging,

$$\Delta {\xi }_{1t}=\Delta {\xi }_{1t-1}+u_t,$$

so $g_t={\xi }_{1t}$ . The measurement equation takes the form

$$y_t=\left(10\right){\xi }_t+w_t,$$

which puts the HP filter in state-space form.

A.2.2.3 Missing Observations

One way to handle missing observations is to “fill out” the data using the Kalman filter to infer our best estimate of the missing data point. This can be done by treating the series that would have been observed as the state variable and relating this to the variables that are actually observed. The two must be identical for data that are observed but can otherwise differ. This means that the H matrix multiplying ${\xi }_t$ in the measurement equation is not constant over time.

For example, consider a bivariate VAR(1) where some of the observations are missing for the first variable:

$$\left(\begin{array}{l}{y}_{1t}\\ y_{2t}\end{array}\right)=\left(\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{l}{y}_{1t-1}\\ y_{2t-1}\end{array}\right)+\left(\begin{array}{l}{v}_{1t}\\ v_{2t}\end{array}\right).$$

Written in state-space form, we have

$$\left(\begin{array}{l}{\xi }_{1t}\\ {\xi }_{2t}\end{array}\right)=\left(\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{l}{\xi }_{1t-1}\\ {\xi }_{2t-1}\end{array}\right)+\left(\begin{array}{l}{v}_{1t}\\ v_{2t}\end{array}\right),$$

$$\begin{array}{r}{y}_t=H_t{\xi }_t.\end{array}$$

In periods where we observe the entire vector $y_t,H_t=I_2.$ When we do not observe $y_t.$ the measurement equation can be set to $y_{2t}={\xi }_{2t},i.e.,y_{2t}=H_t{\xi }_t$ , where $H_t=$ [0, 1]. Both $H_t$ and the dimension of the measurement equation change. This presents no problem for the Kalman filter, which adjusts accordingly.

When ${\gamma }_t$ is fully observed, $H_t=I$ (and R = 0), so the prediction equations are

$$\begin{array}{rcl}& & y_{t|t-1}={\xi }_{t|t-1},\\ & & \\ & & G_{t|t-1}=P_{t|t-1},\end{array}$$

while the updating equations become

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

Hence the interpolated value is the observed value. When ${\gamma }_{1t}$ is not observed, the updating equations provide an estimate of the missing observation.

A.2.3 Estimating Parameters

Many applications contain unknown parameters in the coefficient matrices F , H, $Q.$ and R. These can be estimated using the Kalman filter which provides a convenient way to construct the likelihood or pseudo likelihood for the data.

Suppose that $v_t$ and $w_t$ are normally distributed. From equations (A.13) and (A.14) we then have that, given time t − 1 information,

$$\begin{array}{r}{y}_{t|t-1}\sim \mathrm{N}(H{\xi }_{t|t-1},G_{t|t-1}),\end{array}$$

which depends on the unknown parameter matrices F , H, Q, and R since

$$\begin{array}{rl}{G}_{t|t-1}=H{P}_{t|t-1}{H}^{\prime }+R,& \\ P_{t|t-1}=F{P}_{t-1|t-1}{F}^{\prime }+Q.& \end{array}$$

Using the normality assumption, we can write down the likelihood for an observation, $y_t.$ , given the past elements:

\begin{array} { r l } {f _ {y _ { t }}} ( y _ { t }| y _ { t - 1} , \dots ) = ( 2 \pi ) ^ { - n / 2 } | G _ { t | t - 1 } | ^ { - 1 / 2 } } & { } \\ { \quad } & { \times \exp \left\{ - \displaystyle \frac { 1 } { 2 } ( y _ { t } - H ^ { \prime } \xi _ { t | t - 1 } ) ^ { \prime } G _ { t | t - 1 } ^ { - 1 } ( y _ { t } - H ^ { \prime } \xi _ { t | t - 1 } ) \right\} . } \end{array}\tag{A.23}

With an additional assumption on the distribution of the initial value, ${\nu }_1$ , the likelihood for the observed data becomes

L = \log f _ {y _ { 1}} ( y _ { 1}) + \sum _ { t = 2 } ^ { T } f _ {y _ { t}}} ( y _ { t}| y _ { t - 1} , . . . ) .\tag{A.24}

This expression can be maximized to obtain estimates of the unknown parameter matrices. Of course, we need to make sure that the model is identified and also need to pay attention to the numerical method used to construct the maximum of (A.24). Complicated models may result in likelihood surfaces that are difficult to maximize over. Also, parameters such as the variances are bounded. One approach is to use regressions on the state-space model. Suppose we have “starting values” for $\{{\xi }_{t\mid t}\}$ and use seemingly unrelated regression on the model

$$\begin{array}{l}{\xi }_t=F{\xi }_{t-1}+v_t,\\ {\gamma }_t=H{\xi }_t+w_t,\end{array}$$

resulting in estimates for F , Q, H, and R. These estimates are next used to generate new values for $\{{\xi }_{t\mid t}\}$ . This is a special case of the EM algorithm (Dempster et al., 1977) and has been applied by authors such as Watson and Engle (1983).