19.5.3 Time-Varying Parameter Models

19.5.3 Time-Varying Parameter Models

The simple time-varying parameter (TVP) random walk model takes the form

$$\begin{array}{l}{y}_t=x_{t-1}^{\prime }{\beta }_t+{\epsilon }_{yt},\\ {\beta }_t={\beta }_{t-1}+{\epsilon }_{\beta t}.\end{array}\text{\hspace{1em}(19.23)}$$

This model is in state-space form with ${\gamma }_t$ being the observable process and ${\beta }_t$ being the latent “state,” so we can rewrite (19.23) using the notation from the Kalman filter appendix:

$$y_t=H_t{\xi }_t+w_t,\text{\hspace{1em}(19.24)}$$

$${\xi }_t={\xi }_{t-1}+v_t.\text{\hspace{1em}(19.25)}$$

Moreover $w_t\sim \mathrm{N}(0,R_t),v_t\sim \mathrm{N}(0,Q_t)$ . Assuming that ${\gamma }_t$ and $H_t=x_{t-1}^{\prime }$ are observable while ${\xi }_t$ is a latent variable, (19.24) is the observation equation and (19.25) is the state equation. The model can readily be estimated using Kalman filter methods.

Several variations exist on how to estimate the TVP model and are routinely used to forecast vector-valued variables. As discussed in chapter 9, Sims (1993) proposed a Bayesian approach to computing forecasts. Assuming normally distributed innovations ${\epsilon }_{yt}$ and ${\epsilon }_{\beta t}$ in (19.23), it is standard to use conjugate priors with a normal prior on the initial value for ${\beta }_t$ and an inverse Wishart prior on the variance–covariance matrix. This yields a model that can easily be estimated by Gibbs sampling methods; see our discussion of this topic in section 9.3.4. A fuller (book-length) examination of the formulation and estimation of these models is given in West and Harrison (1998).

Let $Z_t=(y_t,y_{t-1},\dots ,x_t,x_{t-1},\dots )$ be the data available at time t. First consider the simple case with known covariance matrix, and suppose a normal prior has been used to initialize the recursion, i.e., ${\xi }_0\sim \mathrm{N}({\xi }_{0|0},P_{0|0})$ in the notation from the Kalman filter appendix. The posterior predictive distribution for $y_{t+1}$ given $Z_t$ is

$$p_Y(y_{t+1}|Z_t)\sim \mathrm{N}(y_{t+1|t},G_{t+1|t}),$$

where formulas for the arguments in the normal distribution are given in the Kalman filter appendix.

In the more realistic case with an unknown covariance matrix, we also require priors over the covariance parameters. Suppose that $Q_t$ is time varying and consider the normalization $Q_t=R{Q}_t^{\ast }$ , where $R$ is an invertible matrix. Using the inverse Gamma priors

$$\begin{array}{r}R|Z_0\sim \mathrm{IG}\left[\frac{1}{2},\frac{1}{2}{S}_0\right],\end{array}$$

where $S_0$ is some preliminary estimate such as the MLE based on past data, the posterior distribution for the outcome is

$$p_Y(y_{t+1}|Z_t)\sim T_{n_t}(y_{t+1|t},G_{t+1|t}).$$

Here $T_{n_t}$ is the multivariate t-distribution with $n_t$ degrees of freedom, mode $\begin{array}{r}{\gamma }_{t+1|t},\end{array}$ and scale matrix ${\mathbf{G}}_{t+1|t}{.}^1$ We replace R in the above Kalman filter formulas by the MLE estimate of the covariance matrix, $S_t$ . West and Harrison (1998) also consider multiperiod forecasts, although these rely on knowing $H_{t+h-1}$ for the h-step-ahead forecast.

A number of studies have applied these methods. Stock and Watson (1996) find that time-varying parameter models with a random walk component, as well as forecasting models based on a rolling estimation window, in many cases produce better out-of-sample forecasts than models with fixed parameters. Models that allow for unstable parameters also tend to reduce the risks of extremely poor forecasting performance such as when large parameter breaks are present. Average gains relative to a recursively estimated constant parameter model are generally very small, however.

Stock and Watson (2007) propose an unobserved components (univariate) stochastic volatility model of the form in (19.24) where $H=1$ . Variances are assumed to follow stochastic volatility processes

$$\begin{array}{r}\ln (R_t)=\ln (R_{t-1})+{\psi }_{rt},\\ \ln (Q_t)=\ln (Q_{t-1})+{\psi }_{qt},\end{array}\text{\hspace{1em}(19.26)}$$

and the volatility shocks are mutually uncorrelated as well as uncorrelated through time: $({\psi }_{rt},{\psi }_{qt}{)}^{\prime }\sim \mathrm{ind}\mathrm{N}(0,I)$ . The forecast of $y_{t+h|t}$ is obtained as the filtered estimate of ${\xi }_t$ . The model can be estimated using MCMC methods. This model generalizes the simple TVP random walk model with an intercept, which does not allow for stochastic volatility.

Engle and Smith (1999) propose a stopbreak model with stochastic permanent breaks:

$$\begin{array}{l}{y}_{t+1}={\xi }_{t+1}+w_{t+1},\\ {\xi }_{t+1}={\xi }_t+q_t{w}_t,\end{array}$$

where $q_t=q(w_t)$ and $w_t$ follows a martingale process. This model differs from the standard setup since the shocks in both equations are the same although $q_t$ determines the effect of a shock, $w_t,\mathrm{on}{y}_{t+1}.\mathrm{If}{q}_t=1$ , we obtain a simple version of the random walk model with perfectly correlated shocks, while if $q_t=0$ , the shock $w_t$ has no permanent effect on the mean of $y.$