9.3.4 Time-Varying Parameter VARs

9.3.4 Time-Varying Parameter VARs

Following empirical evidence on model instability related to the Great Moderation or the global financial crisis, a number of recent papers have modeled VARs with time-varying parameters.7 The simplest version of these models assumes that the parameters of the mean equation, ${\beta }_t$ , follow a random walk, i.e.,

$$\begin{array}{rl}& y_t=(I_n\otimes z_t^{\prime }){\beta }_t+u_t,\\ & \\ & {\beta }_t={\beta }_{t-1}+{\epsilon }_t,\\ & \\ & {\epsilon }_t\sim \mathrm{N}(0,Q),u_t\sim \mathrm{N}(0,\Sigma ),\end{array}\text{\hspace{1em}(9.26)}$$

where the innovations ${\epsilon }_t$ and $u_t$ are uncorrelated. Conventionally it is assumed that the initial state is normally distributed ${\beta }_1\sim \mathrm{N}(s_{1|0},P_{1|0})$ and ${\beta }_{t+1}|{\beta }_t,Q\sim \mathrm{N}({\beta }_t,Q)$ is obtained iteratively.

Primiceri (2005) and Karlsson (2013) discuss implementation of a Gibbs sampler for this model. Since ${\beta }_t$ is an unobserved state variable, estimation and forecasting with the model involves the Kalman filter. The Gibbs sampler for the time-varying parameter (TVP) model first draws ${\beta }_T$ from its full conditional posterior given the full data set and initial values for - and $Q,$ using the Kalman filter and simulation smoother. In turn, $Q$ and - are updated given the states ${\beta }_1,\dots ,{\beta }_T$ Finally, using these updated parameter values, draws from the posterior distribution $y_{T+1},\dots ,y_{T+h}$ are generated by drawing $u_{T+1},\dots ,u_{T+h}$ and iterating on (9.26)

Stochastic volatility models allow for time-varying second moments in the data. There is strong empirical evidence in economics and finance of such features. One way to incorporate time-varying volatility is by assuming that the variance– covariance matrix for the innovations in the VAR, $u_t$ , rather than being constant at $\Sigma$ , takes the form

$${\Sigma }_t=L_t^{-1}{D}_t{\left(L_t^{-1}\right)}^{\prime },\text{\hspace{1em}(9.27)}$$

where the diagonal matrix $D_t=\mathrm{diag}(d_{1t},\dots ,d_{nt})$ has dynamics (see Karlsson, 2013)

$$\begin{array}{r}\begin{array}{c}{d}_{it}=\exp (\frac{1}{2}{h}_{it}),\\ h_{it+1}={\mu }_i+{\varphi }_i(h_{it}-{\mu }_i)+{\eta }_{it},\end{array}\\ {\eta }_t=({\eta }_{1t},\dots ,{\eta }_{nt})\sim \mathrm{ind}\mathrm{N}(0,V_{\eta }).\end{array}\text{\hspace{1em}(9.28)}$$

Hence the log-volatility of the n individual series, $h_{it+1}$ , is assumed to follow either a mean-reverting process $(\mathrm{if}|{\varphi }_i|<1)$ or a random walk specification (if ${\varphi }_i=1$ , in which case typically ${\mu }_i$ is set to 0); see Primiceri (2005).

The matrix $L_t$ can be specified as lower triangular with 1s on the diagonal:

$$\begin{array}{r}{L}_t=\left(\begin{array}{lllll}1& 0& 0& \cdots & 0\\ l_{21t}& 1& 0& \cdots & ⋮\\ l_{31t}& l_{32_t}& \ddots & & \\ ⋮& ⋮& & 1& 0\\ l_{n1t}& l_{n2t}& l_{n(n-1)t}& 1& \end{array}\right),\\ l_{ijt}=l_{ijt-1}+{\vartheta }_{ijt},{\left({\vartheta }_{11t},\dots ,{\vartheta }_{n(n-1)t}\right)}^{\prime }\sim \mathrm{ind}\mathrm{N}(0,V_{\vartheta }),i>j.\end{array}$$

The autoregressive parameters can be restricted to imply a stationary volatility process by using truncated distributions, i.e., ${\varphi }_{ii}\sim \mathrm{N}({\varphi }_0,V_{\varphi })1(|{\varphi }_{ii}|<1)$ , where 1(·) is an indicator function. As discussed in Karlsson (2013, algorithm 13), estimation and forecasting with the model involves a Gibbs sampler with Kalman filter steps to account for the latent state variables in ${\Sigma }_t$

These types of models have been used extensively in recent work and there is much evidence to suggest that allowing for time-varying volatility can improve model performance considerably. Clark (2011) applies stochastic volatility models to density forecasting of a four-variable model with GDP growth, unemployment, inflation, and the Federal funds rate, while Primiceri (2005) uses stochastic volatility models to analyze monetary policy. Clark and Ravazzolo (2014) compare different volatility models’ ability to predict macroeconomic outcomes. Pettenuzzo and Timmermann (2015) consider different models’ ability to forecast US GDP growth and inflation and find that stochastic volatility models generate better density forecasts than Markov switching or change point models. See also Koop and Korobilis (2010).

A closely related literature studies Bayesian Markov switching VAR models, see, e.g., Sims and Zha (2006), Hartmann et al. (2014), and Hubrich and Tetlow (2015). Sims and Zha (2006) use a multivariate regime switching model to analyze US monetary policy. Hartmann et al. (2014) study the effect of systemic stress and lending on the European economy while Hubrich and Tetlow (2015) study the interaction between economic growth and financial distress. These models are mostly used for analyzing economic policy, but can also be used for forecasting.