9.4.1 Estimation and Computation of Forecasts for DSGE Models

9.4.1 Estimation and Computation of Forecasts for DSGE Models

Estimation of DSGE models can proceed using a state-space representation for the linearized DSGE model given assumptions such as normally distributed innovations. Letting $s_t$ denote the vector of state variables, suppose we can approximate the solution to the DSGE model through a log-linearized first-order VAR,

$$s_t={\Phi }_1(\theta )s_{t-1}+{\Phi }_{ϵ}(\theta ){\epsilon }_t,\text{\hspace{1em}(9.43)}$$

where the vector ${\epsilon }_t$ collects structural shocks (innovations) to the model, i. $.{\mathrm{e}}_{\cdot 3}$ ${\mathbf{\epsilon }}_t=({\epsilon }_{Rt},{\epsilon }_{gt},{\epsilon }_{zt}{)}^{\prime }$ in the small-scale model described above. The matrices ${\Phi }_1$ and ${\Phi }_{ϵ}$ depend on the parameters of the DSGE model, θ. Given this representation, the measurement equation for the aggregate output, ${\gamma }_t$ , can be written as

$$y_t={\Psi }_0(\theta )+{\Psi }_1(\theta )t+{\Psi }_2(\theta )s_t.\text{\hspace{1em}(9.44)}$$

Under these assumptions, the Kalman filter can be used to evaluate the likelihood function and estimate the parameters.

A key complication arising from (9.43) and (9.44) is that the parameters of the linearized state and measurement equations, ${\Phi }_1,{\Phi }_{ϵ},{\Psi }_0,{\Psi }_1,{\Psi }_2$ are nonlinearly related to the parameters underlying the DSGE model, $\theta .$ . Standard sampling algorithms can therefore typically not be used since the conditional probability distribution of $\theta$ is not easily available. To address this issue, one can use algorithms such as the random walk metropolis algorithm; see Del Negro and Schorfheide (2013) for details and references.

Using Bayesian methods, draws from the posterior predictive distribution of the dependent variables in the DSGE model can be based on the following useful decomposition (see Del Negro and Schorfheide, 2013):

$$p_Y(y_{T+1}|y_1,\dots ,y_T)={\int }_{s_T,\theta }\left[{\int }_{s_{T+1}}{p}_Y(y_{T+1}|s_{T+1})p(s_{T+1}|s_T,\theta ,y_1,\dots ,y_T)d{s}_{T+1}\right]$$

$$\times p(s_T|\theta ,y_1,\dots ,y_T)p(\theta |y_1,\dots ,y_T)d(s_T,\theta ).\text{\hspace{1em}(9.45)}$$

Such draws are generated using the same sorts of steps described earlier in this chapter. First, suppose that an algorithm is in place to generate draws ${\theta }^j$ from the posterior parameter distribution $p(\theta |y_1,\dots ,y_T)$ . Next, use the Kalman filter to generate draws $s_T^j$ from the conditional posterior distribution $p(s_T|{\theta }^j,y_1,\dots ,y_T)$ Using these draws, we can obtain draws from $p(s_{T+1}|s_T^j,{\theta }^j,y_1,...,y_T)$ by using (9.43) along with draws of ${\epsilon }_{T+1}^j$ . Finally, using (9.44), we can generate draws from $y_{T+1}^j$ :

$$y_t^j={\Psi }_0({\theta }^j)+{\Psi }_1({\theta }^j)t+{\Psi }_2({\theta }^j)s_t^j.\text{\hspace{1em}(9.46)}$$

Given a sufficient number of draws, we can compute the mean forecast, forecast intervals, and even a predictive distribution in a manner that has the advantage of not conditioning on plug-in parameter estimates, $\hat{\theta }.$ Rather, the forecasts will reflect uncertainty about the underlying parameters, θ, the current state, $s_T$ , and future shocks, ${\epsilon }_t$ . For further details, see (Del Negro and Schorfheide, 2013, algorithm 2).