9.5 CONDITIONAL FORECASTS

9.5 CONDITIONAL FORECASTS

Scenario forecasting offers a way to compute forecasts conditional on the future values taken by some of the variables in a forecasting model. Fixing the future values of a subset of the variables has two effects on the forecasts. First, and most obviously, the fixed variables no longer have to be predicted, but are assumed known (conditioned on). Second, because these variables have a joint distribution under the forecasting model, the conditional distribution of the remaining variables given the fixed variables will typically change.

Waggoner and Zha (1999) consider conditional forecasts in situations where restrictions are imposed on the future values of a set of endogenous variables which are part of a $\mathrm{VAR}(p)$ model for an $(n\times 1)$ vector, $y.$ . For the special case where y follows a VAR(1) process, their setup reduces to

$$A_0{y}_t+A_1{y}_{t-1}=d+{\epsilon }_t,t=1,\dots ,T,\text{\hspace{1em}(9.47)}$$

where, for $s>0$

$$\begin{array}{r}E\left[{\epsilon }_t{\epsilon }_t^{\prime }|y_{t-s}\right]=I,\\ E\left[{\epsilon }_t|y_{t-s}\right]=0.\end{array}$$

Forecasts from (9.47) are most easily generated by rewriting the model as

$$y_t=c+B_1{y}_{t-1}+A_0^{-1}{\epsilon }_t,$$

$$\mathrm{where}c=A_0^{-1}d,B_1=-A_0^{-1}{A}_1.$$

Waggoner and Zha show that out-of-sample forecasts for period $T+h$ given information at time T , take the form

$$y_{T+h}=K_{h-1}c+B_1^h{y}_T+\sum _{j=1}^h{M}_{h-j}{\epsilon }_{t+j},\text{\hspace{1em}(9.48)}$$

where K and M can be characterized in recursive form:

$$K_h=\sum _{j=0}^h{B}_1^j,$$

$$M_i=A_0^{-1}{B}_1^i,$$

with $B_1^0=I$ . The first two terms in (9.48) capture the usual predictive dynamics from a $\mathrm{VAR}(1)$ model. For unconditional forecasts, the third term involving the impulse response matrices, $M_i$ , would be 0. For forecasts that condition on future values of ${\gamma }_{T+h}$ , this condition need not hold. For example, if we constrain the j th element of ${\gamma }_{T+h}$ to some range $y_{T+h}(j)\in \left[y(j);\bar{y}(j)\right]$ , it follows from (9.48) that this amounts to constraining the sum of weighted shocks so that

$$\sum _{j=1}^h{M}_{h-j}{\epsilon }_{T+j}\in \left[\underline{y}(j)-K_{h-1}c-B_1^h(j)y_T;\bar{y}(j)-K_{h-1}c-B_1^h(j)y_T\right],$$

and hence the expectation of future values of $\epsilon ,$ , conditional on this range information, is no longer 0.

Alternatively, we can consider hard conditions which restrict $q$ elements of $y_{T+h}^{\prime }$ to single values $r(a)$ ,

$$R(a{)}^{\prime }\epsilon =r(a),\text{\hspace{1em}(9.49)}$$

where $a=(\mathrm{vec}(A_0{)}^{\prime }$ vec $(-A_1,d{)}^{\prime }{)}^{\prime }$ $R(a)$ is a $(q\times k)$ weighting matrix that stacks the impulse responses, and $k=hm$ is the number of future shocks, ε. Under the restrictions in (9.49), Waggoner and Zha (1999, Proposition 2) establishes that the one-step conditional distribution of ${\gamma }_{T+h}$ given the vector of parameters, a; all observed data up to time $T+h-1,Z_{T+h-1};$ and assuming Gaussian shocks, $\epsilon ,$ denoted by ${\psi }_R(y_{T+h}|a,Z{Y}_{T+h-1})$ ; is Gaussian:

$$\begin{array}{r}\begin{array}{rl}& p_R(y_{T+h}^{\prime }|a,Z_{T+h-1})=\mathrm{N}\left(c+B_1{y}_{T+h-1}+A_0^{-1}{E}_R[{\epsilon }_{T+h}],A_0^{-1\prime }{V}_R({\epsilon }_{T+h}^{\prime })A_0^{-1}\right),\end{array}\end{array}\text{\hspace{1em}(9.50)}$$

where $E_R[{\mathcal{E}}_{T+h}^{\prime }]$ and $V_R[{\sigma }_{T+h}^{\prime }]$ are the restricted (conditional) mean and variance of ${\epsilon }_{T+h},\mathrm{i}.\mathrm{e}.$

$$\begin{array}{rl}& {\epsilon }_{T+h}^{\prime }\sim \mathrm{N}(E_R[{\epsilon }_{T+h}^{\prime }],V_R({\epsilon }_{T+h}^{\prime })),\\ & E_R[{\epsilon }_{T+h}^{\prime }]=R(a){\left[R(a{)}^{\prime }R(a)\right]}^{-1}r(a),\\ & V_R({\epsilon }_{T+h}^{\prime })=I-R(a){\left[R(a{)}^{\prime }R(a)\right]}^{-1}R(a{)}^{\prime }.\end{array}\text{\hspace{1em}(9.51)}$$

Figure 9.2: Recursive forecasts of the unemployment rate. The figure shows forecasts of the unemployment rate generated by a Bayesian VAR (BVAR), VARs with lag length selected by the AIC or BIC information criteria, a univariate AR with lag length selected by the AIC, and a VAR with 16 lags.

Waggoner and Zha propose a Gibbs sampler that can account for parameter estimation errors in generating these conditional forecasts. Given a starting value for the parameters, $a^{(0)}$ , the procedure first uses (9.50) and (9.51) to generate values $y_{T+1}^{(i)},\hat{\dots },y_{T+h}^{(i)}$ given $a^{(i-1)}$ . Using these values, it then generates draws $a^{(i)}$ from $\begin{array}{r}p(a|{\gamma }_{T+1}^{(i)},\dots ,{\gamma }_{T+h}^{(i)},y_T,\dots ,y_1)\end{array}$ . This is repeated a large number of times to obtain a set of out-of-sample forecasts that account for estimation error.