9.2.4 Multiperiod Forecasts with VARs

9.2.4 Multiperiod Forecasts with VARs

VARs are ideally designed for generating multiperiod forecasts. For the $\mathrm{VAR}(1)$ specification,

$$y_{t+1}=c+A{y}_t+u_{t+1},u_{t+1}\sim (0,\Sigma ),\text{\hspace{1em}(9.14)}$$

with serially uncorrelated innovations $u_{t+1},$ the h-step-ahead value can be written

$$y_{t+h}=(I-A^h)(I-A{)}^{-1}c+A^h{y}_t+\sum _{i=1}^h{A}^{h-i}{u}_{t+i},\text{\hspace{1em}(9.15)}$$

and so the forecast under MSE loss becomes

$$f_{t+h\mid t}=(I-A^h)(I-A{)}^{-1}c+A^h{y}_t.\text{\hspace{1em}(9.16)}$$

Higher-order dynamics can easily be handled in this setup by writing the VAR in companion form. For example, the VAR(2) model,

$$y_{t+1}=c+A_1{y}_t+A_2{y}_{t-1}+u_{t+1},$$

can be rewritten as a VAR(1) for ${\tilde{y}}_{t+1}=(y_{t+1}^{\prime },y_t^{\prime }{)}^{\prime }$

$$\begin{array}{c}\genfrac{}{}{0ex}{}{{\tilde{y}}_{t+1}=\left(\genfrac{}{}{0ex}{}{y_{t+1}}{y_t}\right)=\left(\genfrac{}{}{0ex}{}{A_1}{I}\right)A_2}{0}\end{array}\left(y_{t-1}\right)+\left(\genfrac{}{}{0ex}{}{u_{t+1}}{0}\right).$$

When we are interested in forecasting y multiple periods ahead, an important issue is whether it is best to estimate the model at the highest frequency (corresponding to a one-period horizon) and then iterate the model h steps ahead as in (9.16), or whether it is better to use a loss function defined directly on the h-step forecast error. The latter entails estimating a VAR with the variables lagged h periods, i.e., in the VAR( p) case,

$$y_{t+h}=c+\sum _{i=1}^P{A}_{hi}{y}_{t+1-i}+u_{t+h}.\text{\hspace{1em}(9.17)}$$

Using maximum likelihood estimation on data at the highest available frequency tends to generate less sampling variation and so is more efficient if the model is correctly specified than the loss-function-based approach (9.17) that fits the VAR to observations h periods ahead. Conversely, if the VAR is misspecified, the direct forecasting approach based on (9.17) is potentially more robust as it is simply a linear projection model that avoids the cumulated effect of iterating multiple periods ahead on a misspecified model as in (9.16).

A particular source of misspecification for the VAR is the choice of lag order. Schorfheide (2005) develops a model selection criterion that can be used to select both the lag order of the VAR and the parameter estimation method—maximum likelihood or loss based.

Pesaran, Pick, and Timmermann (2011) propose SUR (seemingly unrelated) estimation methods and a modified AIC for model selection that accounts for serial correlation in the residuals from multistep direct forecasts induced by the resulting overlaps. When applied to 170 macroeconomic and financial variables, they find empirically that both SUR estimation and their modifications to the AIC can help improve the predictive accuracy of direct multistep forecasts. Moreover, they find that forecasts from a factor-augmented VAR tend to outperform univariate forecasts (with the exception of variables such as prices and wages), suggesting that multivariate (factor) information helps improve the benchmark autoregressive forecasts.

Example 9.2.1 (Log-linearized present value model for stock prices). Campbell and Shiller (1988) use a first-order Taylor expansion to express the continuously compounded stock return in period $t+1,r_{t+1},$ as an approximate linear function of the logarithms of current and future stock prices, $\mathbf{P}\mathbf{t},\mathbf{P}\mathbf{t}+1,$ and the future log-dividend, $d_{t+1}:$

$$r_{t+1}\approx k+\rho p_{t+1}+(1-\rho )d_{t+1}-p_t,\text{\hspace{1em}(9.18)}$$

where $\rho$ is some parameter close to (but below) 1, and k is a constant. Rearranging, we get a recursive equation for log-prices:

$$p_t=k+\rho p_{t+1}+(1-\rho )d_{t+1}-r_{t+1}.$$

Iterating forward under the transversality condition that lim ${}_{j\infty }{\rho }^j{E}_t[{\rho }_{t+j}]=0$ and taking expectations conditional on current information, we have

$${\mathbf{p}}_t=\frac{k}{1-\rho }+(1-\rho )E_t\left[\sum _{j=0}^{\infty }{\rho }^j{d}_{t+1+j}\right]-E_t\left[\sum _{j=0}^{\infty }{\rho }^j{\mathbf{r}}_{t+1+j}\right].\text{\hspace{1em}(9.19)}$$

According to this expression, stock prices depend on an infinite sum of expected future dividends and returns and will be higher, the higher the expected future dividends and the lower the expected future returns. Key to the present value model is therefore how such expectations are formed. VARs are ideally suited to address this question since they provide a framework for iteratively generating multiperiod forecasts.

To illustrate this point, let $z_{t+1}$ be a vector of state variables with $z_{1t}=p_t,z_{2t}=d_t,$ $z_{3t}=r_t$ and assume that

$$\begin{array}{r}{z}_{t+1}=A{z}_t+{\epsilon }_{t+1},\end{array}$$

so that

$$E_t[z_{t+1+j}]=A^{j+1}{z}_t.$$

Further, define selection vectors $e_1=(1000\dots 0{)}^{\prime },e_2=(0100\dots 0{)}^{\prime },e_3=$ $(0010\dots 0{)}^{\prime }$ so $p_t=e_1^{\prime }{z}_t,d_t=e_2^{\prime }{z}_t,r_t=e_3^{\prime }{z}_t$ . Assuming that $(\rho A{)}^{-1}$ exists, we have (up to a constant) from (9.19),

$$\begin{array}{l}{p}_t=(1-\rho )\sum _{j=0}^{\infty }{\rho }^j{e}_2^{\prime }{A}^{j+1}{z}_t-\sum _{j=0}^{\infty }{\rho }^j{e}_3^{\prime }{A}^{j+1}{z}_t\\ =\left[(1-\rho )e_2^{\prime }A(I-\rho A{)}^{-1}-e_3^{\prime }A(I-\rho A{)}^{-1}\right]z_t.\end{array}$$

This example illustrates how useful VARs are for generating an infinite sequence of internally consistent forecasts.