7.3.3 Direct versus Iterated Multiperiod Forecasts

7.3.3 Direct versus Iterated Multiperiod Forecasts

In many situations, multiperiod forecasts $(h>1)$ may be of interest and so there is a question of whether an iterated or a direct forecasting approach should be used. Under the iterated approach the forecasting model is estimated at a frequency higher than the forecast horizon and iterated upon to obtain multistep forecasts. For example, monthly data could be used to estimate a model used to provide quarterly forecasts. Under the direct approach, the forecasting model is matched with the desired forecast horizon. Hence, the dependent variable is dated period $t+h$ , while all predictor variables are dated period $t$.

It is common to simply plug in the parameter estimates of the single-period model and iterate forward to the desired horizon using chain rules such as (7.25). For the $\mathrm{AR}(1)$ model this entails using the estimated value, ${\hat{\varphi }}_1$ , to obtain a forecast $f_{t+1|t}={\hat{\varphi }}_1{y}_t,f_{t+2|t}={\hat{\varphi }}_1^2{y}_t$ , or, more generally, $f_{t+h|t}={\hat{\varphi }}_1^h{y}_t$

An alternative approach is to estimate the parameters directly by projecting y on information lagged by h or more periods, using the recursively substituted model. Returning to the AR(1) example, by recursive substitution,

$$y_{t+h}={\varphi }_1^h{y}_t+\sum _{s=0}^{h-1}{\varphi }_1^s{\epsilon }_{t+h-s}.$$

The direct forecasting method entails regressing ${\gamma }_t$ on $y_{t-h}$ to obtain an estimate of ${\varphi }_1^h$ , which can then be used to construct the forecast. Note that if overlapping data are used and a model is fitted to every data point, then the error term will follow an $\mathrm{MA}(h-1)$ process even if the underlying innovations ${\epsilon }_t$ are serially uncorrelated. This can potentially be exploited for efficiency gains in the estimation. Alternatively, if nonoverlapping data are used, i.e., if the data are sampled every $h$ periods, then the errors in this regression are serially uncorrelated provided that ${\epsilon }_t$ is serially uncorrelated. More generally, for an $\mathrm{AR}(p)$ regression model,

$\begin{array}{r}{y}_t={\sum }_{s=1}^p{\varphi }_s{y}_{t-s}+{\epsilon }_t,\end{array}$ we have

$$y_{t+h}={\varphi }_h(L)y_t+{\theta }_h(L){\epsilon }_{t+h}=\sum _{s=1}^p{\varphi }_s^h{y}_t+\sum _{s=0}^{h-1}{\theta }_s{\epsilon }_{t+h-s},\text{\hspace{1em}(7.31)}$$

where ${\varphi }_s^h(L)$ and ${\theta }_s$ are nonlinear functions of the original $\mathrm{AR}(p)$ coefficients ${\varphi }_s$ Equation (7.31) suggests regressing ${\gamma }_t$ on $\left\{y_{t-h},y_{t-h-1},\dots ,y_{t-h-p+1}\right\}$ to directly obtain a forecasting model.7

A large theoretical and empirical literature compares the predictive accuracy of the iterated and direct forecasting methods.8 Theoretical analysis suggests a basic trade-off. When the autoregressive model is correctly specified, the iterated approach makes more efficient use of the data and so tends to produce good forecasts.9 Conversely, by virtue of being a linear projection, the direct approach tends to be more robust towards misspecification. Which approach performs best will therefore depend on the true data-generating process, the degree of model misspecification, both of which are unknown, as well as on the extent of parameter estimation error which reflects the sample size. Iterating on the parameters to obtain a multistep forecast generally leads to good forecasts when the model is not grossly misspecified. However, when the model is misspecified, iteration on the misspecified model can exacerbate biases and may result in a larger MSE, although the effects appear to be small in the Monte Carlo study in Bhansali (2002). Empirical evidence presented in Marcellino, Stock, and Watson (2006) suggests that the iterated approach typically works best across a range of variables and improves at longer horizons, although the results can depend on how the lag length of the AR polynomial is selected.