20.3 MULTIVARIATE FORECASTING MODELS

20.3 MULTIVARIATE FORECASTING MODELS

When modeling a vector of persistent variables, a natural starting point is to use a vector autoregression, assuming MSE loss. We could simply allow the estimation method to determine the parameters, ignoring the possibility of unit roots. Such VARs in levels have a long history in forecasting; the original Litterman VAR used a prior that each of the variables followed an AR(1) process with unit roots (see section 9.3.2).

Recall that we can write the VAR in levels as $y_t=A(L)y_{t-1}+{\epsilon }_t$ . As for univariate models, the effect of estimation error on model risk differs from the stationary case but is of the same order and will thus disappear for short (fixed) horizons at rate T. To see this, note that the vector of one-step-ahead forecast errors is

$$y_{T+1}-f_{T+1|T}={\epsilon }_{T+1}+[A(L)-\hat{A}(L)]y_T.\text{\hspace{1em}(20.7)}$$

As in the univariate case, the term due to estimation error will disappear at rate $T^{1/2}$ and so the squared error term disappears at rate T.

When each of the n variables in yt has a unit root, in the absence of cointegration between the variables we can simply impose the known unit roots on the system and estimate the VAR in differences. In this situation the results of chapter 9 are directly relevant. When the n variables are cointegrated and there are only $n-r$ unit roots $(n>r)$ in the model, the Granger representation theorem (Engle and Granger, 1987) shows that the correct specification is an error correction model (ECM). For this case, the VAR can be written as

$$\Delta y_t=\alpha {\beta }^{\prime }{y}_{t-1}+\tilde{A}(L)\Delta y_{t-1}+{\epsilon }_t,\text{\hspace{1em}(20.8)}$$

where ${\beta }^{\prime }{y}_t$ are the r cointegrating vectors. These are linear combinations of ${\gamma }_t$ that have all roots outside the unit circle. Here α is an $(n\times r)$ vector of “impact” coefficients.

It is natural to ask how useful it is to impose the cointegrating restrictions for purposes of forecasting h steps ahead. This is similar to asking how different are forecasts using the VAR in levels (20.7) versus using the error correction model (20.8). A second question is how valuable the cointegrating vector is for forecasting h periods into the future, i.e., how different are forecasts using the VAR in differences— thus omitting information in the cointegrating vector—versus using the ECM. We address each question in turn.

The Granger representation theorem shows that r cointegrating relations between the n unit root variables imply $(n-r{)}^2$ restrictions on the parameters of the VAR in levels, including normalizing the cointegrating vector; these restrictions result in the error correction model. Hence, differences between forecasts based on the VAR and

ECM specification arise due to difference from imposing versus not imposing these restrictions. For the univariate model examined above, the analogy is the difference between imposing the unit root or estimating this parameter. In the univariate case we saw that such a constraint results in better forecasts when there is exactly a unit root, but can increase the expected loss when the restriction is false since any reduction in estimation error gets overwhelmed by additional biases. In the multivariate case considered here, a similar trade-off occurs. However, there are now a larger number of restrictions (allowing the gains to be bigger) and additional dimensions in which the restrictions might not hold exactly, making it more difficult to examine potential losses. Abadir, Hadri, and Tzavalis (1999) examine the impact of estimating the $r^2$ coefficients on estimation error and find that the effects can be large, essentially because the dimension of the restrictions is also large. They do not examine how forecasts are affected, however.

The second question relates to the value of using the information in the cointegrating vectors for forecasting versus simply ignoring cointegration but still imposing unit roots, i.e., compared to the misspecified model in differences. Using the misspecified model was (and to some extent remains) a standard approach to forecasting with persistent variables. This approach essentially asks whether omitting $\alpha {\beta }^{\prime }{y}_{t-1}$ from the forecasting model (20.8) results in any loss. It would seem that if α is close to 0 and/or the cointegrating vector ${\beta }^{\prime }{y}_{t-1}$ is not very persistent, omitting these variables is not going to have much effect. Conversely, if α is large and ${\beta }^{\prime }{y}_{t-1}$ is very persistent, omitting the error correction term might have an effect on the forecast. This turns out to be true. To see this, consider the error correction model for a bivariate system with a single cointegrating relationship $(i.e.,n=2,r=1)$ and no additional dynamics $(i.e.,\tilde{A}(L)=0)$ . Through recursive substitution, we can write

$$y_{t+h}-y_t=\left(\frac{1-{\rho }_c^h}{1-{\rho }_c}\right)\alpha {\beta }^{\prime }{y}_t+\sum _{i=1}^h(\alpha {\beta }^{\prime }+I{)}^{h-i}{\epsilon }_{t+i},\text{\hspace{1em}(20.9)}$$

where ${\rho }_c=1+{\beta }^{\prime }\alpha$ is the autoregressive parameter for the AR(1) cointegrating vector ${\beta }^{\prime }{y}_t$ and so measures the persistence in the cointegrating vector. This example helps clarify the earlier discussion. The bigger is α, the larger is the contribution of the cointegrating vector to the h-step-ahead forecast. Similarly, the bigger is ${\rho }_c$ , the larger is the impact of the cointegrating vector on the forecast. This explains the seeming anomaly between results suggesting a small impact on the forecast from including the cointegrating vector versus results suggesting a larger impact. Engle and Yoo (1987) present Monte Carlo evidence which shows that the cointegrating term can be useful at moderate horizons—their choice for parameters has $\alpha =(-0.4,0.1)$ and ${\rho }_c=0.4$ , so the cointegrating vector is moderately persistent. Christoffersen and Diebold (1998) set ${\rho }_c=0$ so the cointegrating vector is not serially correlated in their simulations and so they obtain little value from including the cointegrating vector. Both Engle and Yoo (1987) and Christoffersen and Diebold (1998) make the point that as h gets large, the relative value of using the cointegrating vector decreases. This can easily be seen in the above model. Provided that $|{\rho }_c|<1,{\beta }^{\prime }{y}_t$ will be stationary with finite asymptotic variance and so the contribution of the error correction term has a bounded variance as h gets large. In contrast, the variance of the unpredictable component (the last term in (20.9)) is increasing in h and so the relative cost from ignoring the cointegrating vector goes to 0 as the forecast horizon increases.

The above analysis assumed that the cointegrating model is correctly specified. There are many ways in which a practitioner can misspecify the model. First, the persistence in the variables may not mean that they have exact unit roots. Second, we may not know the exact number of cointegrating relations in the model. As in the univariate case, one can still impose the misspecified model, or the forecaster could use pre-tests to attempt to estimate the most appropriate model. Such choices involve trade-offs in which imposing the restriction works well if it is close enough to being true. Pre-tests work well when either the null of the pre-test is correct or, alternatively, if the departure of the model from this null is large enough that the test has power close to 1; pre-tests perform poorly between these possibilities, however. Reinsel and Ahn (1992) provide a small Monte Carlo analysis with four variables and two cointegrating vectors. They find that underdifferencing (imposing too few cointegrating vectors) works well in the short run, while overdifferencing works well in the long run since the effect of the cointegrating vectors will to some extent have petered out. They also evaluate a pre-testing approach, although it is difficult to derive general conclusions from this. Lin and Tsay (1996) conduct a larger set of simulations, finding some gains from imposing cointegration constraints on the model; however they do not find any gains in an analysis of actual data.