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686 Chapter 16 Additional Topics in Time Series Regression

Inference in VARs. Under the VAR assumptions, the OLS estimators are consis-
tent and have a joint normal distribution in large samples. Accordingly, statistical
inference proceeds in the usual manner; for example, 95% confidence intervals
on coefficients can be constructed as the estimated coefficient {1.96 standard
errors.

One new aspect of hypothesis testing arises in VARs because a VAR with k
variables is a collection, or system, of k equations. Thus it is possible to test joint
hypotheses that involve restrictions across multiple equations.

For example, in the two-variable VAR(p) in Equations (16.1) and (16.2), you
could ask whether the correct lag length is p or p - 1; that is, you could ask
whether the coefficients on Yt - p and Xt−p are zero in these two equations. The
null hypothesis that these coefficients are zero is

H0: b1p = 0, b2p = 0, g1p = 0, and g2p = 0 . (16.3)

The alternative hypothesis is that at least one of these four coefficients is nonzero.
Thus the null hypothesis involves coefficients from both of the equations, two
from each equation.

Because the estimated coefficients have a jointly normal distribution in large
samples, it is possible to test restrictions on these coefficients by computing an
F-statistic. The precise formula for this statistic is complicated because the nota-
tion must handle multiple equations, so we omit it. In practice, most modern
software packages have automated procedures for testing hypotheses on coeffi-
cients in systems of multiple equations.

How many variables should be included in a VAR? The number of coefficients in
each equation of a VAR is proportional to the number of variables in the VAR.
For example, a VAR with 5 variables and 4 lags will have 21 coefficients (4 lags
each of 5 variables, plus the intercept) in each of the 5 equations, for a total of 105
coefficients! Estimating all these coefficients increases the amount of estimation
error entering a forecast, which can result in deterioration of the accuracy of the
forecast.

The practical implication is that one needs to keep the number of variables in
a VAR small and, especially, to make sure the variables are plausibly related to
each other so that they will be useful for forecasting one another. For example,
we know from a combination of empirical evidence (such as that discussed in
Chapter 14) and economic theory that the growth rate of GDP, the term spread,
and the rate of inflation are related to one another, suggesting that these variables
could help forecast one another in a VAR. Including an unrelated variable in a