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16.1 Vector Autoregressions 687

VAR, however, introduces estimation error without adding predictive content,
thereby reducing forecast accuracy.

Determining lag lengths in VARs. Lag lengths in a VAR can be determined using

either F-tests or information criteria.

The information criterion for a system of equations extends the single-equation

information criterion in Section 14.5. To define this information criterion, we

need to adopt matrix notation. Let Σu be the k * k covariance matrix of the VAR
errors and let Σn u be the estimate of the covariance matrix where the i, j element
Σn u 1 T
of is T g t= 1 un it un jt, where un it is the OLS residual from the ith equation and un jt is

the OLS residual from the jth equation. The BIC for the VAR is

BIC(p) = ln[det(Σn u)] + k(kp + 1)ln(TT ), (16.4)

where det(Σn u) is the determinant of the matrix Σn u. The AIC is computed using
Equation (16.4), modified by replacing the term “ln(T)” with “2.”

The expression for the BIC for the k equations in the VAR in Equation (16.4)
extends the expression for a single equation given in Section 14.5. When there is
a single equation, the first term simplifies to ln[SSR(p)>T]. The second term in
Equation (16.4) is the penalty for adding additional regressors; k(kp + 1) is the
total number of regression coefficients in the VAR. (There are k equations, each
of which has an intercept and p lags of each of the k time series variables.)

Lag length estimation in a VAR using the BIC proceeds analogously to the
single equation case: Among a set of candidate values of p, the estimated lag
length pn is the value of p that minimizes BIC(p).

Using VARs for causal analysis. The discussion so far has focused on using VARs for
forecasting. Another use of VAR models is for analyzing causal relationships among
economic time series variables; indeed, it was for this purpose that VARs were first
introduced to economics by the econometrician and macroeconomist Christopher
Sims (1980). (See the box “Nobel Laureates in Time Series Econometrics” at the end
of this chapter.) The use of VARs for causal inference is known as structural VAR
modeling, “structural” because in this application VARs are used to model the
underlying structure of the economy. Structural VAR analysis uses the techniques
introduced in this section in the context of forecasting, plus some additional tools.
The biggest conceptual difference between using VARs for forecasting and using
them for structural modeling, however, is that structural modeling requires very