Vector Autoregressions (VAR and VEC)
The structural approach to simultaneous equations modeling uses economic theory to describe the
relationships between several variables of interest. The resulting model is then estimated, and used
to test the empirical relevance of the theory.
Unfortunately, economic theory is often not rich enough to provide a tight specification of the dynamic
relationship among variables. Furthermore, estimation and inference are complicated by the fact that
endogenous variables may appear on both the left and right sides of the equations.
These problems lead to alternative, non-structural, approaches to modeling the relationship between
several variables. Here we describe the estimation and analysis of vector autoregression (VAR) and
the vector error correction (VEC) models. We also describe tools for testing for the presence of
cointegrating relationships among several variables.
The vector autoregression (VAR) is commonly used for forecasting systems of interrelated time
series and for analyzing the dynamic impact of random disturbances on the system of variables.
The VAR approach sidesteps the need for structural modeling by modeling every endogenous
variable in the system as a function of the lagged values of all of the endogenous variables in the
The mathematical form of a VAR is
is a k vector of endogenous variables,
is a d vector of exogenous variables,
and B are matrices of coefficients to be estimated, and
is a vector of innovations that may be
contemporaneously correlated with each other but are uncorrelated with their own lagged values and
uncorrelated with all of the right-hand side variables.
Since only lagged values of the endogenous variables appear on the right-hand side of each
equation, there is no issue of simultaneity, and OLS is the appropriate estimation technique. Note that
the assumption that the disturbances are not serially correlated is not restrictive because any serial
correlation could be absorbed by ad