Vector autoregression

Vector autoregression (VAR) is an econometric model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregressive model (AR model) by allowing for more than one evolving variable. All variables in a VAR enter the model in the same way: each variable has an equation explaining its evolution based on its own lagged values, the lagged values of the other model variables, and an error term. VAR modeling does not require as much knowledge about the forces influencing a variable as do structural models with simultaneous equations: The only prior knowledge required is a list of variables which can be hypothesized to affect each other intertemporally.

A VAR model describes the evolution of a set of k variables (called endogenous variables) over the same sample period (t = 1, ..., T) as a linear function of only their past values. The variables are collected in a k × 1 vector y_t, which has as the i ^th element, y_i,t, the observation at time "t" of the i ^th variable. For example, if the i ^th variable is GDP, then y_i,t is the value of GDP at time t.

A p-th order VAR, denoted VAR(p), is

where the l-periods back observation y_t−l is called the l-th lag of y, c is a k × 1 vector of constants (intercepts), A_i is a time-invariant k × k matrix and e_t is a k × 1 vector of error terms satisfying

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