I. Simultaneity, Systems of Equations and Vector Autoregression Models(VAR)As we have seen, in examining the dynamic relationships between two(or more) variables, the causality may be mutual rather than simply unidirectional. This situation often occurs among macroeconomic variables. Vector autoregression modeling is an attempt to deal with this situation. We can capture the simultaneity in the following specification, referred to as the primitive VAR.A. Primitive VAR(1) y(t) = | -：- ”w(t) + | y(t-1) + :w(t-1) +、. x(t) + ey (t)(2) w(t) = - xy(t) + V y(t-1) + -w(t-1) + ：x(t) + ew (t)In equation one, the endogenous variable y(t) depends upon the contemporaneous value of the other endogenous variable w(t), i.e. they are contemporaneously correlated, and depends as well on lagged values of itself, y(t-1), and lagged values of the other dependent variable, w(t-1). In addition, y(t) depends upon an exogenous variable, x(t), and an error termy (t). The latter is presumed to be orthogonal. Similarly, in equation two, the dependent variable w(t) depends contemporaneously on the other endogenous variable, y(t), and on lagged values of itself and the other dependent variable. There is also dependence on an exogenous variable and its own orthogonal error, e (t), in this illustration. The errors for each equation e(t) and e (t) are presumed to be independent of one another.Both dependent variables can be expressed as distributed lags of the endogenous variables, lagged one, by eliminating w(t) in equation one, and y(t) in equation two, through substitution. For example, use the expression for w(t) in equation two to substitute for w(t) in equation one. The resulting equations are called the VAR in standard form.B. Standard VAR(1 ) y(t) E十，i%(1- E以 + (：i+ &i 0)/(1-盼y(t-1) + (.+ 6 1 知)/(1- 1 盼w(t-1)+ (、+ : (1- L 逆 x(t) + (ey(t) + J ew (t)/(1-(2 ) w(t)母i+(1-如Pj +(Eii+ 功/(1-钏y(t-1) + (吼+ 吻/(1-印时w(t-1)+ ( ： ： +(1- | ：, 1 j x(t) + ( ： ：ey (t) + ew (t)/(1-The things to note in equation one prime is that the dependent variable y(t) depends only onlagged values of the endogenous variables, y(t-1) and w(t-1), called predetermined variables because they predate y(t) by one period. In this example, since there is only one lag in the dependence, the structure is autoregressive of order one. Note that y(t) also depends contemporaneously on the exogenous variable x(t). Note also that the error term in equation one prime, (ey (t) + - - e (t)/(1-I ), is a linear combination of e (t) and ew (t), but is still orthogonal.Equation two prime has a similar structure: w(t) depends on lagged y(t-1) and w(t-1), and hence is first order autoregresssive, and depends contemporaneously on the exogenous variable x(t). The error term, (e (t) + ew (t)/(1-I ), is a linear combination of e (t) and ew (t), is orthogonal, but contemporaneously correlated with the error term in equation one prime. The two error terms for equations one prime and two prime will be contemporaneously correlated unless both Pand E?are zero, whichwill be the case only if y(t) does not depend on w(t) in equation one and w(t) does not depend on y(t) in equation two in the primitive system.Note that if : is zero, the error for equation one prime equalsye(t), i.e. it only depends on the shock or error in y(t) in equation one. In contrast, the error for equation two prime will equal :ey (t) + ew (t), and depends on both the shock to y(t) in equation oney et), and the shock to w(t) in equation two, e (t).These two equations in the standard VAR can be estimated by ordinary least squares(OLS), since the error term in each equation is independent of the regressors, y(t-1), w(t-1), and x(t), and is not autocorrelated, i.e. is orthogonal. This is the econometric motivation for expressing the VAR in standard form since the VAR in primitive form can not be estimated by OLS unless both : and :are zero, in which case the primitive form is equivalent to the standard form, i.e. equation one prime reduces to equation one and equation two prime reduces to equation two.The parameter notation for the standard VAR can be simplified now that we have demonstrated the relationship between the primitive VAR and the standard VAR.(1 ) y(t) = ai + bii y(t-1) + cii w(t-1) + di x(t) + ei(t)(2 ) w(t) = a2 + b2i y(t-1) + c2i w(t-1) + d2 x(t) + e2(t)C. Identification and the Choleski Decomposition of the Variance of the Estimated ResidualsThere are twelve parameters in the primitive VAR, two : s, two s, four s, two s , the variance of (t), and the variance of e (t). There are only eleven parameters to be estimated in the standard VAR: two a s, two b s, two c s, two d i(ts, Hhe variiance of 金(t), and the covariance of et) and e(t). Consequently, there is not enough information from the estimation of the VAR in standard form to uniquely identify the parameters of the primitive VAR. An additional restriction is nec