Note that equation 3 is an identity and does not have any parameters to be estimated. Hence we do not need to discuss its identification. The number of endogenous variables minus one is 6 in this model. By the order condition we get the result that equations 1 and 4 are exactly identified and equations 2, 5, 6, and 7 are over-identified.

Let us look at the rank condition for equation 1. The procedure is similar for other equations. Delete the first row and gather the columns for the missing variables /, N, P, W, G, M. We get

1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 10 0 0 0 0 1110 0 0 1110 0

Since we have six rows (and six columns) whose elements are not all zero, the equation is identified. It can be checked thai the same is true for equations 2, 4, and 5. However, for equations 6 and 7, we cannot find six rows whose elements are not all zero. Thus, equations 6 and 7 are not identified by the rank condition even though they are over-identified by the order condition.

9.5 Methods of Estimation: The Instrumental Variable Method

In previous sections we discussed the identification problem. Now we discuss some methods of estimation for simultaneous equations models. Actually, we have already discussed one method of estimation: the indirect least squares method. However, this method is very cumbersome if there are many equations and hence it is not often used. Here we discuss some methods that are more

The question is which of these equations is under-identified, exactly identified, and over-identified. To answer this question, we prepare the following table. A 1 denotes that the variable is present, a 0 denotes that it is missing.

-z{y - fix) = Q

This gives

S E zifix + u) J,zu

But 5! zu/ zx can be written as lln X zul l/n zx. The probability limit of this expression is

coviz, u)

COv(2, x)

since ( , x) 0. Hence plim fi = fi, thus proving that p is a consistent estimator for p. Note that we require z to be correlated with x so that cov(z, jc) #0.

generally applicable. These methods of estimation can be classified into two categories:

1. Single-equation methods (also called "limited-information methods").

2. System methods (also called "full-information methods").

We will not discuss the full-information methods here because they involve algebraic detail beyond the scope of this book. We discuss single-equation methods only. In these methods we estimate each equation separately using only the information about the restrictions on the coefficients of that particular equation. For instance, in the illustrative example in Section 9.4 when we estimate the consumption function (the first equation), we just make use of the fact that the variables /, N, P, W, G, M are missing and that the last two variables are exogenous. We are not concerned about what variables are missing from the other equations. The restrictions on the other equations are used only to check identification, not for estimation. This is the reason these methods are called limited-information methods. In full-information methods we use information on the restrictions on all equations.

A general method of obtaining consistent estimates of the parameters in simultaneous-equations models is the instrumental variable method. Broadly speaking, an instrumental variable is a variable that is uncorrelated with the error term but correlated with the explanatory variables in the equation.

For instance, suppose that we have the equation

= fix +

where x is correlated with u. Then we cannot estimate this equation by ordinary least squares. The estimate of fi is inconsistent because of the correlation between X and M. If we can find a variable z that is uncorrelated with u, we can get a consistent estimator for fi. We replace the condition cov(z, ) = 0 by its sample counterpart

Now consider the simultaneous-equations model

y, = + CyZx + CjZn + = 2 , + cZi + U2

where y„ are endogenous variables, z,, Zi, , are exogenous variables, and M, , are error terms. Consider the estimation of the first equation. Since , and Z2 are independent of ,, we have cov(zi, ,) = 0 and cov {zi, Mi) = 0. However, 2 is not independent of ,. Hence cov( V2, «i) 7 0. Since we have three coefficients to estimate, we have to find a variable that is independent of M. Fortunately, in this case we have Z3 and cov(z„ «i) = 0. z is the instrumental variable for yj. Thus, writing the sample counterparts of these three co-variances, we have the three equations

S Zilji - 2 - CZ - C2Z2) = 0

- S z - , 2 - c,Zi - C2Z2) = 0 (9.8)

S , - 2 - c,z, - C2Z2) = 0

The difference between the normal equations for the ordinary least squares method and the instrumental variable method is only in the last equation.

Consider the second equation of our model. Now we have to find an instrumental variable for y, but we have a choice of , and Z2- This is because this equation is over-identified (by the order condition).

Note that the order condition (counting rule) is related to the question of whether or not we have enough exogenous variables elsewhere in the system to use as instruments for the endogenous variables in the equation with unknown coefficients. If the equation is under-identified we do not have enough instrumental variables. If it is exactly identified, we have just enough instrumental variables. If it is over-identified, we have more than enough instrumental variables. In this case we have to use weighted averages of the instrument variables available. We compute these weighted averages so that we get the most efficient (minimum asymptotic variance) estimators.

It has been shown (proving this is beyond the scope of this book) that the efficient instrumental variables are constructed by regressing the endogenous variables on all the exogenous variables in the system (i.e., estimating the reduced-form equations). In the case of the model given by equations (9.8), we first estimate the reduced-form equations by regressing y, and Vj on Zu Z2, Z3. We obtain the predicted values y, and 2 and use these as instrumental variables. For the estimation of the first equation we use 2, and for the estimation of the second equation we use y,. We can write y, and 2 as linear functions of Zl, Z2, Zv Let us write