2.4 Endogenous explanatory variables

The error terms in these three equations are omitted because they are irrelevant to determining if an equation is identified—remember, identification is an algebraic problem, not a statistical issue. There are 3 endogenous variables in the system and 3 equations in the system. Also, there are 5 exogenous variables in the system of equations. Equation (13) is exactly identified; Equation (14) is over-identified; and Equation (15) is under-identified. What this means is (1) Equation (13) can be estimated directly from the reduced form equation (using indirect-least-squares) or using TSLS; (2) Equation (14) must be estimated using TSLS; and Equation (15) cannot be estimated. Table 2 summarizes how to determine if an equation is or is not identified. Basically, if the number in column 2 equals the number in column 3, the equation is exactly identified. If the number in column 2 is less than the number in column 3, the equation is over-identified. Finally, if the number in column 2 is greater than the number in column 3, the equation is under-identified. In these notes I discuss only what is known in the literature as the order condition for identification. The order condition is necessary for identification. Another condition—the rank condition —is a sufficient condition. See Greene (1990: Chapter 19, especially pp. 600-609) for a fuller discussion of simultaneous-equation models and the identification problem.

One other thing to notice is the similarity of TSLS to IV estimation. The exogenous variables play the role of instruments in TSLS estimation. By implication, the instruments in an IV estimation must not include any of the exogenous variables in the equation. Using one of the exogenous variables in an equation as an instrument will create perfect multicollinearity in the first stage regression. Similarly, one of the

ways to isolate potential instruments in a regression is to think of what system of equation the equation is and then ask what exogenous variables in that system are not included in the equation. These excluded exogenous variables are potential instruments.

Tsls and iv in stata

The command for estimating an equation in Stata using two-stages least squares (TSLS) is a bit tricky. Assume that you want to estimate equations (13) and (14) in the model discussed above. We exclude Equation (15) from this discussion because it is under-identified and, thus, cannot be estimated. For simplicity assume that each variable assumes the name for it in Table 2. Thus, in our Stata commands Y1 refers to variable Thus, in our Stata commands Y1 refers to variable $y_1$ and so on. The command to estimate either a TSLS or an IV regression is the same. The advantage of the ivreg command is that it allows you to estimate a single equation of a system of equations without fully specifying the equations in the rest of the model. Use the command reg3 if you want to specify the whole model or use Three-Stage Least Squares. The command, ivreg , consists of three major parts—(1) the name of the dependent variable is followed by (2) a list of the names of the exogenous variables that are being used as explanatory variables and then followed in parentheses by (3) the information needed to estimate the first stage (the list of the endogenous variables that are explanatory variables along with the names of the exogenous variables in the system that are excluded from the equation or, in the case of IV, a list of the instruments). The description of the command “ivreg depvar [varlist1] (varlist2=varlist_iv)” in the Stata help file is “ivreg fits a linear regression model using instrumental variables (or two-stage least squares) of depvar on varlist1 and varlist2 using varlist_iv (along with varlist1) as instruments for varlist2. In the language of two-stage least squares, varlist1 and varlist_iv are the exogenous variables and varlist2 the endogenous variables.”