(2SLS) estimator (which we will describe in Section 9.6) are the same if the equation is exactly identified. This implies that we can get the IV estimator by the 2SLS method by changing the model so that the supply function is exactly identified. In this case, a standard computer program like the SAS can be used to get the estimates as well as their standard errors. In fact, this is the procedure we have followed.

For instance, to get the IV estimator of the supply function using Y as the instrumental variable, we can specify the model as

g = Po + 1 , + lY + demand function

(2 = «0 + a,P„, + «2 + supply function

Now the supply function is identified exactly.

In this particular case, this does not seem to be an unreasonable model.

Before we leave this example, we will present a case where the 7? from the IV method is negative. Consider the model

= Po + Pi, + P2A + w demand function (Q = ao + aP„, + «2 + " supply function

The first equation is exactly identified. We will use 5 as the instrumental variable for P„. (The IV estimator is the same as the 2SLS estimator.) The OLS estimation of the demand function gave the following results (figures in parentheses are /-ratios):

Q = - 15.30 + 2.064P,, + 1.418A E} = 0.9431

(-14.50) (6.60) (6.76)

The IV estimation gave the following results:

= -304.13 + 119.035P,. - 52.177 = -467.3

(-0.03) (0 03) (-0.03)

What is the problem? To see what the problem is, let us look at the reduced-form equations:

= -(10.42) + 1.316A + 1.0855 R = 0.8208

(-8.12) (1.75) (1.50)

j = 2.47 + 0.449A + 0.009125 R = 0.4668

(4 40) (1.37) (0.03)

The coefficient of 5 in the equation for P„ is almost zero. From the relationships (9.3) and (9.4) between the parameters in the reduced form and structural form discussed in Section 9.3, we note that this impUes that Pi -» 00 and «2 is indeterminate. Thus the demand function is not identified. Whenever the R from the IV estimator (or the 2SLS estimator) is negative or very low and the R from the OLS estimation is high, it should be concluded that something is wrong with the specification of the model or the identification of that particular equation.

9.6 Methods of Estimation: The Two-Stage Least Squares Method

The two-stage least squares (2SLS) method differs from the instrumental variable (IV) method described in Section 9.5 in that the fs are used as regressors rather than as instruments, but the two methods give identical estimates. Consider the equation to be estimated:

1 = + cyz, + (9.9)

The other exogenous variables in the system are Zi, Zi. and Z4.

Let 2 be the predicted value of 2 from a regression on 2 on „ Zi, z, Z4 (the reduced-form equation). Then

2 = 2 + v2

where v,, the residual is uncorrelated with each of the regressors Zi, Z2, Z4 and hence with y, as well. (This is the property of least squares regression that we discussed in Chapter 4.) The normal equations for the efficient IV method are

S 2( \ - , 2 - C,Z,) = 0

2 ,( , - , 2 - c,zi) = 0 Substituting y, = 2 + v2 we get

2 2( - , 2 - C,Z,) - btZ 2 2 = 0 g-y

2 ,( 1 - , 2 - , ,) - b, z,v2 = 0

But Z1V2 = 0 and 2 22 = 0 since Zi and 2 are uncorrelated with . Thus equations (9.11) give

2 yibi - 2 - CZ,) = 0 9 j2)

S zfy, - , 2 - c,Zi) = 0

But these are the normal equations if we replace y, by 2 in (9.9) and estimate the equation by OLS. This method of replacing the endogenous variables on the right-hand side by their predicted values from the reduced form and estimating the equation by OLS is called the two-stage least squares (2SLS) method. The name arises from the fact that OLS is used in two stages:

Stage I. Estimate the reduced-form equations by OLS and obtain the predicted ys.

Stage 2. Replace the right-hand-side endogenous variables by ys and estimate the equation by OLS.

Note that the estimates do not change even if we replace y, by y, in equation (9.9). Take the normal equations (9.12). Write

, = , + V,

where v, is again uncorrelated with each of Z, Zb z, Z4. Thus it is also uncorrelated with y, and y,, which are both linear functions of the zs. Now substitute y, = y, + V in equations (9.12). We get

2 Slifl - , 2 - CiZ,) + 2 21 = 0 2 2i(yi - 1 2 - CZ,) + 2 iV, = 0

The last terms of these two equations are zero and the equations that remain are the normal equations from the OLS estimation of the equation

y, = , 2 + C,Z + w

Thus in stage 2 of the 2SLS method we can replace all the endogenous variables in the equation by their predicted values from the reduced forms and then estimate the equation by OLS.

What difference does it make? The answer is that the estimated standard errors from the second stage will be different because the dependent variables is 1 instead of y,. However, the estimated standard errors from the second stage are the wrong ones anyway, as we will show presently. Thus it does not matter whether we replace the endogenous variables on the right-hand side or all the endogenous variables by ys in the second stage of the 2SLS method.

The preceding discussion has been in terms of a simple model, but the arguments are general because all the ys are uncorrelated with the reduced form residuals vs. Since our discussion has been based on replacing by - v, the arguments all go through for the general models.

Computing Standard Errors

We will now show how the standard errors we obtain from the second stage of the 2SLS method are not the correct ones and how we can obtain the correct standard errors. Consider the very simple model

y. = 2 + « (9.13)

where y, and yj are endogenous variables. There are some (more than one) exogenous variables in the system. We first estimate the reduced-form equation for 2 and write

2 = 2 + V2

The IV estimator of p is obtained by solving 2 2( 1 ~ ) = 0» or

2 2 , Piv ~

The 2SLS estimator of is obtained by solving X ! ~ 3*2) = 0» or

P2SLS ~ V -9

z yl