- 3 =

where n is the sample size. Since is independent of the exogenous variables in the system and is a linear function of the exogenous variables, we have

pUm - 2 2« = 0 n

If we assume that

plim S 0

we see that plim = p and thus 0 is a consistent estimator for p.

To compute the asymptotic variance of p, note that as / <», var(p) 0. Hence it is customary to consider n var(p). We have

n var(P) = plim (P - P)

,. IWn) S ) S 2«]

= -limYM-

,. (1 )(2: 2 )(2 ) = [(IM)SyF

erf, plim Ijl/n) S yj] {plim [(l z)2;y!F

= crplim 2 y) (9.14)

where al = var{u). In practice, we get an estimate of var(P) as J, yl, where al is a consistent estimator of of,. This is obtained by plugging in the consistent estimator p of p in (9.13) and computing the sum of squares of residuals. Thus

Since X = 2 + ) = 2 the two estimators are identical and we will drop the subscripts IV and 2SLS and just write 0.

l.yl in

(Note: S 2 2 = S Si) Hence

(l l) 2 >2«

V2

Comparing this with (9.15), we note that what we need is of, and what we get from the second stage of 2SLS is al. The denominator is correct. It is the estimate of the error variance that is wrong. We can correct this by multiplying the standard errors obtained by aja,,, where

= - 2 (y. -

The latter is the expression we get from the OLS program at the second stage. Most computer programs make this correction anyway, so when using the standard programs we do not have to worry about this problem. The discussion above is meant to show how the correct standard errors are obtained.

The preceding results are also valid when there are exogenous variables in equation (9.13). We have considered a simple model for ease of exposition.

Illustrative Example

In Table 9.3 data are provided on commercial banks loans to business firms in the United States for 1979-1984. The following demand-supply model has been estimated: Demand for loans by business firms:

"The model postulated here is not necessarily the right model for the problem of analyzing the commercial loan market It is adequate for our purpose of a two-equation model with both equations over-identified

There is the question of whether we should use n as divisor or {n - 1). Since the expression is asymptotic, the choice does not matter. Note that if we ignored the fact that is a random variable, then using (9.14) we would have written

as in a simple regression model. In effect, this is the expression we use in getting the asymptotic standard errors (after substituting aj for ctJ), but the correct derivation will have to use probability limits.

Now what is wrong with the standard errors obtained at the second stage of the 2SLS method? To see what is wrong, let us write equation (9.13) as

1 = 2 + (w + Pvj) When we estimate this equation by OLS, the standard error of p is obtained as

e, = Po + + iRD, + , + U,

and supply by banks of commercial loans

(Q, = tto + , + aiRS, + « , + v,

where Q, = total commercial loans (billions of dollars) R, = average prime rate charged by banks RS, = 3-month Treasury bill rate (represents an alternative)

rate of return for banks) RD, = AAA corporate bond rate (represents the price of alternative financing to firms) X, = industrial production index and represents firms

expectation about future economic activity y, = total bank deposits (represents a scale variable) (billions of dollars)

Both the equations are over-identified. So we chose to estimate them by 2SLS. R, is expected to have a negative sign in the demand function and a positive sign in the supply function. The coefficient of RS, is expected to be negative. The coefficients of RD„ X„ and y, are expected to be positive. Both the OLS and 2SLS estimates of the parameters had the expected signs. These estimates are presented in Table 9.4. Note that the R might increase or decrease when we use 2SLS as compared to OLS.

There are only minor changes in the 2SLS estimates compared to the OLS estimates for the demand function. As for the supply function, the only changes we see are in the parameters a, and (coefficients of R, and RS,). In this example this is quite important because what this shows is that quantity supplied is more responsive to changes in interest rates than is evidenced from the OLS estimates.

If the R values from the reduced-form equations are very high, the 2SLS and OLS estimates will be almost identical. This is because the ys that we substitute in the 2SLS estimation procedure are very close to the corresponding ys in case the Rs are very high. This is often the case in large econometric models with a very large number of exogenous variables. An example of this is the quarterly model of T. C. Liu. Liu presents 2SLS and OLS estimates side by side and for some equations they are identical to three decimals. Many studies however, do not report OLS and 2SLS estimates at the same time.

9.7 The Question of Normalization

Going back to equations (9.8), we notice that the coefficient of y, in the first equation and that of y, in the second equation are both unity. This is expressed by saying that the first equation is normalized with respect to y, and the second

T. C. Liu, "An Exploratory Quarterly Model of Effective Demand in the Post-war U.S. Economy," Econometrica, Vol. 31, July 1963.