Measuring

When we are estimating an equation with instrumental variables, a question arises as to how to report the goodness-of-fit measure R. This question also arises with the two-stage-least squares (2SLS) method described in Section 9.6. We can think of two measures.

1. R = squared correlation between and y.

2. Measures based on residual sum of squares:

= 1 (y. - sr

In the results we present we will be reporting the second measure.

Note that the R from the instrumental variable method will be lower than the R from the OLS method. It is also conceivable that the R would be negative, but this is an indication that something is wrong with the specification- perhaps the equation is not identified. We illustrate this point with an example.

Many computer programs present an R- for simultaneous-equations models. But since there is no unique measure of R in such models, it is important to

Si = fliiZi + «1222 + ayjZi

Si = «212, + «222 + o23z3

where the os are obtained from the estimation of the reduced-form equations by OLS. In the estimation of the first equation in (9.8) we use yj, ,, and Zi as instruments. This is the same as using Zj, Z2, and z as instruments because

2 2 1 =02 («2i2i + «222 + anZi)u, = 0

o2i S 2ml + O22 S 22«1 + «23 S 23«1 = 0

But the first two terms are zero by virtue of the first two equations in (9.8). Thus X 1 1 = 0 X ZfU, = 0. Hence using 2 as an instrumental variable is the same as using 23 as an instrumental variable. This is the case with exactly identified equations where there is no choice in the instruments.

The case with the second equation in (9.8) is different. Earlier, we said that we had a choice between , and 2 as instruments for y,. The use of y, gives the optimum weighting. The normal equations now are

2 1 2 = 0 and 2 "2 = 0

5] yi"2 = 0 2 + OnZi + , ) 2 = 0

S + azdui = 0 since 2! 232 = 0. Thus the optimal weights for , and Zi are o,, and 0,2.

"Data are not in logs.

check the formula that the program uses. For instance, the SAS program gives an R, but it appears to be neither one of the above-mentioned measures.

Illustrative Example

Table 9.2 provides data on some characteristics of the wine industry in Australia for 1955-1956 to 1974-1975. It is assumed that a reasonable demand-supply model for the industry would be (where all variables are in logs)

Q, = Oo + a,P + aJP*; + 03 , + 04A, + u, demand Q, = bo + biPl + 8, + V, supply

where Q, = real per capita consumption of wine F; = price of wine relative to CPI = price of beer relative to CPI Y, = real per capita disposable income A, = real per capita advertising expenditure S, = index of storage costs

I would like to thank Kim Sawyer for providing me with these data and the example.

Table 9.2 Data for Wine Industry in Australia"

There are considerable differences between the estimates obtained by the different instrumental variables methods. Particularly, the use of Pg seems to produce very different rseults. F appears to be the best of all the instrumental variables.

The IV estimates can be obtained by the procedures outlined earlier. But an easier way is to note that the IV estimator and the two-stage least squares

Q, and P; are the two endogenous variables. The other variables are exogenous. For the estimation of the demand function we have only one instrumental variable 5,. But for the estimation of the supply function we have available three instrumental variables: F*, Y„ and A,.

The OLS estimation of the demand function gave the following resuks (all variables are in logs and figures in parentheses are r-ratios):

Q = -23.651 + 1.158P„, - 0.275Fb - 0.603A + 3.212F = 0.9772

(-6.04) (4.0) (-0.45) (-(.3) (4.50)

All the coefficients except that of F have the wrong signs. The coefficient of F„. not only has the wrong sign but is also significant.

Treating F„, as endogenous and using 5 as an instrument, we get the following results:

Q = -26.195 + 0.643F,, - . - 0.985A + 4.082F R = 0.9724

(-5.09) (0.98) (-0.20) (-1.51) (3.28)

The coefficient of F„. still has a wrong sign but it is at least not significant. In any case the conclusion we arrive at is that the quantity demanded is not responsive to prices and advertising expenditures but is responsive to income. The income elasticity of demand for wine is about 4.0 (significantly greater than unity).

Turning next to the supply function, there are three instrumental variables available for F„.: Pg, A, and F. Also, the efficient instrumental variable is obtained by regressing F„ on P, A, Y, and S. The results obtained by using the OLS method and the different instrumental variables are as follows (figures in parentheses are asymptotic r-ratios for the instrumental variable methods; the Rs for the IV methods are computed as explained eariier):

Instrumental Variables