MS INSTRUMENTAL VARIABLE METHODS 463

are serially uncorrelated but the true v, are serially correlated, then X, , can be used as an instrumental variable This suggestion was made as early as 1941 by Riersol " As an illustration, consider the model

Y, = , + e, X, = X, + u, X, = px, i + V,

(There is no error of observation in Y,.) Then

\ds{X,) = cr„ + a„„ var(y,) = pV„ + a„ cov(Jf„ Y.) = pa„ cov(A„Af, ,) = pa„ cov(y„ J-,-.) =

cos(Y„X, y) = cov(Jf,. Y.y = ppa„

Thus we can estimate p by

& ( .,-,) - ( ,. r,,,)

( „ ,. ) cov{X„Y, ,)

The former amounts to using X, , as the instrumental variable, and the latter amounts to using Y, , as the instrumental variable

One other instrumental variable method is the method of groupmg Three main grouping methods have been suggested in the literature, by Wald, Bart-lett, and Durbin * In Walds method we rank the A"s and form those above the median X into one group and those below the median into another group If the means in the two groups are. respectively, F,, X and f, X-i, we estimate the slope p by

This amounts to using the instrumental variable

1 if A", > median -1 if Af, < median

"O Riersol. "Confluence Analysis by Means of Lag Moments and Other Methods of Confluence Analysis " Economeinca 1941 pp 1-24

"A Wald "The Fitting of Straight Lines If Both Variables Are Subject to Errors Annals of Malhemaiual Slalistus 1940. pp 284-300 M S Bartlett Fitting of Straight Lines When Both Variables Are Subject to Error Biometrus 1949 pp 207-212, J Durbin. Errors in VanabSts."" Review of International Staltstical Institute 1954 pp 23-32

+1 for the top 1 observations - I for the bottom n/3 observations

Durbin suggests using the ranks of X, as the instrumental variables. Thus

where the X, are ranked in ascending order and the Y, are the values of Y corresponding to the X,. If the errors are large, the ranks will be correlated with the errors and the estimators given by Durbins procedure will be inconsistent. Since the estimators given by Walds procedure and Bartletts procedure also depend on the ranking of the observations, these estimators can also be expected to be inconsistent. Pakes investigates the consistency properties of the grouping estimators in great detail and concludes that except in very few cases they are inconsistent, quite contrary to the usual presumption that they are consistent (although inefficient).

Thus the grouping estimators are of not much use in errors in variable models.

11.6 Proxy Variables

Often, the variables we measure are surrogates for the variables we really want to measure. It is customary to call the measured variable a "proxy" variable- it is a proxy for the true variable. Some commonly used proxies are years of schooling for level of education, test scores for ability, and so on. If we treat the proxy variable as the true variable with a measurement error satisfying the assumptions of the errors in variables model we have been discussing, the analysis of proxy variables is just the same as the one we have been discussing. However, there are some other issues associated with the use of proxy variables that we need to discuss.

Sometimes, in multiple regression models it is not the coefficient of the proxy variable that we are interested in but the other coefficients. An example of this is the earnings function in Section 11.5, where our interest may be in estimating the effect of years of schooling on income and not the coefficient of ability (for

"Ariel Pakes, "On the Asymptotic Bias of Wald- Estimators of a Straight Line When Both Variables Are Subject to Error," InternationalEionomu Review, Vol. 23. 1982. pp. 491-497.

and using the estimator p* = 2 JY, Z,- Bartlett suggested ranking forming three groups, and discarding the /3 observations in the middle group. His estimator of p is

This amounts to using the instrumental variable

tm Wioxy.vAWABtEs 465

which we use the test score as a proxy). In such cases one question that is often asked is whether it is not better to omit the proxy variable altogether. To simplify matters, consider the two-regressor case;

• * Pjc + + (11.16)

where is observed but z is unobserved. We make the usual assumption about X, z and the error term u. Instead of z we observe a proxy p = z + e. As in the usual errors-in-variables literature, we assume that e is uncorrelated with Z, X and u. Let the population variances and covariances be denoted by M„, M„, /, , and so on. .-•: LT we omit the variable z altogether from (11.16) and the estimator of is then using the omitted-variable formula [Equation (4.16)], we get

. 0«,« +7 ( .17)

On the other hand, if we substitute the proxy p for z in equation (11.16) and the estimator of is p, it can be shown that

where p is the correlation between and . The bias due to omission of the variable z is y(M,JM,J. The bias due to the use of the proxy p for z is the same multiplied by the ratio in brackets of (11.18). Since this ratio is less than 1. it follows that the bias is reduced by using the proxy. McCallum and Wickens have euguedon the basis of this result that it is desirable to use even a poor proxy.

However, there are four major qualificatioins to this conclusion:

1. One should look at the variances of the estimators as well and not just at bias. Aigner"* studied the mean-square errors of the two estimators and 3 and derived a sufficient condition under which MSE(0p) MSE(poy). This condition is

?(i-Xn)pX

"C. S. Maddala, Econometrics (New York: McGraw-Hill. 1977), p. 160. This result was derived m B. T. McCallum, "Relative Asymptotic Bias from Errors of Omission and Measurement," Econometnca. July 1972, pp 757-758, and M. R. Wickens, "A Note on the Use of Proxy Variables," Econometrica, July 1972, pp. 365-372.

*D. J. Aigner. "MSE Dominance of Lxast Squared with Errors of Observation," Journal of EcommetHci. . 1,1974. 365-372: