plim P2 = P2 - , „2

These equations correspond to (11.4) considered earlier (which were for x2 = 0).

We will not present the detailed calculations here (because they are tedious), but the equations corresponding to (11.5) obtained from a regression of X on Y and Xl and normalizing with respect to Y are

plim p, = p, + I [P2MP2 + ppi - 22) + ol

(11.10)

I r.

where"

. A.*».pXl - 1- X,) + 2 2 = p,(l - - (p,X, - pp2x2)

[Substituting X2 « 0, we get the results in (11.5).]

If p, and p, are the estimators of p, and p, obtained from a regression of A2 on X, and Y, then plim p, and plim 02 are obtained by just interchanging the subscripts 1 and 2 in equations (11.10).

It is easy to see from equations (11.9) and (11.10) that the direction of biases are rather difficult to evaluate. Further, we saw eariier in the case of equations (11.4) and (11.5) that the derived bounds are in many cases rather too wide to be of any practical use and it is better to use equations (11.4) to get a range of consistent estimates based on different assumptions about X. The problem with equations (11.5) or (11.10) is that they depend on a further unknown parameter CT*.

where Thus

cov(„ w) = -P.Xi

2. W) = - 2 2

We thus have

a,, = cov(y, X,) = , + p2p -

- coVCF.y = p,p -bp, - P2X2 (11-8)

= ( ) = p? + Pi + 2pp,p2 + a - p?X, - p?X,

As before, we can derive the probability limits of p, and pj, the least squares estimators of p, and P2. We get

• Pl.mp.-P,-

p,X: - pp,X, I - p

In this case, we cannot get any bounds on the parameters using the procedures of calculating probability limits as in equations (11.9) and (11.10). There is an alternative procedure due to Klepper and Leamer" but its discussion is beyond our scope.

Here we will illustrate the range of consistent estimates of p, and p: using equations (11.8) or (11.9) with different assumptions about \, and Xj- This method is useful when we have some rough idea about the range of possible values for X, and X,. There is the problem, however, that not all these estimates are valid because the implied estimate of a- obtained from the third equation in (11.8) should be positive and also var(x,) var(jC2) - [cov(X, jc,)]- > 0. This last condition implies that (I - X,)(l - X2) - p- > 0. Using equations (11.9) we obtain consistent estimates pj and P2 of p, and p, by solving the equations

(1 - p - X,)p; + pX,32 = (1 - p2)p,

px.p; -1- (1 - p- - X2)p: = (1 - p)P2

The estimate of is obtained from the last equation in (11.8). It is

ff = - (pp + p; + 2pp;p; - x,p;- - X2P2-) (11.12)

As an example, let us consider the data in Table 7.3 and a regression of Y on X, and A",. The correlation coefficient between X, and X2 is 0.2156. Hence p = 0.2156. Also, a„ = 11.9378 and a,, = 3.2286.

It is reasonable to assume that A", (gross domestic production) has smaller measurement errors than A2 (stock formation). Hence we consider

X, = 0.01,0.02,0.05,0.10 •

X2 = 0.10.0.20, 0.30.0.40

Note that for all these values, the condition (I - X,)(I - X;) - p- > 0 is satisfied. The results of solving equations (II.11) are presented in Table I I.I. The range of consistent estimates of p, is (0.128, 0.201) and the range of consistent estimates of p, is (0.447. 0.697). Note that the OLS estimates were Pi = 0.191 and p, = 0.405. Thus with the assumptions about error variances we have made, the OLS estimate P2 is not within the consistent bounds for p, that we have obtained.

Note that assuming only A2 (and not A",) was measured with error we obtained earlier the bounds

0.169 < p, < 0.191 and 0.405 < P2 < 4.184

The OLS estimates are at the extremes of these bounds. With the assumptions about error variances we have made, the bounds for p, are wider but the bounds for p2 are much narrower.

The important conclusion that emerges from all this discussion and illustrative calculations is that in errors in variables models, making the most general

"S. Klepper and E. E. Leamer, "Consistent Sets of Estimates for Regressions with Errors in All Variables," Econometrica. Vol. 52, 1984, pp. 163-183.

assumptions about error variances leads to very wide bounds for the parameters, thus making no inference possible. On the other hand, making some plausible assumptions about the variances of the errors in the different variables, one could get more reasonable bounds for the parameters. A sensitivity analysis based on reasonable assumptions about error variances would be more helpful than obtaining bounds on very general assumptions.

11.4 Reverse Regression

In Sections 11.2 and 11.3 we considered two types of regressions. When we have the variables and x both measured with error (the observed values being Y and X), we consider two regression equations:

1. Regression of on X. which is called the "direct" regression.

2. Regression of X on Y. which is called the "reverse" regression.

Reverse regression has been frequently advocated in the case of analysis of salary discrimination." Since the problem is one of the usual errors in vari-

"There is a large amount of literature on this issue, many papers arguing in favor of "reverse regression." For a survey and difTerent alternative models, see A. S. Goldberger, "Reverse Regression and Salary Discrimination," Journal of Human Resources. Vol. 19. No. 3, 1984, pp. 293-318.

Table 11.1 Estimates of Regression Parameters and Error Variance Based on Different Assumptions About Error Variances