11.2 The Classical Solution for a Single-Equation Model with One Explanatory Variable

Suppose that the true model is >

y + e ai.I)

Part of the neglect of errors in variables is due to the fact that consistent estimation of parameters is rather difficult. During the 1970s this attitude changed because of the following developments:

1. It was realized that sometimes we can use some extra information available in the dynamics of the equation or the structure of other equations in the model to get consistent estimates of the parameters.

2. The fact that we cannot get consistent point estimates of the parameters does not imply that no inference is possible. It was realized that one can obtain expressions for the direction of the biases and one can also obtain consistent "bounds" for the parameters.

We will now discuss how consistent estimation of equations which contain variables that are measured with errors can be accomplished. We start our discussion with a single-equation model with one explanatory variable measured with error. This is known as the "classical" errors in variable model. We then consider a single equation with two explanatory variables. Next we apply these results to the problem of "reverse regression," which has been a suggested technique in the analysis of wage discrimination by race and sex.

Before we proceed, we should make it clear what we mean by "errors in variables." Broadly speaking, there are two types of errors that we can talk about.

1. Recording errors.

2. Errors due to using an imperfect measure of the true variable. Such imperfect measures are known as "proxy" variables. The true variables are often not measurable and are called "latent" variables.

What we will be concerned with in this chapter are errors of type 2.

A commonly cited example of a proxy variable is years of schooling (5), which is a proxy for education (£). In this case, however, the error S - £ is likely to be independent of S rather than of E. Another example is that of expectational variables, which must be replaced by proxies (either estimates generated from the past or measures obtained from survey data). Examples of this were given in Chapter 10.

We start our discussion with the classical errors-in-variables model and then progressively relax at least some of the restrictive assumptions of this model.

plim brx =

2X 2 (Jf +

( : ) a„

vmix) + var(H) <

since all cross products vanish. Since p = cj,/*, we have

Thus brx will underestimate p. The degree of underestimation depends on If we run a reverse regression (i.e.. regress X on Y), we have

; T E(y + v)

Hence

Instead of and x, we measure

Y = + V and "X x +

and v are measurement errors, x and are called the systematic components. We will assume that the errors have zero means and variances ctJ and ctJ, respectively. We will also assume that they are mutually uncorrelated and are uncorrelated with the systematic components. That is.

E{u) = £(v) = 0 var(m) = ct; var(v) = ctJ

cov(m, ) = cov(m, y) = cov(v, ) = cov(v, y) = 0

Equation (11.1) can be written in terms of the observed variables as

Y-v = X-u) + e

Y = X + w (11.2)

where w = e + v - Pu. The reason we cannot apply the OLS method to equation (11.2) is that cov(»v. AO 0. In fact,

cov(»i, X) = cov(-Pm, x + u) = -Paj

Thus one of the basic assumptions of least squares is violated. If only is measured with error and x is measured without error, there is no problem because cov(»v, AO = 0 in this case. Thus given the specification (11.1) of the true relationship, it is errors in x that cause a problem. If we estimate p by OLS applied to (11.2) we have

IXY 2 (Jf + ")(3 + v)

brx =

al = ? + 1 and CT„ = fial

Hence

Thus l/bxr overestimates p. and we have

phm < .; J .

plim

We can use the two regression coefficients byx and bxr to get bounds on p (at least in large samples).

In the preceding discussion we have implicitly assumed that p > 0. If p < 0, then

plim byx > P and (plim bxy)~ < P

and thus the bounds have to be reversed, that is,

(pHm bxy)* < P < plim byx

Note that since bxy • byx = Rxy ihe higher the value of R- the closer these bounds are. Consider, for instance, the data in Table 7.3.- We have the equation

Y = 0.193 , - 15.86 = 0.969

a:, = 5.088 r + 86.87 R = 0.969

The second equation when solved for gives

Y = 0.199, - 17.32

Hence we have the bounds 0.193 < P, < 0.199. Such consistent estimates of the bounds have been studied by Frisch and Schultz.

The general conclusion from (11.3) is that the least squares estimator of P is biased toward zero and if (11.1) has a constant term, a, the least squares estimator of a is biased away from zero.*

If we define

These data are from E. Malinvaud, Statistical Methods of Econometrics, 3rd ed. (Amsterdam: North-Holland, 1980), p 19

"R. Frisch. Staitstiial Confluence Analysis h\ Means of Complete Regression Systems, Publication 5 (Oslo: University Economics Institute. 1934); H. Schultz, The Theory und Measurement of Demand (Chicago- University of Chicago Press, 1938)

In the discussion that follows we mean by bias, asymptotic bias or more precisely plim $ -e. where 8 is the estimator of 6.

But from (11.1)