where

Hence we get

var(F) = Pi + Pi + 2pp,p2 + - P?X cov(A„ = P, + P2P - covCAT,, = p,p -t- p.

then from (11.3) we get the result that the asymptotic bias in the least squaict estimator of p is - . t

113 The Single-Equation Model with Two Explanatory Variables

In Section 11.2 we derived the bias in the least squares estimator and also some consistent bounds for the true parameter. We will now see how these two results can be extended to the case of several explanatory variables. Since it is instructive to consider some simple models, we discuss a model with two explanatory variables in some detail:

Two Explanatory Variables: One Measured with Error

Let us consider the case of two explanatory variables only one of which is measured with error. The equation is

= + + e

The observed variables are

Y = + V X, = X, + =

As before we will assume that the errors u, v, e are mutually uncorrelated and also uncorrelated with y, x,, x,. Note that we can combine the errors e and v into a single composite error in Y. Let var(w) = and var(v -¥ e) = a-. The regression equation we can estimate in terms of observables is

Y = , , -)- , - tv

where

w = e + V - "

To save on notation (and without any loss in generality) we will normalize the observed variables A, and A, so that var(A,) = 1, var(Aj) = 1, and cov(A,. A2) = p. Note that

plimp. = p.- = P.(l-)-

( .4)

plim P2 = p2 +

I - p=

Thus bias in p, = -p(bias in p,). This result also applies if there is more than one explanatory variable with no errors. For instance, if the regression equation is

= , :, -I- + • • + + f

Hhese results are derived in Z. Griliches and V. Ringstad, Economies of Scale and the Form of the Production Function (Amsterdam: North-Holland, 1971), App. C.

It is important to note that in this model there are some limits on X that need be imposed for the analysis to make sense. This is because the condition

varU,) varUj) - [covix,, :,)] > 0

needs to be satisfied.

Since varlA",) = I we have ( :,) = 1 - X. Also. varCAj) = varUz) = • and cov(a:„ ) = cov(A,, A2) = p. Hence we have the condition

1 - X - p a: 0 or X < I - p2

This condition will turn out to be very stringent if p is large or there is high collinearity between and A",. For instance, if p = 0.99, we cannot assume that X > 0.01. If X = 0.01, this implies that , and , are perfectly correlated.

What this implies is that we cannot use the classical errors in variables model if A", and A2 are highly correlated and we believe that the variance of the error in ;, is high.

We will see the implications of this condition when we derive the probability limits of the least squares estimators , and p,. The least squares estimators P, and p, of p, and P2 are obtained by solving the equations

Yx}fi, + Yx,xA = lx,Y

S . + S Xlfi, = 2 X,Y Dividing by the sample size throughout and taking probability limits, we get var(Ar,) plim p, + cov(X„ X.) plim p, = cov(A„ Y) •

cov(Ar, 2) plim p, + \ar(X plim P2 = covCAi, That is, • - .

plim p, + p plim p, = p, + - p,X p plim Pi + plim P2 = PiP + P2 Solving these equations, we get

and only , is measured with error, the first formula in (11.4) remains the same except that p- is now the square of the multiple correlation coefficient between A", and (A2, A2.....AJ. As for the biases in the other coefficients, we have

bias in = -7,<bias in p,)

where -y, are the regression coefficients from the "auxiliary" regression of A, on A2, .3, ...»Xl,:

E{X,\X„ Xy,..., X,) = 722 + + • +

Note that we are normalizing all the observed variables to have a unit variance.*

Returning to equations (11.4), note that if p = 0, the bias in p, is -pX as derived in (11.3). Whether or not p, is biased toward zero as before depends on whether or not X < (I - p-). As we argued earlier, this condition has to be imposed for the classical errors in variables model to make sense. Thus, even in this model we can assume that p, is biased toward zero. As for p, the direction of bias depends on the sign of pip. The sign of p is known from the data and the sign of p, is the same as the sign of p,.

Consider now the regression of A", on Y and A2. Let the equation be

Then

The least squares estimators -y, and 7, for 7, and 7, are obtained from the equations

(S 7. +(S 12?= (S.JO

(S YX.Yl, + (2 A)2 = CS )

Divide throughout by the sample size and take probability limits. That is, we substitute population variances for sample variances and covariances. We get

Oi + 2 + 2ppp2 + a- - piX) plim 7, + (PiP + 2) plim 72 = P. + P2P - P,X (PiP + 2) pHm 7, -I- plim 7: = P

Similar formulas have Ibeeri deftved In M. h. LeW. "Errors in Variables Bias in the Presence of Correctly Measured Variables." Economeirua. Vol 41, 1973. pp 985-986. and Steven Gar-ber and Steven Klepper. "Extending the Classical Normal Errors in Variables Model." Econometnca. Vol, 48. No 6. 1980. pp 1541-1546 However, these papers discuss the biases in

terms of the variances and covariances of the true variables, ,, x.....as well as auxiliary

regression of the true variable », on , jt,. The formulas stated here are in terms of the

correlations of the observed variables The only unknown parameter is thus The formulas presented here are practically more useful since they are in terms of the correlations of the observed variables.