pUm-, = -0,(1 - - X)]

plim 72 = [- , 2(1 - P- - X) + P<T-1

where

= pf(l - p2 - X) + Hence the estimates (0,, Pj) of (pi, P2) from the equations

have the probability limits plim p, = p, +

P,(l - p- - X)

= P.

PKI - p - X)

J (11.5)

phm p2 = P2 -

P,(l - p- - X)

Again the bias in p, = -p(bias in p,). Also, if X < (1 - p) or (1 - p - X) > 0, we have the result that p, is biased away from zero. Hence

plim p, < p, < plim p, if p, > 0

and (11.6)

plim p, > p, > plim P, if p, < 0

As for p; the bias will depend on the sign of pp,. If pp, > 0, we have

plim p, < p, < plim p2 (11.7a)

and if pp, < 0, we have

plim p2 > p2 > phm p, (11.7b)

Illustrative Example

Consider the data in Table 7.3 and a regression of imports (Y) on domestic production (A",) and consumption (Xy). Note that X, and X are very highly correlated, p- = 0.99789 or 1 - p- = 0.00211. Let us assume that X has no measurement error but that there is measurement error in A",. As mentioned earlier, we have to have X < 1 - p or X < 0.( 2 . This implies that the variance of the measurement error in A", cannot be greater than 0.211% of the variance of A",. Thus we have to make a very stringent assumption on X to do any sensible analysis of the data.

Hence

The regression equation in this case is

Y = 0.043A, + 0.2305 - 18.63 = 0.970

(O.IW) (0 2913)

(Figures in parentheses are standard errors.) The reverse regression is

X, = 0.078 -I- 1.5033 - 16.37 = 0.998

Which, when normalized with respect to Y, gives

= 12.81 , - l9.2SXi + 2.09

If X < 0.00211, we come to the conclusion that , > 0, and hence from (11.6) the consistent estimates of bounds are

0.043 < < 12.81

Also, from (11.7) the consistent bounds for are (since pP, > 0)

-19.28 < Pi < 0.23

All this merely indicates that the coefficients pi and p, are not estimable with any precision, which is what one would also learn from the high multicoUinearity between A", and A", and the large standard errors reported. What the errors in variables analysis shows is that even very small errors in measurement can make a great deal of difference to the results, that is, the multicoUinearity may be even more serious than what it appears to be and that the confidence intervals are perhaps wider than those reported.

Note that the consistent bounds are not comparable to a confidence interval. The estimated bounds themselves have standard errors since p, and p, are both subject to sampling variation. If there are no errors in variables, even with multicoUinearity, the least squares estimators are unbiased. But with errors in measurement they are not. and the estimated bounds yield (large sample) estimates of the biases.

One important thing to note is that the bounds are usually much too wide and making some special assumptions about the errors, one can get a range of consistent estimates of the parameters. We will illustrate this point.

Consider the regression of Y on A", and A, with the data in Table 7.3. Let us assume that stock formation is subject to error but X, is not.

In this case we have rjj = 0.046. The regression of Y on A", and A2 gives

Y - 0.191X, + 0.405 2 - 16.78 = 0.972

( .00«7) (0.)l»)

= 155.8

Tilte4tdr«e<«lirtBSm pves (note we use A2 as the regressand) 0.mY - 0.040Xt + 6.07 R = 0.139

Wc will apply (he formulas in (11.6) and (11.7) that we have to interchange the subscripts I and 2. .

which, when normahzed with respect to Y. gives « <

Y = 0.169, + 4.184, - 25.35 We thus get the bounds

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

The bounds for p2 are very wide and we can learn more by studying the first equation in (11.4)." Suppose that X = 0.477, that is, the error variance is 47.7% of the variance of A",, which is a very generous assumption. In this case, a consistent estimate of p2 is 2 x 0.405 = 0.81, which is far below the upper bound of 4.184. If X = 0, that is, there is no error in A2 a consistent estimate of p2 is, of course, 0.405. Since (1 - p-) = 0.954 in this case, we have to have X = 0.86 or the error variance about 86% of the variance of X2 to say that a consistent estimate of p, is 4.184.

Thus, in many problems, any analysis of bounds should be supplemented by some estimates based on some plausible assumptions of error variances, especially when the bounds are very wide. Many of the comments made here regarding the use of bounds in errors in variables models apply to more general models. The purpose of analyzing a specialized model is to show some of the uses and shortcomings of the analysis in terms of bounds and these would not be as transparent when we consider a A;-variable model with all variables measured with error.

Two Explanatory Variables: Both Measured with Error

Consider the case where both , and , are measured with error. Let the observed variables be

= + V = Xf + , X2 = X2 + U2

We continue to make the same assumptions as before, that is, ,, U2. v are mutually uncorrelated and uncorrelated with ,, Xi. y. As before, we use the normalization varlA",) = I. varCAj) = I, and covCA",, X2) p. Define

var(«i var(«2)

X, = -jrr: X2 = -777; = varf -I- v)

var(A,) var(A2)

We have the equation in observable variables:

Y = fi,X, + 2X2 + w

•Note that since it is x, that we are assuming to have an error, we have to interchange the subscripts I and 2.

The case of only one variable measured with error has also been considered by M. D. Levi, "Measurement Errors and Bounded OLS Estimates." Journal of Econometrics. Vol. 6, 1977, pp. 165-171. However, considering the two variable case discussed here, it is easy to see that the formulae he gives depend on the variances and covariance of the true variables x, and ., and <t;. We have presented the results in terms of the sample variances and covariance of A*, and A*,, as done by Griliches and Ringstad. In this case all we have to do is to make some assumption about X, the proportion of error variance to total variance in A*, to derive the bounds.