Test of measurement errors:

Suppose we are estimating the two-variable regression model

There is a possibility that x might be measured with error.

If then the actual least squares regression would be  

Where 

When (9.18) is estimated using least squares, we obtain

where G,  are the regressioll  residuals. Substituting  the value of eq. (9.19) into eq (9.  i  7) we have

Whether or  not there is meftsmsnt  error, the coefficient of ite  will be consistently estimated by ordinary  lawt squares,  since

In fact,  the least-squares  estimator  of  the coefficient  of x* in eq.  (9.20)  is  identically equal  to the instrumental-variables  estimator, which is given by

To  look at the coefficient of the variable ,  note that

When there is no meburernent error, σ_v² = 0, so that OLS applied to eq. (9.20) will generate a consistent estimator of the coefficient  of  .  However, when there is measurement error,  the coefficient will be estimated inconsistently.  ' This suggests  a  relatively  easy measurement error test. Let δ represent  the coefficient of the variable I?,  in eq. (9.20).

With no measurement error, δ= β, so  that the coefficient of ,  should equal zero. However, with measurement error, δ ≠ β, and the coefficient will (in general) be different from zero. We can  test for measurement error by doing a simde  two-stage procedure. First, we regress x*  on z to obtain the residuals . Then, we regress y on x* and,  and perform a  t test on the coefficient of the  ii,  variable. If we are concerned with measurement  error  in more  than one variable of a multiple  regression model, &I  equivalent F  test could be applied.

The test just described is a special case of a more general test for specification error proposed by Hausman. The Hausman specification test relies on the fact  that under the null hypothesis the ordinary least-squares estimator of the parameters of the original eq. (9.17) is  consistent and (for large samples) eRcient, but is  inconsistent if  the alternative hypothesis is  true.

However,  the instrumental-variables estimator [the least-squares estimator of eq. (9.20)]  is consistent whether or not  the null hypothesis is true, although  it is  inefficient if  the null hypothesis is not valid. The Hausman specification test is as  follows: If there are two estimators β₁, and β₂ that converge to the true value p under the null but converge to different values under the alternative, the null hypothesis can be verified by  testing whether the probability limit of the difference of the two estimators is zero.