As you will most likely recall, one of the assumptions of regression is that the predictor variables are measured without error.

Measurement error in the response variable does not bias the regression coefficient. It does however increase the the standard error of the coefficient, thereby weakening tests of statistical significance.

Measurement error in the predictor variables, on the other hand, has an impact of the estimates of the regression coefficients. With a single predictor, measurement error leads to underestimation of the coefficient. The degree of attenuation is a function of the reliability of the predictor.

Let's look at a regression using the **hsbdemo** dataset.

use http://www.philender.com/courses/data/hsbdemo, clear regress write read femaleSource | SS df MS Number of obs = 200 ---------+------------------------------ F( 2, 197) = 77.21 Model | 7856.32118 2 3928.16059 Prob > F = 0.0000 Residual | 10022.5538 197 50.8759077 R-squared = 0.4394 ---------+------------------------------ Adj R-squared = 0.4337 Total | 17878.875 199 89.843593 Root MSE = 7.1327 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- read | .5658869 .0493849 11.459 0.000 .468496 .6632778 female | 5.486894 1.014261 5.410 0.000 3.48669 7.487098 _cons | 20.22837 2.713756 7.454 0.000 14.87663 25.58011 ------------------------------------------------------------------------------

The predictor **read** is a standardized test score. Every test has measurement error. I
don't know the exact reliability of **read**, but I would say that using .9 for reliability
would not be far off. We will now estimate the same regression model with the Stata **eivreg**
command, which stands for errors-in-variables regression.

eivreg write read female, r(read .9)assumed errors-in-variables regression variable reliability ------------------------ Number of obs = 200 read 0.9000 F( 2, 197) = 83.41 * 1.0000 Prob > F = 0.0000 R-squared = 0.4811 Root MSE = 6.86268 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- read | .6289607 .0528111 11.910 0.000 .524813 .7331085 female | 5.555659 .9761838 5.691 0.000 3.630548 7.48077 _cons | 16.89655 2.880972 5.865 0.000 11.21504 22.57805

Note that the F-ratio and the R^{2} increased along with the regression
coefficient for **read**. Additionally, there was an increase in the standard
error for **read**.

Now, let's try a model with multiple predictors, **read math** and **socst**. First,
we will run a standard multiple regression.

regress write read math socst female Source | SS df MS Number of obs = 200 ---------+------------------------------ F( 4, 195) = 64.37 Model | 10173.7036 4 2543.42591 Prob > F = 0.0000 Residual | 7705.17137 195 39.5136993 R-squared = 0.5690 ---------+------------------------------ Adj R-squared = 0.5602 Total | 17878.875 199 89.843593 Root MSE = 6.286 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- read | .2065341 .0640006 3.227 0.001 .0803118 .3327563 math | .3322639 .0651838 5.097 0.000 .2037082 .4608195 socst | .2413236 .0547259 4.410 0.000 .133393 .3492542 female | 5.006263 .8993625 5.566 0.000 3.232537 6.77999 _cons | 9.120717 2.808367 3.248 0.001 3.582045 14.65939 ------------------------------------------------------------------------------

Now, let's try to account for the measurement error by using the following reliabilities:
**read** - .9, **math** - .9, **socst** - .8.

eivreg write read math socst female, r(read .9 math .9 socst .8)assumed errors-in-variables regression variable reliability ------------------------ Number of obs = 200 read 0.9000 F( 4, 195) = 70.17 math 0.9000 Prob > F = 0.0000 socst 0.8000 R-squared = 0.6047 * 1.0000 Root MSE = 6.02062 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- read | .1506668 .0936571 1.609 0.109 -.0340441 .3353776 math | .350551 .0850704 4.121 0.000 .1827747 .5183273 socst | .3327103 .0876869 3.794 0.000 .159774 .5056467 female | 4.852501 .8730646 5.558 0.000 3.13064 6.574363 _cons | 6.37062 2.868021 2.221 0.027 .7142973 12.02694 ------------------------------------------------------------------------------

Note that the overall F and R^{2} went up, but that the coefficient for **read**
is no longer significant.

Phil Ender, 22dec00