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 female
Source | 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
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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
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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
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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 R2 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
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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
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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
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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
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Note that the overall F and R2 went up, but that the coefficient for read is no longer significant.