When u is low in figure 3 less than zero lnwagerun is

Note the correspondence between figure three and figure four. When u is low in figure 3
(less than zero), lnwagerun is low (below the solid line) in figure four. When u is high
(greater than zero) in figure three, lnwagerun is high (above the solid line) in figure four.
reg lnwageran education
Source | SS df MS Number of obs = 19
-------------+------------------------------ F( 1, 17) = 1.44
Model | 1.54899602 1 1.54899602 Prob > F = 0.2470
Residual | 18.3173689 17 1.07749229 R-squared = 0.0780
-------------+------------------------------ Adj R-squared = 0.0237
Total | 19.866365 18 1.10368694 Root MSE = 1.038
------------------------------------------------------------------------------
lnwageran | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
education | .05213 .043478 1.20 0.247 -.0396006 .1438606
_cons | 4.906743 .4957256 9.90 0.000 3.860854 5.952633
The slope for this line is .05213, which implies that the return to education is 5.213% (see Wooldridge, p.
49, line 3.
Now let’s introduce a new value for the random component:
gen u2 = invnormal(uniform())
Note that the values of u and u2 are not equal
u u2

.1952169 -.9470641
-1.23091 -.5522633
1.953314 .3275578
-.7849584 1.255616
-.6400359 -.036478
-.9740305 -.7476848
.7082468 1.534772
-.3281993 1.252978
-.3066868 1.081464
1.85042 -.8742716
.2078537 .3231856
.8249497 .6407216
-1.335853 .1043822
.9023688 -1.358528
-.6263756 -.2949743
-.1184899 -1.684372
-1.121178 .8746579
-.7535513 -.2219705
-1.787513 .0124836
Furthermore, they are not correlated:
. correlate u u2
(obs=19)
| u u2
-------------+------------------
u | 1.0000
u2 | -0.0847 1.0000
Generate a new variable that incorporates the values of u2:
gen lnwageran2 = lnwage + u2
reg lnwageran2 education
Source | SS df MS Number of obs = 19
-------------+------------------------------ F( 1, 17) = 3.90
Model | 3.4676034 1 3.4676034 Prob > F = 0.0648
Residual | 15.1145584 17 .889091669 R-squared = 0.1866
-------------+------------------------------ Adj R-squared = 0.1388
Total | 18.5821618 18 1.03234232 Root MSE = .94292
------------------------------------------------------------------------------
lnwageran2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
education | .0779969 .0394944 1.97 0.065 -.0053291 .1613229
_cons | 4.861528 .4503058 10.80 0.000 3.911466 5.81159
Here we observe a point estimate of .0779969*100 = 7.7999% as the rate-of-return for an additional year of
education.

Note too that the confidence interval estimate for education extends from -.005 to .1613229. This range
includes what we know to be the true value, .1. Therefore our confidence interval estimate for
1


includes the true value of
1

, .1.
If we were to repeat this process 100 times, that is generate u3, u4…u100, and then run the regression
where the dependent variable is the sum of the error + lnwage, I would expect that 95 of the 100 confidence
interval estimates for
1

would include the true value, .1.