Predictions in the Log-linear Model

In this example, you use the regression to make predictions about the log wage and the level of the wage for a person having 12 years of schooling. The naive prediction of wage merely takes the antilog of the predicted ln(wage). This can be improved upon by using properties of log­normal random variables. It can be shown that if ln(w) N(y.,a2) then E(w) = e^+a2/2 and

var(w) = e2^2 (ef2 — 1).

That means that the corrected prediction is yc = exp(bi + b2x + <t2/2) = e(bl+b2a:)e<j2/2. The script to generate these is given below.

1 open "@gretldirdatapoecps4_small. gdt"

2 logs wage

3 ols l_wage const educ

4 scalar l_wage_12 = $coeff(const)+$coeff(educ)*12

5 scalar nat_pred = exp(l_wage_12)

6 scalar corrected_pred = nat_pred*exp($sigma"2/2)

7 print l_wage_12 nat_pred corrected_pred

The results from the script are

l_wage_12 = 2.6943434

nat_pred = 14.795801

corrected_pred = 16.996428

That means that for a worker with 12 years of schooling the predicted wage is $14.80/hour using the natural predictor and $17.00/hour using the corrected one. In large samples we would expect the corrected predictor to be a bit better. Among the 1000 individuals in the sample, 328 of them have 12 years of schooling. Among those, the average wage is $15.99. Hence the corrected prediction overshoots by about a dollar/hour. Still, it is closer than the uncorrected figure.

To get the average wage for those with 12 years of schooling, we can restrict the sample using the script below:

smpl educ=12 —restrict summary wage smpl full

The syntax is relatively straightforward. The smpl command instructs gretl that something is being done to the sample. The second statement educ=12 is a condition that gretl looks for within the sample. The –restrict option tells gretl what to do for those observations that satisfy the condition. The summary wage statement produces

Подпись: Minimum Maximum 2.50000 72.1300Подпись: Ex. kurtosis 9.08474

Подпись: Mean 15.9933 Std. Dev. 8.84371 Подпись: Median 14.2050 C. V. 0. 552963
Подпись: Skewness 2.31394

Summary Statistics, using the observations 1-328
for the variable wage (328 valid observations)

which shows that the mean for the 328 observations is almost $16.00. The last line smpl full restores the full sample.

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