Applying Indicator Variables

In this section a number of examples will be given about estimation and interpretation of regressions that include indicator variables.

7.2.1 Interactions

Consider the simple wage equation

wage = ві + e2educ + S1black + S2 female
+Y(female x black) + e

where black and female are indicator variables. Taking the expected value of ln(wage) reveals each of the cases considered in the regression

в1 + e2educ White, Males

Подпись: E[wage]Подпись: (7.1)в1 + Si + e2educ Black, Males

в1 + S2 + e2educ White, Females

в1 + Si + S2 + y + e2educ Black, Females

The reference group is the one where all indicator variables are zero, i. e., white males. The parameter S1 measures the effect of being black, relative to the reference group; S2 measures the effect of being female relative to the reference group, and y measures the effect of being both black and female.

The model is estimated using the cps4~small. gdt data which is from 2008. The results appear below:

Model 3: OLS, using observations 1-1000
Dependent variable: wage

Coefficient

Std. Error

t-ratio

p-value

const

-5.28116

1.90047

-2.7789

0.0056

educ

2.07039

0.134878

15.3501

0.0000

black

-4.16908

1.77471

-2.3492

0.0190

female

-4.78461

0.773414

-6.1863

0.0000

blk_fem

3.84429

2.32765

1.6516

0.0989

Mean dependent var 20.61566 S. D. dependent var 12.83472

Sum squared resid 130194.7 S. E. of regression 11.43892

R2 0.208858 Adjusted R2 0.205677

F(4,995) 65.66879 P-value(F) 2.53e-49

Log-likelihood -3853.454 Akaike criterion 7716.908

Schwarz criterion 7741.447 Hannan-Quinn 7726.234

Holding the years of schooling constant, black males earn $4.17/hour less than white males. For the same schooling, white females earn $4.78 less, and black females earn $5.15 less. The coefficient on the interaction term is not significant at the 5% level however.

A joint test of the hypothesis that S1 = S2 = y = 0 is performed via the script

1 open "@gretldirdatapoecps4_small. gdt"

2 series blk_fem = black*female

3 ols wage const educ black female blk_fem

4 restrict

5 b[3]=0

6 b[4]=0

7 b[5]=0

8 end restrict and the result is

Restriction set 1: b[black] = 0 2: b[female] = 0 3: b[blk_fem] = 0

Test statistic: F(3, 995) = 14.2059, with p-value = 4.53097e-009 Restricted estimates:

coefficient

std. error

t-ratio

p-value

const

-6.71033

1.91416

-3.506

0.0005 ***

educ

1.98029

0.136117

14.55

1.25e-043 ***

black

0.000000

0.000000

NA

NA

female

0.000000

0.000000

NA

NA

blk_fem

0.000000

0.000000

NA

NA

Standard error of the regression = 11.6638

The F-statistic is 14.21 and has a p-value less than 5%. The null hypothesis is rejected. At least one of the coefficients is nonzero. The test could be done even more easily using the omit statement after the regression since each of the coefficients in the linear restrictions is equal to zero.

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