Q3

Question 3

a)

We are testing whether an extra year of mother’s education has the same effect on wage as an extra year of father’s education and twice the effect of an extra IQ point. Allow the subscript i to denote how the equation applies for each individual sample.

Null and alternative hypotheses

\[ H_0: \beta_{meduc} = \beta_{feduc} + 2\beta_{IQ} \\ H_1: \beta_{meduc} \neq \beta_{feduc} + 2\beta_{IQ} \]

We reparameterise the model using the method taught last week:

\[ \text{restrict_test}_i = meduc_i - feduc_i - 2 \cdot IQ_i \]

This gives the regression model:

\[ wage_i = \beta_0 + \beta_{IQ} \cdot IQ_i + \beta_{feduc} \cdot feduc_i + \lambda \cdot restrict\_test_i + u_i \]

putting in the values from the regression, we obtain the following estimate

\[ \hat{wage}_i = 24.764 + 27.550 \cdot IQ_i + 24.545 \cdot feduc_i + 10.373 \cdot restrict\_test_i \]

with standard errors in parentheses:

\[ \hat{wage}_i = \underset{(102.387)}{24.764} + \underset{(12.419)}{27.550} \cdot IQ_i + \underset{(5.765)}{24.545} \cdot feduc_i + \underset{(6.281)}{10.373} \cdot restrict\_test_i \]

\[ R^2 = 0.1199 \quad \text{and} \quad n = 722 \]

library(wooldridge)

Warning: package 'wooldridge' was built under R version 4.3.3

data(wage2)
m3a <- lm(wage ~ IQ + feduc + I(meduc - feduc - 2 * IQ), data = wage2)
summary(m3a)

Call:
lm(formula = wage ~ IQ + feduc + I(meduc - feduc - 2 * IQ), data = wage2)

Residuals:
    Min      1Q  Median      3Q     Max 
-904.33 -255.18  -44.87  214.46 2030.56 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                 24.764    102.387   0.242   0.8090    
IQ                          27.550     12.419   2.218   0.0268 *  
feduc                       24.545      5.765   4.258 2.34e-05 ***
I(meduc - feduc - 2 * IQ)   10.373      6.281   1.651   0.0991 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 383.6 on 718 degrees of freedom
  (213 observations deleted due to missingness)
Multiple R-squared:  0.1199,    Adjusted R-squared:  0.1162 
F-statistic:  32.6 on 3 and 718 DF,  p-value: < 2.2e-16

print(qt(0.975, df = 718))

[1] 1.963273

(b)

We are testing the hypothesis that:

\[ H_0: \beta_{meduc} = \beta_{feduc} + 2\beta_{IQ} \\ H_1: \beta_{meduc} \neq \beta_{feduc} + 2\beta_{IQ} \]

The test statistic is t= 10.373/6.281 = 1.651, following a t distribution with 718 degrees of freedom, and at a 5% double tail two sided significance test, the critical value of t is equal to 1.963.

Since |1.651| < 1.963, we fail to reject the null hypothesis. The p value for restrict_test is also 0.0991 which is greater than 0.05, confirming the notion that we do not have sufficient evidence to reject the null hypothesis.

(c)

library(wooldridge)
data(wage2)

m3c <- lm(wage ~ IQ + feduc + I(meduc - feduc), data = wage2)
summary(m3c)

Call:
lm(formula = wage ~ IQ + feduc + I(meduc - feduc), data = wage2)

Residuals:
    Min      1Q  Median      3Q     Max 
-904.33 -255.18  -44.87  214.46 2030.56 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)        24.764    102.387   0.242   0.8090    
IQ                  6.805      1.047   6.500 1.51e-10 ***
feduc              24.545      5.765   4.258 2.34e-05 ***
I(meduc - feduc)   10.373      6.281   1.651   0.0991 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 383.6 on 718 degrees of freedom
  (213 observations deleted due to missingness)
Multiple R-squared:  0.1199,    Adjusted R-squared:  0.1162 
F-statistic:  32.6 on 3 and 718 DF,  p-value: < 2.2e-16

# crit value for one-sided test at 5% significance level
print(qt(0.95, df = 718))

[1] 1.646979

\[ \hat{wage}_i = 24.764 + 6.805 \cdot IQ_i + 24.545 \cdot feduc_i + 10.373 \cdot (meduc_i - feduc_i) \]

with standard errors:

\[ \hat{wage}_i = \underset{(102.387)}{24.764} + \underset{(1.047)}{6.805} \cdot IQ_i + \underset{(5.765)}{24.545} \cdot feduc_i + \underset{(6.281)}{10.373} \cdot (meduc_i - feduc_i) \]

\[ R^2 = 0.1199 \quad \text{and} \quad n = 722 \]

(d) We are testing the hypothesis that:

\[ H_0: \beta_{meduc} = \beta_{feduc} \\ H_1: \beta_{meduc} < \beta_{feduc} \]

This is a one tail test where the alternative states that the father’s education has a larger effect on wage than the mother’s education.

The output finds that the standard error is 6.281 and it has a restriction term of 10.373 being its coefficient, following a t distribution with 718 DOF yielding a t statistic of 1.651. Since this is greater than the t statistic t=1.646979, we can reject the null hypothesis in favour of the alternative and that the father’s education has a larger effect on wage than the mother’s education. Interestingly enough we also note that the 2IQ term doesn’t change the overall regression output much, likely due to them being highly col-linear.