Question 1:

# Load the wooldridge library
library(wooldridge)

# Load the WAGE2 dataset
data("wage2")

# View the first few rows of the dataset to check the variables
head(wage2)
##   wage hours  IQ KWW educ exper tenure age married black south urban sibs
## 1  769    40  93  35   12    11      2  31       1     0     0     1    1
## 2  808    50 119  41   18    11     16  37       1     0     0     1    1
## 3  825    40 108  46   14    11      9  33       1     0     0     1    1
## 4  650    40  96  32   12    13      7  32       1     0     0     1    4
## 5  562    40  74  27   11    14      5  34       1     0     0     1   10
## 6 1400    40 116  43   16    14      2  35       1     1     0     1    1
##   brthord meduc feduc    lwage
## 1       2     8     8 6.645091
## 2      NA    14    14 6.694562
## 3       2    14    14 6.715384
## 4       3    12    12 6.476973
## 5       6     6    11 6.331502
## 6       2     8    NA 7.244227
##(I)

# Simple regression of IQ on education
model_iq_educ <- lm(IQ ~ educ, data = wage2)

# Display summary to get the slope coefficient
summary(model_iq_educ)
## 
## Call:
## lm(formula = IQ ~ educ, data = wage2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -50.228  -7.262   0.907   8.772  37.373 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  53.6872     2.6229   20.47   <2e-16 ***
## educ          3.5338     0.1922   18.39   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.9 on 933 degrees of freedom
## Multiple R-squared:  0.2659, Adjusted R-squared:  0.2652 
## F-statistic:   338 on 1 and 933 DF,  p-value: < 2.2e-16
# Extract the slope coefficient (delta_1)
# This extracts the coefficient for "educ", which is the slope
delta_1 <- coef(model_iq_educ)["educ"]
delta_1
##     educ 
## 3.533829

Question 2:

##(II)
# Simple regression of log(wage) on education
model_logwage_educ <- lm(log(wage) ~ educ, data = wage2)

# Display the regression output to verify it works
summary(model_logwage_educ)
## 
## Call:
## lm(formula = log(wage) ~ educ, data = wage2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.94620 -0.24832  0.03507  0.27440  1.28106 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.973062   0.081374   73.40   <2e-16 ***
## educ        0.059839   0.005963   10.04   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4003 on 933 degrees of freedom
## Multiple R-squared:  0.09742,    Adjusted R-squared:  0.09645 
## F-statistic: 100.7 on 1 and 933 DF,  p-value: < 2.2e-16
# Extract the slope coefficient (beta_1)
beta_1 <- coef(model_logwage_educ)["educ"]
beta_1  # This should now display the slope coefficient
##       educ 
## 0.05983921

Question 3:

# Load the wooldridge library
library(wooldridge)

# Load the wage2 dataset
data("wage2")



# Multiple regression of log(wage) on education and IQ
model_logwage_educ_iq <- lm(log(wage) ~ educ + IQ, data = wage2)

# Display the regression summary
summary(model_logwage_educ_iq)
## 
## Call:
## lm(formula = log(wage) ~ educ + IQ, data = wage2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.01601 -0.24367  0.03359  0.27960  1.23783 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 5.6582876  0.0962408  58.793  < 2e-16 ***
## educ        0.0391199  0.0068382   5.721 1.43e-08 ***
## IQ          0.0058631  0.0009979   5.875 5.87e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3933 on 932 degrees of freedom
## Multiple R-squared:  0.1297, Adjusted R-squared:  0.1278 
## F-statistic: 69.42 on 2 and 932 DF,  p-value: < 2.2e-16
# Extract the slope coefficients (beta_1 for educ and beta_2 for IQ)
beta_1_multi <- coef(model_logwage_educ_iq)["educ"]
beta_2 <- coef(model_logwage_educ_iq)["IQ"]

# Print the slope coefficients
beta_1_multi
##      educ 
## 0.0391199
beta_2
##          IQ 
## 0.005863132

Question 4

# Calculate beta_1 from the right-hand side: beta_1_multi + beta_2 * delta_1
beta_1_rhs <- beta_1_multi + beta_2 * delta_1

# Compare it with the beta_1 from simple regression (question ii)
beta_1_rhs
##       educ 
## 0.05983921
beta_1  # From the simple regression
##       educ 
## 0.05983921