exersice1

install.packages("wooldridge") 

## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.4'
## (as 'lib' is unspecified)

library(wooldridge)             
data("wage2")    

##(i) Simple Regression of IQ on educ:

model_educ_iq <- lm(IQ ~ educ, data = wage2)
summary(model_educ_iq)$coefficients

##              Estimate Std. Error  t value     Pr(>|t|)
## (Intercept) 53.687154  2.6229329 20.46837 3.359659e-77
## educ         3.533829  0.1922095 18.38530 1.164253e-64

##(ii) Simple Regression of log(wage) on educ:

model_wage_educ <- lm(log(wage) ~ educ, data = wage2)
summary(model_wage_educ)$coefficients

##               Estimate  Std. Error  t value     Pr(>|t|)
## (Intercept) 5.97306245 0.081373684 73.40288 0.000000e+00
## educ        0.05983921 0.005963095 10.03493 1.422861e-22

##(iii) Multiple Regression of log(wage) on educ and IQ:

model_wage_educ_iq <- lm(log(wage) ~ educ + IQ, data = wage2)
summary(model_wage_educ_iq)$coefficients

##                Estimate   Std. Error   t value      Pr(>|t|)
## (Intercept) 5.658287588 0.0962408433 58.792997 7.797742e-316
## educ        0.039119901 0.0068382065  5.720784  1.428152e-08
## IQ          0.005863132 0.0009979098  5.875413  5.866622e-09

##(iv) Verify

# Get coefficients from the models
delta_1 <- coef(model_educ_iq)["educ"]
beta_1_hat <- coef(model_wage_educ_iq)["educ"]
beta_2_hat <- coef(model_wage_educ_iq)["IQ"]
beta_1_tilde <- coef(model_wage_educ)["educ"]

# Verification
beta_1_tilde_calculated <- beta_1_hat + beta_2_hat * delta_1
cat("tilde_beta_1: ", beta_1_tilde, "\nCalculated_beta_1_tilde: ", beta_1_tilde_calculated, "\n")

## tilde_beta_1:  0.05983921 
## Calculated_beta_1_tilde:  0.05983921