in-class exercise 1

library(wooldridge)
data("wage2")

# (i) Simple regression of IQ on educ
reg1 <- lm(IQ ~ educ, data = wage2)
summary(reg1)

## 
## 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

beta_IQ_educ <- coef(reg1)["educ"]
beta_IQ_educ

##     educ 
## 3.533829

# (ii) Simple regression of log(wage) on educ
reg2 <- lm(log(wage) ~ educ, data = wage2)
summary(reg2)

## 
## 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

beta_logwage_educ <- coef(reg2)["educ"]
beta_logwage_educ

##       educ 
## 0.05983921

# (iii) Multiple regression of log(wage) on educ and IQ
reg3 <- lm(log(wage) ~ educ + IQ, data = wage2)
summary(reg3)

## 
## 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

beta_logwage_educ_mult <- coef(reg3)["educ"]
beta_IQ_mult <- coef(reg3)["IQ"]

beta_logwage_educ_mult

##      educ 
## 0.0391199

beta_IQ_mult

##          IQ 
## 0.005863132

# (iv) Verify 𝞫1 = 𝞫1 + 𝞫2𝞭
check_equation <- beta_logwage_educ_mult - (beta_logwage_educ + beta_IQ_mult * beta_IQ_educ)
check_equation

##        educ 
## -0.04143861