
  library(AER)
library(MASS)
data("Boston")
attach(Boston)
bh_mod <- lm(medv ~ lstat, data = Boston)
R2_res <- summary(bh_mod)$r.squared


  # conduct the regression
  
  
  # obtain a robust coefficient summary
  
  
  # compare both coefficients of determination


  # conduct the regression
  mod <- lm(medv ~ lstat + crim + age, data = Boston)

# obtain a robust coefficient summary
coeftest(mod, vcov. = vcovHC)

# compare both coefficients of determination
R2_unres <- summary(mod)$r.squared
R2_unres < R2_res


  
  test_or({
    ex() %>% check_function("lm") %>% check_result()
  },{
    ex() %>% override_solution("lm(Boston$medv ~ Boston$lstat + Boston$crim + Boston$age)") %>% check_function("lm") %>% check_result()
  },{
    ex() %>% override_solution("lm(medv ~ lstat + crim + age)") %>% check_function("lm") %>% check_result()
  })


test_or({
  test_function("coeftest")
},{
  f <- ex() %>% override_solution("coeftest(lm(Boston$medv ~ Boston$lstat + Boston$crim + Boston$age), vcov. = vcovHC)") %>% check_function("coeftest") 
  f %>% check_arg("x") %>% check_equal()
},{   
  f <- ex() %>% override_solution("coeftest(lm(medv ~ lstat + crim + age), vcov. = vcovHC)") %>% check_function("coeftest") 
  f %>% check_arg("x") %>% check_equal()
})

test_student_typed("vcov. = vcovHC")

test_predefined_objects("R2_res")
test_object("R2_unres")

test_or({
  test_student_typed("R2_unres > R2_res")
},{
  test_student_typed("R2_unres < R2_res")
},{
  test_student_typed("R2_res < R2_unres")
},{
  test_student_typed("R2_res > R2_unres")
})

success_msg("Right. Remember that R2 generally grows when a regressor is added to the model but this does not necessarily mean that the model improves upon a model with less regressors in any case.")