
           library(AER)
           data(CollegeDistance)


           # complete the function `TSLS()`
           TSLS <- function(Y, X, W = NULL, Z, data) {
           fs_model <- lm(as.formula(paste(..., collapse = "+"))), data = data)
           X_fitted <- ...
           
           ss_model <- lm(as.formula(paste(..., paste(..., collapse = "+"))),  data = data)

           return(coefficients(...))}
          
            # use `TSLS()` to reproduce the estimates from Exercise 3


           # complete the function `TSLS()`
           TSLS <- function(Y, X, W = NULL, Z, data) {
            # first stage regression & fitted values
            fs_model <- lm(as.formula(paste(X, "~", paste(c(Z, W), collapse = "+"))), data = data)
            X_fitted <- fs_model$fitted.values
  
            # second-stage regression
            ss_model <- lm(as.formula(paste(Y, "~", paste(W, collapse = "+"), "+ X_fitted")),  data = data)

            # return coefficients of second stage
            return(
              coefficients(ss_model)
            )}
          
           # use `TSLS()` to reproduce the estimates from Exercise 3
           TSLS(Y = "log(wage)", X = "education", Z = "distance", data = CollegeDistance)
           
           TSLS(Y = "log(wage)", 
                X = "education", 
                W = c("unemp", "ethnicity", "gender", "urban"), 
                Z = "distance", 
                data = CollegeDistance)


            ex() %>% check_fun_def("TSLS") %>% {
             check_arguments(.)
             check_call(., Y = "log(wage)", X = "education", Z = "distance", data = CollegeDistance) %>% check_result %>% check_equal
            }
          
            ex() %>% check_function("TSLS", index = 1) %>% {
              check_arg(., "Y") %>% check_equal()
              check_arg(., "X") %>% check_equal()
              check_arg(., "Z") %>% check_equal()
            } 
            ex() %>% check_function("TSLS", index = 2) %>% {
              check_arg(., "Y") %>% check_equal()
              check_arg(., "X") %>% check_equal()
              check_arg(., "W") %>% check_equal()
              check_arg(., "Z") %>% check_equal()
            } 

           success_msg("Nicely done!")