* Compute covariance matrix of y, x, z
corr y x z, covariance

* Store covariance matrix
matrix C = r(C)

* Extract covariances
scalar cov_y_z  = C[1,3]
scalar cov_x_z = C[2,3]

* Compute β1_IV
scalar beta1_IV = cov_y_z / cov_x_z
display "beta1_IV = " beta1_IV

* Compute means of y and x
summ y, meanonly
scalar mean_y = r(mean)

summ x, meanonly
scalar mean_x = r(mean)

* Compute β0_IV = mean(y) – β1_IV * mean(x)
scalar beta0_IV = mean_y - beta1_IV * mean_x
display "beta0_IV = " beta0_IV

********************************************************************************
* Step 1: Regress x on z
reg x z

* Step 2: Obtain fitted values x_hat
predict x_hat, xb

* Step 3: Regress y on x_hat using OLS
reg y x_hat
*******************************************************************************
*==================================================
* Manual Computation of the Variance of the IV (2SLS) Estimator
*==================================================

*--------------------------------------------------
* First stage: x on z
*--------------------------------------------------
reg x z
predict x_hat, xb

* Store first-stage R-squared
scalar R2_xz = e(r2)

*--------------------------------------------------
* Second stage: y on x_hat
*--------------------------------------------------
reg y x_hat

* Estimate error variance from second stage
scalar sigma2_u = e(rss) / e(df_r)

*--------------------------------------------------
* Compute sum of squared deviations of x
*--------------------------------------------------
summ x, meanonly
scalar mean_x = r(mean)

gen x_dev2 = (x - mean_x)^2
summ x_dev2, meanonly
scalar Sxx = r(sum)

*--------------------------------------------------
* Variance and standard error of IV estimator beta1
*--------------------------------------------------
scalar Var_beta1_IV = sigma2_u / (R2_xz * Sxx)
scalar SE_beta1_IV  = sqrt(Var_beta1_IV)

display "Variance of beta1_IV = " Var_beta1_IV
display "Standard Error of beta1_IV = " SE_beta1_IV
********************************************************************************
*==============================================================
* R-squared Equals the Squared Correlation in Simple Regression
*==============================================================

*--------------------------------------------------------------
* Regress x on z
*--------------------------------------------------------------
reg x z

* Store R-squared from the regression
scalar R2_xz = e(r2)

*--------------------------------------------------------------
* Compute sample correlation between x and z
*--------------------------------------------------------------
corr x z

* Store correlation coefficient
matrix C = r(C)
scalar rho_xz = C[1,2]

*--------------------------------------------------------------
* Compare R-squared and squared correlation
*--------------------------------------------------------------
scalar rho_xz_sq = rho_xz^2

display "R-squared from regression (x on z)      = " R2_xz
display "Squared correlation between x and z     = " rho_xz_sq