To prove that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model, we need to follow these steps:

Step 1: Start with the linear regression model:
$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$

Step 2: In this model, $\boldsymbol{y}$ is the dependent variable, $X$ is the matrix of independent variables, $\boldsymbol{\beta}$ is the vector of coefficients we want to estimate, $\hat{U}$ is the matrix of first-stage residuals, $\boldsymbol{\gamma}$ is the vector of coefficients for the instrumental variables, and $\boldsymbol{\eta}$ is the error term.

Step 3: The OLS estimator of $\boldsymbol{\beta}$ is obtained by minimizing the sum of squared residuals. In other words, we want to find the values of $\boldsymbol{\beta}$ that minimize the expression:
$$\sum_{i=1}^{n}\left(y_{i}-X_{i} \boldsymbol{\beta}\right)^{2}$$

Step 4: To derive $\hat{\boldsymbol{\beta}}_{G I V E}$, we need to consider the first-stage residuals $\hat{U}$ in the linear regression model. These residuals are obtained by regressing the endogenous variable on the instrumental variables.

Step 5: By including the first-stage residuals $\hat{U}$ in the linear regression model, we can rewrite the model as:
$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$

Step 6: Now, we can estimate $\boldsymbol{\beta}$ using the OLS estimator by minimizing the sum of squared residuals:
$$\sum_{i=1}^{n}\left(y_{i}-X_{i} \boldsymbol{\beta}-\hat{U}_{i} \boldsymbol{\gamma}\right)^{2}$$

Step 7: The resulting estimator, denoted as $\hat{\boldsymbol{\beta}}_{G I V E}$, is the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model when the first-stage residuals $\hat{U}$ are included.

Therefore, we have proven that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model.

Question

To prove that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model, we need to follow these steps:

Step 1: Start with the linear regression model:
$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$

Step 2: In this model, $\boldsymbol{y}$ is the dependent variable, $X$ is the matrix of independent variables, $\boldsymbol{\beta}$ is the vector of coefficients we want to estimate, $\hat{U}$ is the matrix of first-stage residuals, $\boldsymbol{\gamma}$ is the vector of coefficients for the instrumental variables, and $\boldsymbol{\eta}$ is the error term.

Step 3: The OLS estimator of $\boldsymbol{\beta}$ is obtained by minimizing the sum of squared residuals. In other words, we want to find the values of $\boldsymbol{\beta}$ that minimize the expression:
$$\sum_{i=1}^{n}\left(y_{i}-X_{i} \boldsymbol{\beta}\right)^{2}$$

Step 4: To derive $\hat{\boldsymbol{\beta}}_{G I V E}$, we need to consider the first-stage residuals $\hat{U}$ in the linear regression model. These residuals are obtained by regressing the endogenous variable on the instrumental variables.

Step 5: By including the first-stage residuals $\hat{U}$ in the linear regression model, we can rewrite the model as:
$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$

Step 6: Now, we can estimate $\boldsymbol{\beta}$ using the OLS estimator by minimizing the sum of squared residuals:
$$\sum_{i=1}^{n}\left(y_{i}-X_{i} \boldsymbol{\beta}-\hat{U}_{i} \boldsymbol{\gamma}\right)^{2}$$

Step 7: The resulting estimator, denoted as $\hat{\boldsymbol{\beta}}_{G I V E}$, is the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model when the first-stage residuals $\hat{U}$ are included.

Therefore, we have proven that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model.

Knowee AI · Accepted Answer