Problem 2. Consider TSLS estimation with a single included endogenous variable and a single instrument. Then the predicted value from the first-stage regression is X₁ = fo + 1 Z₁. (i) Use the definition of the sample variance and covariance to show that sxy #1szy and (ii) Remember that ₁ = 8xz/s2 and that the TSLS estimator TSLS is the OLS estimator of the slope parameter in the regression of Y on X. Use these facts and the result of part (i) to prove that obtained by the two-stage least squares procedure can be written as TSLS
Problem 2. Consider TSLS estimation with a single included endogenous variable and a single instrument. Then the predicted value from the first-stage regression is X₁ = fo + 1 Z₁. (i) Use the definition of the sample variance and covariance to show that sxy #1szy and (ii) Remember that ₁ = 8xz/s2 and that the TSLS estimator TSLS is the OLS estimator of the slope parameter in the regression of Y on X. Use these facts and the result of part (i) to prove that obtained by the two-stage least squares procedure can be written as TSLS
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Problem 2.** Consider TSLS (Two-Stage Least Squares) estimation with a single included endogenous variable and a single instrument. The predicted value from the first-stage regression is \( \hat{X}_i = \hat{\pi}_0 + \hat{\pi}_1 Z_i \).
(i) Use the definition of the sample variance and covariance to show that \( s_{X\hat{Y}} = \hat{\pi}_1 s_{ZY} \) and \( s^2_{X} = \hat{\pi}_1^2 s^2_{Z} \).
(ii) Remember that \( \hat{\pi}_1 = s_{XZ} / s^2_{Z} \) and that the TSLS estimator \( \hat{\beta}^{TSLS} \) is the OLS (Ordinary Least Squares) estimator of the slope parameter in the regression of \( Y \) on \( \hat{X} \). Use these facts and the result of part (i) to prove that \( \hat{\beta}^{TSLS} \), obtained by the two-stage least squares procedure, can be written as
\[
\hat{\beta}^{TSLS} = \frac{s_{ZY}}{s_{ZX}}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b8a4ad9-b66e-4dbe-8b3f-39bf0adb3cae%2F800d8de8-ce09-464c-931e-aac1b26f1d4a%2F10htlrj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2.** Consider TSLS (Two-Stage Least Squares) estimation with a single included endogenous variable and a single instrument. The predicted value from the first-stage regression is \( \hat{X}_i = \hat{\pi}_0 + \hat{\pi}_1 Z_i \).
(i) Use the definition of the sample variance and covariance to show that \( s_{X\hat{Y}} = \hat{\pi}_1 s_{ZY} \) and \( s^2_{X} = \hat{\pi}_1^2 s^2_{Z} \).
(ii) Remember that \( \hat{\pi}_1 = s_{XZ} / s^2_{Z} \) and that the TSLS estimator \( \hat{\beta}^{TSLS} \) is the OLS (Ordinary Least Squares) estimator of the slope parameter in the regression of \( Y \) on \( \hat{X} \). Use these facts and the result of part (i) to prove that \( \hat{\beta}^{TSLS} \), obtained by the two-stage least squares procedure, can be written as
\[
\hat{\beta}^{TSLS} = \frac{s_{ZY}}{s_{ZX}}.
\]
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