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 X₂ = 0 + 1 Zi. (i) Use the definition of the sample variance and covariance to show that sxy = 1szy and X = is the OLS estimator of (ii) Remember that 1 sxz/s2 and that the TSLS estimator the slope parameter in the regression of Y on X. Use these facts and the result of part (i) to prove that TSLS obtained by the two-stage least squares procedure can be written as TSLS TSLS SZY SZX

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**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 \( \hat{X}_i = \hat{\pi}_0 + \hat{\pi}_1 Z_i \).

(i) Use the definition of the sample variance and covariance to show that \( s_{\hat{X}Y} = \hat{\pi}_1 s_{ZY} \) and \( s^2_{\hat{X}} = \hat{\pi}^2_1 s^2_{Z} \).

(ii) Remember that \( \hat{\pi}_1 = s_{XZ}/s^2_Z \) and that the TSLS estimator \( \hat{\beta}^{TSLS} \) is the OLS 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

\[
\hat{\beta}^{TSLS} = \frac{s_{ZY}}{s_{ZX}}.
\]
Transcribed Image Text:**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 \( \hat{X}_i = \hat{\pi}_0 + \hat{\pi}_1 Z_i \). (i) Use the definition of the sample variance and covariance to show that \( s_{\hat{X}Y} = \hat{\pi}_1 s_{ZY} \) and \( s^2_{\hat{X}} = \hat{\pi}^2_1 s^2_{Z} \). (ii) Remember that \( \hat{\pi}_1 = s_{XZ}/s^2_Z \) and that the TSLS estimator \( \hat{\beta}^{TSLS} \) is the OLS 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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