Consider the regression model with a single regressor: (Y_{i}=beta_{0}+beta_{1} X_{i}+u_{i}). Suppose the least squares assumptions in Key
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Consider the regression model with a single regressor: \(Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i}\). Suppose the least squares assumptions in Key Concept 4.3 are satisfied.
a. Show that \(X_{i}\) is a valid instrument. That is, show that Key Concept is satisfied with \(Z_{i}=X_{i}\).
b. Show that the IV regression assumptions in Key Concept 12.4 are satisfied with this choice of \(Z_{i}\).
c. Show that the IV estimator constructed using \(Z_{i}=X_{i}\) is identical to the OLS estimator.
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