Consider the following population regression model relating the dependent variable Y_i and regressor X_i,
Y_i = β₀ + β₁X_i + u_i, i = 1, …, n.
X_i ≡ Y_i + Z_i
where Z is a valid instrument for X.
(a)Explain why you should not use OLS to estimate β₁.
(b)To generate a consistent estimator for β₁, what should you do?
(c)The two equations above make up a system of equations in two unknowns. Specify the two reduced form equations in terms of the original coefficients. (Hint: substitute the identity into the first equation and solve for Y. Similarly, substitute Y into the identity and solve for X.)
(d)Do the two reduced form equations satisfy the OLS assumptions? If so, can you find consistent estimators of the two slopes? What is the ratio of the two estimated slopes? This estimator is called "Indirect Least Squares." How does it compare to the TSLS in this example?

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The Answer of Consider the following population regression model relating the dependent variable Y_i and regressor X_i, Y_i = β₀ + β₁X_i + u_i, i = 1, …, n. X_i ≡ Y_i + Z_i where Z is a valid instrument for X. (a)Explain why you should not use OLS to estimate β₁. (b)To generate a consistent estimator for β₁, what should you do? (c)The two equations above make up a system of equations in two unknowns. Specify the two reduced form equations in terms of the original coefficients. (Hint: substitute the identity into the first equation and solve for Y. Similarly, substitute Y into the identity and solve for X.) (d)Do the two reduced form equations satisfy the OLS assumptions? If so, can you find consistent estimators of the two slopes? What is the ratio of the two estimated slopes? This estimator is called "Indirect Least Squares." How does it compare to the TSLS in this example?

Consider the Following Population Regression Model Relating the Dependent Variable