Consider the Following Population Regression Model Relating the Dependent Variable

question 49

Essay

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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Definitions:

Average Variable Cost

The total variable costs of production divided by the number of units produced, indicating the variable cost per unit.

Economies of Scale

Cost advantages that enterprises obtain due to their scale of operation, with cost per unit of output generally decreasing with increasing scale as fixed costs are spread out over more units of output.

Diseconomies of Scale

The phenomenon where production costs per unit increase as a firm or production process scales up, contrary to economies of scale where costs decrease.

Boeing

An American multinational corporation that designs, manufactures, and sells airplanes, rotorcraft, rockets, satellites, telecommunications equipment, and missiles worldwide.