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:

Demand Options

Different strategies or plans a company may use to adapt or respond to varying demand levels for its products or services.

Aggregate Plan

An aggregate plan is a strategic document that outlines the production targets, inventory levels, and workforce activities of a company over a specific period.

Demand Smoothing

Techniques used to reduce volatility in demand by managing and anticipating consumer or production needs.

Chase Strategy

A planning strategy that sets production equal to forecasted demand.