Consider the Following Population Regression Model Relating the Dependent Variable

question 48

Essay

Consider the following population regression model relating the dependent variable Yi and regressor Xi,
Yi = β0 + β1Xi + ui,i = 1,…,n.
Xi ≡ Yi + Zi
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:

Price Reduction

A decrease in the selling price of a product or service, often to attract more customers or to sell excess inventory.

Trade Discount

A trade discount is a reduced price offered by suppliers to retailers or other professionals in the industry, encouraging bulk purchases or to compensate for distribution services.

Cumulative Quantity Discount

A discount policy where the price reduction is based on the total volume of purchases made over time, encouraging repeated business by rewarding larger orders.

Stated Period

A specific, predefined duration of time mentioned in a document or agreement.