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:

Residual Value

The estimated salvage value or the value of an asset at the end of its useful life.

Accumulated Depreciation

A cumulative account that records the depreciation expense of an asset, reducing its book value over time.

Initial Cost

The original acquisition cost of an asset or investment, prior to any adjustments for depreciation, improvement, or amortization.

Fixed Asset

A Fixed Asset is a long-term tangible piece of property that a company owns and uses in its operations to generate income, and not expected to be consumed or converted into cash within a year.