Question

Suppose an econometrician had specified the following regression:
\[ y_t = \beta_1 + \beta_2 z_2t + \beta_3 z_3t + \beta_4 z_4t + \epsilon_t \] but a researcher estimated the following regression:
\[ y_t = \beta_1 + \beta_2 z_2t + \beta_3 z_3t + \beta_4 z_4t + \beta_5 z_5t + \epsilon_t \] What will be the consequence of including the irrelevant variable on the estimated coefficients?

• A. Coefficient estimates will be unbiased, consistent but inefficient
• B. Coefficient estimates will be consistent, asymptotically efficient but biased
• C. Coefficient estimates will be inconsistent and efficient
• D. Coefficient estimates will be biased, consistent and efficient

Hint:

Including irrelevant variables in regression models makes the estimates inefficient, but they remain unbiased and consistent.

Answer 1

The Correct Option is A

Step 1: Understanding the issue.
Including an irrelevant variable (such as \( z_5 \)) in the regression model will not affect the unbiasedness of the coefficient estimates. The estimates of the included variables will still be unbiased and consistent. However, the inclusion of an irrelevant variable introduces inefficiency because it increases the variance of the estimated coefficients, making them less precise. Step 2: Analyzing the options.
- (A) Coefficient estimates will be unbiased, consistent but inefficient: This is the correct answer, as adding an irrelevant variable does not affect unbiasedness or consistency, but it introduces inefficiency. - (B) Coefficient estimates will be consistent, asymptotically efficient but biased: This is incorrect because the inclusion of an irrelevant variable does not introduce bias. - (C) Coefficient estimates will be inconsistent and efficient: This is incorrect because the estimates remain consistent and inefficient, not inconsistent. - (D) Coefficient estimates will be biased, consistent and efficient: This is incorrect because the estimates are unbiased and consistent, but not efficient. Step 3: Conclusion.
The correct answer is (A), as including an irrelevant variable does not affect unbiasedness or consistency, but it reduces efficiency.