Question

Consider a simple pooled regression model: y_it = β₀ + β₁x_it + vit where v_it = μ_i + ∈_it and Cov(x_it, μ_i) ≠ 0. Here, μ_i captures the unknown individual specific effects and it is the idiosyncratic error uncorrelated with both x_it and μ_i. If the parameters of this model are estimated using the ordinary least squares (OLS) method, then the estimated slope coefficient will be

• A. biased
• B. inconsistent
• C. unbiased but consistent
• D. unbiased but efficient

Answer 1

The Correct Option is A, B

In the given pooled regression model, the equation is: y_it = β₀ + β₁x_it + v_it, where v_it = μ_i + ∈_it. The variable μ_i represents individual-specific effects, and ∈_it is the idiosyncratic error. A crucial detail provided is Cov(x_it, μ_i) ≠ 0, indicating that x_it is correlated with μ_i.

When estimating parameters using OLS, a key assumption is that the regressors are uncorrelated with the error term. However, Cov(x_it, μ_i) ≠ 0 violates this assumption, meaning the regressor x_it is correlated with the composite error term v_it.

Due to this correlation, the OLS estimator will not consistently estimate the true parameter values. This leads to:

• Bias: The estimated coefficients will be systematically off from the true population parameters because the omitted individual-specific effects μ_i correlate with x_it.
• Inconsistency: As the sample size increases, an inconsistent estimator will not converge to the true parameter values, reflecting the persistent bias in the presence of correlated regressors and error terms.

In conclusion, under these circumstances, the OLS estimated slope coefficient is both biased and inconsistent.