Questions: Asymptotic Normality of Regression Estimators

Question 1

A researcher runs OLS on a dataset where the error terms are clearly right-skewed (not normal). With n = 800 observations, can they still conduct valid t-tests on the coefficients?

• A. No — t-tests require normally distributed errors, so the results are invalid
• B. Yes — the CLT ensures the OLS estimator is approximately normally distributed in large samples regardless of error distribution
• C. Yes — but only if heteroskedasticity-robust standard errors are used
• D. No — asymptotic normality only applies when errors have zero skewness

Answer: (B)

This is the core insight: asymptotic normality does NOT require normally distributed errors. The OLS estimator can be written as a weighted sum of the y_i values, and by the CLT, this sum converges to a normal distribution as n grows — regardless of the error distribution — provided errors have finite variance. With n = 800, the large-sample approximation is reliable. Option A reflects the common misconception that normal errors are a prerequisite for standard inference.

Question 2

The statement '√n(β̂ − β) converges in distribution to N(0, V)' means which of the following?

• A. β̂ equals the true β plus normally distributed noise in every sample
• B. β̂ is exactly normally distributed around β for any sample size n
• C. The standardized deviation of β̂ from β follows approximately a normal distribution in large samples
• D. β̂ is unbiased in large samples, meaning E[β̂] = β

Answer: (C)

Asymptotic normality is a large-sample approximation about the *standardized* estimator, not an exact finite-sample result. The √n scaling is needed to produce a non-degenerate limit — without it, β̂ − β collapses to zero (consistency). The claim is that the shape of the sampling distribution approaches normal as n grows, enabling approximate inference via t- and F-statistics. Options A and B confuse asymptotic with exact results; option D describes consistency, not normality.

Question 3

Asymptotic normality of OLS requires that the error terms ε_i are normally distributed.

• T. True
• F. False

Answer: False

This is the central misconception the topic corrects. Asymptotic normality holds because the OLS estimator is a sum of terms x_i·ε_i, and the CLT applies to sums of independent random variables with finite variance — regardless of the underlying distribution. The normal-errors assumption matters for *exact* finite-sample inference (e.g., exact t-distributions with small n), but in large samples, the CLT takes over and normality of errors is not needed.

Question 4

In small samples, the assumption of normally distributed errors is more important for justifying OLS inference than it is in large samples.

• T. True
• F. False

Answer: True

In large samples, asymptotic normality via the CLT validates t- and F-statistics without requiring normal errors. In small samples, the CLT approximation may be poor, and the exact finite-sample justification for t-tests relies on the error terms actually being normal (so that β̂ is exactly normal, and t-statistics follow exact t-distributions). As n grows, the distinction vanishes because the CLT approximation improves.

Question 5

Why does the OLS estimator become approximately normally distributed in large samples even when the error term is not normally distributed?

Model answer: The OLS estimator can be written as β̂ = β + (X'X)⁻¹X'ε, where the key term X'ε is a sum of n terms x_i·ε_i. By the central limit theorem, the standardized sum of many independent random variables with finite variance converges in distribution to a normal — regardless of the individual distribution of each term. It is the averaging and summing process, not the shape of ε_i, that produces the asymptotic normality.

The insight is that normality emerges from the mathematical structure of the estimator (a weighted sum), not from any property of the underlying data-generating process. This is why asymptotic inference is so powerful: it requires only weak conditions (finite variance, well-behaved regressors) rather than strong distributional assumptions.