A researcher has 12 observations and uses OLS to estimate a regression coefficient. She reports a p-value of 0.04 from a t-test. Under which condition is this p-value exact rather than an asymptotic approximation?
AWhen the sample is a simple random sample from the population of interest
BWhen the error terms follow a normal distribution, as assumed by the normal linear regression model
CWhen the OLS estimator is consistent and the Gauss-Markov assumptions hold
DWhen n > 30, because the Central Limit Theorem guarantees approximate normality at that threshold
The Gauss-Markov assumptions (zero-mean, homoskedastic, uncorrelated errors) make OLS BLUE but say nothing about the shape of β̂'s sampling distribution. Without knowing that distribution, you cannot compute p-values exactly. The normality assumption (u ~ N(0,σ²)) is what produces the exact result: since β̂ is a linear combination of normal errors, β̂ is exactly normal, and the t-statistic follows an exact t-distribution with (n-k) degrees of freedom. With only 12 observations, the CLT has not kicked in, and asymptotic results are unreliable. Only the normality assumption provides valid inference here.
Question 2 Multiple Choice
An econometrician fits a regression on 800 observations where the error terms are visibly right-skewed — clearly not normal. She uses standard OLS t-statistics for inference. What is the most accurate characterization?
AInvalid — t-statistics require exact normality of errors, so all her inference is meaningless
BValid exactly — OLS estimators are unbiased regardless of error distribution, and unbiasedness implies valid inference
CApproximately valid — with 800 observations, the CLT ensures the sampling distribution of β̂ is approximately normal, making t-tests approximately correct
DValid exactly — skewness only affects standard error estimation, not the t-statistic distribution
With a large sample, the Central Limit Theorem ensures that the OLS estimator β̂ is approximately normal regardless of the error distribution, under mild regularity conditions. This asymptotic normality makes t and F tests approximately valid even when errors are not normally distributed. The approximation is excellent at n = 800. The key contrast with a small sample (12 observations): asymptotic justification is reliable with large n, but in small samples, if normality fails, p-values may be meaningfully wrong. Unbiasedness (option B) is a property of the estimator's expected value, which says nothing about the shape of its sampling distribution.
Question 3 True / False
The OLS estimator β̂ follows an exact normal distribution in finite samples if and only if the error terms are normally distributed (given a fixed X matrix).
TTrue
FFalse
Answer: True
β̂ = β + (X'X)⁻¹X'u — it is a linear function of the error vector u. A fundamental property of normal distributions is that any linear combination of normal random variables is also normal. So if u ~ N(0, σ²I), then β̂ ~ N(β, σ²(X'X)⁻¹) exactly. Conversely, if u is not normally distributed, β̂ is not normally distributed in finite samples (though it converges to normal asymptotically via CLT). This is why the normality assumption is the precise ingredient that converts Gauss-Markov efficiency into exact distributional results.
Question 4 True / False
Because large samples make the normality assumption unnecessary, the normal linear regression model is purely a teaching tool with no practical relevance in applied econometrics.
TTrue
FFalse
Answer: False
In large samples, asymptotic theory (CLT) makes normality less critical — t and F tests are approximately valid regardless. But in many applied settings — macroeconomics with quarterly data over 30 years (n ≈ 120), natural experiments with limited treatment groups, clinical trials — samples are genuinely small. In these cases, normality is load-bearing: without it, there is no exact justification for t-test critical values, and inference can be wrong in practice, not just in theory. Additionally, the normal model provides the clean finite-sample framework on which asymptotic theory is built — understanding it precisely is essential for knowing when you can safely relax it.
Question 5 Short Answer
Why does adding the normality assumption for error terms enable exact finite-sample inference, when the Gauss-Markov assumptions alone cannot provide this?
Think about your answer, then reveal below.
Model answer: The Gauss-Markov assumptions specify the mean and variance of the errors (zero mean, constant variance, no serial correlation) and guarantee that OLS is the best linear unbiased estimator. But they say nothing about the shape of the error distribution. Without knowing the shape, you cannot determine the sampling distribution of β̂, and without the sampling distribution you cannot compute probabilities — i.e., p-values and confidence intervals. The normality assumption adds u ~ N(0,σ²). Since β̂ is a linear combination of u, and linear combinations of normal variables are normal, β̂ ~ N(β, σ²(X'X)⁻¹) exactly. This exact distribution produces exact t and F statistics valid in any sample size.
The distinction between Gauss-Markov and the normal linear model is between efficiency claims and distributional claims. Gauss-Markov: 'OLS is best among linear unbiased estimators.' Normal model: 'β̂ has this exact distribution.' The first claim is about competing estimators; the second is about the probability calculus needed for inference. They are logically independent — you can have efficiency without normality (and thus no exact inference), or normality without Gauss-Markov (and thus exact inference with a possibly inefficient estimator). The normal linear model combines both.