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Multiple Regression & Inferenceintermediate

Consistency and Asymptotic Normality

Consistency means β^j\hat{\beta}_j converges in probability to the true βj\beta_j as n→∞n\to\infty, so estimates settle on the truth with enough data. It is a weaker and more general property than unbiasedness and holds under MLR.1 to MLR.4. OLS is also asymptotically normal, so even when normality (MLR.6) fails, the usual t and F statistics are approximately valid in large samples when homoskedasticity (MLR.5) holds, and otherwise with heteroskedasticity-robust versions.

Why it matters

Unbiasedness asks for the estimator to be centered correctly in every sample size; consistency only asks it to home in on the truth as data accumulate. The central limit theorem then makes the sampling distribution bell-shaped without assuming the errors were normal. This is what rescues standard inference when MLR.6 is unrealistic.

Formulas

Consistency
plim n→∞ β^j=βj\text{plim}_{\,n\to\infty}\ \hat{\beta}_j=\beta_j
The probability limit of the estimator equals the true parameter.
Asymptotic normality
n (β^j−βj) →d N(0, σ2/aj)\sqrt{n}\,(\hat{\beta}_j-\beta_j)\ \xrightarrow{d}\ \mathcal{N}(0,\,\sigma^2/a_j)
Holds without normal errors, by the central limit theorem. Here aja_j is the population variance of the part of xjx_j left unexplained by the other regressors.

Worked examples

Scenario

Your dependent variable is a heavily skewed count, so errors are clearly non-normal, but n=4000n=4000.

Solution

With MLR.1 to MLR.4 holding, OLS is consistent and asymptotically normal, so the t statistics from `regress y x1 x2 x3` are approximately valid despite the non-normal errors. The large sample does the work that MLR.6 would otherwise require.

NoteAsymptotic justification is why applied work rarely tests for normal errors in big samples.

Common mistakes

  • ✗Treating consistency and unbiasedness as the same thing. An estimator can be biased in small samples yet still consistent.
  • ✗Thinking consistency repairs omitted variable bias. If MLR.4 fails, OLS is inconsistent, so more data does not help.
  • ✗Believing you always need normal errors for t tests. Asymptotic normality makes them approximately valid in large samples regardless of the error distribution.
  • ✗Assuming a consistent estimator is automatically unbiased. The bias only has to vanish in the limit, not at every sample size.

Revision bullets

  • •Consistency: plim β^j=βj\text{plim}\,\hat{\beta}_j=\beta_j, the estimate converges to the truth as nn grows.
  • •Consistency is weaker and more general than unbiasedness.
  • •OLS is consistent under MLR.1 to MLR.4; OVB breaks consistency.
  • •Asymptotic normality validates t and F tests in large samples without MLR.6.
  • •A biased small-sample estimator can still be consistent.

Quick check

An estimator is consistent if, as the sample size grows without bound, it:

Why can we still use t statistics when the error term is not normally distributed?

Connected topics

UnbiasednessOmitted var biasMLR assumptionsGauss-MarkovCLM normalityt test

Sources

  1. Wooldridge, Introductory Econometrics, Ch. 5
    Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Ch. 5 (asymptotics)
  2. Greene, Econometric Analysis
    Greene (2018), Econometric Analysis, 8th ed., Ch. 4 (large-sample properties)
How to cite this page
Dr. Phil's Quant Lab. (2026). Consistency and Asymptotic Normality. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-asymptotics-consistency
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