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An estimator θ̂ₙ is consistent if θ̂ₙ converges in probability to θ as n → ∞. Consistency is a minimum requirement for reasonable estimators—as sample size grows, the estimator should approach the truth. Under regularity conditions, MLEs and method of moments estimators are consistent.
An estimator is a rule for turning data into a guess about an unknown parameter. For that rule to be useful, it should at minimum do better with more data — intuitively, collecting millions of observations should get you very close to the truth. Consistency formalizes this requirement using the language of convergence in probability that you already know.
Recall that θ̂ₙ converges in probability to θ means: for any ε > 0, the probability P(|θ̂ₙ − θ| > ε) → 0 as n → ∞. In words, the chance that your estimate is far from the truth becomes negligible as the sample grows. This is weaker than almost-sure convergence (which says the estimate *will* eventually be close with probability 1 along every path), but it is the standard benchmark for estimators. A consistent estimator might produce a bad estimate for any specific sample — you could get unlucky — but the probability of a bad estimate vanishes as n grows.
The most important consistency results are for the sample mean and for MLEs. The sample mean X̄ₙ is consistent for the population mean μ by the Weak Law of Large Numbers, which is itself a direct consequence of convergence in probability. For MLEs, consistency follows from general regularity conditions (differentiability of the log-likelihood, identifiability of the model, compactness arguments) and is one reason MLEs are the default estimator in most settings. A useful sufficient condition: if an estimator is unbiased (E[θ̂ₙ] = θ) and its variance vanishes (Var(θ̂ₙ) → 0), then by Chebyshev's inequality it is consistent. But note that consistency does not require unbiasedness — a biased estimator can still be consistent if the bias shrinks to zero with n.
What consistency does *not* guarantee is equally important. Consistency is an asymptotic property — it says nothing about performance at any finite sample size. An estimator could be badly biased for small n yet perfectly consistent. And consistency gives no rate: it does not tell you how quickly the estimate approaches the truth. That information lives in asymptotic normality (the next topic), which tells you √n(θ̂ₙ − θ) converges in distribution to a normal, quantifying the speed of convergence and enabling confidence intervals. Think of consistency as the entry requirement for an estimator — necessary but far from sufficient for a complete understanding of its behavior.