ECMT6002 Chap.4 Asymptotic Reasoning
Asymptotic Reasoning
Asymptotic theory replaces exact finite sample claims with approximations that improve as information accumulates. The law of large numbers supports probability limits and consistency, while a central limit theorem supports approximate normal inference. This chapter treats those results as an argument with conditions, not a magic appeal to a large sample.
It also separates asymptotic bias, variance and distribution so that each conclusion is attached to the theorem that supports it. Probability convergence says an estimator becomes arbitrarily close to a target with high probability. Consistency applies that concept to the parameter of interest, but does not claim exact unbiasedness in any fixed sample.
A useful proof usually decomposes estimation error into sample averages, applies a law of large numbers to their probability limits and uses continuous mapping or Slutsky reasoning to combine them. Approximate inference needs more: a central limit theorem describes the scaled fluctuation around the target, and a consistent variance estimator converts it into a standardised statistic.
The square-root sample-size rate reflects how averaging reduces variance. Dependence, unbounded moments or weak identification can disrupt this familiar limit even when a dataset appears large. Asymptotics also clarifies persistent bias. If a regressor remains correlated with the disturbance, the estimator converges to a biased probability limit; collecting more observations sharpens concentration around the wrong value.
The chapter therefore asks not only what happens as n grows, but which stochastic and identification conditions make that limiting argument valid.
What this chapter covers
- 01
Random samples and sample moments
- 02
Probability limits
- 03
Law of large numbers
- 04
Consistency
- 05
Central limit theorem
- 06
Asymptotic normality and standard errors
Worked example · free
Finding a probability limit
- +1Apply the law of large numbers to the sample average of disturbances.
- +1The sample average converges in probability to its expectation, which is zero.
- +1Use the probability limit algebra to obtain beta plus zero, so the estimator is consistent.
Key terms
- Probability limit
- The value to which a random sequence converges in probability.
- Consistency
- Convergence in probability of an estimator to the target parameter.
- Law of large numbers
- A result giving conditions under which sample averages converge to population expectations.
- Asymptotic normality
- Convergence of a suitably centred and scaled estimator to a normal distribution.
Asymptotic Reasoning FAQ
Does consistency imply unbiasedness?
No. Consistency is a limiting property, while unbiasedness is an exact expectation statement at a given sample size. An estimator can be biased in finite samples and still have bias that vanishes asymptotically.
Why is scaling by the square root of sample size used?
A sample average typically has variance shrinking at the inverse sample size rate. Multiplying its centred error by the square root of sample size produces a nondegenerate limiting distribution.
Is any large sample automatically safe?
No. Dependence, heavy tails, weak identification or changing data generating processes can invalidate familiar approximations. Sample size does not replace the conditions of the relevant limit theorem.
What is the cleanest way to prove consistency?
Rewrite estimation error in terms of sample averages, take their probability limits under an appropriate law of large numbers, and combine the limits using valid algebra. Check that any denominator converges to a nonzero value. Conclude with the target parameter and state the assumptions that made each limiting step possible.
Exam move
For every asymptotic result, write a three line proof skeleton: decomposition, theorem and conclusion. Maintain separate vocabulary for probability limits and limiting distributions. Practise explaining why a condition matters, especially independence or weak dependence, finite moments and identification, rather than stating only the theorem name.
Build a symbol glossary that distinguishes convergence in probability, convergence in distribution and equality of expectations. For a sample mean, derive the probability limit first and the standardised limiting distribution second; do not mix their scaling. Next, practise ratio estimators by taking probability limits of numerator and denominator separately and checking that the denominator limit is nonzero.
Use Slutsky's theorem to justify replacing an unknown variance with a consistent estimate. Construct counterexamples where a large dataset does not rescue the method, such as a common shock shared by every observation or a weak first stage whose identifying covariance remains near zero. When reading a proof, circle the exact line where each assumption is used.
Translate the final result into inference: name the approximate reference distribution, the estimated standard error and the sample-size qualification. End with one sentence contrasting finite-sample bias with inconsistency so those concepts never become interchangeable.
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