TOC
Asymptotic Properties
When we have MLR.1 - MLR.6, for any sample of data, the finite sample properties hols:
- Unbiasedness of OLS.
- OLS is BULE.
- Sampling distribution of the OLS estimators.
Asymptotic properties or large sample properties are defined as the sample size grows without bound and is valid even without MLR.6. And
It can be shown that even without MLR.6, t and F statistics have approximately t and F distributions, in large sample sizes.
1. Consistency
Definition๏ผ
Let be an estimator of based on sample of size . Then is a consistent estimator of if for any
That is
Intuitively, consistency means when the sample size becomes larger, the estimator get closer to the true value
There are no relations between the unbiasedness and consistency of an estimator
2. The Consistency of the OLS estimator
Consistency is a desired property for an estimator.
Under MLR.1 - MLR.4, The OLS estimator is consistent for
It means that as sample size gets large, gets closer and closer to .
Proof
In simple linear regression model:
Under the law of large numbers:
thus
Under MLR.4 (),
Then , which means that is consistent.
3. Asymptotic Normality of OLS
Under the Gauss-Markov Assumptions MLR.1 - MLR.5, for each
- The theorem implies that we can relax the normality assumption of . It means that, no matter how is distributed, the OLS estimators are approximately normal when the sample size is large enough.
- And then we can use the same methods in ch4 to construct confidence intervals and test hypothesis even without MLR.6.
- The Proof is based on the Central Limit Theorem , which goes that
- Let be a random sample with mean and variance , then
- It means that no matter how is distributed, when the sample size gets larger, the sample average has approximately a normal distrtbution.
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