Large Sample Theory and Properties of the OLS Estimator

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Large Sample Theory and Properties of the OLS
Estimator

• ECON 222

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Goals of Large Sample Theory

• Establish consistency does the estimator
  approach the true value as the sample size gets
  large?
• Asymptotic normality what is the limiting
  distribution of the estimator?
• Asymptotic efficiency does the estimator have
  the lowest variance within its class?

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Four notions of convergence

• Convergence in probability

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• Convergence in mean-square

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• Convergence almost-surely

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• Convergence in distribution

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How the different notions of convergence relate
to one another
Conv in Mean-square
Conv in distribution
Conv in prob
Almost sure conv
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Slutsky Theorems (proof due to Rao)
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Mann-Wald Theorems (proof due to Rao)
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Laws of Large Numbers

• Commonly used to show that when you average a
  bunch of random variables, the randomness
  essentially goes away.
• Two forms
• Weak
• Strong

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Chebyshevs Weak Law of Large Numbers
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Weak consistency
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Analyze Bias of the OLS Estimator (Stochastic
regressor case)
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Show Consistency
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Asymptotic Normality Lindberg-Levy Central Limit
Theorem
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Distribution of the ols estimator