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Asymptotics

Asymptotics. What do you need to know?. Need to understand concept of a plim Need to be able to do things like prove consistency (or inconsistency of an estimator). What else do you need to know?. Understand the idea of convergence in distribution

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Asymptotics

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  1. Asymptotics

  2. What do you need to know? • Need to understand concept of a plim • Need to be able to do things like prove consistency (or inconsistency of an estimator)

  3. What else do you need to know? • Understand the idea of convergence in distribution • That we can , subject to certain regularity conditions, show that, in :

  4. What you do not need to know • Behind these results are various theorems • Laws of Large Numbers for plims • Central Limit Theorems for asymptotic normality • You do not have to know which theorems you are using • You do not have to be able to prove them

  5. Plims • Simplest example – yi is iid with mean μ and variance σ2 • Interested in properties of sample mean • as N→∞ • We will have:

  6. So as N→∞ • the mean of the sample mean is μ • The variance goes to zero • Implies the limit of the sample mean is non-stochastic, simply a number μ

  7. This is convergence in probability • We write this as: • Can think of this as two conditions: • Note: this is not formal definition but ‘works’ in most cases

  8. A More complicated example • Plim (X’X)/N where:

  9. Can take plims of individual components to get:

  10. Manipulating plims • Where plims exist, they are just numbers so can be manipulated like numbers • plim(AB)=plim(A)plim(B) • plim(A/B)=plim(A)/plim(B) • plim(A-1)=plim(A)-1 • So:

  11. But have to know where to stop… • Catch is that can only do this where plims exist: • Can do: • But can’t do:

  12. How do you know where to stop? • Think about dimensions of matrices • If X is Nxk then X’X is kxk – does not depend on N – will typically have plim • But dimension of X is Nxk so depends on N – does not make sense to talk about its plim

  13. Asymptotic Distributions/ Convergence in Distribution • So far we have looked at random variables whose limiting distribution is degenerate i.e. it collapses on a point • But, we can scale things by a factor that depends on N to stop the variance going to zero

  14. An Example: the Sample Mean • Define: • For all N this has mean zero and variance one • But what is its distribution?

  15. The Central Limit Theorem • The CLT says that, under certain regularity conditions, the limiting distribution of ZN as N→∞ is a standard normal distribution • There are different notations for denoting this but you might see things like:

  16. For estimators might see…. • Where the asymptotic variance would be worked out as:

  17. Example: the OLS estimator

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