1 / 17

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

carlow
Download Presentation

Asymptotics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related