1. Large Sample Theory
EC 532
Burak Saltoğlu

2. Additional readings
W Greene (Econometric Analysis), pg:
Hayashi (Econometrics), 2000

3. Outline Convergence in Probability Laws of Large Numbers
Convergence of Functions
Convergence in Distribution: Limiting Distributions
Central Limit Theorems
Asymtotic Distributions

4. Why large sample
We have studied the finite (exact) distribution of OLS estimator and its associated tests.
If the regressors are endogeneous (i.e. X and u are correlated) we won’t be able to handle the estimation.
Rather than making assumptions on a sample of a given size, large sample theory makes assumptions on the stochastic processes that generates the sample.

5. Asymptotics
What happens to a rv or a distribution as n tends to infinity.
What is the approximate distribution under these limiting conditions.
What is the rate of convergence to a limit.
2 Critical Laws of statistics are studied under large sample theory.
Law of Large Numbers
Central Limit Theorem

6. Convergence in Probability
Definition : Let xn be a sequence random variable where n is sample size,
the random variable xn converges in probability to a constant c if
the values that the x may take that are not close to c become
increasingly unlikely as n increases.
If xn converges to c, then we say,
All the mass of the probability distribution concentrates around c.

7. mean square Convergence
Definition mean square convergence:

8. Convergence in Probability
Definition : An estimator of a parameter is a consistent estimator iff

9. Almost sure convergence
İntiutively: once the sequence
gets closer to c it stays that way .

10. Law of large numbers: Strong versus Weak Law of Large Numbers
Based on convergence in probability
Strong form of large numbers
Based on Almost sure convergence

11. Laws of Large Numbers Weak Law of Large Numbers: (Khinchine) Remarks:
1) No finite variance assumption .
2) Requires i.i.d sampling

12. Strong law of large numbers: Kolmogorov
Remarks:
İt is stronger because of in AS convergence iid’ness is not required
. But with iid’ness we will obtain a convergence to a constant mean.

13. Convergence of Functions
Theorem (Slutsky): For a continious function, g(xn)
Using Slutsky theorem, we can write some rules of plim.

14. Convergence of Functions
Rules for Probability Limits
1) For plimx=c and plimy=d
i) plim (x+y) = c+d
ii) plim xy=cd
iii) plim (x/y)=c/d
2) For matrices X and Y with plimX=A and plimY=B
i) plim X-1=A-1
ii) plim XY=AB

15. Convergence in Distribution: Limiting Distributions
Definition 6: xn with cdf Fn(x) converges in distribution to a random variable with cdf, F(x) if
then F(x) is the limiting distribution of xn and can be shown as

16. Example: Limiting Probability Distributions
Student’s t Distribution: Given that
Properties:
(i) It is symmetric, positive everywhere and leptokurtic(has fatter tails
than normal dist.)
(ii) Only parameter is n, degrees of freedom.
(iii)

17. Convergence in Distribution: Limiting Distributions
Rules for Limiting Distributions:
1) If and plim yn=c, then
2) As a corrolary to Slutsky theorem, if and g(x) is a cont. function

18. Convergence in Distribution: Limiting Distributions
3) If yn has a limiting distribution and plim(xn-yn)=0 then xn has the same limiting distribution with yn

19. Central Limit Theorems
Lindberg-Levy Central Limit Theorem:
CLT states that any sampling distribution from any distribution would converge to normal distribution
the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed

20. Example: distirbution of sample mean
So X’s are said to be identically and independently distributed (iid) random variables. derive the distribution of

22. Central limit idea

23. A simple monte carlo for CLT See Matlab: clt.m
Let X1,X2…Xn are ~N(0,1),
As E(Xbar)=mu
ninfinity Stdev(Xbar)0

24. X=(x1) x1~N(0,1): E(Xbar)=-0.0182, sigma=0.98

25. CLT when n=5, aver(Xbar)=0.01, stdev=0.47

26. N=1000,s.aver=0.002, st.dev=0.031

27. Stdev of sample average disappears

28. End of the Lecture