Econometrics Lecture Notes 10
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Econometrics Lecture Notes 10

Course: ECON 4820, Winter 2012

School: U. Memphis

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Econometrics 10 1.1. var( ) is inversely proportional to n 1.1.1. the spread (standard deviation) of the sampling distribution is proportional to 1/ 1.1.2. Thus the sampling uncertainty associated with is proportional to 1/ (larger samples, less uncertainty, but squareroot law) The sampling distribution of when n is large (note 1-33) For small sample sizes, the distribution of will usually be complicated (unless....

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10 1.1. var( Econometrics ) is inversely proportional to n 1.1.1. the spread (standard deviation) of the sampling distribution is proportional to 1/ 1.1.2. Thus the sampling uncertainty associated with is proportional to 1/ (larger samples, less uncertainty, but squareroot law) The sampling distribution of when n is large (note 1-33) For small sample sizes, the distribution of will usually be complicated (unless. . . what is true about the distribution of the Yi values in the population?) But if n is large, the sampling distribution is simple! 1.1. As n increases, the distribution of becomes more tightly centered around Y (the Law of Large Numbers) 1.2. Moreover, the distribution of both become normal (the Central Limit Theorem) 1.2.1. 1.2.2. The Law of Large Numbers: (note 1-34) An estimator is consistent if the probability that its falls within an interval of the true population value tends to one as the sample size increases. If (Y1,,Yn) are i.i.d. and < , then is a consistent estimator of Y, that is, Pr[| Y| < ] 1 as n which can be written, ( means converges in probability to Y). The Central Limit Theorem (CLT): (note 1-35) If (Y1,,Yn) are i.i.d. and 0 < < , then when n is large the distribution of is well approximated a by normal distribution. is approximately distributed N(Y, ) (normal distribution with mean Y and variance /n) AND. . . ( Y)/Y is Y approximately distributed N(0,1) (standard normal) That is, standardized = = is approximately distributed as N(0,1) VIP: The larger is n, the better is the approximation. Fig. 2.8 Sampling distribution of when Y is Bernoulli, p = 0.78 (n = 2, 5, 25, 100) Fig. 2.8 Sampling distribution of (cont.) (note 1-36) In figure on previous slide (fig. 2.8), when n = 100, it might not be easy to see that the distribution of is normal. Its easier to see this if we examine the distribution of standardized = See next slide Same example: sampling distribution of (n = 2, 5, 25, 100) (Fig. 2.9 in book) Summary: The Sampling Distribution of For Y1,,Yn i.i.d. with 0 < < , The exact (finite sample) sampling distribution of has mean Y ( is an unbiased estimator of Y) and variance /n Other than its mean and variance, the exact distribution of is complicated and depends on the distribution of Y (the population distribution) When n is large, the sampling distribution simplifies:

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