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Unformatted text preview: 1 Review of Statistics (SW Chapter 3) 1. The probability framework for statistical inference (Lecture 1) 2. Estimation 3. Testing 4. Confidence Intervals 2 Estimation Y is the natural estimator of the mean. But: (a) What are the properties of Y ? (b) Why should we use Y rather than some other estimator? Y 1 (the first observation) maybe unequal weights not simple average median( Y 1 ,, Y n ) The starting point is the sampling distribution of Y 3 (a) The sampling distribution of Y Y is a random variable, and its properties are determined by the sampling distribution of Y The individuals in the sample are drawn at random. Thus, the values of ( Y 1 ,, Y n ) are random Thus functions of ( Y 1 ,, Y n ), such as Y , are random: had a different sample been drawn, they would have taken on a different value The distribution of Y over different possible samples of size n is called the sampling distribution of Y . The mean and variance of Y are the mean and variance of its sampling distribution, E ( Y ) and var( Y ). The concept of the sampling distribution underpins all of econometrics. 4 The sampling distribution of Y , ctd. Example : Suppose Y takes on 0 or 1 (a Bernoulli random variable) with the probability distribution, Pr[ Y = 0] = .22, Pr( Y =1) = .78 Then E ( Y ) = p 1 + (1 p ) 0 = p = .78 2 Y = E [ Y E ( Y )] 2 = p (1 p ) [remember this?] = .78 (1.78) = 0.1716 The sampling distribution of Y depends on n . Consider n = 2. The sampling distribution of Y is, Pr( Y = 0) = .22 2 = .0484 Pr( Y = ) = 2 .22 .78 = .3432 Pr( Y = 1) = .78 2 = .6084 5 The sampling distribution of Y when Y is Bernoulli ( p = .78): 6 Things we want to know about the sampling distribution: What is the mean of Y ? o If E ( Y ) = true = .78, then Y is an unbiased estimator of What is the variance of Y ? o How does var( Y ) depend on n (see below for the formula) Does Y become close to when n is large? o Law of large numbers: Y is a consistent estimator of Y appears bell shaped for n largeis this generally true? o In fact, Y is approximately normally distributed for n large (Central Limit Theorem) 7 The mean and variance of the sampling distribution of Y General case that is, for Y i i.i.d. from any distribution, not just Bernoulli: mean: E ( Y ) = E ( 1 1 n i i Y n = ) = 1 1 () n i i EY n = = 1 1 n Y i n = = Y Variance: var( Y ) = E [ Y E ( Y )] 2 = E [ Y Y ] 2 = E 2 1 1 n i Y i Y n =  = E 2 1 1 ( ) n i Y i Y n =  8 so var( Y ) = E 2 1 1 ( ) n i Y i Y n =  = 1 1 1 1 ( ) ( ) n n i Y j Y i j E Y Y n n = =   = 2 1 1 1 ( )( ) n n i Y j Y i j E Y Y n = =  = 2 1 1 1 cov(,) n n...
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This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.
 Spring '11
 j

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