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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 σ = 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 = ½) = 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 μ = .78, then Y is an unbiased estimator of μ • What is the variance of Y ? o How does var( Y ) depend on n (see below for the formula) • Does Y become close to μ when n is large? o Law of large numbers: Y is a consistent estimator of μ • Y – μ appears bell shaped for n large…is this generally true? o In fact, Y – μ 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 μ = ∑ = μ Y Variance: var( Y ) = E [ Y – E ( Y )] 2 = E [ Y – μ Y ] 2 = E 2 1 1 n i Y i Y n μ =  ∑ = E 2 1 1 ( ) n i Y i Y n μ =  ∑ 8 so var( Y ) = E 2 1 1 ( ) n i Y i Y n μ =  ∑ = 1 1 1 1 ( ) ( ) n n i Y j Y i j E Y Y n n μ μ = =  × ∑ ∑ = 2 1 1 1 ( )( ) n n i Y j Y i j E Y Y n μ μ = =  ∑∑ = 2 1 1 1 cov(,) n n...
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 Spring '11
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 Normal Distribution, µY

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