Lecture 3

Lecture 3 - Why Use Y To Estimate Y ? Y is unbiased: E ( Y...

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Unformatted text preview: Why Use Y To Estimate Y ? Y is unbiased: E ( Y ) = Y ; consistent: Y p Y Y is the least squares estimator of Y ; Y solves, 2 1 min ( ) n m i i Y m =- so, Y minimizes the sum of squared residuals optional derivation (also see App. 3.2) 2 1 ( ) n i i d Y m dm =- = 2 1 ( ) n i i d Y m dm =- = 1 2 ( ) n i i Y m =- Set derivative to zero and denote optimal value of m by m : 1 n i Y = = 1 n i m = = nm or m = 1 1 n i i Y n = = Y Y has a smaller variance than all other linear unbiased estimators: consider the estimator, 1 1 n Y i i i a Y n = = , where { a i } are such that Y is unbiased; then var( Y ) var( Y ) Hypothesis Testing The hypothesis testing problem (for the mean): make a provisional decision, based on the evidence at hand, whether a null hypothesis is true, or instead that some alternative hypothesis is true. That is, test H : E ( Y ) = Y ,0 vs. H 1 : E ( Y ) > Y ,0 (1-sided, >) {belief} H : E ( Y ) = Y ,0 vs. H 1 : E ( Y ) < Y ,0 (1-sided, <) {alternative} H : E ( Y ) = Y ,0 vs. H 1 : E ( Y ) Y ,0 (2-sided) { p- value = probability of drawing a statistic (e.g. Y ) at least as extreme as the one that was actually computed with your data, assuming that the null hypothesis is true. Calculating the p-value based on Y : p-value = ,0 ,0 Pr [| | | |] act H Y Y Y Y -- where act Y is the value of Y actually observed (nonrandom) To compute the p-value, you need the to know the sampling distribution of Y , which is complicated if n is small. If n is large, you can use the normal approximation (CLT): p-value = ,0 ,0 Pr [| | | |] act H Y Y Y Y -- , = ,0 ,0 Pr [| | | |] / / act Y Y H Y Y Y Y n n -- = ,0 ,0 Pr [| | | |] act Y Y H Y Y Y Y -- 2245 probability under left+right N (0,1) tails where Y = std. dev. of the distribution of Y = Y / n . Calculating the p-value with Y known: For large n , p-value = the probability that a N (0,1) random variable falls outside |( act Y Y ,0 )/ Y | Estimator of the variance of Y : In practice, Y is unknown it must be estimated 2...
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This note was uploaded on 11/23/2009 for the course FIN 5290 taught by Professor Li during the Fall '09 term at Temple.

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Lecture 3 - Why Use Y To Estimate Y ? Y is unbiased: E ( Y...

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