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∙ intuitively we compute the expected value of y

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Unformatted text preview: ∙ Intuitively, we compute the expected value of Y for each slice of the population determined by the values x 1 , x 2 ,... and then we weight these by the share of the population accounted for by each x r . 41 ∙ To show LIE, suppose Y and X are both discrete. Then, using f X , Y x , y f Y | X y | x f X x , E Y ∑ s 1 y s f Y y s ∑ s 1 ∑ r 1 y s f X , Y x r , y s ∑ s 1 ∑ r 1 y s f Y | X y s | x r f X x r ∑ r 1 ∑ s 1 y s f Y | X y s | x r f X x r ∑ r 1 E Y | X x r f X x r ∑ r 1 x r f X x r E X . 42 ∙ There is a hand-waving step when the order of the summations is switched, but it is valid under the assumption that E | Y | . With a finite number of outcomes there is no issue about switching the order. ∙ A similar argument works in the continuous case, but we need to switch the order of the integration. 43 EXAMPLE : Suppose hourly wage Y is related to education X as E Y | X 5.8 .40 X If the average education level in the population is 12.5, then we can find the average hourly wage: E Y E 5.8 .40 X 5.8 .40 E X 5.8 .40 12.5 10.8, or $10.80. 44 ( ce4 ) (General Version of the LIE) Let X and Z be random vectors and assume E | Y | , so that E Y | X and E Y | X , Z exist. Call these 1 X E Y | X and 2 X , Z E Y | X , Z . Then E 2 X , Z | X 1 X , which we more commonly write in a simpler form: E E Y | X , Z | X E Y | X 45 EXAMPLE : Suppose E Y | X , Z 1 X 2 Z and E Z | X 1 X . Then E Y | X E 1 X 2 Z | X 1 X 2 E Z | X 1 X 2 1 X 2 1 2 1 X This simple example is fundamental to understanding “omitted variables bias” in regression analysis. 46 ( ce5 ) If Y and X are independent then E Y | X E Y . This is obvious because D Y | X D Y under independence. Somewhat less obvious is that if X , Y is independent of Z , then E Y | X , Z E Y | X ∙ It is not enough to assume X is independent of Z and Y is independent of Z . Generally, to ensure E Y | X , Z E Y | X , we need D X , Y | Z D X , Y , and this does not follow from D X | Z D X and D Y | Z D Y ....
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∙ Intuitively we compute the expected value of Y for each...

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