Now y − y 2 y − x x − y y − x 2 2 y − x x

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Unformatted text preview: Now Y − Y 2 Y − X X − Y Y − X 2 2 Y − X X − Y X − Y 2 Taking the expected value conditional on X E Y − Y 2 | X E Y − X 2 | X 2 X − Y X − Y 2 Var Y | X E Y | X − Y 2 59 Now, Y E E Y | X by LIE. Take expectations with respect to X , use LIE again, and the definition of a variance: E Y − Y 2 E Var Y | X E E Y | X − Y 2 E Var Y | X Var E Y | X 60 EXAMPLE : Suppose Y | Z ~ Poisson Z where Z 0 is a continuous random variable. Then E Y | Z Z Var Y | Z Z The unconditional mean and variance are E Y E Z Var Y E Z Var Z Without completely specifying D Z , we cannot find D Y . But we know the first two moments of Y . 61 ( cv4 ) We can write a variance conditional on a smaller set of variables in terms of conditional means and variances on a larger set of variables: Var Y | X E Var Y | X , Z | X Var E Y | X , Z | X ∙ This feature is very useful for generating overdispersion in conditional models. Often Z is an unobserved random variable (or variables) that need to be “integrated out” to obtain usable conditional mean and variance functions. 62 EXAMPLE : Suppose D Y | X , U Poisson U exp X , where U 0 is independent of X with E U 1 and Var U U 2 . Such models arise often in econometrics, where X are observable factors (for a firm, say, spending on research and development, sales, profitability) and U are unobserved attributes (managerial “talent,” workplace environment). We want to find features of D Y | X . 63 ∙ We can obtain the conditional mean and conditional variance: E Y | X E E Y | X , U | X E U | X exp X exp X Var Y | X E Var Y | X , U | X Var E Y | X , U | X E U exp X | X Var U exp X | X E U | X exp X Var U | X exp X 2 exp X U 2 exp X 2 E Y | X U 2 E Y | X 2 ....
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Now Y − Y 2 Y − X X − Y Y − X 2 2 Y − X X − Y X...

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