Cd3 let f y y be the cdf of a random variable y and

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Unformatted text preview: ( cd3 ) Let F Y y be the CDF of a random variable Y , and assume Y is independent of the random vector X . Then for any function a : X → , P Y ≤ a X | X x F Y a x . 22 2 . Conditional Moments and Medians ∙ Naturally, with any conditional distribution comes the usual features of distributions we are interested in. Probably we are most commonly interested in the conditional mean, but other conditional moments – such as the variance – are also important. ∙ If Y is a random variable and X is a random vector, the conditional expectation of Y given X is the mean of the distribution D Y | X . We write this conditional expectation as E Y | X . ∙ We are interested in how the expected value changes as the outcome on X changes. So we often write E Y | X x ≡ Y | X x . 23 ∙ Notice how Y | X x is a function of the argument x . Econometrics is often about estimating this function given data on Y and X . ∙ Sometimes we will just write x . 24 EXAMPLE : Let Y be unemployment duration, measured in years, for the population of people who became unemployed during, say, January 2010. Let X be years of schooling. Suppose we know Y | X x Exponential exp 1/4 − x /8 , x 0, which implies E Y | X x exp 1/4 − x /8 . ∙ The average unemployment duration falls as education level increases, initially quite rapidly. 25 ∙ We can plug in specific values to see this: E Y | X exp 1/4 ≈ 1.284 E Y | X 4 exp − 1/4 ≈ .779 E Y | X 8 exp − 3/4 ≈ .472 E Y | X 12 exp 1/4 − 4/3 ≈ .338 E Y | X 16 exp 1/4 − 2 ≈ .174 ∙ At no years of schooling (unrealistic), the expected unemployment duration is about 1.3 years. For someone with a high school degree, it is about 4 months (.388 12 ≈ 4.06). It is about two months with a college education. 26 .5 1 1.5 E(Y|X = x) 4 8 12 16 20 x Graph of E(Y|X = x) = exp(1/4 - x/8) . range x 0 20 1000 obs was 0, now 1000 . gen eyx exp(1/4 - x/8) . twoway (line eyx x) 27 ∙ As a practical matter, it is too simplistic to act as if we know E Y | X x exp 1/4 − x /8 . At a minimum we would start with a model such as E Y | X x exp x , where we treat and as unknown parameters (constants) – rather than assuming 1/4 and − 1/8 – and then figure out how to estimate and using statistical methods. In probability, we assume we know the relationship in the population....
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cd3 Let F Y y be the CDF of a random variable Y and assume...

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