This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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(YX = x) 4 8 12 16 20 x Graph of E(YX = 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....
View
Full Document
 Fall '12
 Jeff
 Probability theory, CDF, conditional distribution, conditional density

Click to edit the document details