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handout2 - Introduction to Econometrics Andrew Sweeting1 1...

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Introduction to Econometrics Andrew Sweeting 1 1 Department of Economics Duke University Spring 2011 Econ 139 Handout 2 (Duke) Probability Spring 2011 1 / 48
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An Unusual Case In this course, we will usually work with distributions that are well-behaved, and have well-de&ned moments There are cases where this would not be true, e.g., the Cauchy Distribution -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 for which neither mean nor variance are de&ned (but the median is) Econ 139 Handout 2 (Duke) Probability Spring 2011 2 / 48
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Fat Tails Can Be Relevant Econ 139 Handout 2 (Duke) Probability Spring 2011 3 / 48
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Relationships Between Two Random Variables In economics we are typically interested in the relationship between random variables education and salary quantity and price tax rates and hours worked stock price and earnings R&D policy and innovation rate Econ 139 Handout 2 (Duke) Probability Spring 2011 4 / 48
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Joint and Marginal Distributions The joint probability distribution is the probability that two discrete random variables ( X and Y ) simultaneously take on particular values ( x and y ) The joint probability distribution function can be written as P ( X = x , Y = y ) = f ( X , Y ) The various probabilities should (of course) sum to 1 With a third variable one could construct a more complicated table Econ 139 Handout 2 (Duke) Probability Spring 2011 5 / 48
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Joint and Marginal Distributions We can recover the distribution of each variable individually from their joint distribution The marginal probability distribution of a r.v. Y is just its own probability distribution It can be computed from the joint probability distribution of X and Y as: P ( Y = y ) = all values of X P ( X = x i , Y = y ) or (more compactly) P ( Y = y ) = n i = 1 P ( X = x i , Y = y ) Econ 139 Handout 2 (Duke) Probability Spring 2011 6 / 48
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Joint and Marginal Distributions: Example Long Commute: P ( Y = 0 ) = P ( X = 0 , Y = 0 ) + P ( X = 1 , Y = 0 ) = 0 . 15 + . 07 = . 22 Econ 139 Handout 2 (Duke) Probability Spring 2011 7 / 48
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Conditional Distribution Knowing something about one random variable may tell us something about another e.g., a commute is likely to be longer when it is raining The distribution of a random variable Y conditional on another random variable X taking on a speci&c value is called the conditional distribution of Y given X The conditional distribution can be calculated P ( Y = y j X = x ) = P ( X = x , Y = y ) P ( X = x ) This formula should be intuitive "out of the times when X = x how often is Y = y ? " Econ 139 Handout 2 (Duke) Probability Spring 2011 8 / 48
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Conditional Distribution: Example P ( Y = y j X = x ) = P ( X = x , Y = y ) P ( X = x ) P ( M = 1 j A = 1 ) = P ( M = 1 , A = 1 ) P ( A = 1 ) = 0 . 035 0 . 5 = 0 . 07 Econ 139 Handout 2 (Duke) Probability Spring 2011 9 / 48
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Conditional Expectation A conditional expectation (i.e., the expected value of one random variable conditional on the realized value of another random variable) is calculated like any expectation, but using conditional probabilities E [ Y j X = x ] = n i = 1 y i P ( Y = y i j X = x ) Example: E ( M j A = 1 ) = 0 & 0 . 9 + 1 & 0 . 07 + 2 & 0 . 02 + 3 & 0 . 01 + 4 & 0 = 0 . 14 Similarly E ( M j A = 0 ) = 0 . 56 Econ 139 Handout 2 (Duke) Probability Spring 2011 10 / 48
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Conditional Expectation: Further Example Suppose a long commute takes 25 minutes and a short commute 10 minutes (so Y is now minutes).
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handout2 - Introduction to Econometrics Andrew Sweeting1 1...

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