Lecture_3_Prof_Arkonac's_slides_(Ch3_and_Ch4)

Lecture_3_Prof_Arkonac's_slides_(Ch3_and_Ch4) -...

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Introduction to Econometrics W3412, Fall 2010 Lecture 3 Prof: Seyhan Arkonac, PhD
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Stata sessions: Thursday 9/23, 11-11:50am, SCH 558 (Naihobe) Thursday 9/23, 12-12:50pm, SCH 558 (Ran) Thursday 9/23, 4:10-6pm, SCH 558 (WooRam) Friday 9/24, 2:10-3pm, SCH 558 (Ju Hyun) Every week starting 9/24 11-1150am SCH 558 (Shreya)
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2-
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Last time we stopped at correlation coefficient What was correlation coefficient between two random variables? Say the correlation coefficient between (1) class size and (2) student performance
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5 The correlation coefficient is defined in terms of the covariance: corr( X , Z ) = cov( , ) var( )var( ) XZ XZ  = r XZ –1 corr( X , Z ) 1 corr( X , Z ) = 1 mean perfect positive linear association corr( X , Z ) = –1 means perfect negative linear association corr( X , Z ) = 0 means no linear association
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What was covariance? Remember?
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7 2 random variables: covariance The covariance between X and Z is cov( X , Z ) = E [( X X )( Z Z )] = XZ The covariance is a measure of the linear association between X and Z ; its units are units of X units of Z cov( X , Z ) > 0 means a positive relation between X and Z If X and Z are independently distributed, then cov( X , Z ) = 0 (but not vice versa!!) The covariance of a r.v. with itself is its variance cov( X , X ) = E [( X X )( X X )] = E [( X X ) 2 ] = 2 X
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Now, let’s see again “pictures” of some possible correlations between 2 random variables: Remember: 1 corr( X , Z ) 1 corr( X , Z ) = 1 mean perfect positive linear association corr( X , Z ) = 1 means perfect negative linear association corr( X , Z ) = 0 means no linear association
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9 The correlation coefficient measures linear association
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Here was the plan (from last lecture): Review of Statistical Theory: (1) The Probability Framework for Statistical Inference (2) Estimation (3) Testing (4) Confidence Intervals
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(1) The Probability Framework for Statistical Inference (a) Population, random variable and distribution (b)Moments of distribution (mean, variance, standard deviation, covariance, correlation (c) Conditional distributions and conditional means (d)Distribution of a sample of data randomly from a population: Y 1 …….Y n ( we did part (a) and part (b) last time, now let’s continue from part (c))
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12 (c) Conditional distributions and conditional means Conditional distributions The distribution of Y , given value(s) of some other random variable, X Ex: the distribution of test scores, given that STR < 20 Conditional expectations and conditional moments conditional mean = mean of conditional distribution = E ( Y | X = x ) ( important concept and notation ) conditional variance = variance of conditional distribution Example : E ( Test scores | STR < 20) = the mean of test scores among districts with small class sizes The difference in means is the difference between the means of two conditional distributions:
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