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handout5 - Econ 139 Introduction to Econometrics Andrew...

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Econ 139: Introduction to Econometrics Andrew Sweeting 1 Department of Economics Duke University Spring 2011 Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 1 / 38

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Relating Two Variables Econometrics is concerned with understanding the relationships between variables economists care about. e.g., education and wages, investment and innovation, advertising and sales, class size and test scores We are particularly interested in causal relationships So far we°ve only used covariance and correlation to describe the relationship between variables We° ll refresh our ideas on this, explain why we can°t do everything with just these concepts and then study regression analysis Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 2 / 38
Covariance and Correlation Covariance measures how 2 variables move together on average Cov ( X , Y ) = E [( X ° μ X ) ( Y ° μ Y )] But how can we estimate this? Suppose ( X i , Y i ) ± iid (pairs of observations are iid ) Then we can use s XY = 1 n ° 1 ° X i ° X ± ° Y i ° Y ± Furthermore, we can show that s XY p ! σ XY (i.e., consistent) Correlation also measures how two variables move together (a normalized measure of covariance) In particular, r XY = s XY s X s Y (it°s also true that r XY p ! ρ XY ) Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 3 / 38

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Correlation and Causation But does r XY > 0 mean that high values of X cause the values of Y to be high? No, correlation is not causation. Examples Umbrella demand & rain P Microsoft & P Wal-Mart Years of schooling & wage To identify causation we often need to control for confounding factors. Regression analysis will allow us to do so. Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 4 / 38
Regression Analysis Furthermore, we usually care about more than just correlation. In many cases we want to know If we increase X by a certain amount, what is the expected e/ect on Y ? i.e., the size of the causal e/ect To motivate regression analysis let°s start with a case where X is discrete and simply compare E ( Y j X ) for two values of X . Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 5 / 38

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Motivation: Regression Analysis Table 3.1 has data on average earnings for men and women. Is there a signi±cant gender gap? Looking at 1998, the wage gap ° Y m ° Y w ± is \$2.45 per hour. The standard error SE ° Y m ° Y w ± = 0 . 29 so the t -stat for H 0 : Y m ° Y w = 0 is 2 . 45 ° 0 . 29 = 8 . 45 , which has a p -value (two-sided) that°s very close to 0 (2 Φ ( ° 8 . 45 ) ² 0 ) . Indeed, a 99% CI for the wage gap is 2 . 45 ³ 2 . 58 ´ 0 . 29 = ( 1 . 7 , 3 . 2 ) Econ 139 Handout 5 (Duke) Univariate Regression Spring 2011 6 / 38
Aside: Testing for Di/erences in the Means of Two RVs Previously we talked about testing whether the expected value of one random variable was di/erent from some value. We can apply the same logic to test whether the di/erence between expected values of random variables is di/erent from some value (often zero).

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