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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 covariance and correlation to describe the relationship between variables 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 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 P Microsoft P Wal-Mart 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 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. 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 ) ± 0 ) . Indeed, a 99% CI for the wage gap is 2 . 45 ² 2 . 58 ³ 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

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## This note was uploaded on 08/02/2011 for the course ECON 139 taught by Professor Alessandrotarozzi during the Spring '08 term at Duke.

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handout5 - Econ 139: Introduction to Econometrics Andrew...

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