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Unformatted text preview: PAM 3100 Multiple Regression Analysis Bivariate Linear Regression Model Fall 2010 Michael Lovenheim mfl55@cornell.edu In both social sciences and hard sciences, we often want to explain one variable in terms of another. We want to draw inference about this relationship in the population (which is unobserved) from a sample (which is observed). The goal is to see how two variables vary with each other. In certain circumstances, we can make causal statements, such as changing X by 1 unit will change Y by a given amount. Causality is difficult to establish and is more fully dealt with in subsequent courses. Crowding and Time to Degree AL AR AZ CA CO CT DE FL GA IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT VA VT WA WI WV.2 .2 .4 Percentage Change in Time to Degree.4.2 .2 .4 .6 Percentage Change in Population 18 How can we tell whether changes in the 18 year old population is systematically related to changes in time to degree? Cigarette Taxes and Sales How would you go about getting the average relationship between sales and taxes from this graph? AL AR AZ CA CO CT DC DE FL GA IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT VA VT WA WI WV WY 3.5 4 4.5 5 5.5 Cigarette Sales 3 4 5 6 Cigarette Taxes Cigarette Taxes and Sales Bivariate linear regression is about finding the slope and intercept of the red line above that is, of the line that best fits the data. What does it mean to best fit? Minimize distance between each point and the best fit line. AL AR AZ CA CO CT DC DE FL GA IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT VA VT WA WI WV WY 3.5 4 4.5 5 5.5 Cigarette Sales 3 4 5 6 Cigarette Taxes We want a formula for the red line that tells us how cigarette sales and taxes vary on average. In other words, we want an expression of the average of cigarette sales at any level of taxes. This is the conditional expectation function, E(Y  X). In the previous example, the intercept is 6.2 and the slope is 0.46. This means that as taxes increase by 1, sales decrease by 0.46 If we call the intercept and the slope , we can write the conditional mean as: This expression tells us average sales for any given level of taxes. Where we are going: we want to be able to learn something about the relationship between 2 variables (like cigarette sales and taxes) in the population through what we observe in the sample . Example: can we rule out from the previous graph that cigarette taxes are unrelated to sales? 1 Taxes Taxes Sales E 1 ]  [ + = To make such determinations, we need to formally model the relationship among variables we think is true in the population....
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This note was uploaded on 01/30/2012 for the course PAM 3100 at Cornell University (Engineering School).
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