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Unformatted text preview: 1 Regression Analysis with One Variable Introduction to Econometrics (18) Econ 281 Spring Quarter 2007 2 Introduction Remember our two examples Student Test Scores and StudentTeacher Ration Wages and Education Both are Relationships between two variables Dependent Variable: Test Scores, Wage Independent Variable (Regressors): StudentTeacher Ration, Education Simple Regression Analysis is all about investigating the relationship between those two variables 3 Introduction We start out by looking at the population relationship between the two variables Note: We are not really able to do that The population regression function: m(x i ) is a general function Example: ( 29 ( 29 i i i x m x X Y E = =  ( 29 ( 29 i i i edu m edu Education wage E = =  4 Introduction We will simplify things by assuming that m(x i ) is linear Note: It is important to remember that this is an assumption and that we NEVER actually observe the TRUE relationship between those variables i i X x m + = 1 ) ( 5 Introduction Now we confront this model with sample data taken from the population. Would we expect the above relationship to hold perfectly? No! Data differs because other variables influence Y i Data differs because of random variation i i X Y + = 1 6 Introduction Solution: We add an ERROR TERM (u i ) to the equation The error term captures the effect of other variables random deviations from Y i This gives us the linear regression model i i i u X Y + + = 1 7 The Linear Regression Model i i i u X Y + + = 1 Slope Coefficient Intercept Coefficient Dependent Variable Independent Variable (Regressor) Error Term 8 (X 1 ,Y 1 ) 600 620 640 660 680 700 15 20 25 Test Scores and StudentTeacher Ratio StudentTeacher Ratio Test Scores (Unobserved) Population Regression Function =u 1 (X 2 ,Y 2 ) (X 3 ,Y 3 ) =u 2 =u 3 9 Interpretation of the Coefficients Lets look at the Test Score example Where TestScore = Average Test Score of a class STR = StudentTeacher Ration of class Knowledge of the slope coefficient would enable us to predict the effect on Test Scores of an change in the Class Size STR + = 1 TestScore STR = = TSC Ratio acher Student/Te in change Score Test in change 1 10 Interpretation of the Coefficients Assume the studentteacher ratio changes Then we can calculate the predicted change in Test Scores according to If we know both the intercept AND the slope coefficient we can also predict the average test score (and the change) for a school district tio StuTeachRa = 1 TestScore tio StuTeachRa + = 1 TestScore 11 The Linear Regression Model Note that is the POPULATION regression function!!...
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This note was uploaded on 04/07/2008 for the course ECON 281 taught by Professor Habermalz during the Spring '08 term at Northwestern.
 Spring '08
 HABERMALZ
 Econometrics

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