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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 ) ( β β 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 β β 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 β β 7 The Linear Regression Model i i i u X Y + ⋅ + = 1 β β 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 β β STR ∆ ∆ = = TSC Ratio acher Student/Te in change Score Test in change 1 β 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 β tio StuTeachRa ⋅ + = 1 TestScore β β 11 The Linear Regression Model • Note that is the POPULATION regression function!!...
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 Spring '08
 HABERMALZ
 Econometrics, Linear Regression, Regression Analysis, Yi, Ordinary least squares

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