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Econ 281 Regression Analysis with One Variable

# Econ 281 Regression Analysis with One Variable - Regression...

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1 Regression Analysis with One Variable Introduction to Econometrics Econ 281

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2 Introduction Remember our two examples Student Test Scores and Student-Teacher Ration Wages and Education Both are Relationships between two variables Dependent Variable: Test Scores, Wage Independent Variable (Regressors): Student-Teacher 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: () ( ) i i i x m x X Y E = = | ( ) i i i edu m edu Education wage E = = |

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4 Saturated Models One can keep the population regression function completely flexible by including dummy (binary) variables and their interactions Dummy Variables = indicator variable that is equal to one if a condition is met and otherwise equal to zero 1 0 male if individual is male D otherwise =
5 Saturated Models Interaction = the product of two or more variables Let’s assume that we only have two variables: gender and race (white, non- white) A saturated model looks like ( ) 01 2 3 |, i male gender male gender male gender EY D D D D DD ββ β =+ + +⋅

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6 Saturated Models Let’s see why this is a perfectly general model There are four outcomes ( ) 0 |0 , 0 i male gender EY D D β == = ( ) 01 |1 , 0 i male gender D D = + ( ) 02 , 1 i male gender D D = + ( ) 012 3 , 1 i male gender D D ββ = + + +
7 Saturated Models Why is this perfectly general? => Every possible constellation has its own effect => No need to combine effects However, we usually don’t use saturated models. Why? How many possible outcomes do you have when there are 10 possible educational outcomes 10 possible experience outcomes Gender Race Too many!! We need to simplify things!

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8 Linearity Assumption 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 0 ) ( β
9 Linearity Assumption Now we “confront” this model with 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 0 β

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10 Linearity Assumption 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 0 β
11 The Linear Regression Model i i i u X Y + + = 1 0 β Slope Coefficient Intercept Coefficient Dependent Variable Independent (Regressor) Error Term

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12 (X 1 ,Y 1 ) 600 620 640 660 680 700 15 20 25 Test Scores and Student-Teacher Ratio Student-Teacher Ratio Test Scores (Unobserved) Population Regression Function =u 1 (X 2 ,Y 2 ) (X 3 ,Y 3 ) =u 2 =u 3 (X 4 ,Y 4 ) =u 4
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Econ 281 Regression Analysis with One Variable - Regression...

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