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handout7_1 - 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 7 (Duke) Univariate Regression (3) Spring 2011 1 / 38
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Regression When X is a Binary Variable So far, we have only looked at examples where the regressor ( X ) is a &continuous±variable (e.g. dosage, class size). Regression Analysis can also be used when X is a binary or dummy variable (i.e. can only take on the values 0 and 1). gender, drug treatment, democrat. .. Although the coe¢ cients are calculated in exactly the same way when X is binary, the interpretation of β 1 di/ers. Why? Because a regression with a binary regressor is equivalent to performing a di/erence of means analysis. So β 1 isn²t really a slope anymore. .. Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 2 ³ 38
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Regression When X is a Binary Variable For example, suppose we look at the CPS data on earnings from Chapter 3. Let&s focus only on 1998. Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 3 / 38
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Regression When X is a Binary Variable Let Y i be average hourly earnings in 1998 and D i equal 1 if the worker is male and 0 if the worker is female. The population regression model with D i as the regressor is Y i = β 0 + β 1 D i + u i Since D i is not continuous, we can&t really think of β 1 as a slope (because D i only takes on 2 values, there&s no ±line²). For this reason, we just call β 1 the coe¢ cient on D i , instead of the slope. So how do we interpret β 1 if it&s not a slope? Let&s look at what we have for each value of D i Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 4 / 38
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Regression When X is a Binary Variable Y i = β 0 + β 1 D i + u i When D i = 0 (the worker is female) Y i = β 0 + β 1 & 0 + u i = β 0 + u i Since E ( Y i j D i = 0 ) = β 0 , β 0 is the population mean value of earnings for women. Whereas when D i = 1 (the worker is male) Y i = β 0 + β 1 & 1 + u i = β 0 + β 1 + u i So E ( Y i j D i = 1 ) = β 0 + β 1 , the population mean value of earnings for men. β 1 is then the di/erence between the two population means. Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 5 ± 38
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Regression When X is a Binary Variable Earnings i = β 0 + β 1 Male i + u i Here&s the result of the regression above using the 1998 data: \ Earnings = 15 . 49 ( . 20 ) + 2 . 45 ( 0 . 29 ) & Male Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 6 / 38
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\ Earnings = 15 . 49 ( . 20 ) + 2 . 45 ( 0 . 29 ) & Male b β 0 = 15 . 49 is the average value of earnings for women. b β 0 + b β 1 = 17 . 94 is the average value of earnings for men. b β 1 = 2 . 45 is the di/erence between the two sample averages. Recall that, using Table 3.1, a 95% CI for the wage gap is 17 . 94 ± 15 . 49 ² 1 . 96 & r 7 . 86 2 1393 + 6 . 80 2 1210 = 2 . 45 ² 1 . 96 & . 29 = ( 1 . 89 , 3 . 02 ) What is the 95% CI for β 1 ? b β 1 ² 1 . 96 & SE & b β 1 ± = 2 . 45 ² 1 . 96 & . 29 = ( 1 . 89 , 3 . 02 ) Econ 139 Handout 7 (Duke) Univariate Regression (3) Spring 2011 7 ± 38
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Regression When X is a Binary Variable \ Earnings = 15 . 49 ( . 20 ) + 2 . 45 ( 0 . 29 ) & Male We can test the hypothesis H 0 : β 1 = 0 H A : β 1 6 = 0 by calculating the t- statistic t act = b β 1 ± 0 SE & b β 1 ± = 2 . 45 . 287 = 8 . 53 and then calculating the p-value p-value = 2 Φ ² ± ³ ³ t ³ ³ ´ = 2 Φ ( ± 8 . 53 ) ² 0 We can reject the null hypothesis at any positive level of signi&cance (just as before).
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handout7_1 - Econ 139: Introduction to Econometrics Andrew...

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