Lecture 7 Prof. Arkonac's Slides (Ch 5.3 - Ch 6.3) for ECO 4000

Lecture 7 Prof. Arkonac's Slides (Ch 5.3 - Ch 6.3) for ECO 4000

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Introduction to Multiple Regression ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 7 Prof: Seyhan Erden Arkonac, PhD 1
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Where we stopped last time, and what will we do today? Regression when X is a binary (dummy) variable (i.e. X=0 or X=1) Heteroskedasticity and Homoskedasticity (variance of the error term is constant? Or not?) The theoretical foundation of the OLS (not in detail!) Omitted variable bias THE MULTIPLE REGRESSION MODEL 2
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3 Regression when X is Binary (Section 5.3) Sometimes a regressor is binary: X = 1 if small class size, = 0 if not X = 1 if female, = 0 if male X = 1 if treated (experimental drug), = 0 if not Binary regressors are sometimes called “dummy” variables. So far, 1 has been called a “slope,” but that doesn’t make sense if X is binary. How do we interpret regression with a binary regressor?
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4 Interpreting regressions with a binary regressor Y i = 0 + 1 X i + u i , where X is binary ( X i = 0 or 1): When X i = 0, Y i = 0 + u i the mean of Y i is 0 that is, E ( Y i | X i =0) = 0 When X i = 1, Y i = 0 + 1 + u i the mean of Y i is 0 + 1 that is, E ( Y i | X i =1) = 0 + 1 so : 1 = E ( Y i | X i =1) – E ( Y i | X i =0) = population difference in group means
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5 Example : Let D i = 1 if 20 0 if 20 i i STR STR OLS regression : TestScore = 650.0 + 7.4 D (1.3) ( 1.8 ) Tabulation of group means : Class Size Average score ( Y ) Std. dev. ( s Y ) N Small ( STR ≤ 20) 657.4 19.4 238 Large ( STR > 20) 650.0 17.9 182 Difference in means: small large YY = 657.4 – 650.0 = 7.4 Standard error: SE = 22 sl ss nn = 19.4 17.9 238 182 = 1.8
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6 Summary: regression when X i is binary (0/1) Y i = 0 + 1 X i + u i 0 = mean of Y when X = 0 0 + 1 = mean of Y when X = 1 1 = difference in group means, X =1 minus X = 0 SE( 1 ˆ ) has the usual interpretation t -statistics, confidence intervals constructed as usual This is another way (an easy way) to do difference-in-means analysis The regression formulation is especially useful when we have additional regressors ( as we will very soon )
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7 Heteroskedasticity and Homoskedasticity, and Homoskedasticity-Only Standard Errors (Section 5.4) What…? Consequences of homoskedasticity Implication for computing standard errors What do these two terms mean? If var( u | X = x ) is constant – that is, if the variance of the conditional distribution of u given X does not depend on X then u is said to be homoskedastic . Otherwise, u is heteroskedastic .
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To understand heteroskedasticity better; Wage i = β 0 + β 1 Male i + u i Wage i = β 0 + u i for women Wage i = β 0 + β 1 + u i for men For women, u i is the deviation of the i th woman’s wage from the population mean wage for women (β 0 ) and for men, u i is the deviation of the i th man’s wage from the population mean wage for men (β 0 1 ) The following 3 statements are the same: “The variance does not depend on the regressor Male”= “the variance of wage is the same for men as it is for women” = “error term is homoskedastic” 8
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9 Example : hetero/homoskedasticity in the case of a binary regressor (that is, the comparison of means) Standard error when group variances are unequal : SE = 22 sl ss nn Standard error when group variances are equal : SE = 11 p s where 2 p s = ( 1) ( 2 s s l l n s n s (SW, Sect 3.6) s p = “pooled estimator of 2 ” when 2 l = 2 s
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This note was uploaded on 05/05/2011 for the course ECON 4000 taught by Professor Arkonac during the Spring '11 term at CUNY Baruch.

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Lecture 7 Prof. Arkonac's Slides (Ch 5.3 - Ch 6.3) for ECO 4000

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