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# 3100 notes - Introductory Econometrics Lecture 6 Multiple...

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Introductory Econometrics Lecture 6 - Multiple Regression II Xiaoxia Shi University of Wisconsin - Madison 09/21/2010 Lecture (6) Intro Metrics 09/21/2010 1 / 47

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Outline A "Partialling Out" Interpretation of Multiple Regression Unbiasedness of OLS Estimators Variance of OLS Estimators Irrelevant Variables and Omitted Variables Changing Units and Functional Forms . Lecture (6) Intro Metrics 09/21/2010 2 / 47
Outline A "Partialling Out" Interpretation of Multiple Regression Unbiasedness of OLS Estimators Variance of OLS Estimators Irrelevant Variables and Omitted Variables Changing Units and Functional Forms . Lecture (6) Intro Metrics 09/21/2010 3 / 47

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Multiple Regression Regression with many regressors Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ... + β K X Ki + ε i . Help solving the omitted variable bias Can study the e/ects of many variables at the same time Can use the prediction power of many variables simultaneously . Lecture (6) Intro Metrics 09/21/2010 4 / 47
Estimation of Multiple Regression We use the sample moment conditions: 1 n N i = 1 ° Y i ° ˆ β 0 ° ˆ β 1 X 1 i ° ... ° ˆ β Ki X Ki ± = 0 1 n N i = 1 ° Y i ° ˆ β 0 ° ˆ β 1 X 1 i ° ... ° ˆ β Ki X Ki ± X 1 i = 0 ... 1 n N i = 1 ° Y i ° ˆ β 0 ° ˆ β 1 X 1 i ° ... ° ˆ β Ki X Ki ± X Ki = 0. Explicit formula for each ˆ β j is complicated. . Lecture (6) Intro Metrics 09/21/2010 5 / 47

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A Partialling-Out Form The complicated formula actually has a simple, intuitive form: ˆ β 1 = n i = 1 ˆ r i 1 Y i n i = 1 ˆ r 2 i 1 , where ˆ r i 1 are the OLS residuals from regressions of X 1 on the rest of the explanatory variables, i.e., X 2 , ..., X K . . Lecture (6) Intro Metrics 09/21/2010 6 / 47
A Partialling-Out Form In fact, ˆ β 1 = n i = 1 ˆ r i 1 Y i n i = 1 ˆ r 2 i 1 = n i = 1 ° ˆ r i 1 ° ˆ r 1 ± Y i n i = 1 ° ˆ r i 1 ° ˆ r 1 ± ˆ r i 1 because ˆ r 1 = n ° 1 n i = 1 ˆ r i 1 = 0. (why?) Interpretation: ˆ β 1 equals the slope coe¢ cient from the simple regression of Y on ˆ r 1 . . Lecture (6) Intro Metrics 09/21/2010 7 / 47

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A Partialling-Out Form The residuals ˆ r i 1 are part of X 1 i that is uncorrelated with X 2 i , ..., X Ki . The multiple regression slope coe¢ cient, ˆ β 1 , is the sample relationship between X 1 and Y after the other explanatory variables are "partialled out." . Lecture (6) Intro Metrics 09/21/2010 8 / 47
Example Let°s verify this using STATA. . Lecture (6) Intro Metrics 09/21/2010 9 / 47

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Outline A "Partialling Out" Interpretation of Multiple Regression Unbiasedness of OLS Estimators Variance of OLS Estimators Irrelevant Variables and Omitted Variables Changing Units and Functional Forms . Lecture (6) Intro Metrics 09/21/2010 10 / 47
Assumptions Assumption MLR.1 (Linear in Parameters). The population model can be written as Y = β 0 + β 1 X 1 + ... + β K X K + U , where β 0 , β 1 , ..., β K are unknown parameters of interest and U is an error term (unobserved). . Lecture (6) Intro Metrics 09/21/2010 11 / 47

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Assumptions Assumption MLR.2 (Random Sampling). We have a random sample of n observations f ( Y i , X 1 i , ..., X Ki ) : i = 1 , ..., n g following the population model in MLR.1 .
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