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Ch. 6, 7, 18 Regression with Multiple Regressors

# Ch. 6, 7, 18 Regression with Multiple Regressors - 4...

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1 | Multiple Regression 4. REGRESSION WITH MULTIPLE REGRESSORS (CH. 6, 7, and 18) The recommended exercise questions from the textbook: Chapter 6: All except (6.10), and except (6.11.f) from the 3 rd ed. Chapter 7: All except (7.8) c, and except (7.11) from the 3 rd ed. What do we learn? Multiple regression. Omitted Variable bias. Matrix formulas for OLS. F -tests. Adjusted R 2 .

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2 | Multiple Regression (1) Omitted Variable Bias 01 YX u  . The error u contains all of the factors that influence Y , but are not included in the regression function ( Omitted variables ). 1 measures the effect of X on Y holding others being equal . Sometimes, the omission of those variables can lead to bias in the OLS estimator. Omitted variable bias occurs if an omitted factor “ Z ” is: 1. A determinant of Y (i.e. Z is part of u ); and 2. Correlated with the regressor X ( i.e. corr( Z , X ) 0).
3 | Multiple Regression In the test score example: Y = Average test score; X = STR. 1. English language ability ( Z ) may affect standardized test scores: Z is a determinant of Y . 2. Immigrant communities tend to be less affluent and higher STR : Z is correlated with X .

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4 | Multiple Regression What is the direction of the bias in 1 ˆ ? 01 1 1 22 1 1 () ( ) [ ( ) ] ˆ ( )[( ) ( )] ( )( ) ii i i i o i i i X XY Y X X X u X u XX XXXX u u u u    11 2 ( ) 1 ˆ 1 ˆ cov( , ) cov( , ) i p u u n n X uX u s  
5 | Multiple Regression The omitted variable bias formula : 11 2 cov( , ) ˆ p X Xu   . Suppose that * 01 2 _ TestScore STR EL PCT u  . But estimate TestScore STR u , * 2 _ uE L P C T u * 22 2 cov( , ) cov( , _ )c o v ( , _ ) c o v ( , _ )( ) ( ) X u STR EL PCT u STR EL PCT STR EL PCT   “downward biased”.

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6 | Multiple Regression [Example] Dependent Variable: TESTSCR Included observations: 420 White Heteroskedasticity-Consistent Standard Errors & Cov Variable Coefficient Std. Error t-Statistic Prob. C 698.9 10.36 67.44 0.000 STR -2.280 0.520 -4.389 0.000 R-squared 0.051 Mean dependent var 654.16 Dependent Variable: TESTSCR Variable Coefficient Std. Error t-Statistic Prob. C 686.0 8.728 78.60 0.000 STR -1.10 0.433 -2.544 0.011 EL_PCT -0.650 0.031 -20.94 0.000 R-squared 0.426 Mean dependent var 654.2
7 | Multiple Regression Correlation Matrix STR EL_PCT STR 1.000 0.188 EL_PCT 0.188 1.000

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8 | Multiple Regression [ EXAMPLE] Data: (WAGE2.WF1 or WAGE2.TXT – from q:\drive) # of observations (T): 935 1. wage monthly earnings (\$) 2. hours average weekly hours 3. IQ IQ score 4. KWW knowledge of world work score 5. educ years of education 6. exper years of work experience 7. tenure years with current employer 8. age age in years 9. married =1 if married 11. south =1 if live in south 12. urban =1 if live in SMSA 13. sibs number of siblings 14. brthord birth order 15. meduc mother's education 16. feduc father's education 17. lwage natural log of wage
9 | Multiple Regression Mincerian wage equation: lwage = 0 + 1 educ + 2 exper + error Dependent Variable: LWAGE Sample: 1 935 Included observations: 935 Variable Coefficient Std. Error t-Statistic Prob.

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Ch. 6, 7, 18 Regression with Multiple Regressors - 4...

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