Title: Ch 6: Multiple Regression
1Ch 6 Multiple Regression
- Omitted variable bias
- Causality and regression analysis
- Multiple regression and OLS
- Measures of fit
- Sampling distribution of the OLS estimator
- Multicollinearity
2Omitted Variable Bias
3Omitted variable bias, ctd.
4Omitted variable bias, ctd.
5Omitted variable bias, ctd.
6Omitted variable bias formula
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8Omitted variable bias formula two Xs case
- is slope coefficient from regression
of excluded X2 on included X1 -
- Bias term
9Omitted variable bias formula two Xs case
application
. reg prate mrate age, r Linear regression
Number of obs
1534
F( 2, 1531) 98.18
Prob gt F
0.0000
R-squared 0.0922
Root MSE 15.937 -------------------------
--------------------------------------------------
--- Robust
prate Coef. Std. Err. t Pgtt
95 Conf. Interval ---------------------------
--------------------------------------------------
mrate 5.521289 .4498478 12.27
0.000 4.638906 6.403672 age
.2431466 .0393743 6.18 0.000 .1659133
.3203798 _cons 80.11905 .846797
94.61 0.000 78.45804
81.78005 -----------------------------------------
-------------------------------------
- prate participation rate in companys 401(k)
plan - mrate match rate (amount firm contributes for
each 1 worker contributes) - age age of the 401(k) plan
10Omitted variable bias formula two Xs case
application
. reg prate mrate, r Linear regression
Number of obs
1534
F( 1, 1532) 157.77
Prob gt F
0.0000
R-squared 0.0747
Root MSE 16.085 -------------------------
--------------------------------------------------
--- Robust
prate Coef. Std. Err. t Pgtt
95 Conf. Interval ---------------------------
--------------------------------------------------
mrate 5.861079 .4666276 12.56
0.000 4.945783 6.776376 _cons
83.07546 .6112819 135.90 0.000 81.87642
84.27449 --------------------------------------
----------------------------------------
11Omitted variable bias formula two Xs case
application
. reg age mrate, r Linear regression
Number of obs 1534
F( 1, 1532) 18.75
Prob gt F
0.0000
R-squared 0.0141
Root MSE
9.1092 ----------------------------------
--------------------------------------------
Robust age
Coef. Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
mrate 1.39747 .322743 4.33 0.000
.7644054 2.030535 _cons 12.15896
.3132499 38.82 0.000 11.54451
12.7734 ------------------------------------------
------------------------------------
12Digression on causality and regression analysis
13Ideal Randomized Controlled Experiment
- Ideal subjects all follow the treatment protocol
perfect compliance, no errors in reporting,
etc.! - Randomized subjects from the population of
interest are randomly assigned to a treatment or
control group (so there are no confounding
factors) - Controlled having a control group permits
measuring the differential effect of the
treatment - Experiment the treatment is assigned as part of
the experiment the subjects have no choice, so
there is no reverse causality in which subjects
choose the treatment they think will work best.
14Back to class size
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163 solutions to Omitted Variable Bias
- Run a randomized controlled experiment in which
treatment (STR) is randomly assigned. - Use the cross tabulation approach, but
- Include the variable as an additional covariate
in the multiple regression.
17The Population Multiple Regression Model (SW
Section 6.2)
18Interpretation of coefficients in multiple
regression
19The OLS Estimator in Multiple Regression (SW
Section 6.3)
20Example the California test score data
21Multiple regression in STATA