Ch 6: Multiple Regression

1
Ch 6 Multiple Regression

• Omitted variable bias
• Causality and regression analysis
• Multiple regression and OLS
• Measures of fit
• Sampling distribution of the OLS estimator
• Multicollinearity

2
Omitted Variable Bias
3
Omitted variable bias, ctd.
4
Omitted variable bias, ctd.
5
Omitted variable bias, ctd.
6
Omitted variable bias formula
7
(No Transcript)
8
Omitted variable bias formula two Xs case

• is slope coefficient from regression
  of excluded X2 on included X1
• Bias term

9
Omitted 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

10
Omitted 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 --------------------------------------
----------------------------------------
11
Omitted 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 ------------------------------------------
------------------------------------

• Conclusion?

12
Digression on causality and regression analysis
13
Ideal 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.

14
Back to class size
15
(No Transcript)
16
3 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.

17
The Population Multiple Regression Model (SW
Section 6.2)
18
Interpretation of coefficients in multiple
regression
19
The OLS Estimator in Multiple Regression (SW
Section 6.3)
20
Example the California test score data
21
Multiple regression in STATA