PS 233

1
PS 233

• Intermediate Statistical Methods
• Lecture 13
• Omitted Variable Bias, part 2

2
Choosing Your Variables

• Variable selection should be based on your
  theoretical understanding of the causal process
  underlying the dependent variable
• This is a theoretical rather than a statistical
  exercise
• There are rules of thumb for including or
  excluding variables from an analysis

3
Reasons to Include a Variable

• X2 is a potential Source of Omitted Variable Bias
• Variable is a cause of Y and correlated with X1
• Methodological Control variables
• Selection parameters, temporal dependence, etc
• You must include these Xs as controls
• X2 is a benchmark for evaluating the substantive
  impact of X1
• Selected at your discretion

4
Reasons to Exclude a Variable

• X2 is NOT a cause of Y, but it creates
  multicollinearity problems
• X2 is an intervening variable in between X1 and Y
• With intervening variables, run model both with
  and without X2
• Examine change in coefficient for X1

5
WARNING !!!

• ALL DATA IN THE FOLLOWING EXAMPLES ARE COMPLETELY
  ARTIFICIAL!
• PLEASE DONT DRAW ANY SUBSTANTIVE CONCLUSIONS
  FROM THESE RESULTS!!!

6
Omitted Variable Bias Trade as a Cause of
Democracy
Trade Openness
B0.5
r 0.6
Democracy
B0.4
Economic Wealth
7
Regressing Democracy on Trade, Omitting Wealth
reg democ tottrade Source SS df
MS Number of obs
100 ---------------------------------------
F( 1, 98) 34.98 Model
3446.51643 1 3446.51643 Prob gt
F 0.0000 Residual 9656.70632 98
98.5378196 R-squared
0.2630 ---------------------------------------
Adj R-squared 0.2555 Total
13103.2227 99 132.355785 Root
MSE 9.9266 -------------------------------
-----------------------------------------------
democ Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- tottrade 1.010671 .1708917 5.914
0.000 .6715418 1.3498 _cons
2.372788 2.009129 1.181 0.240
-1.614264 6.35984 ----------------------------
--------------------------------------------------
8
Regressing Democracy on Trade AND Wealth
. reg democ tottrade wealth Source
SS df MS Number of
obs 100 -----------------------------------
---- F( 2, 97) 28.27
Model 4825.11958 2 2412.55979
Prob gt F 0.0000 Residual 8278.10316
97 85.3412697 R-squared
0.3682 ---------------------------------------
Adj R-squared 0.3552 Total
13103.2227 99 132.355785 Root
MSE 9.238 ------------------------------
------------------------------------------------
democ Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- tottrade .5284415 .1992195
2.653 0.009 .1330461 .923837 wealth
.3886304 .0966934 4.019 0.000
.1967208 .5805399 _cons 1.199239
1.892422 0.634 0.528 -2.556694
4.955173 -----------------------------------------
-------------------------------------
9
Irrelevant Controls and Multicollinearity
Trade Openness
B0.5
r 0.6
Democracy
r .75
B0.4
Economic Wealth
B 0
r .95
Military Power
10
Regressing Democracy on Wealth, Trade and Power
. reg democ tottrade wealth milpow Source
SS df MS Number
of obs 100 --------------------------------
------- F( 3, 96) 18.70
Model 4833.01576 3 1611.00525
Prob gt F 0.0000 Residual 8270.20698
96 86.1479894 R-squared
0.3688 ---------------------------------------
Adj R-squared 0.3491 Total
13103.2227 99 132.355785 Root
MSE 9.2816 -------------------------------
-----------------------------------------------
democ Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- tottrade .6306823 .3925667 1.607
0.111 -.1485564 1.409921 wealth
.5127502 .4213262 1.217 0.227
-.3235757 1.349076 milpow -.0619981
.2047824 -0.303 0.763 -.468488
.3444917 _cons 1.088159 1.936422
0.562 0.575 -2.755609
4.931927 -----------------------------------------
-------------------------------------
11
Auxiliary Regression on Wealth, Trade and Power
reg wealth tottrade milpow Source SS
df MS Number of obs
100 ---------------------------------------
F( 2, 97) 1382.91 Model
13837.6049 2 6918.80243 Prob
gt F 0.0000 Residual 485.297631 97
5.00306836 R-squared
0.9661 ---------------------------------------
Adj R-squared 0.9654 Total
14322.9025 99 144.675783 Root
MSE 2.2368 -------------------------------
-----------------------------------------------
wealth Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- tottrade -.7139595 .0607854 -11.746
0.000 -.8346018 -.5933173 milpow
.4729452 .0113791 41.562 0.000
.4503608 .4955297 _cons 1.007912
.4552951 2.214 0.029 .1042778
1.911547 -----------------------------------------
-------------------------------------
12
Irrelevant Controls and Extraneous (but harmless)
Bs
Trade Openness
B0.5
r 0.6
Democracy
r .03
B0.4
Economic Wealth
B 0
r .10
Olympic Medals
13
Regressing Democracy on Wealth, Trade and Medals
reg democ tottrade wealth medals Source
SS df MS Number
of obs 100 --------------------------------
------- F( 3, 96) 19.03
Model 4886.75371 3 1628.9179
Prob gt F 0.0000 Residual 8216.46904
96 85.5882191 R-squared
0.3729 ---------------------------------------
Adj R-squared 0.3533 Total
13103.2227 99 132.355785 Root
MSE 9.2514 -------------------------------
-----------------------------------------------
democ Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- tottrade .5201341 .1997475 2.604
0.011 .1236385 .9166298 wealth
.3988365 .0975772 4.087 0.000
.2051473 .5925257 medals -.1334243
.1572285 -0.849 0.398 -.4455204
.1786718 _cons 2.471675 2.416604
1.023 0.309 -2.325246
7.268597 -----------------------------------------
-------------------------------------
14
Control Variables for Comparisons
GNP Growth
B1.0
r 0.05
Presidential Approval
B0.5
International Threats
15
Regressing Approval on GNP Growth, Omitting
Threats
reg approve gnpgro Source SS
df MS Number of obs
100 ---------------------------------------
F( 1, 98) 13.70 Model
4419.43882 1 4419.43882 Prob gt
F 0.0004 Residual 31604.8543 98
322.498513 R-squared
0.1227 ---------------------------------------
Adj R-squared 0.1137 Total
36024.2931 99 363.881748 Root
MSE 17.958 ------------------------------
------------------------------------------------
approve Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- gnpgro 1.144467 .3091601
3.702 0.000 .5309485 1.757985 _cons
27.2151 3.634715 7.488 0.000
20.00213 34.42807 -----------------------------
-------------------------------------------------
16
Regressing Approval on GNP Growth AND Threats
reg approve gnpgro threat Source SS
df MS Number of obs
100 ---------------------------------------
F( 2, 97) 162.55 Model
27745.7297 2 13872.8648 Prob
gt F 0.0000 Residual 8278.56343 97
85.3460147 R-squared
0.7702 ---------------------------------------
Adj R-squared 0.7655 Total
36024.2931 99 363.881748 Root
MSE 9.2383 -------------------------------
-----------------------------------------------
approve Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- gnpgro 1.015053 .1592343
6.375 0.000 .6990165 1.331089 threat
.4972377 .0300769 16.532 0.000
.4375434 .556932 _cons 4.292249
2.327818 1.844 0.068 -.3278251
8.912323 -----------------------------------------
-------------------------------------
17
Controlling for Intervening Variables
Similarity of Regime Type
B 0
B 1.5
Bilateral Trade Flows
B2.0
Similarity of Security Interests
18
Regressing Trade Flows Directly on Regime
Similarity
reg tradeflo regimsim Source SS
df MS Number of obs
100 ---------------------------------------
F( 1, 98) 46.42 Model
21021.1745 1 21021.1745 Prob gt
F 0.0000 Residual 44375.3387 98
452.809579 R-squared
0.3214 ---------------------------------------
Adj R-squared 0.3145 Total
65396.5132 99 660.570841 Root
MSE 21.279 -------------------------------
----------------------------------------------- tr
adeflo Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- regimsim 2.49602 .3663341 6.814
0.000 1.769042 3.222998 _cons
10.20431 4.306895 2.369 0.020
1.657422 18.75121 -----------------------------
-------------------------------------------------
19
Regressing Trade Flows on Regime Security
Similarity
reg tradeflo allysim regimsim Source
SS df MS Number of
obs 100 -----------------------------------
---- F( 2, 97) 334.65
Model 57118.4095 2 28559.2047
Prob gt F 0.0000 Residual 8278.10377
97 85.341276 R-squared
0.8734 ---------------------------------------
Adj R-squared 0.8708 Total
65396.5132 99 660.570841 Root
MSE 9.238 -------------------------------
----------------------------------------------- tr
adeflo Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- allysim 1.98863 .0966934 20.566
0.000 1.796721 2.18054 regimsim
.0284416 .1992195 0.143 0.887
-.3669539 .423837 _cons 4.199239
1.892422 2.219 0.029 .4433061
7.955173 -----------------------------------------
-------------------------------------
20
Regressing Security Similarity on Regime
Similarity
reg tradeflo allysim regimsim Source
SS df MS Number
of obs 100 --------------------------------
------- F( 2, 97) 334.65
Model 57118.4095 2 28559.2047
Prob gt F 0.0000 Residual 8278.10377
97 85.341276 R-squared
0.8734 ---------------------------------------
Adj R-squared 0.8708 Total
65396.5132 99 660.570841 Root
MSE 9.238 -------------------------------
----------------------------------------------- tr
adeflo Coef. Std. Err. t Pgtt
95 Conf. Interval ----------------------
--------------------------------------------------
----- allysim 1.98863 .0966934 20.566
0.000 1.796721 2.18054 regimsim
.0284416 .1992195 0.143 0.887
-.3669539 .423837 _cons 4.199239
1.892422 2.219 0.029 .4433061
7.955173 -----------------------------------------
-------------------------------------