Omitted Variable Bias Example

CPP 523

Class Size Data From Lab 02

	Dependent Variable: Test Scores
	Model 1	Model 2	Model 3	Model 4	Model 5
	(1)	(2)	(3)	(4)	(5)
Classroom Size	-4.22***	-3.91***		-2.67	-2.22***
	(0.18)	(0.03)		(1.63)	(0.23)
Teacher Quality		55.01***	55.03***		55.01***
		(0.25)	(0.26)		(0.25)
Socio-Economic Status			40.94***	16.34	17.77***
			(0.27)	(17.10)	(2.40)
Intercept	738.34***	456.70***	272.91***	665.29***	377.26***
	(4.88)	(1.48)	(1.39)	(76.57)	(10.82)
Observations	1,000	1,000	1,000	1,000	1,000
Adjusted R2	0.36	0.99	0.99	0.36	0.99
Standard errors in parentheses	p<0.1; p<0.05; p<0.01

Lecture Notes

Tables and Math

Consider the effect of omitting SES from the full model of CS + SES:

EQUATION	MODEL
\(TestScore = B_0 + B_1 \cdot ClassSize + B_2 \cdot SES + e_1\)	\((Full \ Model)\)
\(TestScore = b_0 + b_1 \cdot ClassSize + e_2\)	\((Naive \ Model)\)
\(SES = a_0 + a_1 \cdot ClassSize + e_3\)	\((Auxiliary \ Regression)\)

	Dependent Variables
	TS	TS	SES
	(1)	(2)	(3)
Classroom Size	-2.671	-4.222***	-0.095***
	(1.632)	(0.176)	(0.0003)
Socio-Economic Status	16.344
	(17.098)
Intercept	665.289***	738.337***	4.469***
	(76.574)	(4.879)	(0.009)
Observations	1,000	1,000	1,000
Adjusted R2	0.365	0.365	0.988
Standard errors in parentheses	p<0.1; p<0.05; p<0.01

Code for the Example

PDF

URL <- "https://raw.githubusercontent.com/DS4PS/cpp-523-fall-2019/master/labs/class-size-seed-1234.csv"
dat <- read.csv( URL )

m1 <- lm( test ~ csize, data=dat  ) 
m2 <- lm( test ~ csize + tqual, data=dat )
m3 <- lm( test ~ tqual + ses, data=dat ) 
m4 <- lm( test ~ csize + ses, data=dat  )
m5 <- lm( test ~ csize + tqual + ses, data=dat  )


# FULL MODEL
summary( m4 )
# lm(formula = test ~ csize + ses, data = dat)
# Coefficients:
# ----------------------------------
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  665.289     76.574   8.688   <2e-16 ***
# csize         -2.671      1.632  -1.637    0.102    
# ses           16.344     17.098   0.956    0.339    
# ----------------------------------

# NAIVE MODEL 
summary( m1 )

# lm(formula = test ~ csize, data = dat)
# Coefficients:
# ----------------------------------
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept) 738.3366     4.8788  151.34   <2e-16 ***
# csize        -4.2221     0.1761  -23.98   <2e-16 ***
# ----------------------------------



# AUXILIARY REGRESSION
m.auxiliary <- lm( ses ~ csize, data=dat )
summary( m.auxiliary )

# lm(formula = ses ~ csize, data = dat)
# Coefficients:
# ----------------------------------
#              Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  4.469458   0.009033   494.8   <2e-16 ***
# csize       -0.094876   0.000326  -291.0   <2e-16 ***
# ----------------------------------


# b1 = B1 + bias
# b1 - B1 = bias
b1 <- -4.22
B1 <- -2.67
b1 - B1

# bias = a1*B2
a1 <- -0.0949
B2 <- 16.34
a1*B2

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