Classwork 2

Omitted Variable Bias; Bad Controls

Author

Byeong-Hak Choe

Published

January 26, 2026

Modified

January 26, 2026

Omitted Variable Bias

Code

library(tidyverse)
library(broom)
library(stargazer)

oj <- read_csv('https://bcdanl.github.io/data/dominick_oj_ad.csv')

reg_short <- lm(log(sales) ~ log(price),
                data = oj)

reg_long <- lm(log(sales) ~ log(price) + brand,
                data = oj)

stargazer(reg_short, reg_long,
          type = 'html')


	Dependent variable:

	log(sales)
	(1)	(2)

log(price)	-1.601^***	-3.139^***
	(0.018)	(0.023)

brandminute_maid		0.870^***
		(0.013)

brandtropicana		1.530^***
		(0.016)

Constant	10.423^***	10.829^***
	(0.015)	(0.015)


Observations	28,947	28,947
R²	0.208	0.394
Adjusted R²	0.208	0.394
Residual Std. Error	0.907 (df = 28945)	0.794 (df = 28943)
F Statistic	7,608.212^*** (df = 1; 28945)	6,275.074^*** (df = 3; 28943)

Note:	p<0.1; p<0.05; p<0.01

Code

reg_1st <- lm(log(price) ~ brand,
              data = oj)

aug_m2 <- augment(reg_1st)    # broom::augment()

reg_2nd <- lm(log(sales) ~ aug_m2$.resid,
              data = oj)

stargazer(reg_long, reg_2nd,
          type = 'html')


	Dependent variable:

	log(sales)
	(1)	(2)

log(price)	-3.139^***
	(0.023)

brandminute_maid	0.870^***
	(0.013)

brandtropicana	1.530^***
	(0.016)

.resid		-3.139^***
		(0.023)

Constant	10.829^***	9.168^***
	(0.015)	(0.005)


Observations	28,947	28,947
R²	0.394	0.392
Adjusted R²	0.394	0.392
Residual Std. Error	0.794 (df = 28943)	0.795 (df = 28945)
F Statistic	6,275.074^*** (df = 3; 28943)	18,683.570^*** (df = 1; 28945)

Note:	p<0.1; p<0.05; p<0.01

Bad Controls

Education, Occupation, and Wages

Code

set.seed(1)

tb <- tibble( 
  highschool_degree = ifelse(runif(10000)>=0.5, 1,0), # highschool_degree indicator variable
  ability = rnorm(10000), # e.g., talent, usually unobserved.
  discrimination = highschool_degree, # highschool_degree discrimination variable
  occupation_index = 1 + 2*ability + 0*highschool_degree - 2*discrimination + rnorm(10000), # true data generating process for occupation_index variable
  wage = 1 - 1*discrimination + 1*occupation_index + 2*ability + rnorm(10000) # true data generating process for wage variable
)

lm_1 <- lm(wage ~ highschool_degree, tb)
lm_2 <- lm(wage ~ highschool_degree + occupation_index, tb)
lm_3 <- lm(wage ~ highschool_degree + occupation_index + ability, tb)

stargazer::stargazer(lm_1,lm_2,lm_3, 
                     column.labels = c("Biased Unconditional", 
                                       "Biased",
                                       "Unbiased Conditional"),
                     type = 'html')


	Dependent variable:

	wage
	Biased Unconditional	Biased	Unbiased Conditional
	(1)	(2)	(3)

highschool_degree	-3.021^***	0.557^***	-1.039^***
	(0.084)	(0.030)	(0.029)

occupation_index		1.789^***	0.981^***
		(0.006)	(0.010)

ability			2.028^***
			(0.023)

Constant	1.968^***	0.201^***	1.021^***
	(0.059)	(0.020)	(0.018)


Observations	10,000	10,000	10,000
R²	0.115	0.907	0.948
Adjusted R²	0.115	0.907	0.948
Residual Std. Error	4.192 (df = 9998)	1.357 (df = 9997)	1.013 (df = 9996)
F Statistic	1,297.822^*** (df = 1; 9998)	48,922.550^*** (df = 2; 9997)	61,140.750^*** (df = 3; 9996)

Note:	p<0.1; p<0.05; p<0.01

Fertilizer, Plant Height, and Crop Yield

Code

set.seed(2026)

# Simulate 10,000 plots
tb3 <- tibble(
  fertilizer    = rbinom(10000, 1, 0.5),       # 0 = no fertilizer, 1 = fertilizer applied
  soil_quality  = rnorm(10000),                # unobserved “plot fertility”
  # Height (cm) is boosted by fertilizer *and* by soil quality
  height        = 50 + 8 * fertilizer + 5 * soil_quality + rnorm(10000),
  # Yield (bushels) depends on fertilizer, height, and soil quality
  yield         = 200 
  + 10 * fertilizer      # direct effect of fertilizer
  + 3  * height          # return to plant height
  + 6  * soil_quality    # underlying soil fertility
  + rnorm(10000)
)

# 1) Omitting height: picks up both direct and indirect effects
lm1 <- lm(yield ~ fertilizer,         data = tb3)
# 2) Controlling for height: "blocks" the mediation channel
lm2 <- lm(yield ~ fertilizer + height, data = tb3)
# 3) Also control soil_quality to recover the true direct effect
lm3 <- lm(yield ~ fertilizer + height + soil_quality, data = tb3)

stargazer(
  lm1, lm2, lm3,
  column.labels   = c("Omit Height (Biased)", 
                      "Control Height (More Biased)",
                      "Also Soil (Unbiased)"),
  dep.var.labels  = "yield",
  keep            = c("fertilizer", "height", "soil_quality"),
  covariate.labels= c("Fertilizer", "Plant Height", "Soil Quality"),
  type            = "html"
)


	Dependent variable:

	yield
	Omit Height (Biased)	Control Height (More Biased)	Also Soil (Unbiased)
	(1)	(2)	(3)

Fertilizer	34.657^***	0.789^***	10.025^***
	(0.426)	(0.040)	(0.081)

Plant Height		4.156^***	2.999^***
		(0.003)	(0.010)

Soil Quality			6.002^***
			(0.050)


Observations	10,000	10,000	10,000
R²	0.398	0.997	0.999
Adjusted R²	0.398	0.997	0.999
Residual Std. Error	21.323 (df = 9998)	1.551 (df = 9997)	0.996 (df = 9996)
F Statistic	6,604.218^*** (df = 1; 9998)	1,563,217.000^*** (df = 2; 9997)	2,532,263.000^*** (df = 3; 9996)

Note:	p<0.1; p<0.05; p<0.01