Control Variables

Stijn Masschelein

Causal Graphs

For more information see Chapter 8 in Huntington-Klein (2021).

An Example of a Causal Graph

Difference with equilibrium models

Differences

• All the qualitative information about causal relations is in the graph.
• The equilibrium model directly gives the relation between the variables of interest.

e.g.: Signaling model

• Is there a relation between Performance and Return?
• There is a parallel with voluntary disclosure of information that the company does not have.
• What happens if firms are no longer allowed to donate? What happens if firms are all forced to donate?

Assignment: CSR report

• What is the effect of an increase in scandals?
• What happens if we keep the number of scandals constant?

Measurement error and control variables

Causal Graph

Assume that we are interested in the role of the GC on performance.

Simulation

set.seed(230383)
N <- 1000
ds <- tibble(CG = runif(N, 0, 10),
             TI = rbinom(N, 1, .25)) %>%
  mutate(Performance =
           rnorm(N, CG * .15 + TI * 10, 5))

lm1 <- lm(Performance ~ CG, data = ds)
lm2 <- lm(Performance ~ CG + TI, data = ds)

gof_omit <- "Adj|IC|Log|Pseudo|RMSE"
stars <- c('*' = .1, '**' = .05, '***' = .01)
msummary(list(lm1, lm2), stars = stars,
         gof_omit = gof_omit, output = "markdown")

	(1)	(2)
(Intercept)	2.976***	0.036
	(0.408)	(0.310)
CG	0.081	0.130**
	(0.073)	(0.052)
TI		10.433***
		(0.344)
Num.Obs.	1000	1000
R2	0.001	0.480

Note: ^^ * p < 0.1, ** p < 0.05, *** p < 0.01

Measurement error typically decreases the effect and this is also what happened in the assignment. For instance, the difference between the donation and no donation should be 3 but it is less.

Confounders and control variables

Causal Graph

There is only 1 difference between this causal graph and the previous one.

Simulation

N <- 1000
ds <- tibble(TI = rbinom(N, 1, .25)) %>%
  mutate(CG = rnorm(N, .5 - TI, .2),
         Performance = rnorm(N, TI + 0 * CG, 1))

lm1 <- lm(Performance ~ CG, data = ds)
lm2 <- lm(Performance ~ CG + TI, data = ds)

msummary(list(lm1, lm2), stars = stars,
         gof_omit = gof_omit, output = "markdown")

	(1)	(2)
(Intercept)	0.473***	0.002
	(0.036)	(0.087)
CG	-0.945***	-0.090
	(0.067)	(0.159)
TI		1.018***
		(0.172)
Num.Obs.	1000	1000
R2	0.165	0.193

Note: ^^ * p < 0.1, ** p < 0.05, *** p < 0.01

In the simulation, we set the effect of GC equal to 0 i.e. there is not effect. The reason to do that is to show why it’s necessary to adjust for TI.

Fixed effects as a special case

Definition

Effects that are the same for every industry, year, firm, or individual can be adjusted for by using fixed effects.

Benefits

We do not need to measure the specific variables and can just use indicators variables for each category (e.g. for each different industry).

See more in chapter 16 of Huntington-Klein (2021)

Fixed effects (for industry)

• Causal Diagram
• Simulation
• Regressions
• Simulation with correlated fixed effects
• Regressions with correlated fixed effects

Nind <- 20
N <- 5000
di <- tibble(
  ind_number = 1:Nind,
  ind_CG = rnorm(Nind, 0, 1),
  ind_performance = rnorm(Nind, 0, 1)
)
ds <- tibble(
    ind_number = sample(1:Nind, N, replace = TRUE)) %>%
  left_join(
    di, by = "ind_number") %>%
  mutate(
    CG = rnorm(N, .5 + ind_CG, .2),
    Performance = rnorm(N, 0 * CG + ind_performance, 1)
  )

glimpse(di, width = 50)

Rows: 20
Columns: 3
$ ind_number      <int> 1, 2, 3, 4, 5, 6, 7, 8, …
$ ind_CG          <dbl> 0.23567083, -0.34180999,…
$ ind_performance <dbl> 1.1335103, 1.2873377, 0.…

glimpse(ds, width = 50)

Rows: 5,000
Columns: 5
$ ind_number      <int> 9, 12, 5, 7, 6, 8, 15, 1…
$ ind_CG          <dbl> 1.91243572, 0.16031769, …
$ ind_performance <dbl> 0.1773941, -0.1250858, 0…
$ CG              <dbl> 2.3076069, 0.6604549, 1.…
$ Performance     <dbl> 0.09219279, -0.37244401,…

lm1 <- lm(Performance ~ CG, data = ds)
lm2 <- lm(Performance ~ CG + factor(ind_number), data = ds)
library(fixest)
fe <- feols(Performance ~ CG | ind_number, data = ds)

msummary(list(lm1, lm2, fe), gof_omit = gof_omit, stars = stars, output = "markdown")

	(1)	(2)	(3)
(Intercept)	0.490***	1.054***
	(0.019)	(0.080)
CG	-0.076***	0.030	0.030
	(0.018)	(0.071)	(0.061)
factor(ind_number)2		0.207**
		(0.097)
factor(ind_number)3		-0.490***
		(0.090)
factor(ind_number)4		-0.372**
		(0.161)
factor(ind_number)5		-0.410***
		(0.099)
factor(ind_number)6		-1.907***
		(0.169)
factor(ind_number)7		-0.083
		(0.134)
factor(ind_number)8		-1.191***
		(0.178)
factor(ind_number)9		-0.935***
		(0.148)
factor(ind_number)10		-1.712***
		(0.087)
factor(ind_number)11		-1.162***
		(0.122)
factor(ind_number)12		-1.118***
		(0.090)
factor(ind_number)13		0.765***
		(0.091)
factor(ind_number)14		-0.564***
		(0.104)
factor(ind_number)15		0.730***
		(0.129)
factor(ind_number)16		0.689***
		(0.158)
factor(ind_number)17		-2.056***
		(0.098)
factor(ind_number)18		-0.503***
		(0.091)
factor(ind_number)19		-1.907***
		(0.093)
factor(ind_number)20		0.608***
		(0.086)
Num.Obs.	5000	5000	5000
R2	0.003	0.443	0.443
R2 Within			0.000
Std.Errors			by: ind_number
FE: ind_number			X

Note: ^^ * p < 0.1, ** p < 0.05, *** p < 0.01

• Why do we need the factor(ind_number) formulation?

Nind <- 20
N <- 5000
correl <- -0.5
di <- tibble(
    ind_number = 1:Nind,
    ind_CG = rnorm(Nind, 0, 1)) %>%
  mutate(
    ind_performance = sqrt(1 - correl^2) * rnorm(Nind, 0, 1) + correl * ind_CG)
ds <- tibble(
    ind_number = sample(1:Nind, N, replace = TRUE)) %>%
  left_join(
    di, by = "ind_number") %>%
  mutate(
    CG = rnorm(N, .5 + ind_CG, .2),
    Performance = rnorm(N, 0 * CG + ind_performance, 1)
  )

glimpse(di, width = 50)

Rows: 20
Columns: 3
$ ind_number      <int> 1, 2, 3, 4, 5, 6, 7, 8, …
$ ind_CG          <dbl> -0.82999044, 0.44908313,…
$ ind_performance <dbl> -1.12284290, -0.57326559…

glimpse(ds, width = 50)

Rows: 5,000
Columns: 5
$ ind_number      <int> 20, 17, 2, 1, 9, 20, 5, …
$ ind_CG          <dbl> -1.1358960, 0.4833522, 0…
$ ind_performance <dbl> 1.0252155, -0.6131181, -…
$ CG              <dbl> -0.79273388, 1.44367931,…
$ Performance     <dbl> 1.75927497, -1.39745179,…

lm1 <- lm(Performance ~ CG, data = ds)
fe <- feols(Performance ~ CG | ind_number, data = ds)

msummary(list(lm1, fe), gof_omit = gof_omit, stars = stars)

	(1)	(2)
(Intercept)	−0.033
	(0.022)
CG	−0.279***	−0.012
	(0.015)	(0.046)
Num.Obs.	5000	5000
R2	0.067	0.321
R2 Within		0.000
RMSE	1.17	1.00
Std.Errors		by: ind_number
FE: ind_number		X
* p < 0.1, ** p < 0.05, *** p < 0.01

• Why do we need the factor(ind_number) formulation?
• The trick with the correlated ind_CG and ind_performance

What do fixed effects do?

• Full Sample
• Highlight Industry
• Remove Industry Effects

Code

fe_plot <-
  ggplot(ds, aes(y = Performance, x = CG)) +
  geom_point()
plot(fe_plot)

Code

fe_colour <-
  ggplot(ds, aes(y = Performance, x = CG,
                colour = factor(ind_number))) +
  geom_point() + theme(legend.position="none") 
plot(fe_colour)

Code

fe_demean <- group_by(ds, ind_number) %>%
  mutate(Performance2 = Performance - mean(Performance),
         CG2 = CG - mean(CG)) %>%
  ggplot(aes(y = Performance2, x = CG2,
             colour = factor(ind_number))) +
  geom_point() + theme(legend.position="none") 
plot(fe_demean)

Speedboat Racing Example (Booth and Yamamura 2017)

• Mixed-sex and single-sex races determined by lottery (Randomisation)
• 7 race courses
• Multiple races in the same month and location

Results of Speedboat Races

Code

load(here("data", "booth_yamamura.Rdata"))
table <- as_tibble(table) %>%
  select(p_id, women_dat, time, ltime, mix_ra, course,
         race_id, yrmt_locid)
table_clean <- filter(table, complete.cases(table)) %>%
  select(ltime, women_dat, mix_ra, course, p_id, race_id,
         yrmt_locid)
ltime_reg <- feols(ltime ~ women_dat : mix_ra + mix_ra
                   | course + p_id + yrmt_locid,
                   cluster = "race_id",
                   data = table_clean)
msummary(ltime_reg, gof_omit = gof_omit, stars = stars)

	(1)
mix_ra	−0.002***
	(0.000)
women_dat × mix_ra	0.007***
	(0.001)
Num.Obs.	142346
R2	0.361
R2 Within	0.001
Std.Errors	by: race_id
FE: course	X
FE: p_id	X
FE: yrmt_locid	X
* p < 0.1, ** p < 0.05, *** p < 0.01

This requires an explanation of interactions. Luckily, it’s relatively simple with two discrete variables.

| ltime | man    | woman  |
|-------|:------:|:------:|
| same  | 0      | 0      |
| mixed | -0.002 | 0.005  |

Colliders and bad controls

Warning

Equilibrium models are very good at incorporating these effects!

Bad Controls, Survival Bias, Selection Bias, Self-Selection Bias

Example in the assignment

Simulation Bad Control

• Causal Graph
• Simulate
• Results

d <- tibble(corp_gov = rnorm(N, 0, 1)) %>%
  mutate(acc_profit = rnorm(N, corp_gov, sd = 3),
         market_return = rnorm(N, 2 * corp_gov + acc_profit,
                               sd = 3))
lm1 <- lm(acc_profit ~ corp_gov, data = d)
lm2 <- lm(acc_profit ~ corp_gov + market_return, data = d)

msummary(list(lm1, lm2),
         gof_omit = gof_omit, stars = stars)

	(1)	(2)
(Intercept)	0.039	0.012
	(0.042)	(0.030)
corp_gov	1.000***	−0.489***
	(0.042)	(0.037)
market_return		0.498***
		(0.007)
Num.Obs.	5000	5000
R2	0.101	0.551
RMSE	2.98	2.11
* p < 0.1, ** p < 0.05, *** p < 0.01

Survival Bias

• Simulate
• Results

d <- mutate(d, survival = if_else(market_return > 5, 1, 0))

lm1 <- lm(acc_profit ~ corp_gov, data = filter(d, survival == 1))
lm2 <- lm(acc_profit ~ corp_gov * survival, data = d)
msummary(list(lm1, lm2), gof_omit = gof_omit, stars = stars, output = "markdown")

	(1)	(2)
(Intercept)	3.518***	-0.549***
	(0.115)	(0.043)
corp_gov	-0.137	0.606***
	(0.095)	(0.045)
survival		4.067***
		(0.135)
corp_gov × survival		-0.743***
		(0.115)
Num.Obs.	853	5000
R2	0.002	0.262

Note: ^^ * p < 0.1, ** p < 0.05, *** p < 0.01

One interpretation is that a firm can survive by being lucky (and having high returns) and by having good corporate governance (which translates in high returns). The survivors all are more likely to be having good corporate governance and there is little that can be explained further.

Visualisation of Colliders (and Interactions)

• Full Sample
• Survival Highlighted
• Survival Only

Pitching Template

Pitching Format

1. Description (Important)
   • Title
   • Research Question
   • Key Paper
   • Motivation
2. THREE (IDioT) (Important)
   • Idea
   • Data
   • Tools

Pitching Format

1. Description (Important)
   • Title
   • Research Question
   • Key Paper
   • Motivation
2. THREE (IDioT) (Important)
   • Idea
   • Data
   • Tools

3. TWO
   • What’s new?
   • So what?
4. ONE contribution
5. Other considerations.

References

Booth, Alison, and Eiji Yamamura. 2017. “Performance in Mixed-Sex and Single-Sex Competitions: What We Can Learn from Speedboat Races in Japan.” The Review of Economics and Statistics 100 (4): 581–93. https://doi.org/10.1162/rest_a_00715.

Huntington-Klein, Nick. 2021. The Effect: An Introduction to Research Design and Causality. First. Boca Raton: Chapman and Hall/CRC. https://doi.org/10.1201/9781003226055.