34.3 Neutral Controls

34.3.1 Good Predictive Controls

Good for precision

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; z->y}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, y=2),
  y = c(x=1, z=2, y=1))

## ggplot
ggdag(model) + theme_dag()

Controlling for \(Z\) does not help or hurt identification, but it can increase precision (i.e., reducing SE)

n <- 1e4
z <- rnorm(n)
x <- rnorm(n)
y <- x + 2 * z + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 34.13:
Model 1Model 2
(Intercept)0.01    0.01    
(0.02)   (0.01)   
x1.00 ***1.01 ***
(0.02)   (0.01)   
z       2.00 ***
       (0.01)   
N10000       10000       
R20.17    0.83    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Similar coefficients, but smaller SE when controlling for \(Z\)

Another variation is

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->m; z->m; m->y}")

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, m=2, y=3),
  y = c(x=1, z=2, m=1, y=1))

## ggplot
ggdag(model) + theme_dag()

n <- 1e4
z <- rnorm(n)
x <- rnorm(n)
m <- 2 * z + rnorm(n)
y <- x + 2 * m + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 34.14:
Model 1Model 2
(Intercept)-0.00    -0.00    
(0.05)   (0.02)   
x0.97 ***0.99 ***
(0.05)   (0.02)   
z       4.02 ***
       (0.02)   
N10000       10000       
R20.04    0.77    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Controlling for \(Z\) can reduce SE

34.3.2 Good Selection Bias

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->z; z->w; u->w;u->y}")

# set u as latent
latents(model) <- "u"

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, w=3, u=3, y=5),
  y = c(x=3, z=2, w=1, u=4, y=3))

## ggplot
ggdag(model) + theme_dag()

  1. Unadjusted estimate is unbiased
  2. Controlling for Z can increase SE
  3. Controlling for Z while having on W can help identify X
n <- 1e4
x <- rnorm(n)
u <- rnorm(n)
z <- x + rnorm(n)
w <- z + u + rnorm(n)
y <- x - 2*u + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + w), lm(y ~ x + z + w))
Table 34.15:
Model 1Model 2Model 3
(Intercept)0.01    0.01    0.03    
(0.02)   (0.02)   (0.02)   
x0.99 ***1.65 ***0.99 ***
(0.02)   (0.02)   (0.02)   
w       -0.67 ***-1.01 ***
       (0.01)   (0.01)   
z              1.02 ***
              (0.02)   
N10000       10000       10000       
R20.16    0.39    0.50    
*** p < 0.001; ** p < 0.01; * p < 0.05.

34.3.3 Bad Predictive Controls

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; z->x}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=1, y=2),
  y = c(x=1, z=2, y=1))

## ggplot
ggdag(model) + theme_dag()

n <- 1e4
z <- rnorm(n)
x <- 2 * z + rnorm(n)
y <- x + 2 * rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 34.16:
Model 1Model 2
(Intercept)-0.02    -0.02    
(0.02)   (0.02)   
x0.99 ***1.00 ***
(0.01)   (0.02)   
z       -0.00    
       (0.04)   
N10000       10000       
R20.55    0.55    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Similar coefficients, but greater SE when controlling for \(Z\)

Another variation is

rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->z}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=1, y=2),
  y = c(x=1, z=2, y=1))

## ggplot
ggdag(model) + theme_dag()

set.seed(1)
n <- 1e4
x <- rnorm(n)
z <- 2 * x + rnorm(n)
y <- x + 2 * rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 34.17:
Model 1Model 2
(Intercept)0.02    0.02    
(0.02)   (0.02)   
x1.00 ***0.99 ***
(0.02)   (0.05)   
z       0.00    
       (0.02)   
N10000       10000       
R20.20    0.20    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Worse SE when controlling for \(Z\) (\(0.02 < 0.05\))

34.3.4 Bad Selection Bias

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->z}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, y=2),
  y = c(x=1, z=2, y=1))

## ggplot
ggdag(model) + theme_dag()

Not all post-treatment variables are bad.

Controlling for \(Z\) is neutral, but it might hurt the precision of the causal effect.