The Rubin Causal Model

EC655

Justin Smith

Wilfrid Laurier University

Fall 2022

Introduction

Introduction

• The workhorse model in econometrics is linear regression

• Two key uses of this model

  • Prediction

  • Modelling (causal) effect of one variable on another

• In many empirical application the focus is causal effects

  • The independent effect of a particular variable on the outcome

• We will study linear regression with a focus on causality

• First, we attempt to understand the underlying concept of causality

• For this we use the Rubin Causal Model

  • Defines causality and under what conditions we can measure it

Model Basics

Potential Outcomes

• Start with a binary framework

  • There is a “treatment” or “no treatment”

  • An individual theoretically has an outcome with treatment and without

• Treatment is defined generally

  • Getting a drug

  • Going to university

  • Being in a large class

• Define the following potential outcomes

  • \(y_{1}\) is the outcome with treatment

  • \(y_{0}\) is the outcome without treatment

  • \(w\) is a binary variable with 1 denoting treatment, and 0 no treatment

Treatment Effects

• We would like to know the treatment effect \(y_{1} - y_{0}\) for an individual

  • This is the .causal effect of the treatment

  • Effect differs from person to person in the population

• Fundamental problem of causal inference: we never observe both \(y_{1}\) and \(y_{0}\)

• We only observe \((y, w)\), where

  \[y = y_{0} + (y_{1} -y_{0})w\]

  • We observe treatment status, potential outcome given that treatment status

• The counterfactual outcome with opposite treatment is never observed

Simple Differences in Average Outcomes

• What if we naïvely compute difference in average outcomes between treated and control? \[E(y|w=1) - E(y|w=0)\]

• Using the definition of \(y\) above,

  \[E(y|w=1) - E(y|w=0)\] \[= E(y_{1}|w=1) - E(y_{0}|w=0)\] \[= \left [ E(y_{1}|w=1) - E(y_{0}|w=1) \right ] + E(y_{0}|w=1) - E(y_{0}|w=0)\]

• The first term is called the Average Treatment Effect on the Treated (ATT)

  • Average effect of the treatment for those in the treatment group

Simple Differences in Average Outcomes

• The second term is Selection Bias

  • Baseline difference between treatment and control groups

• Simple average differences will not identify a treatment effect

  • It is partly a treatment effect, partly differences in who gets treated

• Ex: Comparing average incomes of university grads to high school grads

  • Will be partly average causal effect of university

  • Also difference in baseline earning ability without the degree

• Lesson is that simple differences in averages do not reveal causal effects

• Under what conditions can we measure the causal effect of \(w\) on \(y\)?

Randomizing Treatment Status

Randomization and Independence of Treatment

• A common way to isolate treatment effects is to randomize \(w\)

  • Blindly put people into treatment or control group

  • Ensures that on average the two groups are similar at baseline

• Mathematically, potential outcomes are independent from treatment \[(y_{0}, y_{1}) \perp w\]

• Independence means conditioning on \(w\) has no effect on expectation

  \[E(y_{0}|w=1) =E(y_{0}|w=0)\] \[E(y_{0}|w) = E(y_{0})\] \[E(y_{1}|w) = E(y_{1})\]

Randomization and Treatment Effects

• With randomization, selection bias is zero

  \[E(y_{0}|w=1) - E(y_{0}|w=0) = E(y_{0}|w=1) - E(y_{0}|w=1) = 0\]

• As a result the difference in mean \(y\) is \[E(y|w=1) - E(y|w=0)\] \[= E(y_{1}|w=1) - E(y_{0}|w=0)\] \[= E(y_{1}) - E(y_{0})\]

• The first term is the ATT we saw before

• The second term is the Average Treatment Effect (ATE)

  • The treatment effect across the whole population

Recent Example in Economics Literature

• When we randomize treatment we can measure causal effects

• Randomization is the standard way to measure the effects of medical treatments

• It is becoming more popular in economics

• Ex: Bangladesh mask study (Abaluck et. al., 2021)

  • Randomized promoting mask use in rural Bangladesh

  • Compare COVID rates between treatment and control

  • Find some positive effect of masks, especially for age 50+

• Next we will show how to model this in a regression framework

Link to Regression Models

• In randomized experiments, you can get causal effects with difference in means

• Economists typically work with regression models

• Remember the outcome we observe \(y\) is

  \[y = (y_{1} - y_{0})w + y_{0}\]

• If we define the potential outcomes as \[y_{0} = \alpha + \eta\] \[y_{1} = y_{0} + \rho\]

Link to Regression Models

• If we combine the three equations we get

  \[y = \alpha + \rho w + \eta\]

• The difference in means for \(y\) by treatment status is now

  \[E(y|w=1) - E(y|w=0)\] \[\rho + E(\eta |w=1) - E(\eta |w=0)\]

• Treatment effect is the regression slope

• Selection bias is correlation between error term and treatment

  \[E[y_{0}|w=1] - E[y_{0}|w=0] = E[\eta|w=1] - E[\eta|w=0]\]

Link to Regression Models

• Randomization means no selection bias

• In a regression framework, this means the error is unrelated to treatment

  \[E[\eta|w=1] - E[\eta|w=0] =0\]

• So a regression of \(y\) on \(w\) would measure the causal effect

• In reality, we cannot always randomize \(w\)

  • Economists deal mostly with observational data

• Next we look at causal effects when we cannot randomize

Causal Effects without Randomization

Mean Independence of Treatment

• We cannot always randomize into treatment and control

• Can we uncover causal effects without experiments?

• The answer is yes, depending on assumptions

• One possible assumption is Mean Independence

  \[E(y_{0}|w) = E(y_{0})\] \[E(y_{1}|w) = E(y_{1})\]

• Says conditional means do not depend on treatment status

  • Weaker assumption than full statistical independence

  • Full independence means one event has no effect on probability of another

Mean Independence of Treatment

• With mean independence, we get

  \[E(y|w=1) - E(y|w=0)\] \[= \left [ E(y_{1}|w=1) - E(y_{0}|w=1) \right ] + E(y_{0}|w=1) - E(y_{0}|w=0)\] \[= \left [ E(y_{1}|w=1) - E(y_{0}|w=1) \right ]\] \[= E(y_{1}) - E(y_{0})\]

• This identifies ATT = ATE

• Is this assumption realistic?

  • Means both potential outcomes unrelated to treatment

  • On average, people in treatment and control have similar treated and non-treated outcomes

  • Whether this is realistic depends on context

Mean Independence of \(y_{0}\)

• A variation if this assumption is mean independence of \(\mathbf{y_{0}}\)

  \[E(y_{0}|w) = E(y_{0})\]

• With this assumption

• Meaning that \[E(y|w=1) - E(y|w=0)\] \[= \left [ E(y_{1}|w=1) - E(y_{0}|w=1) \right ] + E(y_{0}|w=1) - E(y_{0}|w=0)\] \[= \left [ E(y_{1}|w=1) - E(y_{0}|w=1) \right ]\]

• With this assumption, we only measure the ATT (Not ATE)

• Is this realistic?

  • Means untreated outcome is same between groups on average

  • Puts no restriction on differences in treated outcome

  • Intuitively, there are no baseline differences between groups

Conditional Mean Independence

• We can also use other variables to help with our assumptions

• Suppose we observe a set of pre-treatment characteristics \(\mathbf{x}\)

  • Ex: gender, parental education, school test scores, etc.

  • Key is they are determined before treatment

• With this information you could assume Conditional Independence

  \[(y_{0}, y_{1}) \perp w |\mathbf{x}\]

• Conditional on \(\mathbf{x}\), treatment is independent of outcomes

• Write this mathematically as \[E(y_{0}|w=1, \mathbf{x}) =E(y_{0}|w=0, \mathbf{x})\] \[E(y_{0}|w, \mathbf{x}) = E(y_{0}| \mathbf{x})\] \[E(y_{1}|w, \mathbf{x}) = E(y_{1}|\mathbf{x})\]

Conditional Mean Independence

• This implies that we can get treatment effects at each \(\mathbf{x}\) \[E(y|w=1, \mathbf{x}) - E(y|w=0, \mathbf{x})\] \[= E(y_{1}|w=1, \mathbf{x}) - E(y_{0}|w=1, \mathbf{x})= E(y_{1} | \mathbf{x}) - E(y_{0}| \mathbf{x})\] \[= ATT( \mathbf{x}) =ATE( \mathbf{x})\]

• These treatment effects are functions of \(\mathbf{x}\)

  • They will differ across values of \(\mathbf{x}\)

  • So there are multiple treatment effects

• Finally, a variation on this is Conditional Mean Independence

  \[E(y_{0}|w, \mathbf{x}) = E(y_{0}| \mathbf{x})\] \[E(y_{1}|w, \mathbf{x}) = E(y_{1}|\mathbf{x})\]

• Gives you the same \(ATE( \mathbf{x}) = ATT( \mathbf{x})\) as above

Link with Regression

• We can also model this scenario with regression

• Remember again the following equations for \(y\), \(y_{0}\), and \(y_{1}\)

  \[y = (y_{1} - y_{0})w + y_{0}\] \[y_{0} = \alpha + \eta\] \[y_{1} = y_{0} + \rho\]

• This time, the random term \(\eta\) depends on \(\mathbf{x}\)

  \[\eta = \mathbf{x}\boldsymbol{\gamma} + \epsilon\]

Link with Regression

• Substitute this into the equation for \(y_{0}\)

  \[y_{0} = \alpha + \mathbf{x}\boldsymbol{\gamma} + \epsilon\]

• Now if we combine the equations for \(y\), \(y_{0}\), and \(y_{1}\)

  \[y = \alpha + \rho w + \mathbf{x}\boldsymbol{\gamma} + \epsilon\]

• The difference in means for \(y\) at each \(\mathbf{x}\) between groups is now \[E(y| \mathbf{x}, w=1) - E(y| \mathbf{x}, w=0)\] \[\rho + E(\epsilon | \mathbf{x}, w=1) - E(\epsilon | \mathbf{x}, w=0)\]

Link with Regression

• Again, the difference in the mean conditional error term is selection bias

  \[E[y_{0}|\mathbf{x}, w=1] - E[y_{0}|\mathbf{x}, w=0] = E[\epsilon|\mathbf{x},w=1] - E[\epsilon|\mathbf{x},w=0]\]

• If we assume Conditional Independence (or Conditional Mean Independence) then \[E(y_{0} | \mathbf{x}, w) = E(y_{0} | \mathbf{x})\]

• and as a result

  \[E(y_{0} | \mathbf{x}, w=1) = E(y_{0} | \mathbf{x}, w=0)\]

• and

  \[E[\epsilon|\mathbf{x},w=1] - E[\epsilon|\mathbf{x},w=0] =0\]

Link with Regression

• Intuition is as follows

  • Participation in treatment and control might be related to baseline factors

  • But, for those with the same baseline factors, treatment is as good as random

  • Computing treatment effects for each \(\mathbf{x}\) eliminates selection bias

• In this model, adding “control variables” identifies the causal effect of \(w\)

• Whether this is actually true depends on assumptions

  • Often we cannot prove those assumptions

Summary of Rubin Model

Summary of Rubin Model

• The Rubin model defines what is a causal effect

• Roughly speaking, it an Average Treatment Effect

  • They will differ across values of \(\mathbf{x}\)

  • Difference in potential outcomes, on average in population

  • Depending on context, it might be an Average Treatment Effect for the Treated

• We can express the Rubin model in a regression framework

• The slope in a linear regression is the causal effect if we can assume one of

  • Randomization of treatment

  • “As good as” randomization

    • Mean Independence, Conditional Independence, Conditional Mean Indpendence

Summary of Rubin Model

• When our regression model identifies an underlying causal effect, we call it a structural model

• In many econometric applications, this is what we want

• Next, we discuss in more detail linear regression

• First we discuss the population model

  • We will define the parameters we are measuring

  • Some of this might be new

• Then we discuss estimation by OLS

  • Focus is on when OLS consistently estimates the parameters

Simulation

Data Setup

• To help understand the Rubin model we will demonstrate with simulated data

• Code to the right creates potential outcomes

• For simplicity the treatment effect is set to 5 for everyone

• Outcomes \(y_{0}\) and \(y_{1}\) have a Normal distribution because of \(\eta\)

data <- data.frame(eta=rnorm(100000,0,1)) %>%
  mutate(y0 = 2 + eta, y1 = y0 + 5, 
         treat_eff = y1 - y0)

sumtable(data, summ=c('notNA(x)','mean(x)','sd(x)'))

Summary Statistics

Variable	NotNA	Mean	Sd
eta	1e+05	0.004	1.004
y0	1e+05	2.004	1.004
y1	1e+05	7.004	1.004
treat_eff	1e+05	5	0

Random Assignment to Treatment

• Next assign treatment \(w\) using randomization

• In the code, \(w=1\) randomly with probability 0.5

• Compute observed \(y\) based on treatment status

data %<>% mutate(w = if_else(runif(100000) > .5,1,0), 
                 y = y0 + (y1-y0)*w) %>% 
  group_by(w)

head(data)

# A tibble: 6 × 6
# Groups:   w [2]
      eta    y0    y1 treat_eff     w     y
    <dbl> <dbl> <dbl>     <dbl> <dbl> <dbl>
1 -0.471   1.53  6.53         5     0  1.53
2 -0.253   1.75  6.75         5     0  1.75
3  1.17    3.17  8.17         5     1  8.17
4  1.28    3.28  8.28         5     1  8.28
5 -0.0665  1.93  6.93         5     1  6.93
6 -0.873   1.13  6.13         5     1  6.13

Random Assignment to Treatment

• With random assignment we know

  • \(y_{0}\) is independent of \(w\)

  • \(y_{1}\) is independent of \(w\)

• So the distributions of \(y_{0}\) and \(y_{1}\) are the same when \(w=0\) and when \(w=1\)

• To the right we show the distribution of \(y_{0}\)

ggplot(data, aes(x=y0, color=as.factor(w))) +
  geom_density(alpha = .4, size=2) +
  theme_pander(nomargin=FALSE, boxes=TRUE) +
  labs(title = "Distribution of Y0")

Random Assignment to Treatment

• Randomization ensures difference in average \(y\) between groups equals the ATE and ATT

• On the right we show the difference in mean of \(y\) equals 5

  • (it’s not exactly 5 due to sampling error)

summarize(data, mean(y))

# A tibble: 2 × 2
      w `mean(y)`
  <dbl>     <dbl>
1     0      2.00
2     1      7.01

summarize(data,mean(y))$`mean(y)`[2] - 
  summarize(data,mean(y))$`mean(y)`[1]

[1] 5.003349

Random Assignment to Treatment

• Can implement difference in means as a regression

• Recall slope in OLS regression of \(y\) on dummy variable is difference in means of \(y\)

lm(y ~ w, data)

Call:
lm(formula = y ~ w, data = data)

Coefficients:
(Intercept)            w  
      2.002        5.003  

Selection into Treatment

• Now simulate selection into treatment based on \(y_{0}\)

  • Treatment now related to value of \(y_{0}\)

• We know \(\eta\) determines value of \(y_{0}\)

  • If we make \(w=\) with higher values of \(\eta\) then \(w\) is related to \(y_{0}\)

data2 <- data %>% 
  ungroup() %>% 
  select(eta, y0,y1) %>%
  mutate(w = if_else(eta + runif(100000,-1,1) > 0,1,0), 
         y = y0 + (y1-y0)*w) %>%
  group_by(w)

sumtable(data2, 
         summ=c('notNA(x)','mean(x)','sd(x)'), 
         group="w",
         group.long = TRUE)

Selection into Treatment

• The means of \(y_{0}\) and \(y_{1}\) are now different by group

• Because of selection bias

  • Treated group has better non-treated outcomes

Summary Statistics

Variable	NotNA	Mean	Sd
w: 0
eta	49794	-0.684	0.733
y0	49794	1.316	0.733
y1	49794	6.316	0.733
y	49794	1.316	0.733
w: 1
eta	50206	0.686	0.734
y0	50206	2.686	0.734
y1	50206	7.686	0.734
y	50206	7.686	0.734

Selection into Treatment

• The distribution of \(y_{0}\) differs by \(w\)

  • Treated group has better baseline outcomes

• This creates selection bias

ggplot(data2, aes(x=y0, color=as.factor(w))) +
  geom_density(alpha = .4, size=2) +
  theme_pander(nomargin=FALSE, boxes=TRUE) +
  labs(title = "Distribution of Y0")

Selection into Treatment

• Selction bias shows up when you take difference in mean \(y\)

  • We know the true treatment effect is 5

  • But difference in \(y\) is larger

  • There is positive selection bias

  • Bias is about 1.4

summarize(data2, mean(y))

# A tibble: 2 × 2
      w `mean(y)`
  <dbl>     <dbl>
1     0      1.32
2     1      7.69

summarize(data2,mean(y))$`mean(y)`[2] - 
  summarize(data2,mean(y))$`mean(y)`[1]

[1] 6.370304

Selection into Treatment

• You can implement this as a regression

• OLS estimates biased treatment effect

• Remember the intercept is mean of \(y\) when \(w=0\)

lm(y ~ w, data2)

Call:
lm(formula = y ~ w, data = data2)

Coefficients:
(Intercept)            w  
      1.316        6.370  

Mean Independence of \(y_{0}\)

• Randomization ensures entire distribution of \(y_{0}\) is the same for treatment and control

• We do not need this to estimate average treatment effects

• If the mean of \(y_{0}\) is the same between treatment and control we can estimate treatment effect with difference in means

• Code makes mean of \(y_{0}\) the same, but variance bigger for \(w=1\)

data3 <- data %>% 
  ungroup() %>% 
  select(eta, y0,y1) %>%
  mutate(w =  if_else(between(percent_rank(y0),.25,.75),0,1), 
         y = y0 + (y1-y0)*w) %>%
  group_by(w)

sumtable(data3, 
         summ=c('notNA(x)','mean(x)','sd(x)'), 
         group="w",
         group.long = TRUE)

Mean Independence of \(y_{0}\)

• Randomization ensures entire distribution of \(y_{0}\) is the same for treatment and control

• We do not need this to estimate average treatment effects

• If the mean of \(y_{0}\) is the same between treatment and control we can estimate treatment effect with difference in means

• Code makes mean of \(y_{0}\) the same, but variance bigger for \(w=1\)

Summary Statistics

Variable	NotNA	Mean	Sd
w: 0
eta	50000	0.004	0.378
y0	50000	2.004	0.378
y1	50000	7.004	0.378
y	50000	2.004	0.378
w: 1
eta	50000	0.003	1.369
y0	50000	2.003	1.369
y1	50000	7.003	1.369
y	50000	7.003	1.369

Mean Independence of \(y_{0}\)

• The distribution of \(y_{0}\) is plotted on the right

• The spread is larger for \(w=1\)

• This does not affect estimate of the average treatment effect

ggplot(data3, aes(x=y0, color=as.factor(w))) +
  geom_density(alpha = .4, size=2) +
  theme_pander(nomargin=FALSE, boxes=TRUE) +
  labs(title = "Distribution of Y0")

Mean Independence of \(y_{0}\)

• Take difference in mean \(y\)

  • This equals the treatment effect

  • Difference in variance did not create bias

summarize(data2, mean(y))

# A tibble: 2 × 2
      w `mean(y)`
  <dbl>     <dbl>
1     0      1.32
2     1      7.69

summarize(data2,mean(y))$`mean(y)`[2] - 
  summarize(data2,mean(y))$`mean(y)`[1]

[1] 6.370304

Mean Independence of \(y_{0}\)

• Running regression produces same result

• The variance in \(y_{1}\) affects the standard error

  • But we are not concerned with that right now

lm(y ~ w, data3)

Call:
lm(formula = y ~ w, data = data3)

Coefficients:
(Intercept)            w  
      2.004        4.999  

Conditional Mean Independence

• Finally consider conditional mean independence

• Treatment is related to \(y_{0}\), but only through \(x\)

  • For people with the same \(x\), \(y_{0}\) is unrelated to \(w\)

• Ex: Education and wages

  • Smart people \((x = 1)\) earn higher wages regardless of schooling \((y_0)\)

  • Smart people are more likely to go to university \((w = 1)\)

  • People at university will have higher \(y_0\)

data4 <- data %>% 
  ungroup() %>% 
  select(eta) %>%
  mutate(x = if_else(runif(100000) > .5,1,0),
         w = if_else(x + runif(100000, -1,1) > .5,1,0),
         y0 = 2 + 3*x + eta,
         y1 = y0 + 5,
         y = y0 + (y1-y0)*w) %>%
  group_by(w)

sumtable(data4, 
         summ=c('notNA(x)','mean(x)','sd(x)'), 
         group="w",
         group.long = TRUE)

Conditional Mean Independence

• Comparing treatment and control, \(y_{0}\) is bigger when \(w=1\)

• This is because

  • \(y_{0}\) is bigger when \(x=1\)

  • \(w\) more likely to be \(1\) when \(x=1\)

Summary Statistics

Variable	NotNA	Mean	Sd
w: 0
eta	49882	0.011	1.005
x	49882	0.25	0.433
y0	49882	2.763	1.646
y1	49882	7.763	1.646
y	49882	2.763	1.646
w: 1
eta	50118	-0.003	1.003
x	50118	0.752	0.432
y0	50118	4.253	1.636
y1	50118	9.253	1.636
y	50118	9.253	1.636

Conditional Mean Independence

• What if we focus only on people with \(x=1\)?

• No difference in \(y_{0}\) between treated and control

  • Because \(x\) is only reason why they differed

  • This is holding \(x\) fixed

sumtable(filter(data4, x==1), 
         summ=c('notNA(x)','mean(x)','sd(x)'), 
         group="w")

Summary Statistics

w	0			1
Variable	NotNA	Mean	Sd	NotNA	Mean	Sd
eta	12493	0.017	1	37694	-0.005	1.004
x	12493	1	0	37694	1	0
y0	12493	5.017	1	37694	4.995	1.004
y1	12493	10.017	1	37694	9.995	1.004
y	12493	5.017	1	37694	9.995	1.004

Conditional Mean Independence

• Same result if we hold \(x=0\)?

  • Again because \(x\) is only reason why they differed

sumtable(filter(data4, x==0), 
         summ=c('notNA(x)','mean(x)','sd(x)'), 
         group="w")

Summary Statistics

w	0			1
Variable	NotNA	Mean	Sd	NotNA	Mean	Sd
eta	37389	0.009	1.007	12424	0.001	0.999
x	37389	0	0	12424	0	0
y0	37389	2.009	1.007	12424	2.001	0.999
y1	37389	7.009	1.007	12424	7.001	0.999
y	37389	2.009	1.007	12424	7.001	0.999

Conditional Mean Independence

• Regression of \(y\) on \(w\) is biased

  • Because \(w\) is correlated with error

• But regression of \(y\) on \(w\) and \(x\) generates actual treatment effect

• This is conditional mean independence

  • Holding \(x\) fixed, potential outcomes no longer related to treatment

lm(y ~ w, data4)

Call:
lm(formula = y ~ w, data = data4)

Coefficients:
(Intercept)            w  
      2.763        6.490  

lm(y ~ w + x, data4)

Call:
lm(formula = y ~ w + x, data = data4)

Coefficients:
(Intercept)            w            x  
      2.011        4.985        3.001