3.2 — DAGs

ECON 480 • Econometrics • Fall 2022

Dr. Ryan Safner
Associate Professor of Economics

safner@hood.edu
ryansafner/metricsF22
metricsF22.classes.ryansafner.com

Contents

Causation and Correlation

Causal Diagrams

DAG Rules

Causation and Correlation

You Don’t Need an RCT to Talk About Causality

  • Statistics profession is obstinant that we cannot say anything about causality

  • But you have to! It’s how the human brain works!

  • We can’t conceive of (spurious) correlation without some causation

The Causal Revolution

RCTs and Evidence-Based Policy

Should we ONLY base policies on the evidence from Randomized Controlled Trials

Source: British Medical Journal

RCTs and Evidence-Based Policy II

RCTs and Evidence-Based Policy II

Correlation vs. Causation I

What Does Causation Mean?

  • “Correlation does not imply causation”
    • this is exactly backwards!
    • this is just pointing out that exogeneity is violated

What Does Causation Mean?

  • “Correlation does not imply causation”
    • this is exactly backwards!
    • this is just pointing out that exogeneity is violated
  • “Correlation implies causation”
    • for an association, there must be some causal chain that relates \(X\) and \(Y\)
    • but not necessarily merely \(X \rightarrow Y\)
  • “Correlation plus exogeneity is causation.”

Correlation and Causation

  • Correlation:
    • Math & Statistics
    • Computers, AI, Machine learning can figure this out (better than humans)
  • Causation:
    • Philosophy, Intuition, Theory
    • Counterfactual thinking, unique to humans (vs. animals or computers)
    • Computers cannot (yet) figure this out

The Causal Revolution

Causation Requires Counterfactual Thinking

Causal Inference

  • We will seek to understand what causality is and how we can approach finding it

  • We will also explore the different common research designs meant to identify causal relationships

  • These skills, more than supply & demand, constrained optimization models, ISLM, etc, are the tools and comparative advantage of a modern research economist

    • Why all big companies (especially in tech) have entire economics departments in them

“The Credibility Revolution” in Econometrics

]

  • Simultaneous “credibility revolution” in econometrics (c.1990s—2000s)

  • Use clever research designs to approximate natural experiments

  • Note: major disagreements between Pearl & Angrist/Imbens, etc.!

Clever Research Designs Identify Causality

Correlation and Causation

What Then IS Causation?

  • \(X\) causes \(Y\) if we can intervene and change \(X\) without changing anything else, and \(Y\) changes
  • \(Y\) “listens to” \(X\)
    • \(X\) may not be the only thing that causes \(Y\)!

What Then IS Causation?

  • \(X\) causes \(Y\) if we can intervene and change \(X\) without changing anything else, and \(Y\) changes
  • \(Y\) “listens to” \(X\)
    • \(X\) may not be the only thing that causes \(Y\)!

Example

If \(X\) is a light switch, and \(Y\) is a light:

  • Flipping the switch \((X)\) causes the light to go on \((Y)\)
  • But NOT if the light is burnt out (No \(Y\) despite \(X\))
  • OR if the light was already on \((Y\) without \(X\))

Non-Causal Claims

  • All of the following have non-zero correlations. Are they causal?

Examples

  • Greater ice cream sales \(\rightarrow\) more violent crime
  • Rooster crows \(\rightarrow\) the sun rises in the morning
  • Taking Vitamin C \(\rightarrow\) colds go away a few days later
  • Political party \(X\) in power \(\rightarrow\) economy performs better/worse

Counterfactuals

  • The sine qua non of causal claims are counterfactuals: what would \(Y\) have been if \(X\) had been different?

  • It is impossible to make a counterfactual claim from data alone!

  • Need a (theoretical) causal model of the data-generating process!

Counterfactuals and RCTs

  • Again, RCTs are invoked as the gold standard for their ability to make counterfactual claims:

  • Treatment/intervention \((X)\) is randomly assigned to individuals

If person i who recieved treatment had not recieved the treatment, we can predict what his outcome would have been

If person j who did not recieve treatment had recieved treatment, we can predict what her outcome would have been

  • We can say this because, on average, treatment and control groups are the same before treatment

From RCTs to Causal Models

  • RCTs are but the best-known method of a large, growing science of causal inference

  • We need a causal model to describe the data-generating process (DGP)

  • Requires us to make some assumptions

Causal Diagrams

Causal Diagrams/DAGs

  • A surprisingly simple, yet rigorous and powerful method of modeling is using a causal diagram or DAG:
    • Directed: Each node has arrows that points only one direction
    • Acyclic: Arrows only have one direction, and cannot loop back
    • Graph

Causal Diagrams/DAGs

  • A visual model of the data-generating process, encodes our understanding of the causal relationships

  • Requires some common sense/economic intuition

  • Remember, all models are wrong, we just need them to be useful!

Causal Diagrams/DAGs

  • Our light switch example of causality

Drawing a DAG: Example

  • Suppose we have data on three variables

    • IP: how much a firm spends on IP lawsuits
    • tech: whether a firm is in tech industry
    • profit: firm profits

  • They are all correlated with each other, but what’s are the causal relationships?

  • We need our own causal model (from theory, intuition, etc) to sort

    • Data alone will not tell us!

Drawing a DAG

  1. Consider all the variables likely to be important to the data-generating process (including variables we can’t observe!)

  2. For simplicity, combine some similar ones together or prune those that aren’t very important

  3. Consider which variables are likely to affect others, and draw arrows connecting them

  4. Test some testable implications of the model (to see if we have a correct one!)

Drawing a DAG

  • Drawing an arrow requires a direction - making a statement about causality!

  • Omitting an arrow makes an equally important statement too!

    • In fact, we will need omitted arrows to show causality!

  • If two variables are correlated, but neither causes the other, likely they are both caused by another (perhaps unobserved) variable - add it!

  • There should be no cycles or loops (if so, there’s probably another missing variable, such as time)

DAG Example I

Example

what is the effect of education on wages?

  • Education \(X\), “treatment” or “exposure”

  • Wages \(Y\), “outcome” or “response”

DAG Example I

  • What other variables are important?
    • Ability
    • Socioeconomic status
    • Demographics
    • Phys. Ed. requirements
    • Year of birth
    • Location
    • Schooling laws
    • Job connections

DAG Example I

  • In social science and complex systems, 1000s of variables could plausibly be in DAG!

  • So simplify:

    • Ignore trivial things (Phys. Ed. requirement)
    • Combine similar variables (Socioeconomic status, Demographics, Location) \(\rightarrow\) Background

DAG Example II

  • Background, Year of birth, Location, Compulsory schooling, all cause education

  • Background, year of birth, location, job connections probably cause wages

DAG Example II

  • Background, Year of birth, Location, Compulsory schooling, all cause education

  • Background, year of birth, location, job connections probably cause wages

  • Job connections in fact is probably caused by education!

  • Location and background probably both caused by unobserved factor (u1)

DAG Example II

  • This is messy, but we have a causal model!

  • Makes our assumptions explicit, and many of them are testable

  • DAG suggests certain relationships that will not exist:

    • all relationships between laws and conx go through educ
    • so if we controlled for educ, then cor(laws,conx) should be zero!

Let the Computer Do It: Dagitty.net I

  • Dagitty.net is a great tool to make these and give you testable implications

  • Click Model -> New Model

  • Name your “exposure” variable \((X\) of interest) and “outcome” variable \((Y)\)

Let the Computer Do It: Dagitty.net II

  • Click and drag to move nodes around

  • Add a new variable by double-clicking

  • Add an arrow by double-clicking one variable and then double-clicking on the target (do again to remove arrow)

Let the Computer Do It: Dagitty.net II

Let the Computer Do It: Dagitty.net III

  • Tells you how to identify your effect! (upper right)

Minimal sufficient adjustment sets containing background, location, year for estimating the total effect of educ on wage: background, location, year

Let the Computer Do It: Dagitty.net III

  • Tells you some testable implications of your model

  • These are (conditional) independencies:

\[X \perp Y \, | \, Z\]

“X is independent of Y, given Z”

  • Implies that by controlling for \(Z\), \(X\) and \(Y\) should have no correlation

Let the Computer Do It: Dagitty.net III

  • Tells you some testable implications of your model

  • Example: look at the last one listed:

job_connections \(\perp\) year \(\vert\) educ

“Job connections are independent of year, controlling for education”

  • Implies that by controlling for educ, there should be no correlation between job_connections and year — can test this with data!

Causal Effect

  • If we control for background, location, and year, we can identify the causal effect of educ \(\rightarrow\) wage.

You Can Draw DAGs in R

  • New package: ggdag
  • Arrows are made with formula notation: Y ~ X + Z means “Y is caused by X and Z”
library(ggdag)
dagify(wage ~ educ + conx + year + bckg + loc,
       educ ~ bckg + year + loc + laws,
       conx ~ educ,
       bckg ~ u1,
       loc ~ u1,
       exposure = "educ", # optional: define X
       outcome = "wage" # optional: define Y
       ) %>%
  ggdag()+
  theme_dag()

You Can Draw DAGs in R II

  • Or you can just copy the code from dagitty.net!
  • Use dagitty() from the dagitty package, and paste the code in quotes
# install.packages("dagitty")
library(dagitty)
dagitty('dag {
bb="0,0,1,1"
background [pos="0.413,0.335"]
compulsory_schooling_laws [pos="0.544,0.076"]
educ [exposure,pos="0.185,0.121"]
job_connections [pos="0.302,0.510"]
location [pos="0.571,0.431"]
u1 [pos="0.539,0.206"]
wage [outcome,pos="0.552,0.761"]
year [pos="0.197,0.697"]
background -> educ
background -> wage
compulsory_schooling_laws -> educ
educ -> job_connections
educ -> wage
job_connections -> wage
location -> educ
location -> wage
u1 -> background
u1 -> location
year -> educ
year -> wage
}') %>%
  ggdag()+
  theme_dag()

You Can Draw DAGs In R

  • It’s not very pretty, but if you set text = FALSE, use_labels = "name inside ggdag(), makes it easier to read
dagitty('dag {
bb="0,0,1,1"
background [pos="0.413,0.335"]
compulsory_schooling_laws [pos="0.544,0.076"]
educ [exposure,pos="0.185,0.121"]
job_connections [pos="0.302,0.510"]
location [pos="0.571,0.431"]
u1 [pos="0.539,0.206"]
wage [outcome,pos="0.552,0.761"]
year [pos="0.197,0.697"]
background -> educ
background -> wage
compulsory_schooling_laws -> educ
educ -> job_connections
educ -> wage
job_connections -> wage
location -> educ
location -> wage
u1 -> background
u1 -> location
year -> educ
year -> wage
}') %>%
  ggdag(., text = FALSE, use_labels = "name")+
  theme_dag()

ggdag: Additional Tools

  • If you have defined X (exposure) and Y (outcome), you can use ggdag_paths() to have it show all possible paths between \(X\) and \(Y\)!
dagify(wage ~ educ + conx + year + bckg + loc,
       educ ~ bckg + year + loc + laws,
       conx ~ educ,
       bckg ~ u1,
       loc ~ u1,
       exposure = "educ",
       outcome = "wage"
       ) %>%
  tidy_dagitty(seed = 2) %>%
  ggdag_paths()+ #<<
  theme_dag()

ggdag: Additional Tools

  • If you have defined X (exposure) and Y (outcome), you can use ggdag_adjustment_set() to have it show you what you need to control for in order to identify \(X \rightarrow Y\)!
dagify(wage ~ educ + conx + year + bckg + loc,
       educ ~ bckg + year + loc + laws,
       conx ~ educ,
       bckg ~ u1,
       loc ~ u1,
       exposure = "educ",
       outcome = "wage"
       ) %>%
  ggdag_adjustment_set(shadow = T)+ #<<
  theme_dag()

ggdag: Additional Tools

  • You can also use impliedConditionalIndependencies() from the dagitty package to have it show the testable implications from dagitty.net
dagify(wage ~ educ + conx + year + bckg + loc,
       educ ~ bckg + year + loc + laws,
       conx ~ educ,
       bckg ~ u1,
       loc ~ u1,
       exposure = "educ",
       outcome = "wage"
       ) %>%
  impliedConditionalIndependencies() #<<
bckg _||_ conx | educ
bckg _||_ laws
bckg _||_ loc | u1
bckg _||_ year
conx _||_ laws | educ
conx _||_ loc | educ
conx _||_ u1 | bckg, loc
conx _||_ u1 | educ
conx _||_ year | educ
educ _||_ u1 | bckg, loc
laws _||_ loc
laws _||_ u1
laws _||_ wage | bckg, educ, loc, year
laws _||_ year
loc _||_ year
u1 _||_ wage | bckg, loc
u1 _||_ year

DAG Rules

DAG Rules

  • How does dagitty.net and ggdag know how to identify effects, or what to control for, or what implications are testable?

  • Comes from fancy math called “do-calculus”

  • Fortunately, these amount to a few simple rules that we can see on a DAG

DAGs I

  • Typical notation:

  • \(X\) is independent variable of interest

    • Epidemiology: “intervention” or “exposure”

  • \(Y\) is dependent or “response” variable

  • Other variables use other letters

  • You can of course use words instead of letters!

DAGs and Causal Effects

  • Arrows indicate causal effect (& direction)

  • Two types of causal effect:

  1. Direct effects: \(X \rightarrow Y\)

DAGs and Causal Effects

  • Arrows indicate causal effect (& direction)

  • Two types of causal effect:

  1. Direct effects: \(X \rightarrow Y\)

  2. Indirect effects: \(X \rightarrow M \rightarrow Y\)

  • \(M\) is a “mediator” variable, the mechanism by which \(X\) affects \(Y\)

DAGs and Causal Effects

  • Arrows indicate causal effect (& direction)

  • Two types of causal effect:

  1. Direct effects: \(X \rightarrow Y\)

  2. Indirect effects: \(X \rightarrow M \rightarrow Y\)

  • \(M\) is a “mediator” variable, the mechanism by which \(X\) affects \(Y\)
  1. You of course might have both!

Confounders

  • \(Z\) is a “confounder”: it causes both \(X\) and \(Y\)

  • \(cor(X,Y)\) is made up of two parts:

    1. Causal effect of \((X \rightarrow Y)\) 👍
    2. \(Z\) causing both the values of \(X\) and \(Y\) 👎

  • Failing to control for \(Z\) will bias our estimate of the causal effect of \(X \rightarrow Y\)!

Confounders

  • Confounders are the DAG-equivalent of omitted variable bias (next class)

\[Y_i=\beta_0+\beta_1 X_i\]

  • By leaving out \(Z_i\), this regression is biased

  • \(\hat{\beta}_1\) picks up both:

    1. \(X \rightarrow Y\)
    2. \(X \leftarrow Z \rightarrow Y\)

“Front Doors” and “Back Doors”

  • With this DAG, there are 2 paths that connect \(X\) and \(Y\)1:
  1. A causal “front-door” path: \(X \rightarrow Y\)
  • 👍 what we want to measure
  1. A non-causal “back-door” path: \(X \leftarrow Z \rightarrow Y\)
  • At least one causal arrow runs in the opposite direction
  • 👎 adds a confounding bias

Controlling I

  • Ideally, if we ran a randomized control trial and randomly assigned different values of \(X\) to different individuals, this would delete the arrow between \(Z\) and \(X\)

    • Individuals’ values of \(Z\) do not affect whether or not they are treated (\(X\))

  • This would only leave the front-door, \(X \rightarrow Y\)

  • But we can rarely run an ideal RCT

Controlling II

  • Instead of an RCT, if we can just “adjust for” or “control for” \(Z\), we can block the back-door path \(X \leftarrow Z \rightarrow Y\)

  • This would only leave the front-door path open, \(X \rightarrow Y\)

  • “As good as” an RCT!

Controlling II

  • Using our terminology from last class, we have an outcome \((Y)\), and some treatment

  • But there are unobserved factors \((u)\)

\[Y_i = \beta_0 + \beta_1 Treatment + u_i\]

Controlling II

  • Using our terminology from last class, we have an outcome \((Y)\), and some treatment

  • But there are unobserved factors \((u)\)

\[Y_i = \beta_0 + \beta_1 Treatment + u_i\]

  • If we can randomly assign treatment, this makes treatment exogenous:

\[cor(treatment,u)=0\]

Controlling II

  • Using our terminology from last class, we have an outcome \((Y)\), and some treatment

  • But there are unobserved factors \((u)\)

\[Y_i = \beta_0 + \beta_1 Treatment + u_i\]

  • When we (often) can’t randomly assign treatment, we have to find another way to control for measurable things in \(u\)

Controlling II

  • Controlling for a single variable along a long causal path is sufficient to block that path!

  • Causal path: \(X \rightarrow Y\)

  • Backdoor path: \(X \leftarrow A \rightarrow B \rightarrow C \rightarrow Y\)

  • It is sufficient to block this backdoor by controlling either \(A\) or \(B\) or \(C\)!

Controlling II

  • Controlling for a single variable along a long causal path is sufficient to block that path!

  • Causal path: \(X \rightarrow Y\)

  • Backdoor path: \(X \leftarrow A \rightarrow B \rightarrow C \rightarrow Y\)

  • It is sufficient to block this backdoor by controlling either \(A\) or \(B\) or \(C\)!

The Back Door Criterion

  • To identify the causal effect of \(X \rightarrow Y\):

  • “Back-door criterion”: control for the minimal amount of variables sufficient to ensure that no open back-door exists between \(X\) and \(Y\)

  • Example: in this DAG, control for \(Z\)

The Back Door Criterion

  • Implications of the Back-door criterion:
  1. You only need to control for the variables that keep a back-door open, not all other variables!

Example:

  • \(X \rightarrow Y\) (front-door)
  • \(X \leftarrow A \rightarrow B \rightarrow Y\) (back-door)

The Back Door Criterion

  • Implications of the Back-door criterion:
  1. You only need to control for the variables that keep a back-door open, not all other variables!

Example:

  • \(X \rightarrow Y\) (front-door)

  • \(X \leftarrow A \rightarrow B \rightarrow Y\) (back-door)

  • Need only control for \(A\) or \(B\) to block the back-door path

  • \(C\) and \(Z\) have no effect on \(X\), and therefore we don’t need to control for them!

The Back Door Criterion: Colliders

  1. Exception: the case of a “collider”
  • If arrows “collide” at a node, that node is automatically blocking the pathway, do not control for it!
  • Controlling for a collider would open the path and add bias!

Example:

  • \(X \rightarrow Y\) (front-door)
  • \(X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y\) (back-door, but blocked by B!)

The Back Door Criterion: Colliders

  1. Exception: the case of a “collider”
  • If arrows “collide” at a node, that node is automatically blocking the pathway, do not control for it!
  • Controlling for a collider would open the path and add bias!

Example:

  • \(X \rightarrow Y\) (front-door)
  • \(X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y\) (back-door, but blocked by B!)
  • Don’t need to control for anything here!

The Back Door Criterion: Colliders Example

Are you less likely to get the flu if you are hit by a bus?

  • \(Flu\): getting the flu

  • \(Bus\): being hit by a bus

  • \(Hos\): being in the hospital

  • Both \(Flu\) and \(Bus\) send you to \(Hos\) (arrows)

  • Conditional on being in \(Hos\), negative correlation between \(Flu\) and \(Bus\) (spurious!)

The Back Door Criterion: Colliders Example

  • In the NBA, apparently players’ height has no relationship to points scored?

The Back Door Criterion: Colliders Example

  • In the NBA, apparently players’ height has no relationship to points scored?

The Front Door Criterion: Mediators I

  • Another case where controlling for a variable actually adds bias is if that variable is known as a “mediator”.

Example

  • \(X \rightarrow M \rightarrow Y\) (front-door)

  • \(X \leftarrow A \rightarrow Y\) (back-door)

  • \(X \leftarrow B \rightarrow Y\) (back-door)

  • Should we control for \(M\)?

  • If we did, this would block the front-door!

The Front Door Criterion: Mediators II

  • Another case where controlling for a variable actually adds bias is if that variable is known as a “mediator”.

Example

  • \(X \rightarrow M \rightarrow Y\) (front-door)

  • \(X \leftarrow A \rightarrow Y\) (back-door)

  • \(X \leftarrow B \rightarrow Y\) (back-door)

  • If we control for \(M\), would block the front-door!

  • If we can estimate \(X \rightarrow M\) and \(M \rightarrow Y\) (note, no back-doors to either of these!), we can estimate \(X \rightarrow Y\)

  • This is the front door method

The Front Door Criterion: Mediators III

  • Tobacco industry claimed that \(cor(smoking, cancer)\) could be spurious due to a confounding gene that affects both!

    • Smoking gene is unobservable

  • Suppose smoking causes tar buildup in lungs, which cause cancer

  • We should not control for tar, it’s on the front-door path

    • This is how scientific studies can relate smoking to cancer

Summary: DAG Rules for Causal Identification

Thus, to achieve causal identification, control for the minimal amount of variables such that:

  1. Ensure no back-door path remains open
  • Close back-door paths by controlling for any one variable along that path
  • Colliders along a path automatically close that path
  1. Ensure no front-door path is closed
  • Do not control for mediators

ECON 480 — Econometrics