Intro to Casual Inference

Up to Now in Econ 466

Up to Now

Relationship between variables represented by a Linear Model

For instance, summarize the predictive power of schooling’s effect on wages with the conditional expectation function (CEF):

$$\mathbb{E}[log(Salary)_i|X_i]=\beta_0+\beta_1Edu_{i}+\beta_kExper_{i}$$

Up to Now

library(wooldridge)
data("wage2")
model <- lm(lwage ~ educ + exper, wage2)

Also

• Add some assumptions, and then

  • Hypothesis testing
  • ``Casual effect" (?)

• After midterm 1, some assumptions may fail, for instance,

  • Heteroskedasticity
  • Serial correlation

Endogeneity

One relevant part we reviewed was:

Endogeneity

• Recall Omitted Variable Bias,

$$\begin{eqnarray} \mathbb{E}(\hat{\beta}_{1}|X)\equiv\beta_1+\beta_2\frac{Cov(X,Omitted\,Part)}{Var(X)} \end{eqnarray}$$

• Because of endogeneity, the causal effect cannot be identified.

Causality

Again: Correlation and Causation

(there are many correlations that are not causal)

Causal Effect

• How do we know if $X$ causes $Y$?

• E.g., causal relationships

  • Light switch turned on => light to be on
  • College degree => earnings (?)

• Discussion

  • what causality is and how we can find it
  • research designs for uncovering causality

Causal Effect

Causal Effect

Pick up one observation: $i=Luis$

Crazy idea, assume there are two parallel universes (PU):

• In PU1: Luis has a college degree, $X_i^0$, then observe earnings, $Y_i^0$.
• In PU2: (same) Luis has a Ph.D., $X_i^1$, then observe earnings, $Y_i^1$.

EVERYTHING is the same, EXCEPT Schooling.

Hence, we can conclude with respect to differences in earnings.

Causal Effect

In practice there is one issue,

a prediction of what would have happened in the absence of the treatment.

Causal Inference

Outline

1. Notions: Treatment, Control, Experiment, Quasi-Experiment, others.

2. Methods:

• Selection on unobservables
  • Differences-in-differences
  • Panel methods
  • Instrumental Variables
• Selection on observables:
  • Regression
  • Regression Discontinuity Design
  • Matching

Treatment and Control

Let’s use a dummy variable for whether an observation, $i$, received a treatment or not:

• Treatment: $D_i=1$

• Control: $D_i=0$

There is an ACTUAL outcome: $Y_i$

and an UNOBSERVED counterfactual:

$$Unobserved\,outcome= \left\{ \begin{array}{c} Y_i^1&if&D_i=1\\ Y^0_i&if &D_i=0 \end{array} \right.$$

Experiments

Experiments

• We would like to compare two observations that are basically exactly the same except that one has $D=0$ and one has $D=1$.

• Experiments:

  • research design (control the assignment to treatment)
  • direct manipulation of subjects’ values on $X$
  • measure changes in the values on $Y$

• Randomized Controlled Trial, RCT

Experiments

Experiments

The econometric specification could be as simple as,

$$Y_i=\beta_0+\beta_1D_i+u_i$$ which is the same as a two-sample t-test.

$\,$

On the other hand, there are some challenges to consider. For instance,

• Practical issues (e.g., randomizing types of exchange rate systems for developing countries).
• Ethical issues.
• “Too” controlled.

Natural Experiments

• The control and experimental variables of interest are not artificially manipulated by researchers
• Nature does the randomization
• Exogenous source of variation that determine treatment assignment
• Examples:
  • policy changes
  • government randomization

Quasi-experiments

Experiments and Quasi-experiments:

• In an experiment, participants are randomly assigned to treatment or control.

• In a quasi-experiment, observations are not assigned randomly.

• Thus,

  • Groups may differ not only in treatment

    (need to statistically control for differences).

  • There may be several “rival hypotheses” competing with the experimental manipulation as explanations for observed results.

Methods

Some Standard Methods

• Fixed effects
• Differences-in-Differences
• Instrumental Variables
• Regression Discontinuity

Fixed Effects

Fixed Effects

Idea,

• Again, recall Omitted Variable Bias
• We cannot control for many things
  • measurement issues
  • data issues

$\,$

• But, if we observe each observation $i$ multiple times $t$,
  • control for each observation identity.
  • it would be like “trying to control for everything unique” to that observation.

Panel Data

library(foreign)
Panel <- read.dta("http://dss.princeton.edu/training/Panel101.dta") # Source: http://dss.princeton.edu/training, visited Apr 2019.

Exploring Panel Data

library(gplots)
plotmeans(y ~ country, main="Heterogeineity across countries", data=Panel)

Exploring Panel Data

Fixed Effects

For instance, the model could be like

$$\begin{eqnarray} y_{it}&=&\beta_0+\beta_1x_{it}+\underbrace{\boldsymbol{\theta}_{i}}_{FE}+u_{it} \end{eqnarray}$$

$\,$

where, $\theta$ could represent (unobserved) characteristic(s)

Fixed Effects

If we ignore $\theta_i$,

OLS <-lm(y ~ x1, data=Panel)
panderOptions('digits',4)
pander(OLS)

(not stat. significant)

Fixed Effects

F.E. Estimation

• Dummy variable regression, $$y_{it}=\alpha_AD_A+\alpha_BD_B+...+\beta_1x_{it}+u_{it}$$ $\,$

• Within Estimator, $$(y_{it}-\bar{y}_i)\,\,on\,\,(x_{it}-\bar{x}_i)$$

$\,$

• First Differencing, $$(y_{it}-y_{it-1})\,\,on\,\,(x_{it}-x_{it-1})$$

Fixed Effects

Dummy Variable Regression,

FE.D <-lm(y ~ x1 + factor(country) - 1, data = Panel)
pander(coefficients(summary(FE.D)))

(stat. significant)

Fixed Effects

Dummy Variable Regression,

Fixed Effects

Within Estimator,

$$(y_{it}-\bar{y}_i)=\beta_1(x_{it}-\bar{x}_i)+\epsilon_{it}$$

library(plm)
FE <- plm(y ~ x1, data=Panel, index=c("country", "year"), model="within")
pander(coefficients(summary(FE)))

(stat. significant)

Fixed Effects

Discussion (from Wooldridge)

• Strict exogeneity in the original model has to be assumed

• In the case T = 2, fixed effects and first differencing are identical

• For T > 2, fixed effects is more efficient if classical assumptions hold

• If T is very large (and N not so large), the panel has a pronounced time series character and problems such as strong dependence arise

• First differencing may be better in the case of severe serial correlation in the errors

Differences-in-Differences

Differences-in-Differences

Motivation

Doctor John Snow, physician. London, 1813-1858.

• Cholera,

  • in the early 1800s
  • three main waves

• Jhon noticed variations

  • water supply
  • rates of cholera

Differences-in-Differences

Differences-in-Differences

• 1849:

  • London’s worst cholera epidemic
  • Two main companies supplied water:
    • Lambeth Waterworks Co. 
    • Southwark and Vauxhall Water Co

• 1852: Lambeth Waterworks moved their intake upriver

• 1853: London has another cholera outbreak

Differences-in-Differences

Do you want to learn more?

• Watch John Snow and the 1854 Broad Street cholera outbreak

• Or watch England: The Broad Street Pump

Differences-in-Differences

Illustration,

Differences-in-Differences

Differences-in-Differences

Econometric specification,

$$y=\beta_0+\beta_1D_{treatment}+\beta_2D_{post}+\beta_3(D_{treatment}*D_{post})+u$$

$\beta_3$ is the parameter of interest.

Notices that this is related to FE. For instance, if we have more than two groups (T/C = g) and more than two periods (Pre/Post = t):

$$y_{igt}=\beta_0+\beta_1D_{g}+\beta_2D_{t}+\beta_3D_{gt}+\beta_4X_{igt}+u_{igt}$$

Differences-in-Differences

Key Assumptions:

Differences-in-Differences

Do you want to learn more?

• watch ‘Does the Death Penalty Reduce Homicides?’

Instrumental Variables

Instrumental Variables

Idea,

• For the linear model, $y=\beta_0+\beta_1x+u$,

• say that we do not have random assignment

• but, what if we can find another variable $z$ with random assignment that causes $X$

• if this is the case, we coud get a natural experiment!

This variable is called it an “instrument” or “instrumental variable”

Instrumental Variables

Conditions for instrumental variable:

• Does not appear in regression equation
• Is uncorrelated with error term
• Is partially correlated with endogenous explanatory variable

Instrumental Variables

In other words,

• There is an endogeneity issue, $$Cov(x,u)\neq 0$$

• but, what if we can find $z$, such as this variable

  • causes $x$ (relevance):

  $$Cov(z,x)\neq 0$$

  • is randomly assigned, and the only relationship with $y$ is through $x$ (exogeneity):

  $$Cov(z,u)= 0$$

Instrumental Variables

Instrumental Variables

Before moving on, check

Instrumental Variables

Estimation for $k=2$, $y=\alpha+\beta x+u$.

$$If\,\,\,\,Cov(z,u)=0,\,\,\Rightarrow\,\,Cov(z,y)-\beta Cov(z,x)=0$$

thus,

$$\hat{\beta}_{IV} = \frac{\sum{(z_i-\bar{z})(y_i-\bar{y})}}{\sum{(z_i-\bar{z})(x_i-\bar{x})}}$$

Cigarette smoking and child birth weight.

data("bwght") 
reg_ls <- lm(lbwght~packs, data = bwght)

library(AER) 
reg_iv <- ivreg(lbwght~packs | cigprice, data = bwght)

Instrumental Variables

Properties of IV with a poor instrumental variable (for $k=2$),

$$\begin{eqnarray} plim\,\hat{\beta}_{OLS}&=&\beta_1+Corr(x,u)\frac{\sigma_u}{\sigma_x}\\ & &\\ plim\,\hat{\beta}_{IV}&=&\beta_1+\frac{Corr(z,u)}{Corr(z,x)}\frac{\sigma_u}{\sigma_x} \end{eqnarray}$$

IV worse than OLS if $\frac{Corr(z,u)}{Corr(z,x)}>Corr(x,u)$

Instrumental Variables

IV estimation in the multiple regression model ($k>2$),

$$y=\beta_0+\beta_1\underbrace{x_{1}}_{Endog}+\beta_2\underbrace{x_{2}}_{Exog}+u$$

Two Stage Least Squares (2SLS)

• Stage one (reduced form):

  Estimate, $x_1=\gamma_0+\gamma_1z+\gamma_2x_2+\epsilon$ , to get $\hat{x}_1$

• Stage two:

  Estimate, $y=\beta_0+\beta_1\hat{x}_{1}+\beta_2x_{2}+u$

Card(1995), College Proximity as an IV for Education (Example 15.4)

data("card") 
reg.ls  <-lm   (lwage~educ   + exper+expersq+black+smsa+south+smsa66+reg662+reg663+reg664+reg665+reg666+reg667+reg668+reg669, data = card)
reg.iv  <-ivreg(lwage~educ   + exper+expersq+black+smsa+south+smsa66+reg662+reg663+reg664+reg665+reg666+reg667+reg668+reg669|
                      nearc4 + exper+expersq+black+smsa+south+smsa66+reg662+reg663+reg664+reg665+reg666+reg667+reg668+reg669, data = card)

Instrumental Variables

Testing for endogeneity of $x_1$ in

$$y=\beta_0+\beta_1x_1+\beta_2x_2+u$$

• Reduced form, $x_1=\gamma_0+\gamma_1z+\gamma_2x_2+\epsilon$

• $x_1$ is exogenous if and only if $\epsilon$ is correlated with $u$

• Test equation

  $$y=\beta_0+\beta_1x_1+\beta_2x_2+\delta\hat{\epsilon}+e$$

  $H_0$, $exogeneity$ of $x_1$, is rejected if $\delta$ is significantly different from zero.

Regression Discontinuity Design

Regression Discontinuity Design

Motivating Examples

• National Merit Scholarship and GPA.
  • Endogeneity issue:
    • students performing well => more likely to win
    • and continue performing well
  • thus, it’s not clear the causal effect of the schoolarship on GPA

• Class size and Students reading achievement (Angrist and Lavy, 1997).

Regression Discontinuity Design - basic idea

• RDD exploits exogeneity in program characteristics (designing aspects).

• We could think of it as a randomized experiment at cutoff \ Treatment is assigned based on a cutoff (or running variable).

• Intuition: Participants in the program are very different, however, at the margin (cutoff), those just at the threshold are virtually identical.

Regression Discontinuity Design - basic idea

For instance,

• scholarship allocation based on scoring above a certain threshold on the PSAT.
• being admitted to a training program based on a test score.
• class size design using Maimonides’ rule.

RDD - Scholarship and Grades

RDD - Scholarship and Grades (2)

RDD - General Idea

(Animation) Source: Nick C. Huntington-Klein. Twitter @nickchk

Ather example. Alcohol and Mortality (Before 1984, MLDA<21)

library(foreign); library(ggplot2)
RDData <- read.dta("http://masteringmetrics.com/wp-content/uploads/2015/01/AEJfigs.dta"); RDData$over21 <- (RDData$agecell>=21); attach(RDData);
ggplot(RDData,aes(x=agecell, y=all,colour=over21))+geom_point()+ylim(80,110)+stat_smooth(method=loess)+labs(x="Age",y="Mortality Rate (per 100,000)") 

Regression Discontinuity Design

Model

$$y=\beta_0+\beta_1D_{treatment}+\beta_2Running+u$$

Final comments,

• Sharp RDD
• Fuzzy RDD
• Functional form specification
• External / Internal Validity