Estimating Causal Effects with
Experimental Data
Some Basic Terminology
• Start with example where X is binary (though simple to generalize):
– X=0 is control group
– X=1 is treatment group
• Causal effect sometimes called treatment effect
• Randomization implies everyone has same probability of treatment
Why is Randomization Good?
• If X allocated at random then know that X is independent of all pre-treatment
variables in whole wide world
• an amazing claim but true.
• Implies there cannot be a problem of omitted variables, reverse causality etc
• On average, only reason for difference between treatment and control group is different receipt of treatment
Why is this useful?
An Example: Racial Discrimination
• Black men earn less than white men in US
LOGWAGE | Coef. Std. Err. t ---+--- BLACK | -.1673813 .0066708 -25.09 NO_HS | -.2138331 .0077192 -27.70 SOMECOLL | .1104148 .0049139 22.47 COLLEGE | .4660205 .0048839 95.42 AGE | .0704488 .0008552 82.38 AGESQUARED | -.0007227 .0000101 -71.41 _cons | 1.088116 .0172715 63.00
• Could be discrimination or other factors unobserved by the researcher but observed by the employer?
• hard to fully resolve with non-experimental data
An Experimental Design
• Bertrand/Mullainathan “Are Emily and
Greg More Employable Than Lakisha and Jamal”, American Economic Review, 2004
• Create fake CVs and send replies to job adverts
• Allocate names at random to CVs – some given ‘black-sounding’ names, others
‘white-sounding’
• Outcome variable is call-back rates
• Interpretation – not direct measure of
racial discrimination, just effect of having a
‘black-sounding’ name – may have other connotations.
• But name uncorrelated by construction with other material on CV
The Treatment Effect
• Want estimate of:
i i1 i i 0
E y X E y X
Estimating Treatment Effects: the Statistics Course Approach
• Take mean of outcome variable in treatment group
• Take mean of outcome variable in control group
• Take difference between the two
• No problems but:
– Does not generalize to where X is not binary – Does not directly compute standard errors
Estimating Treatment Effects: A Regression Approach
• Run regression:
yi=β0+β1Xi+εi
• Proposition 2.2 The OLS estimator of β1 is an unbiased estimator of the causal effect of X on y:
• Proof: Many ways to prove this but simplest way is perhaps:
– Proposition 1.1 says OLS estimates E(y|X)
– E(y|X=0)= β0 so OLS estimate of intercept is consistent estimate of E(y│X=0)
– E(y|X=1)= β0+β1 so β1 is consistent estimate of E(y│X=1) -E(y│X=0)
• Hence can read off estimate of treatment effect from coefficient on X
• Approach easily generalizes to where X is not binary
• Also gives estimate of standard error
Computing Standard Errors
• Unless told otherwise regression package will compute standard errors assuming errors are homoskedastic i.e.
• Even if only interested in effect of treatment on mean X may affect other aspects of distribution e.g. variance
• This will cause heteroskedasticity
• Heteroskedasticity does not make OLS
regression coefficients inconsistent but does make OLS standard errors inconsistent
‘Robust’ Standard Errors
• Also called:
– Huber standard errors – White standard errors
– Heteroskedastic-consistent standard errors
• Simple to use in practice e.g. in STATA:
. reg y x, robust
• Statistics course approach
– Get variance of estimate of mean of treatment and control group
– Sum to give estimate of variance of difference in means
Bertrand/Mullainathan:
Basic Results
Summary So Far
• Econometrics very easy if all data comes from randomized controlled experiment
• Just need to collect data on
treatment/control and outcome variables
• Just need to compare means of outcomes of treatment and control groups
• Is data on other variables of any use at all?
– Not necessary but useful
Including Other Regressors
• Can get consistent estimate of treatment effect without worrying about other variables
• Reason is that randomization ensures no problem of omitted variables bias
• But there are reasons to include other regressors:
– Improved efficiency
– Check for randomization – Improve randomization
– Control for conditional randomization – Heterogeneity in treatment effects
The Uses of Other Regressors I:
Improved Efficiency
• Don’t just want consistent estimate of causal effect – also want low standard error (or high precision or efficiency).
• Standard formula for standard error of OLS estimate of β is σ2/Var(X)
• σ2 comes from variance of residual in regression – (1-R2)* Var(y)
• Include more variables and R2 rises – formal proof (Proposition 2.4) a bit more complicated but this is basic idea.
The Uses of Other Regressors II:
Check for Randomization
• Randomization can go wrong
– Poor implementation of research design – Bad luck
• If randomization done well then W should be independent of X – this is testable:
– Test for differences in W in treatment/control groups
– Probit model for X on W
The Uses of Other Regressors III:
Improve Randomization
• Can also use W at stage of assigning treatment
• Can guarantee that in your sample X and W are independent instead of it being just probabiliistic
• This is what Bertrand/Mullainathan do when assigning names to CVs
The Uses of Other Regressors IV:
Adjust for Conditional Randomization
• This is case where must include W to get consistent estimates of treatment effects
• Conditional randomization is where probability of treatment is different for people with different values of W, but random conditional on W
• Why have conditional randomization?
– May have no choice
– May want to do it (c.f. stratification)
An Example: Project STAR
• Allocation of students to classes is random within schools
• But small number of classes per school
• This leads to following relationship between probability of treatment and number of kids in school:
.1.2.3.4.5Fraction in Treatment Group
40 60 80 100 120
Number of Kids in School
Controlling for Conditional Randomization
• X can know be correlated with W
• But, conditional on W, X independent of other factors
• But must get functional form of relationship between y and W correct – matching
procedures
• This is not the case with (unconditional) randomization – see class exercize
Heterogeneity in Treatment Effects
• So far have assumed causal (treatment) effect the same for everyone
• No good reason to believe this
• Start with case of no other regressors:
yi=β0+β1iXi+εi
• Random assignment implies X independent of β1i
• Sometimes called random coefficients model
What treatment effect to estimate?
• Would like to estimate causal effect for everyone – this is not possible – Holland’s fundamental
problem of statistical inference
• Can only hope to estimate some average
• Average treatment effect:
• Proposition 2.5: OLS estimates ATE
1i 1ATE E
Observable Heterogeneity
• Full outcomes notation:
– Outcome if in control group:
y0i=γ0’Wi+u0i – Outcome if in treatment group:
y1i=γ1’Wi+u1i
• Treatment effect is (y1i-y0i) and can be written as:
(y1i-y0i )=(γ1- γ0 )’Wi+u1i-u0i
• Note treatment effect has observable and unobservable component
• Can estimate as:
– Two separate equations – One single equation
Combining treatment and control groups into single regression
• We can write:
• Combining outcomes equations leads to:
• Regression includes W and interactions of W with X – these are observable part of treatment effect
• Note: error likely to be heteroskedastic
1 1 0
i i i i i
y X y X y
1 1 0 0
0 1 0 0 1 0
' 1 '
' '
i i i i i i i
i i i i i i i
y X W u X W u
W X W u X u u
Bertrand/Mullainathan
• Different treatment effect for high and low quality CVs:
Units of Measurement
• Causal effect measured in units of
‘experiment’ – not very helpful
• Often want to convert causal effects to more meaningful units e.g. in Project
STAR what is effect of reducing class size by one child
Simple estimator of this would be:
11
00
E y X E y X E S X E S X
• where S is class size
• Takes the treatment effect on outcome variable and divides by treatment effect on class size
• Not hard to compute but how to get standard error?
IV Can Do the Job
• Can’t run regression of y on S – S influenced by factors other than treatment status
• But X is:
– Correlated with S
– Uncorrelated with unobserved stuff (because of randomization)
• Hence X can be used as an instrument for S
• IV estimator has form (just-identified case):
1ˆIV X S' X y'
The Wald Estimator
• This will give estimate of standard error of treatment effect
• Where instrument is binary and no other
regressors included the IV estimate of slope coefficient can be shown to be:
11
00
E y X E y X E S X E S X
Partial Compliance
• So far:
– in control group implies no treatment
– In treatment group implies get treatment
• Often things are not as clean as this
– Treatment is an opportunity
– Close substitutes available to those in control group
– Implementation not perfect e.g. pushy parents
An Example: Moving to Opportunity
• Designed to investigate the impact of living in bad neighbourhoods on outcomes
• Gave some residents of public housing projects chance to move out
• Two treatments:
– Voucher for private rental housing
– Voucher for private rental housing restricted for use in
‘good’ neighbourhoods
• No-one forced to move so imperfect compliance – 60% and 40% did use it
Some Terminology
• Z denotes whether in control or treatment group – ‘intention-to-treat’
• X denotes whether actually get treatment
• With perfect compliance:
– Pr(X=1│Z=1)=1 – Pr(X=1│Z=0)=0
• With imperfect compliance:
1>Pr(X=1│Z=1)>Pr(X=1│Z=0)>0
What Do We Want to Estimate?
• ‘Intention-to-Treat’:
ITT=E(y|Z=1)-E(y|Z=0)
• This can be estimated in usual way
• Treatment Effect on Treated
11
00
E y Z E y Z TOT E X Z E X Z
Estimating TOT
• Can’t use simple regression of y on Z
• But should recognize TOT as Wald estimator
• Can estimated by regressing y on X using Z as instrument
• Relationship between TOT and ITT:
. Pr 1 1 Pr 1 0
ITT TOT X Z X Z
Most Important Results from MTO
• No effects on adult economic outcomes
• Improvements in adult mental health
• Beneficial outcomes for teenage girls
• Adverse outcomes for teenage boys
Sample results from MTO
• TOT approximately twice the size of ITT
• Consistent with 50% use of vouchers
IV with Heterogeneous Treatment Effects (More Difficult)
• If treatment effect same for everyone then TOT recovers this (obvious)
• But what if treatment effect heterogeneous?
• No simple answer to this question
• Suppose model for treatment effect is:
0 1
i i i i
y X
Proposition 2.6
The IV estimate for the heterogeneous
treatment case is a consistent estimate of:
where:
the difference in the probability of treatment for individual i when in treatment and control group
1
ˆ1,
lim IV i i
i
p E
E
1 1
1 0
i Pr Xi Zi Pr Xi Zi
Interpretation
• This is weighted average of treatment effects
• ‘weights’ will vary with instrument –
contrast with heterogeneous treatment case
• Some cases in which can interpret IV estimate as ATE
How will IV estimate differ from ATE
• IV is ATE if no correlation between β1i and πi
• Previous formula says depends on covariance of β1i and πi
• In some situations can sign – but not always
• Example 1: no-one gets treatment in the absence of the programme so
• If those who get treatment when in the treatment group are those with the highest returns then:
• IV>ATE
1
i
p
i
i, 1i 1i, 1i 0
Cov Cov p
• Example 2: treatment is voluntary for those in the control group but compulsory for
those in the treatment group
• This implies
• If those who get treatment in control are those with highest returns then:
• IV<ATE
i, 1i 0i, 1i 0
Cov Cov p
1 0
i p i
Angrist/Imbens Monotonicity Assumption
• Case where IV estimate is not ATE
• Assume that everyone moved in same direction by treatment – monotonicity assumption
• Then can show that IV is average of
treatment effect for those whose behaviour changed by being in treatment group
• They call this the Local Average Treatment Effect (LATE)
Problems with Experiments
• Expense
• Ethical Issues
• Threats to Internal Validity
– Failure to follow experiment
– Experimental effects (Hawthorne effects)
• Threats to External Validity
– Non-representative programme – Non-representative sample
– Scale effects
Conclusions on Experiments
• Are ‘gold standard’ of empirical research
• Are becoming more common
• Not enough of them to keep us busy
• Study of non-experimental data can deliver useful knowledge
• Some issues similar, others different