Introduction to ‘Advanced Quantitative Analysis: Identification and Causality’

Question 1

Missing counterfactuals with observational data

Answer 1

• with observational data we only know the outcome of interest for every subject in its actual situation
• to know the causal effect we would require the outcome of subjects in a hypothetical situation
  ) that’s the missing counterfactual problem

Question 2

Notation of POM

Answer 2

Y = YT if D=T

Y = YU if D=U

The actual causal effect of D on subject i is
Other common notation
Delta i = yiT - yiU.

Question 3

ATE

Answer 3

E[delta] = E[YT - YU] = E[YT] - E[YU].

Question 4

ATT

Answer 4

E[delta]|D = T] = E[YT|D = T] - E[YU|D = T].

Question 5

ATU

Answer 5

E[delta]|D = U] = E[YT |D = U] - E[YU|D = U].

Question 6

Naive estimation of the ATE

Answer 6

We could attempt to estimate the ATE by averaging over the realizations of the outcome for several subjects i
deltaˆnaive = E[yi|di =T] - E[yi|di =U],
which converges to

E [Y T |D = T] E [Y U |D = U]
but not necessarily the ATE
E[delta] = E[YT] - E[YU].

Question 7

ATE ignorability condition

Answer 7

E(YT|D = T) = E(YT|D = U),

and E(YU|D = T) = E(YU|D = U).

Question 8

Systematic expression of bias

Answer 8

The true ATE can be expressed as:
E[delta] = E[YT] - E[YU]
= pi E[YT|D=T]+(1-pi)E[YT|D=U]}
{⇡E[YU|D = T] + (1 -⇡)E[YU|D = U]}. Some re-arrangement yields a new expression for the naive estimator:
E[YT|D = T] - E[YU|D = U] = E[delta] + {E[YU|D = T] -E[YU|D = U]} +(1-⇡){E[delta]D = T] - E[delta]|D = U]}.
Righthand side expression has three components:
1. trueATE E[delta]
2. level difference in Y between the T and U group prior to treatment (first {•})

3. difference in treatment effect between is in T versus C weighted by 1 ⇡ (second {•})
   )delta ˆnaive = ATE + selection bias + treatment effect bias

Question 9

conditional ignorability

Answer 9

I ignorability conditional on some observables allows to estimate unbiased conditional ATE
I conditional ignorability holds if for every value of X (i.e. every stratum)

E(YT|D = T,X = x) = E(YT|D = U,X = x),

and
 E(YU|D=T,X =x) = E(YU|D=U,X =x).

Question 10

SUTVA

stable unit treatment value assumption

Answer 10

I requires that treatment status of i does not affect (potential) outcome of any j
I put differently, the individual causal effect i must be independent of the treatment assignment process.

s simply the a priori assumption that the value of Y for unit u when exposed to treatment t will be the same no matter what mechanism is used to assign treatment t to unit u and no matter what treatments the other units receive (Rubin 1986).

• SUTVA can be thought of as the absence of ‘macro effects’; the treatment must not affect ‘general equilibrium’
• SUTVA is violated if, for example, the proportion of treated versus untreated alters the size of effects

Implications for policy
I effect of a policy may differ from the ATE in a small-scale study once it is rolled out more widely
I SUTVA usually violated if group composition effects are part of ATE

Question 11

Solutions for selection on observables

Answer 11

regression,

matching,

and selection correction

Question 12

SOlutions for selection on unobservables

Answer 12

instrumental variables

fixed effects

diff-in-diff

NEs