F9 Instrumental Variables Flashcards
What is the advantage and disadvantage of IV?
Advantage: It’s possible to deal with unobserved confounders.
Disadvantage: It relies heavily on the exclusion restriction, which is difficult to defend.
Draw the DAG of IV
Z –> D –> Y
+
D <– U –> Y
What is the exclusion restriction?
The instrument can only influence Y through D. It’s important with correlation between the instrument and the outcome (the only through)
The instrument cannot affect confounders or Y directly.
No causal relation.
Cov(Z, epsilon) = 0
What is a good instrument?
Quasi-random (easier to defend the exclusion restriction).
What are the four important groups in IV
Compliers: Treatment status is affected by the instrument in the correct direction (treatment units is treated). Complied with the draft of the lottery.
Defiers: Treatment status is affected by the instrument in the wrong direction (treatment units is not treated). Military-deserters
Never takers: Unit never take treatment regardless of the instrument. Medical exempt for military.
Always takers: Always take the treatment regardless of the instrument. Patriots - military no matter what.
How can an estimate be biased by confounders?
Biased estimate: delta-hat = delta + gamma * Cov(u,x)/Var(x).
Bias: gamma * Cov(u,x)/Var(x).
(1) Gamma: The effect of unobserved confounder - positive or negative - cov(u,y).
(2) Cov(u,x)/Var(x): The relation between x and unobserved confounders
What direction can bias have if affected by confounders?
Negative-negative or positive-positive: Upward bias.
Negative-positive or positive-negative: Downward bias
What is a strong instrument?
Highly correlated with the independent variable (non-zero at first stage)
What is 2SLS?
Two stages least squares regression. Two steps:
(1) X_i = γ + βZ_i + ε_i. The endogenous independent variable as a function of the instrument.
(2) Y_i = α + δX_i-hat + ϵ_i. The fitted values from the first stage is used in the primary regression
The estimate becomes (first stage replaced with fitted values):
Cov(βZ,Y)/Var(βZ) = Cov(X-hat,Y)/Var(X-hat)
What are the assumptions behind 2SLS?
(1) Exclusion restriction (instrument cannot be correlated with the error term). Necessary but not sufficient (must be a strong instrument as well).
(2) Non zero first stage (instrument must be correlated with the endogenous independent variable D).
What is the reduced form?
The correlation between the instrument and the outcome.
Must be different from zero and thus significant (to make sure that it’s different from zero) if we want to estimate an effect.
What is a weak instrument and how can you test whether an instrument is strong or weak?
Weak instrument: Not highly correlated with the endogenous independent variable.
Problem: Inconsistent and large SE
Test: The F-statistic on the first stage of 2SLS.
What is the F-statistic?
The joint significance of all independent variables in a regression. We’re punished by:
Every independent variable added
Weak instruments.
Relevant if you have more instruments.
F > 10 for the first stage.
Can you have more than one instrument?
Yes as long as all instruments meet the exclusion restrictions.
What is the beta-estimate with IV?
Cov(y,z)/Cov(x,z) with z being the instrument.
So, if the effect of Z on Y (the reduced form) is 2 and the effect of Z on X is 3, the effect of X and Y is 2/3 = 0,66
How can relevance and exogeneity/the exclusion restriction be expressed in terms of covariance?
(1) Relevance: Cov(X_i , Z_i) different from 0
(2) Exclusion: Cov(Z_i, u_i) = 0
Can an instrument have a direct effect on the outcome?
NO! But it must be correlated with the outcome if there’s is to be found an effect (through the effect of x on y).
What kind of effect does IV estimate?
A complier ATE (a LATE)
IV only identify a causal effect for any group of units whose behaviors are changed as a result of the instrument (complier).
Homogeneous treatment effects: Compliers have the same treatment effects as non-compliers, so the distinction is irrelevant.
What happens if the treatment is heterogeneous with the effect and assumptions?
It’s LATE for compliers. And there are more assumptions
(3) SUTVA. The potential outcomes for each person
are unrelated to the treatment status of other individuals
(4) Independence: The instrument has to be as if random related to potential outcome. Causality in first stage.
(5) Monotonicity: If an instrument increases the likelihood of receiving treatment for one individual, it cannot decrease the likelihood for another
What is the monotonicity assumption?
Z affect D in the same direction for all i.
When do we use the Wald estimator?
Binary Instrument
No Perfect Compliance
What are two main problems for IV?
Heterogeneous effects (complier ATE/LATE)
Weak instruments (and thus underpowered, type ll)
What is the difference between homogeneous and heterogenous effects?
Treatment has the same causal effect for everybody (“homogeneous treatment effects”)
Treatment effects can differ across the population (“heterogeneous treatment effects”)
How can you write up heterogeneous treatment effects with potential outcomes?
Y_i^1 - Y_i^0 = delta_i
