F9 Instrumental Variables

Question 1

What is the advantage and disadvantage of IV?

Answer 1

Advantage: It’s possible to deal with unobserved confounders.

Disadvantage: It relies heavily on the exclusion restriction, which is difficult to defend.

Question 2

Draw the DAG of IV

Answer 2

Z –> D –> Y
+
D <– U –> Y

Question 3

What is the exclusion restriction?

Answer 3

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

Question 4

What is a good instrument?

Answer 4

Quasi-random (easier to defend the exclusion restriction).

Question 5

What are the four important groups in IV

Answer 5

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.

Question 6

How can an estimate be biased by confounders?

Answer 6

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

Question 7

What direction can bias have if affected by confounders?

Answer 7

Negative-negative or positive-positive: Upward bias.
Negative-positive or positive-negative: Downward bias

Question 8

What is a strong instrument?

Answer 8

Highly correlated with the independent variable (non-zero at first stage)

Question 9

What is 2SLS?

Answer 9

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)

Question 10

What are the assumptions behind 2SLS?

Answer 10

(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).

Question 11

What is the reduced form?

Answer 11

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.

Question 12

What is a weak instrument and how can you test whether an instrument is strong or weak?

Answer 12

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.

Question 13

What is the F-statistic?

Answer 13

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.

Question 14

Can you have more than one instrument?

Answer 14

Yes as long as all instruments meet the exclusion restrictions.

Question 15

What is the beta-estimate with IV?

Answer 15

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

Question 16

How can relevance and exogeneity/the exclusion restriction be expressed in terms of covariance?

Answer 16

(1) Relevance: Cov(X_i , Z_i) different from 0
(2) Exclusion: Cov(Z_i, u_i) = 0

Question 17

Can an instrument have a direct effect on the outcome?

Answer 17

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).

Question 18

What kind of effect does IV estimate?

Answer 18

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.

Question 19

What happens if the treatment is heterogeneous with the effect and assumptions?

Answer 19

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

Question 20

What is the monotonicity assumption?

Answer 20

Z affect D in the same direction for all i.

Question 21

When do we use the Wald estimator?

Answer 21

Binary Instrument
No Perfect Compliance

Question 22

What are two main problems for IV?

Answer 22

Heterogeneous effects (complier ATE/LATE)
Weak instruments (and thus underpowered, type ll)

Question 23

What is the difference between homogeneous and heterogenous effects?

Answer 23

Treatment has the same causal effect for everybody (“homogeneous treatment effects”)

Treatment effects can differ across the population (“heterogeneous treatment effects”)

Question 24

How can you write up heterogeneous treatment effects with potential outcomes?

Answer 24

Y_i^1 - Y_i^0 = delta_i