13: Instrumental Variables

Question 1

basic idea/concept

Answer 1

interested in the causal effect of a particular explanatory variable of a policy interest

BUT you have concerns that OLS is subject to OVB, reverse causality, and/or attenuation bias from measurement error

main idea is that the explanatory variable of interest has both good and bad variation
- instrument helps pick out the exogenous part of the variation in our independent variable of interest

Question 2

2SLS regressions

Answer 2

first-stage regression

second-stage regression

reduced form regression

Question 3

first-stage regression

Answer 3

regression of X on Z (explanatory variable on instrument)
- tells you if instrument is statistically significantly related to the explanatory variable of interest

decomposes the variation in X into the good variation that you want to use for estimation if assumptions hold

Question 4

second-stage regression

Answer 4

regression of Y on predicted variation (outcome of interest on predicted variation i nX)

correctly identifies the causal effect of X on Y if the instrument pushes around X and isn’t related to other things that determine Y

Question 5

reduced-form regression

Answer 5

regression of Y on Z

Question 6

second-stage IV point estimate computation

Answer 6

cov(Y,Z) / cov(X,Z) = point estimate of reduced form / point estimate of first-stage

Question 7

sampling distribution of 2SLS

Answer 7

normal in large samples
- distributed by the CLT

Question 8

three essential assumptions to hold true for estimating causal effects using IVs

Answer 8

relevance

exogeneity

exclusion restriction

Question 9

instrument relevance

Answer 9

instrument needs to be related strongly enough to the endogenous explanatory variable of interest

running regression of X on Z and seeing how strong it is

Question 10

instrument exogeneity

Answer 10

instrument must be uncorrelated with any unobserved variables that also affect Y (in the error term)

cov (Z, error) = 0

Question 11

instrument exclusion restriction

Answer 11

instrument must not have a direct effect on Y itself except through its relationship with X

conditional on X, Z has no effect on Y

again cov (Z, error) = 0

Question 12

F-statistic

Answer 12

used to test a joint hypothesis (involving more than one restriction/equation)

exploits the fact that t-statistics of individual coefficients are normally distributed

rule of thumb in 2SLS is that the F-stat is above 10

if you have one IV for an endogenous regressor, F-stat is just the square of the t-stat from the slope coefficient of the first stage regression

Question 13

over-identification test

Answer 13

assessing exogeneity/exclusion restriction

need one instrument for each endogenous regressor
- with less, equation is under-identified
- with more, equation is over-identified

testing whether two different IVs lead to a very similar 2nd stage IV point estimate or not (should if exogeneity/exclusion restriction holds)

Question 14

IV estimation and the local average treatment effect

Answer 14

instrument doesn’t allow you to estimate the ATE but the local ATE
- causal effect of X on Y where the units affected by instruments with relationship on Xs are the ones who comply with the IV effect

Question 15

3 cases when the IV point estimate is the ATE

Answer 15

when the treatment effect of X on Y is constant
- treatment effect has no variation

when the effect of the instrument on X in the first-stage is constant
- no heterogeneity but still recover what you want to recover

when there is no covariance between heterogeneity in the first and second stages

Question 16

less than full compliance and ITT

Answer 16

can use assignment status as an instrumental variable for treatment to estimate the causal effect

beta = point estimate of reduced form / point estimate of first-stage = difference in mean outcomes between D=1 and D=0 / difference in fraction of treated between D=1 and D=0

here, estimating the ATT
- LATE in potential outcomes is the ATT

ITT is comparing the outcomes in assigned treatment and control groups

Question 17

ATE, ATT, ITT

Answer 17

ATE: what is the expected effect of intervention on a random sample of the population?

ATT: what is the expected effect of intervention on individuals who choose to take up their treatment assignment?

ITT: what is the expected effect of intervention on the assigned treatment group relative to the assigned control group?