Econometrics Exam 2

quasi-experiment / natural experiment

has a source of randomization that is “as if” randomly assigned, but it was not the result of an explicit randomized treatment and control design

how do instrumental variables correct omitted variable bias

it splits the independent variable into the parts correlated and uncorrelated with the error

exogenous regressor

variable that is uncorrelated with the error term

endogenous regressor

variable that is correlated with the error term

which variable is exogenous in IV regression

instrumental variable

which variable is endogenous in IV regression

independent variable

valid instrumental variable

1. relevance (correlated with independent variable)

   1. exogeneity (uncorrelated with error)

beta in IV regression

Cov(Y, Z)/Cov(X, Z)

simultaneous causality

both the independent and dependent variable are causal to each other

two stage least squares

1. isolate the part of X that is uncorrelated with the error by regressing X on Z and get predicted values Xhat

2. replace X by Xhat to estimate the main equation by regressing Y on Xhat

purpose of two stage least squares

make X uncorrelated with the error, so the first least squares assumption holds

2SLS with controls

1. regress X on all exogenous regressors, so Z and controls W, to get predicted Xhat

   1. regress Y on Xhat and W

exactly identified beta in IV regression

number of endogenous variables = number of instruments

overidentified beta in IV regression

number of endogenous variables < number of instruments. you can test whether instruments are valid

underidentified beta in IV regression

number of endogenous variables > number of instruments. IV regression cannot be identified

how to check instrument relevance

check if at least one of the instrument coefficients in first stage regression are nonzero

checking instrument relevance in practice

if first stage F-statistic is less than 10, set of instruments are weak

Wald Estimator

(expected Y of treatment group - expected Y of control group)/(expected X of treatment group - expected X of control group)

compliers

subpopulation that take the treatment when the instrumental variable says they should, and don’t when when it doesn’t

always-takers

take the treatment no matter what

never takers

never take the treatment no matter what

defiers

does the opposite of what their instrumental variables says they should

monotonicity

no defiers, so instrument pushes affected individuals in one direction only

Local Average Treatment Effect (LATE)

for any randomly assigned instrument with a nonzero first stage satisfying both monotonicity and an exclusion restriction, the ratio of reduced form to first stage is the LATE, or average causal effect of treatment on compliers

why is LATE different from treatment on the treated (TOT)?

TOT may include some always-takers in its calculations

how to check instrument exogeneity when coefficients are exactly identified

not possible

how to check instrument exogeneity when beta is overidentified

J-test

J-test

1. estimate equation of interest using 2SLS and all m instruments, computed predicted values using actual X

2. compute residuals uhat

3. regress uhat against all instruments and controls

4. compute homoskedasticity-only F-statistic testing the hypothesis that the coefficients on instruments are all zero

5. J = mF

Intention to Treat Effect (ITT)

causal effect of the offer of treatment, building in that many of those offered declined treatment and is smaller relative to average causal effect on those who were actually treated

how does IV fix ITT smallness

divide ITT by the difference in compliance rates between treatment and control groups as originally assigned

IV regression reduced form

Yi = p0 + p1Zi + p2Wi + vi

IV regression first stage

Xi = pi0 pi1Zi + pi2Wi + ni

IV estimator

p1/pi1

panel data

observation on multiple entities over time

balanced panel

no missing observation in panel data

unbalanced panel

has some missing data for at least one time period for at least one entity

state fixed effect

varies from one state to the next but does not vary over time

state fixed effect variable (alpha)

B0 + B2S2 + B3S3 +… + BnSn

time fixed effect

varies across time but not states

difference-in-difference

comparing the change in a group that received the treatment to the change in a group that did not receive the treatment

counterfactual

what the trend for the treatment group would’ve been if the treatment hadn’t been received

treatment effect

difference between actual treatment group results and counterfactual

parallel trends

without treatment, the treatment and control groups would have followed the same trend

causal effect of interest in DiD setup

average TOT in period 2; E[Yi,2(1) - Yi,2(0)|Di=1]

no anticipation assumption

treatment in period 2 doesn’t affect outcome in period 1

ATT in DiD

change in pop mean for the treated - change in pop mean for control

how to relax parallel trends assumption

control for state-specific trends

staggered DiD (event study)

includes m lags and q leads

useful because we can test whether parallel trends holds prior to treatment and analyze how ATT changes over time

staggered DiD in action

include variable for “time until treatment”, so the exact year of receiving treatment does not need to be the same (control group will have 0 for all values of this variable)

clustered standard errors

extend the OLS variance formula to allow (Yit, Xit) to be correlated across observations in the same cluster

serial autocorrelation

Yi1 will likely be correlated with Yi2

clustering at individual level

Yi1 and Yi2 are dependent, but assume (Yi1, Yi2) are independent of (Yj1, Yj2)

Regression Discontinuity

exploits that some thresholds for treatment are arbitrary and observations just above and below a threshold aren’t inherantly very different, so any change in the outcome of interest is likely due to the assigned treatment

sharp regression discontinuity design

everyone above the threshold receives the treatment of interest, everyone below does not

running variable

variable used for measuring whether an individual receives the treatment

why is treatment status a discontinuous function of the running variable

how matter how it gets to the closer, treatment is unchanged until the cutoff is reached

McCrary test

test to see if there are a similar number of units on both sides of the cutoff (i.e to see if individuals are being bumped over the threshold to get the treatment)

linear specification

formula that accounts for different slopes on either side of the threshold

quadratic specification

same as linear specification, but for nonlinear slopes

nonparametric RD

estimates the model in a narrow window around the cutoff

fuzzy RD

some individuals above the cutoff do not get the treatment, so we estimate the effect of being above the cutoff on the outcome and divide this by the effect on the treatment

when does fuzzy RD give a LATE

when there’s continuity at the cutoff, relevance, exclusion, and monotonicity

fuzzy RD as IV

crossing the threshold has no direct effect on Y, only affect Y by influencing the probability of treatment. so, Z is an exogenous instrument for D

limited dependent variable

instead of being continuous, Y is binary or categorical

linear probability model

predicted beta value is the change in probability that Y=1, with a binary Y, for each additional unit of X

pros of linear probability model

simple to estimate and interpret

inference is the same as for multiple regression

cons of linear probability model

change in probability is the same for all values of X

predicted probabilities could be less than 0 or more than 1

probit model

models binary outcomes, generates Z score

Z score

generated by probit model, and corresponds to probability that Y=1

beta interpretation in probit model

if it’s positive, increasing X increases the probability that Y=1, and vice versa

measures of fit for probit model

fraction correctly predicted

psuedo R² (improvement in value log likelihood relative to having no Xs)

maximum likelihood estimation

beta that estimates Y=1

logit regression

probability of Y=1 given X as cumulative standard logistic distribution function; F(B0 + B1Xi) = 1/(1+e^-(B0 + B1Xi))