Regression Discontinuity:
 Assumptions for RD to be valid:
o All variables, except treatment status, are smooth in running var about the
threshold
o Agents cannot “self-sort” at threshold
o (The “first stage” or relevance condition) Conditional
on a smooth function of midterm scores, scoring above
the median must have a strong correlation with
attendance after the midterm, where “strong”
specifically requires an F-stat larger than 10 (or a
corresponding t-stat larger that the square root of
10.
o Factors besides midterm scores that affect performance
on the final must be a smooth function of the midterm
score in a neighborhood of scoring above the median on
the midterm (in other words, there can be no other
reason scores “jump” at the threshold.)
o There must be no sorting around the median on the
midterm
 Fuzzy vs Sharp:
o An RD is “fuzzy” when being to the right of the threshold does not perfectly
predict the treatment status; instead, it is a “first stage” in an IV regression (e.g.,
birthday relative to school entry cutoff
o The opposite of a “sharp” RD, in which threshold side is perfectly correlated with
treatment status (e.g., legal to drink at age 21)
 What does an RD identify under heterogeneous treatment effects?
o Sharp RD: treatment effect in the neighborhood of the threshold (the effect of
legal drinking at age 21.) You could also interpret the reduced form of a “fuzzy”
RD this way (e.g., the effect of being admitted to charter schools at a particular
test score cutoff – the “intention to treat” effect.)
o Fuzzy RD: IV estimate, so it’s a local average treatment effect (see above). So
the effect in a neighborhood of the threshold, for compliers (e.g., people who do
not hold their kids back in the case of a school entry cutoff.)
 Estimation equation for RD (how is it normally formed?):
o Ma = alpha+rhoDa+ya+ea
 Slope should be interacted with threshold dummy, AND include quartic
terms
 Interaction term ALWAYS needed to differ above and below
 Stata implementation:
o Generate interacted term, running var relative to threshold
o Write out full regression
 Rememember RD only applies for the area about the threshold
 Valid RD does NOT need other controls, beyond smooth function of running var
o Are uncorr w/ threshold, but might be included to reduce standard errors
 If adding other controls, be sure to interact with the WHOLE equation
Difference-in-Differences:
IV Regression:
 Monotonicity – no defiers!
 LATE applies to compliers
 Stat significant:
o Is instrument valid? Check first stage
o Is IV statistically significant? Check reduced form
o Is IV significantly different from OLS? Check Hausman test
 Remember conditions:
o Strongly correlated w/ x (t>3, F>10)
o Unrelated with unobservered error term that directly affects y (placebo, balance)
 IVreg y (X = z) controls, cluster/robust
Other reminders:
 If fixed effects come in, error term becomes 𝜏𝑠 + 𝑢𝑖𝑠
 Placebo test: Correlated with error, uncorrelated with x var or instrument