My questions are about the chapter of Regression Discontinuity
Designs. What criteria are used to determine the neighborhood size in
nonparametric RDD Fuzzy and Sharp?

Great question Vanderson – The bandwidth is indeed at the business end of
nonparametric RD, though until recently we simply would have had to say
“try a few.”

Happily,  a new paper by Imbens and Kalyanaraman provides a better answer
by deriving formulas for an MSE-minimizing choice.

Good luck with your project!

I'd like to estimate the effect of fuel price (which I assume is exogenous)
on distance driven. As a control, I would like to include the fuel
efficiency of the driver's car. Although efficiency is likely to be
endogenous, leaving it out of the specification runs the risk of
imparting omitted bias on my fuel price estimate. But since it is
*just* a control, I'm inclined to leave efficiency as is in the model
and not worry about whether it is endogenous. Wise move?
Any insights would be appreciated!

Before tackling the metrics, think about a likely motivation for the research question.  Suppose the government is considering a rise in the gas tax.  Policy-makers would like to know how this will affect driving habits and fuel consumption.  The government is unlikely to forbid people from buying a new more fuel efficient car in response to the tax, in fact they probably would like to encourage that.  So who needs to know what the causal effect of a price rise is conditional on being locked in to my current vehicle?  I think this observation neatly answers Colin’s  question.  Prices will go up, driving behavior will change for a number of reasons.  There is no scenario where only one response is all that’s allowed (driving in the same car). Then there is the econometric problem that conditioning on fuel efficiency will not actually answer the question of how driving behavior changes for those who don’t buy a more fuel efficient car.  That’s the bad control problem described in MHE – but that’s just metrics.

JA

George: its certainly not a crazy idea. In fact, Dehejia-Wahba (1999) tried
this (Table 5, estimates labeled quadratic in score).  But its not clear
what the theoretical justification is here; once you are using regression,
why do this two-step procedure instead of just sticking the covs you've put
in the score right into the reg (since you're implicitly assuming these are
the only source of OVB)?  Also, as we know from chpt 3, regression does not
estimate the pop ATE or the effect of treatment on the treated except under
constant effects or if the score is constant. Score fiends are often after
those parameters instead of the variance-weighted avg that regression produces.

I am estimating Y = b0+ b1*X1 +b2* X2 + b3*X1*X2 + X3

Many thanks in advance for your response and best regards,

thanks for your question Daniela.  Models with multiple endogenous variables are indeed hard to identify and the results can be hard to interpret.

So we don’t usually like to see them – for one thing it’s not clear why you’re tackling two causal questions at the same time; one is hard enough.
You may have noticed that the only model with more than one endogenous regressor in MHE is the peer effects regression (equation 4.6.6, based on Acemoglu and Angrist, 2000).  Here we have both individual and state-level schooling endogenous in a wage equation.

But we are really only interested in the peer effect in this case – the effect of state average schooling. Individual schooling is there because we realize that any instrument for average schooling must also be correlated with individual schooling.  We therefore try to fix this violation of the exclusion restriction by treating individual schooling as endogenous as well. This is the best reason for having a second endog variable that I can think of.  And the model may work – in the case of schooling we have enough instruments.  But not very often, I would think.

More generally, it doesn’t make sense to think of one endogenous variable as a “control” when looking at the effects of another, at least not a good one (in the sense in which we use the terms good and bad control in chapter 3).  So any time someone shows me a problem with more than one endogenous variable, my first question is always: why?

In sections 5.3-5.4, there is a great discussion of using
fixed effects vs. a lagged dependent variable with panel data. I am
having trouble reconciling some of this discussion with a section in a
recent paper by Imbens and Wooldridge (2008) titled “Recent
Developments in the Econometrics of Program Evaluation.” On page 68 of
their paper (as published by IZA in 2008) they suggest that it might
be better in some circumstances with two periods of data to use first
differencing and a lag of the dependent variable (assuming
unconfoundedness given lagged outcomes). I understand your discussion
of instrumenting for lagged variables if you have more than two
periods, but with two periods, how do you react to adding a lag (the
baseline value of the dependent variable) after first differencing
with only two periods of data? I have had difficulty finding support
for this approach elsewhere and given that you have given much thought
to this issue, I was wondering what your opinion might be.

The way I see it, once you add a lagged dependent variable to a differenced model, you are really doing lagged-dep-var control and not fixed effects.  Steve may disagree (he’s generally less dogmatic than me).  This is not always exactly true but it is a theorem for the simple example we use to contrast f.e. and lagged-dep-var control in Section 5.4

Here’s that again:

two periods
no covariates
the treatment, D_it, is zero for everybody in period 1 and switched on for some in period 2 (think of a training program that some people participate in between periods; period 1 is before, period 2 is after (similar to Ashenfelter and Card, 1985)

ignoring constants,  fixed effects estimation fits

(1) Y_it – Y_it-1 = aD_it + error

lagged dependent variable estimation fits

(2) Y_it = gY_it-1 + bD_it + error

As I understand it, the Imbens-Wooldridge proposal is to throw Y_it-1 into equation (1):

(3) Y_it – Y_it-1 = dY_it-1 + cD_it + error

But in this case, c is (algebraically) the same as b.  Why ? The coefficient c is

c= COV(Y_it – Y_it-1, D_it*)/V(D_it*)

where D_it* is the residual from a regression of D_it on Y_it-1.  But this residual is orthogonal to Y_it-1, hence

c= COV(Y_it – Y_it-1, D_it*)/V(D_it*) = COV(Y_it, D_it*)/V(D_it*) = b in equation (2)

So I say: “You wanna do fixed effects?  no lagged dependent variable, please (or at least be prepared to instrument it if you include one).  You wanna control for  lagged dependent variables?  Then, just do it!

– JDA

Thanks for your question Elias.

–JA