Category Mostly Harmless Econometrics: An Empiricist’s Companion

Fuzzy RD is IV

Fuzzy RD exploits discontinuities in the probability or expected value of treatment conditional on a covariate. The result is a research design where the discontinuity becomes an instrumental variable for treatment status instead of deterministically switching treatment on or off. To see how this works, let D; denote the treatment as before, though here D; is no longer deterministically related to the threshold-crossing rule, x; > xo■ Rather, there is a jump in the probability of treatment at xo, so that

r, I go(xi) if x; > xo

P[D; = 1|x;J = > , where gi (xo ) = go (xo)■

I gi(x;) if x; < xo

The functions go(x;) and g1(x;) can be anything as long as they differ (and the more the better) at xo. We’ll assume g1(xo) > go(xo), so x; > xo makes treatment more likely...

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Local Average Treatment Effects

In an IV framework, the engine that drives causal inference is the instrument, zj, but the variable of interest is still Dj. This feature of the IV setup leads us to adopt a generalized potential-outcomes concept, indexed against both instruments and treatment status. Let Yj(d, z) denote the potential outcome of individual i were this person to have treatment status Dj = d and instrument value Zj = z. This tells us, for example, what the earnings of i would be given alternative combinations of veteran status and draft-eligibility status. The causal effect of veteran status given i’s realized draft-eligibility status is Yj(1,Zj)—Yj(0,Zj), while the causal effect of draft-eligibility status given i’s veteran status is Yj(Dj, 1)—Yj(Dj, 0).

We can think of instrumental variables as ini...

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Serial Correlation in Panels and Difference-in-Difference Models

Serial correlation – the tendency for one observation to be correlated with those that have gone before – used to be Somebody Else’s Problem, specifically, the unfortunate souls who make their living out of time series data (macroeconomists, for example). Applied microeconometricians have therefore long ignored it.[126] But our data often have a time dimension too, especially in differences-in-differences models. This fact combined with clustering can have a major impact on statistical inference.

Suppose, as in Section 5.2, that we are interested in the effects of a state minimum wage. In this context, the regression version of differences-in-differences includes additive state and time effects. We therefore we get an equation like (5.2.3), repeated below:

y ist = 7s + Dst + "ist; (8.2...

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The Bias of 2SLSF

It is a fortunate fact that the OLS estimator is not only consistent, it is also unbiased. This means that in a sample of any size, the estimated OLS coefficient vector has a distribution that is centered on the population coefficient vector.[74] The 2SLS estimator, in contrast, is consistent, but biased. This means that the 2SLS estimator only promises to be close the causal effect of interest in large samples. In small samples, the 2SLS estimator can differ systematically from the population estimand.

For many years, applied researchers have lived with the knowledge that 2SLS is biased without losing too much sleep. Neither of us heard much about the bias of 2SLS in our graduate econometrics classes. A series of papers in the early 1990s changed this, however...

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Appendix: Derivation of the average derivative formula

Begin with the regression of Yi on Si :

Cov(Yj, Si) _ E[h(Si)(Si – E[Si])] V(Si) _ E[Si(Si – E[Si])] ‘

Let К-ж = lim h (t). By the fundamental theorem of calculus, we have:

t—» — OO

s

h (si) = к_ж + / h’ (t) dt.

Substituting for h(Si), the numerator becomes

/

+ 1 ps

/ h’ (t) (s – E[Si)g(s)dtd.

– OO J — OO

where g(s) is the density of si at s. Reversing the order of integration, we have

/

+i p+i

h’ (tW (s – E[Si])g(s)dsdt.

-OO J t

The inner integral is easily seen to be equal to g, t = fE[si|si > t] — E[si|si < t]}{P(si > t)[1 — P(si > t)},

which is clearly non-negative. Setting si =Yi, the denominator can similarly be shown to be the integral of these weights...

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Quantile Regression

Here’s a prayer for you. Got a pencil? . . . ‘Protect me from knowing what I don’t need to know. Protect me from even knowing that there are things to know that I don’t know. Protect me from knowing that I decided not to know about the things I decided not to know about. Amen.’ There’s another prayer that goes with it. ‘Lord, lord, lord. Protect me from the consequences of the above prayer.’

Douglas Adams, Mostly Harmless (1995)

Rightly or wrongly, 95 percent of applied econometrics is concerned with averages. If, for example, a training program raises average earnings enough to offset the costs, we are happy. The focus on averages is partly because obtaining a good estimate of the average causal effect is hard enough...

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