Quantile regression allows us to look at features of the conditional distribution of Yi when part of the distribution is hidden. Suppose you have have data of the form
Yi;obs — Yi * l[Yi < c]; (7T.5)
where Yi;0bs is what you get to see and Yі is the variable you would like to see. The variable Yi;0bs is censored – information about Yi in Yi;0bs is limited for confidentiality reasons or because it was too difficult or time-consuming to collect more information. In the CPS, for example, high earnings are topcoded to protect respondent confidentiality. This means data above the topcode are recoded to have the topcode value... Read More
We’ve seen that, except in special cases, each instrumental variable identifies a unique causal parameter, one specific to the subpopulation of compliers for that instrument. Different valid instruments for the same causal relation therefore estimate different things, at least in principle (an important exception being
instruments that allow for perfect compliance on one side or the other). Although different IV estimates are "weighted-up" by 2SLS to produce a single average causal effect, over-identification testing of the sort discussed in Section 4.2.2, where multiple instruments are validated according to whether or not they estimate the same thing, is out the window in a fully heterogeneous world.
Differences in compliant sub-populations might explain variability in treatment effects... Read More
We’ve made the point that control for covariates can make the CIA more plausible. But more control is not always better. Some variables are bad controls and should not be included in a regression model even when their inclusion might be expected to change the short regression coefficients. Bad controls are variables that are themselves outcome variables in the notional experiment at hand. That is, bad controls might just as well be dependent variables too. Good controls are variables that we can think of as having been fixed at the time the regressor of interest was determined.
The essence of the bad control problem is a version of selection bias, albeit somewhat more subtle than Read More
the selection bias discussed in Chapter (2) and Section (3.2)...
One of the oldest questions in Labor Economics is the connection between union membership and wages. Do workers whose wages are set by collective bargaining earn more because of this, or would they earn more anyway? (Perhaps because they are more experienced or skilled). To set this question up, let Yit equal the (log) earnings of worker i at time t and let Dit denote his union status. The observed Yit is either Yoit or Yiit, depending on union status. Suppose further that
E(y0 it I Dit) — E(Y0itAi;
i. e. union status is as good as randomly assigned conditional on unobserved worker ability, Ai, and other observed covariates Xu, like age and schooling.
The key to fixed-effects estimation is the assumption that the unobserved Ai appears without a time subscript in a linear model for E(Yoit... Read More
The reduced-form equation, (4.1.4b), can be derived by substituting the first stage equation, (4.1.4a), into the causal relation of interest, (4.1.6), which is also called a “structural equation” in simultaneous equations language. We then have:
Yj _ a’Xj + p[Xj^10 + ^11zj + C1j] + Pj
_ Xj[a + p^10] + p^11zj + [p?1j + pj]
_ X W20 + ^21zj + ^2i,
where ^20 = a + p’Kio, ^21 = її, and £2i = pCii + Vi in equation (4.1.4b). Equation (4.1.7) again shows why p = 021. Note also that a slight re-arrangement of (4.1.7) gives
Y i = a’Ni + p[Xi^io + ^11 Zi] + C2i; (4.1.8)
where [Xi^1o + ^11Zi] is the population fitted value from the first-stage regression of Si on Xi and Zi... Read More