Category Mostly Harmless Econometrics: An Empiricist’s Companion

Bad Control

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
the selection bias discussed in Chapter (2) and Section (3.2)...

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Individual Fixed Effects

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...

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Two-Stage Least Squares

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:

Подпись: (4.1.7)Yj _ a’Xj + p[Xj^10 + ^11zj + C1j] + Pj

_ Xj[a + p^10] + p^11zj + [p?1j + pj]

image149

_ 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...

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The Quantile Regression Approximation Property*

The CQF of log wages given schooling is unlikely to be exactly linear, so the assumptions of the original quantile regression model fail to hold in this example. Luckily, quantile regression can also be understood as giving a MMSE linear approximation to the CQF, though in this case the MMSE problem is a little more complicated and harder to derive than for the regression-CEF theorem. For any quantile index т 2 (0, 1), define the quantile regression specification error as:

Дт(Xj, PT) — N’iPT – QT(Yi|Xi).

The population quantile regression vector can be shown to minimize an expected weighted average of the squared specification error, Д^ (Xj, P), as shown in the following theorem from Angrist, Chernozhukov, and Fernandez-Val (2006):

Theorem 7.1...

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Generalizing LATE

The LATE theorem applies to a stripped-down causal model where a single dummy instrument is used to estimate the impact of a dummy treatment with no covariates. We can generalize this in three important ways: multiple instruments (e. g., a set of quarter-of-birth dummies), models with covariates (e. g., controls for year of birth), and models with variable and continuous treatment intensity (e. g., years of schooling). In all three cases, the IV estimand is a weighted average of causal effects for instrument-specific compliers. The econometric tool remains 2SLS and the interpretation remains fundamentally similar to the basic LATE result, with a few bells and whistles...

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Heterogeneity and Nonlinearity

As we saw in the previous section, a linear causal model in combination with the CIA leads to a linear CEF with a causal interpretation. Assuming the CEF is linear, the population regression is it. In practice, however, the assumption of a linear CEF is not really necessary for a causal interpretation of regression. For one thing, as discussed in Section 3.1.2, we can think of the regression of Y; on X; and S; as providing the best linear approximation to the underlying CEF, regardless of its shape. Therefore, if the CEF is causal, the fact that regression approximates it gives regression coefficients a causal flavor. This claim is a little vague, however, and the nature of the link between regression and the CEF is worth exploring further...

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