Category Mostly Harmless Econometrics: An Empiricist’s Companion

Covariates in the Heterogeneous-effects Model

You might be wondering where the covariates have gone. After all, covariates played a starring role in our earlier discussion of regression and matching. Yet the LATE theorem does not involve covariates. This stems from the fact that when we see instrumental variables as a type of (natural or man-made) randomized trial, covariates take a back seat. If, after all, the instrument is randomly assigned, it is likely to be independent of covariates. Not all instruments have this property, however...

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Control for Covariates Using the Propensity Score

The most important result in regression theory is the omitted variables bias formula: coefficients on included variables are unaffected by the omission of variables when the variables omitted are uncorrelated with the variables included. The propensity score theorem, due to Rosenbaum and Rubin (1983), extends this idea to estimation strategies that rely on matching instead of regression, where the causal variable of interest is a treatment dummy.[31]

The propensity score theorem states that if potential outcomes are independent of treatment status conditional on a multivariate covariate vector, Xi, then potential outcomes are independent of treatment status conditional on a scalar function of covariates, the propensity score, defined as p(Xi) = E[Di|Xi]. Formally, we have

Theorem 3.3...

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Fixed Effects versus Lagged Dependent Variables

Fixed effects and differences-in-differences estimators are based on the presumption of time-invariant (or group-invariant) omitted variables. Suppose, for example, we are interested in the effects of participation in a subsidized training program, as in the Dehejia and Wahba (1999) and Lalonde (1986) studies discussed in section (3.3.3). The key identifying assumption motivating fixed effects estimation in this case is

E(y0it&i; Xit, Dit) — E(Y0it&i; Xit), (5.3.1)

where a. i is an unobserved personal characteristic that determines, along with covariates, Xit, whether individual i gets training...

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Asymptotic 2SLS Inference

4.2.1 The Limiting Distribution of the 2SLS Coefficient Vector

We can derive the limiting distribution of the 2SLS coefficient vector using an argument similar to that used

I

Подпись: denote the vector of regressors in the 2SLS secondin Section 3.1.3 for OLS. In this case, let Vi = Xі s,■

Подпись: where Г = Подпись: a' p image160 Подпись: (4.2.1)

stage, equation (4.1.9). The 2SLS estimator can then be written

where the second equality comes from the fact that the first-stage residuals, (Si — S/), are orthogonal to Vi in the sample. The limiting distribution of the 2SLS coefficient vector is therefore the limiting distribution of Ei V/V/] 1 ^0i Vp. This quantity is a little harder to work with than the corresponding OLS quantity, because the regressors in this case involve estimated fitted values, S/...

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The QTE Estimator

Подпись: (aT,PT) = argminE{pT(YІ - aDi - Xib)|Dii > DoJ a.b Подпись: argminE{KipT(YІ - aDi - Х[Ь)}, a.b Подпись: (7.2.3)

The QTE estimator is motivated by the observation that, since the parameters of interest are quantile regression coefficients for compliers, they can (theoretically) be estimated consistently by running quantile regressions in the population of compliers. As always, however, the compliers population is not identifiable; we cannot list the compliers in a given data set. Nevertheless, as in Section 4.5.2, the relevant econometric minimand can be constructed using the Abadie Kappa theorem. Specifically,

image303 Подпись: Z) 1|Xi) Подпись: (1 - Di)Zi P (Zi = 1|Xi);

where

as before. The QTE estimator is the sample analog of (7.2.3).

There are a number of practical issues that arise when implementing QTE...

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Average Causal Response with Variable Treatment Intensity*

An important difference between the causal effects of a dummy variable and a variable that takes on the values {0, 1, 2, . . .} is that in the first case, there is only one causal effect for any one person, while in the latter there are many: the effect of going from 0 to 1, the effect of going from 1 to 2, and so on. The potential-outcomes notation we used for schooling recognizes this. Here it is again: let

Ysl = fi(s),

denote the potential (or latent) earnings that person i would receive after obtaining s years of education. Note that the function fi(s) has an “i” subscript on it while s does not. The function f(s) tells us what i would earn for any value of schooling, s, and not just for the realized value, Si...

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