Category Mostly Harmless Econometrics: An Empiricist’s Companion

і

^XlYl = (X ‘X )-1X ‘y,

i where X is the NXK matrix with rows Xi and y is the N x 1 vector of Yi’s. We saw in Section 3.1.3 that /3 has an asymptotically Normal distribution. We can write:

VN(3 – /) – n(0, n)

where П is the asymptotic covariance matrix. Repeating (3.1.7), the formula for П in this case is

Пг = E [XiXi]-1^ [XiXief] E [XiXi]-1, (8.1.1)

where ei = Yi—Xi/. When residuals are homoskedastic, П simplifies to fic = a2E[XiXi]~1 where a2 = E[e2].

We are concerned here with the bias of robust standard errors in independent samples (i. e., no clustering or serial correlation). To simplify the derivation of bias, we assume that the regressor vector can be treated as fixed in repeated samples, as it would be if we sampled stratifying on Xi...

A vast literature in social science is concerned with peer effects. Loosely speaking, this means the causal effect of group characteristics on individual outcomes. Sometimes regression is used in an attempt to uncover these effects. In practice, the use of regression models to estimate peer effects is fraught with peril. Although this is not really an IV issue per se, the language and algebra of 2SLS helps us understand why peer effects are hard to identify.

Broadly speaking, there are two types of peer effects. The first concerns the effect of group characteristics such as the average schooling in a state or city on individually-measured outcome variable. This peer effect links the average of one variable to individual outcomes as described by another variable...

Many empirical studies involve variables that take on only a limited number of values. An example is the Angrist and Evans (1998) investigation of the effect of childbearing on female labor supply, discussed in

Section 3.4.2 in this chapter and in the chapter on instrumental variables, below. This study is concerned with the causal effects of childbearing on parents’ work and earnings. Because childbearing is likely to be correlated with potential earnings, the study reports instrumental variables estimates based on sibling – sex composition and multiple births, as well as OLS estimates. Almost every outcome in this study is either binary (like employment status) or non-negative (like hours worked, weeks worked, and earnings)...

Sharp RD is used when treatment status is a deterministic and discontinuous function of a covariate, xj. Suppose, for example, that

1 if Xj > xo

. (6.1.

0 if Xj < xo

where xo is a known threshold or cutoff. This assignment mechanism is a deterministic function of Xj because once we know xj we know Dj. It’s a discontinuous function because no matter how close xj gets to xo, treatment is unchanged until xj = xo.

This may seem a little abstract, so here is an example. American high school students are awarded National Merit Scholarship Awards on the basis of PSAT scores, a test taken by most college-bound high school juniors, especially those who will later take the SAT...

The discussion of IV up to this point postulates a constant causal effect. In the case of a dummy variable like veteran status, this means Y^—Yoi = p for all i, while with a multi-valued treatment like schooling, this means Ysi — YS-1,i = p for all s and all i. Both are highly stylized views of the world, especially the multi-valued case which imposes linearity as well as homogeneity. To focus on one thing at a time in a heterogeneous-effects model, we start with a zero-one causal variable. In this context, we’d like to allow for treatment-effect heterogeneity, in other words, a distribution of causal effects across individuals.

Why is treatment-effect heterogeneity important? The answer lies in the distinction between the two types of validity that characterize a research design...