# Category Mostly Harmless Econometrics: An Empiricist’s Companion

## Appendix: Derivation of the simple Moulton factor

Write

and

where Lg is a column vector of ng ones and G is the number of groups. Note that

Let tg = 1 + (ng — 1 )p, so we get

^2X ng t g xg Xg •

With this in hand, we can write

V(b) = (X’X) 1 X’ФХ (X’X) 1

X ngxgx’g) X ngtgxgx’g X ngxgx’g

ggg

We want to compare this with the standard OLS covariance estimator

If the group sizes are equal, ng = n and tg = t = 1 + (n — 1)p, so that

V (b) = ^T ( X nxg x’gI X nxg xg ( X nxg x’g I

g / g V g /

nxgxg

g

which implies (8.2.4).

Table 8.1.1: Monte Carlo results for robust standard errors

 Empirical 5% Rejection Rates Mean Standard Normal t Deviation (1) (2) (3) (4) A. Lots of Heteroskedasticity І і -0.001 0.586 Standard Errors:

## The Omitted Variables Bias Formula

The omitted variables bias (OVB) formula describes the relationship between regression estimates in models with different sets of control variables. This important formula is often motivated by the notion that a longer regression, i. e., one with more controls such as equation (3.2.9), has a causal interpretation, while a shorter regression does not. The coefficients on the variables included in the shorter regression are therefore said to be "biased". In fact, the OVB formula is a mechanical link between coefficient vectors that applies to short and long regressions whether or not the longer regression is causal...

## Parallel Worlds: Fixed Effects, Differences-in-differences, and Panel Data

The first thing to realize about parallel universes… is that they are not parallel.

The key to causal inference in chapter 3 is control for observed confounding factors. If important confounders are unobserved, we might try to get at causal effects using IV as discussed in Chapter 4. Good instruments are hard to find, however, so we’d like to have other tools to deal with unobserved confounders. This chapter considers a variation on the control theme: strategies that use data with a time or cohort dimension to control for unobserved-but-fixed omitted variables. These strategies punt on comparisons in levels, while requiring the counterfactual trend behavior of treatment and control groups to be the same...

## IV and causality

To motivate the constant-effects setup as a framework for the causal link between schooling and wages, suppose, as before, that potential outcomes can be written

Y si = fi (s) ;

and that

fi (s) = ^0 + KlS + Vi, (4.1.1)

as in the introduction ...

## Censored Quantile Regression

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...