Appendix: Mastering Inference

YOUNG CAINE: I am puzzled.

master po: That is the beginning of wisdom.

Kung Fu, Season 2, Episode 25

This is the first of a number of appendices that fill in key econometric and statistical details. You can spend your life studying statistical inference; many masters do. Here we offer a brief sketch of essential ideas and basic statistical tools, enough to understand tables like those in this chapter.

The HIE is based on a sample of participants drawn (more or less) at random from the population eligible for the experiment. Drawing another sample from the same population, we’d get somewhat different results, but the general picture should be similar if the sample is large enough for the LLN to kick in...

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One Mississippi, Two Mississippi

Policymakers facing a bank run can open the flow of credit or turn off the tap. Friedman and Schwartz argued that the Federal Reserve (America’s central bank) foolishly restricted credit as the Great Depression unfolded. Easy money might have allowed banks to meet increasingly urgent withdrawal demands, staving off depositor panic. By lending to troubled banks freely, the central bank has the power to stem a liquidity crisis and obviate the need for a bailout in the first place.

But who’s to say when a crisis is merely a crisis of confidence? Some crises are real. Bank balance sheets may be so sickened by bad debts that no amount of temporary liquidity support will cure ’em. After all, banks don’t lose their liquidity by random assignment...

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Conditional Expectation Functions

Chapter 1 introduces the notion of mathematical expectation, called “expectation” for short. We write E[Yj] for the expectation of a variable, Yt. We’re also concerned with conditional expectations, that is, the expectation of a variable in groups (also called “cells”) defined by a second variable. Sometimes this second variable is a dummy, taking on only two values, but it need not be. Often, as in this chapter, we’re interested in conditional expectations in groups defined by the values of variables that aren’t dummies, for example, the expected earnings for people who have completed 16 years of schooling. This sort of conditional expectation can be written as


and it’s read as “The conditional expectation of Yf given that X{ equals the particular value



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Econometricians Are Known by Their… Instruments

It’s the Law

Economists think people make important choices such as those related to schooling by comparing anticipated costs with expected benefits. The cost of staying in secondary school is determined partly by compulsory schooling laws, which punish those who leave school too soon. Since you avoid punishment by staying in school, compulsory schooling laws make extra schooling seem cheaper relative to the alternative, dropping out. This generates a causal chain reaction leading from compulsory schooling laws to schooling choices to earnings that might reveal the economic returns to schooling. The ’metrics methods behind this idea are those of Chapters 3 and 5: instrumental variables and dif f erences-indif f erences.

As always, IV begins with the first stage...

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The Population Bomb

Population control or race to oblivion?

Paul Ehrlich, 1968

World population increased from 3 billion to 6 billion between 1960 and 1999, a doubling time of 39 years, and about half as long as the time it took to go from 1.5 billion to 3 billion. Only a dozen years passed before the seventh billion came along. But contemporary demographers agree that population growth has slowed dramatically. Projections using current fertility rates point to a doubling time of 100 years or more, perhaps even forever. One widely quoted estimate has population peaking at 9 billion in 2070.— Contemporary hand-wringing about sustainable growth notwithstanding, the population bomb has been defused—what a relief!

The question of how population growth affects living standards has both a macro side and a micro...

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The t-Statistic and the Central Limit Theorem

Having laid out a simple scheme to measure variability using standard errors, it remains to

interpret this measure. The simplest interpretation uses a t-statistic. Suppose the data at hand come from a distribution for which we believe the population mean, E[Y] takes on a particular value, ц (read this Greek letter as “mu”). This value constitutes a working hypothesis. A t-statistic for the sample mean under the working hypothesis that E[Yt] = ц is constructed as


The working hypothesis is a reference point that is often called the null hypothesis. When the null hypothesis is ц = 0, the t-statistic is the ratio of the sample mean to its estimated standard error.

Many people think the science of statistical inference is boring, but in fact it’s nothing short of miraculous...

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