Regresiones

The Inaugural Coase Lecture

An Introduction to Regression Analysis
Alan O. Sykes*
Regression analysis is a statistical tool for the investigation of relationships between variables. Usually, the investigator seeks to
ascertain the causal effect of one variable upon another—the effect of
a price increase upon demand, for example, or the effect of changes
in the money supply upon the inflationrate. To explore such issues,
the investigator assembles data on the underlying variables of
interest and employs regression to estimate the quantitative effect of
the causal variables upon the variable that they infl uence. The
investigator also typically assesses the “statistical significance” of the
estimated relationships, that is, the degree of con fi dence that the
true relationship isclose to the estimated relationship.
Regression techniques have long been central to the fi eld of economic statistics (“econometrics”). Increasingly, they have become
important to lawyers and legal policy makers as well. Regression has
been o ff ered as evidence of liability under Title VII of the Civil
Rights Act of , as evidence of racial bias in death penalty litigation, as evidence ofdamages in contract actions, as evidence of
violations under the Voting Rights Act, and as evidence of damages
in antitrust litigation, among other things.
In this lecture, I will provide an overview of the most basic techniques of regression analysis—how they work, what they assume,
‥

Professor of Law, University of Chicago, The Law School. I thank Donna
Cote for helpful researchassistance.

See, e.g, Bazemore v. Friday,  U.S. ,  ().

See, e.g., McClesky v. Kemp,  U.S.  ().

See, e.g., Cotton Brothers Baking Co. v. Industrial Risk Insurers,  F.d
 ( th Cir. ).

See, e.g., Thornburgh v. Gingles,  U.S.  ().

See, e.g., Sprayrite Service Corp. v. Monsanto Co.,  F.  d  (th Cir.
 ).

Chicago Working Paper inLaw & Economics

and how they may go awry when key assumptions do not hold. To
make the discussion concrete, I will employ a series of illustrations
involving a hypothetical analysis of the factors that determine individual earnings in the labor market. The illustrations will have a
legal fl avor in the latter part of the lecture, where they will
incorporate the possibility that earnings areimpermissibly infl uenced
by gender in violation of the federal civil rights laws.  I wish to
emphasize that this lecture is not a comprehensive treatment of the
statistical issues that arise in Title VII litigation, and that the
discussion of gender discrimination is simply a vehicle for expositing
certain aspects of regression technique. Also, of necessity, there are
many importanttopics that I omit, including simultaneous equation
models and generalized least squares. The lecture is limited to the
assumptions, mechanics, and common diffi culties with singleequation, ordinary least squares regression.
. What is Regression?
For purposes of illustration, suppose that we wish to identify and
quantify the factors that determine earnings in the labor market. A
moment’s reflectionsuggests a myriad of factors that are associated
with variations in earnings across individuals—occupation, age, experience, educational attainment, motivation, and innate ability
come to mind, perhaps along with factors such as race and gender
that can be of particular concern to lawyers. For the time being, let
us restrict attention to a single factor—call it education. Regression
analysiswith a single explanatory variable is termed “simple regression.”


See  U.S.C. §e- (), as amended.
Readers with a particular interest in the use of regression analysis under Title
VII may wish to consult the following references: Campbell, “Regression Analysis
in Title VII Cases—Minimum Standards, Comparable Worth, and Other Issues
Where Law and Statistics Meet,”  Stan. L....