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C h a p t e r Linear Regression

4 with One Regressor

Astate implements tough new penalties on drunk drivers: What is the effect
on highway fatalities? A school district cuts the size of its elementary school
classes: What is the effect on its students’ standardized test scores? You successfully
complete one more year of college classes: What is the effect on your future
earnings?

All three of these questions are about the unknown effect of changing one
variable, X (X being penalties for drunk driving, class size, or years of schooling), on
another variable, Y (Y being highway deaths, student test scores, or earnings).

This chapter introduces the linear regression model relating one variable, X, to
another, Y. This model postulates a linear relationship between X and Y; the slope of
the line relating X and Y is the effect of a one-unit change in X on Y. Just as the mean
of Y is an unknown characteristic of the population distribution of Y, the slope of the
line relating X and Y is an unknown characteristic of the population joint distribution
of X and Y. The econometric problem is to estimate this slope—that is, to estimate the
effect on Y of a unit change in X—using a sample of data on these two variables.

This chapter describes methods for estimating this slope using a random sample
of data on X and Y. For instance, using data on class sizes and test scores from
different school districts, we show how to estimate the expected effect on test scores
of reducing class sizes by, say, one student per class. The slope and the intercept of
the line relating X and Y can be estimated by a method called ordinary least squares
(OLS).

4.1 The Linear Regression Model

The superintendent of an elementary school district must decide whether to hire
additional teachers and she wants your advice. If she hires the teachers, she will
reduce the number of students per teacher (the student–teacher ratio) by two.
She faces a trade-off. Parents want smaller classes so that their children can
receive more individualized attention. But hiring more teachers means spending
more money, which is not to the liking of those paying the bill! So she asks you:
If she cuts class sizes, what will the effect be on student performance?

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