Unit 6 The Simple Regression Model

Throughout this book we have asked whether differences between groups are larger than we would expect from chance alone. Regression asks a broader question: how is one variable related to another?

That shift matters because economic theory usually makes qualitative predictions — higher prices reduce demand, more education increases earnings — and qualitative predictions are hard to act on. Regression lets us estimate how much one variable moves with another, express it in units someone can use, and attach a measure of uncertainty to the answer.

Almost every empirical question in economics has this shape. How do wages change with education? How does consumption change with income? How does crop yield change with rainfall? How does a student’s GPA change with attendance?

Answering such questions turns out to be surprisingly difficult, and three problems arise immediately.

Other things matter. Attendance is not the only determinant of GPA, and education is not the only determinant of wages. The relationship is never exact, so we need a way of describing a pattern that holds on average while individuals depart from it.

We do not know the functional form. The relationship between two variables might be a straight line, or a curve, or something that rises and then flattens. Economic theory rarely specifies which.

An observed relationship need not be causal. Even if attendance and grades move together, attendance may not be responsible for the difference.

The rest of this unit develops a model for studying relationships between variables. Along the way it answers four questions.

  1. What relationship are we trying to describe?
  2. Why do individuals not lie on that relationship?
  3. Under what conditions can we interpret it causally?
  4. How do we estimate it from a sample?

Each section that follows answers one of them.

We build all of it from ideas we already have — variation, sampling and statistical inference. Nothing genuinely new is required; what changes is the object of study, from a mean or a difference of means to a relationship.

The result is the central tool of econometrics — the field concerned with measuring economic relationships from data — though the tool itself is used across the social sciences and beyond.