Tuesday, April 14, 2015

Regression Coefficients & Units of Measurement

A linear regression equation is just that - an equation. This means that when any of the variables - dependent or explanatory - have units of measurement, we also have to keep track of the units of measurement for the estimated regression coefficients.

All too often this seems to be something that students of econometrics tend to overlook.

Consider the following regression model:

yi = β0 + β1X1i + β2x2i + β3x3i + εi    ;    i = 1, 2, ...., n                   (1)

where y and x2 are measured in dollars; x1 is measured in Kg; and x3 is a unitless index.

Because the term on the left side of (1) has units of dollars, every term on the right side of that equation must also be expressed in terms of dollars. These terms are β0, (β1x1i), (β2x2i), (β3x3i), and εi.

In turn, this implies that β0 and β3 have units which are dollars; the units of β1 are (\$ / Kg); and β2 is unitless. In addition, the error term, ε, has units that are dollars, and so does its standard deviation, σ.

What are some of the implications of this?