Linear Regression in a Nonlinear World

TL;DR

This paper explores how linear regression coefficients can be interpreted when the true data generating process is nonlinear, revealing conditions for unbiasedness and the nature of bias when nonlinearity is present.

Contribution

It provides a theoretical analysis of the interpretation of linear regression coefficients under nonlinear data relationships, highlighting when they represent derivatives and when they are biased.

Findings

Regression coefficients represent weighted averages of derivatives under linear relationships.

Nonlinear relationships cause bias in regression coefficients, which can be interpreted similarly to measurement error bias.

Bias magnitude depends on the nonlinearity of the underlying data generating process.

Abstract

The interpretation of coefficients from multivariate linear regression relies on the assumption that the conditional expectation function is linear in the variables. However, in many cases the underlying data generating process is nonlinear. This paper examines how to interpret regression coefficients under nonlinearity. We show that if the relationships between the variable of interest and other covariates are linear, then the coefficient on the variable of interest represents a weighted average of the derivatives of the outcome conditional expectation function with respect to the variable of interest. If these relationships are nonlinear, the regression coefficient becomes biased relative to this weighted average. We show that this bias is interpretable, analogous to the biases from measurement error and omitted variable bias under the standard linear model.