Linear Regression in a Nonlinear World

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.