🤖 AI Summary
Standard interpretations of OLS coefficients in multiple linear regression presume linearity of the conditional expectation function (CEF), yet real-world data-generating processes are often nonlinear. Method: We show that when the CEF is nonlinear, OLS coefficients represent weighted averages of its partial derivatives, with bias arising systematically from the nonlinear structure of covariates. Leveraging Taylor expansion of the CEF, weighted average theory, and linear projection, we derive a closed-form expression for this bias—termed the “weighted derivative bias”—and prove its decomposition into functions of covariate nonlinearity. Contribution/Results: We unify this bias within an interpretable framework analogous to classical measurement error and omitted-variable bias, establishing a new theoretical foundation for coefficient interpretation under nonlinearity. Simulation and empirical analyses consistently validate the predicted direction and magnitude of the bias.
📝 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.