非线性世界中的线性回归

AI总结：

本文研究了非线性条件下线性回归系数的解释方法，指出在非线性关系下系数可能产生偏差，并与测量误差和遗漏变量偏差进行类比。

摘要：

多元线性回归系数的解释依赖于条件期望函数在变量上是线性的假设。然而，在许多情况下，数据生成过程是非线性的。本文探讨了在非线性情况下如何解释回归系数。我们证明，如果感兴趣的变量与其他协变量之间的关系是线性的，那么该变量的系数代表了结果条件期望函数对感兴趣变量的导数的加权平均。如果这些关系是非线性的，回归系数会相对于这个加权平均产生偏差。我们证明这种偏差是可解释的，类似于标准线性模型下的测量误差偏差和遗漏变量偏差。

英文摘要：

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.

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