"残差项作为因变量"难题终于找到解决方法了

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上一日，因果推断研究小组所引荐的看完顶级期刊文章后, 整理了内生性处理小册子很受欢迎，因此咱们商科计量研究小组也希望能够在今后的日子里引荐一些前沿计量方法。

在分析了一个在实证会计和金融研究中常见的步骤：研究者使用OLS将因变量分解成它的预测值和残差余项，并将得到的残差用作第二阶段回归中的因变量。这个两步法步骤常常用于分析一些在会计研究中比较重要的变量的决定因素，如可自由支配的应计利润、盈余管理(社群有一些code可参考)、投资效率和可自由支配帐面税差额等。

但这篇发表在Journal of accounting research的"Incorrect Inferences When Using Residuals as Dependent Variables"文章表明这个过程所得到的系数和标准误都是有偏差的，因此可能会导致不正确的推断。 他们的模拟结果表明，系数和标准误差的偏差幅度是模型中变量之间相关性的函数。

之前有关会计方面所引荐的文章：

以下就是在会计和金融研究中常见的二步法：

在会计和金融研究中，为什么用这么多二阶回归法，以下这个graph给出了清晰的解释。

这张图的注解：

This figure displays the different types of two-stage (or two-step) regression procedures used in accounting and finance research. Box A represents studies that use a firststage regression to generate a predicted value that is then used in a second regression (as in 2SLS). 

Box B represents studies that use a first-step regression to generate a residual which is then used in a second regression as either the dependent variable (box B.1) or an independent variable (box B.2). Box B.1.i represents studies that use the residual from a first-step regression as the dependent variable in a second regression using a pooled OLS regression, which is the primary focus of this study. 

Boxes B.1.ii and B.1.iii are similar to box.1.i: box.1.ii represents studies using residuals that are transformed (e.g., by taking their absolute value) before their use in the second-step regression; box B.1.iii represents studies where the secondstep regression is estimated using something other than a pooled OLS regression (e.g., Fama–MacBeth regressions).

针对这种情况的三种解决办法

There are several straightforward ways to eliminate the biases resulting from this two-step procedure. I. The most basic solution is to simply estimate the coefficients for all the model regressors in a single, as opposed to two-step regression(一步回归法). If the two-step approach generates residuals by estimating the first-step regression by year or industry-year, then a single regression can be estimated by including a set of year or industry-year indicator variables and their interactions with each of the first-step regressors.

II. An alternative way of generating the same coefficients and standard errors as those from a single regression, as indicated by the Frisch–Waugh–Lovell Theorem, is to use a two-step regression model in which the residuals obtained from the first-step regression are then regressed on the residuals from regressions of the second-step regressors on the first-step regressor(残差回归残差法). Referring to equation (10), this two-step procedure consists of regressing y on x1 in the first step, then regressing the residuals from this regression on the residuals obtained by regressing x2 on x1 and x3 on x1. 

III. Another approach is to regress the residual from a first-step regression on the combination of all the second-step regressors and all the first-step regressors(放入所有控制变量法). Both of these two-step procedures generate unbiased estimates of the coefficient of interest and reliable t-statistics, that is, the same coefficients and t-statistics obtained from a single regression (though, as noted above, there can be slight variation in the standard errors across these methods due to differences in the degrees of freedom calculated by the statistical software used to estimate the regressions). Although estimating  a regression model in a single regression is the most straightforward approach for obtaining correct coefficient estimates and t-statistics, if for expositional or pedagogical purposes a two-step framework is preferred, researchers can avoid the bias by using one of these alternative two-step procedures.

五大会计顶级期刊中使用二步回归法在2011-2015期间发表的论文列表

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