It is rather a norm for researchers to directly use the log difference of an asset price to compute returns. Just like using lnX+1 to avoid taking the natural logarithm of zero(s). However, this log returns is but a conditional approximation of the actual returns. Nonetheless, can log difference approximations and the lnX+1 common practices produce BLUE estimates? Using the log return as an example, this study discusses the approximation nature and conditions for using the log difference approximation both for the interest regressor and control variables. These conditions are; that both the sample average and variance of the original series tend to zero. When these conditions are not met, the log difference approximation is, in fact, not a good approximation and biases OLS causal estimators. When the conditions are met, it produces unbiased, consistent but less efficient estimators. Thereby making the estimates less precise and less accurate. Nonetheless, this is true for a log differenced interest regressor(s) and control variables, when it correlates with the interest variable(s) and explains, in part, the dependent variable, even in large samples. Similarly, the common use of lnX+1 biases the estimation of the true causal effect, even the intercept term, except when X tends to infinity. A robust solution of using non-zero subsamples, against lnX+1, produces unbiased and consistent estimators for the true causal effects under the causal assumptions. These biasedness, inconsistencies, and inefficiencies do not disappear in large samples. Finally, both ex-ante and ex-post test statistics are discussed, however, the ex-post estimation test statistic is recommended to confirm both the choice of using log difference approximation and that of using lnX+1, in an empirical data causal regression analysis. Ideally, researchers should ensure the conditions for using the log difference approximation are met. Otherwise, these approximations and practices produce biased, inconsistent, and inefficient results, even in large samples, leading to misinformed policy implications.

中文翻译：

---

温馨提示：返回值是否应该解释为对数差异？


研究人员直接使用资产价格的对数差来计算回报是一种常态。就像使用 lnX+1 来避免取 0（s） 的自然对数一样。但是，此日志返回只是实际返回的有条件近似值。尽管如此，对数差值近似值和 lnX+1 常见做法能否产生 BLUE 估计值？以对数返回为例，本研究讨论了对数差异近似的近似性质和对兴趣回归变量和控制变量使用对数差近似的条件。这些条件是;原始序列的样本平均值和方差都趋于零。当这些条件不满足时，对数差近似实际上不是一个好的近似，并且会使 OLS 因果估计量产生偏差。当条件满足时，它会生成无偏、一致但效率较低的估计器。从而使估计值不那么精确和不准确。尽管如此，当对数差分兴趣回归变量和控制变量与兴趣变量相关并部分解释因变量时，即使在大样本中也是如此。同样，lnX+1 的常见用法会使真实因果效应的估计产生偏差，甚至是截距项，除非 X 趋于无穷大。针对 lnX+1 使用非零子样本的稳健解决方案，为因果假设下的真实因果效应生成无偏且一致的估计量。这些偏倚、不一致和效率低下在大样本中不会消失。 最后，讨论了事前和事后测试统计，但是，在实证数据因果回归分析中，建议使用事后估计测试统计来确认使用对数差近似和使用 lnX+1 的选择。理想情况下，研究人员应确保满足使用对数差近似的条件。否则，即使在大样本中，这些近似值和做法也会产生有偏见、不一致和低效的结果，从而导致错误的政策影响。