A gentle reminder: Should returns be interpreted as log differences?
本文指出，研究者常用资产价格的对数差分近似计算收益率，但这种近似仅在样本均值和方差趋近于零时才有效，否则会导致OLS估计有偏、不一致且低效，并建议使用子样本法或事后检验统计量来确认近似是否合理。
It is rather a norm for researchers to directly use the log difference of an asset price to compute returns. However, this log return is only a conditional approximation of the actual returns. Next, do log difference approximations and the ln X + 1 common practices produce BLUE estimates? Using 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 the sample average and variance of the original series both 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 ln X + 1 biases the estimation of the true causal effect, even the intercept term. A robust solution of using subsamples 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, an ex-post estimation test statistic is suggested to confirm both the choice of using log difference approximation and for using ln X + 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 produce biased, inconsistent, and inefficient results, even in large samples, leading to misinformed policy implications. • Returns are often misinterpreted and misrepresented as log differences in prices. • Systematic Difference; Log return is only but a conditional return approximation. • ln X + 1 is another suicidal common practice in empirical research. • Log return and ln X + 1 may result in biased and inconsistent estimators, nonetheless. • Good news; log returns can approximate returns correctly under some conditions.