A gentle reminder: Should returns be interpreted as log differences?

Abstract

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.