In regression analysis with a continuous and positive dependent variable, a multiplicative relationship between the unlogged dependent variable and the independent variables is often specified. It can then be estimated on its unlogged or logged form. The two procedures may yield major differences in estimates, even opposite signs. The reason is that estimation on the unlogged form yields coefficients for the relative arithmetic mean of the unlogged dependent variable, whereas estimation on the logged form gives coefficients for the relative geometric mean for the unlogged dependent variable (or for absolute differences in the arithmetic mean of the logged dependent variable). Estimated coefficients from the two forms may therefore vary widely, because of their different foci, relative arithmetic versus relative geometric means. The first goal of this article is to explain why major divergencies in coefficients can occur. Although well understood in the statistical literature, this is not widely understood in sociological research, and it is hence of significant practical interest. The second goal is to derive conditions under which divergencies will not occur, where estimation on the logged form will give unbiased estimators for relative arithmetic means. First, it derives the necessary and sufficient conditions for when estimation on the logged form will give unbiased estimators for the parameters for the relative arithmetic mean. This requires not only that there is arithmetic mean independence of the unlogged error term but that there is also geometric mean independence. Second, it shows that statistical independence of the error terms on regressors implies that there is both arithmetic and geometric mean independence for the error terms, and it is hence a sufficient condition for absence of bias. Third, it shows that although statistical independence is a sufficient condition, it is not a necessary one for lack of bias. Fourth, it demonstrates that homoskedasticity of error terms is neither a necessary nor a sufficient condition for absence of bias. Fifth, it shows that in the semi-logarithmic specification, for a logged error term with the same qualitative distributional shape at each value of independent variables (e.g., normal), arithmetic mean independence, but heteroskedasticity, estimation on the logged form will give biased estimators for the parameters for the arithmetic mean (whereas with homoskedasticity, and for this case thus statistical independence, estimators are unbiased, from the second result above).

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连续因变量的乘法模型：对未记录形式与记录形式的估计

在具有连续和正因变量的回归分析中，经常指定未记录的因变量和自变量之间的乘法关系。然后可以对其未记录或已记录的形式进行估计。这两种程序可能会产生估计的重大差异，甚至是相反的迹象。原因是对未记录形式的估计产生未记录因变量的相对算术平均值的系数，而对记录形式的估计给出未记录因变量的相对几何平均值（或算术平均值的绝对差异）的系数记录的因变量）。因此，这两种形式的估计系数可能会有很大差异，因为它们的焦点、相对算术平均值与相对几何平均值不同。本文的第一个目标是解释为什么会出现系数的主要差异。尽管在统计文献中很好理解，但这在社会学研究中并没有被广泛理解，因此具有重要的实际意义。第二个目标是推导出不会发生分歧的条件，其中对记录形式的估计将给出相对算术平均值的无偏估计。首先，它推导出了当对数形式的估计将给出相对算术平均参数的无偏估计量的充分必要条件。这不仅需要未记录误差项的算术平均独立性，而且还需要几何平均独立性。第二，它表明回归量上的误差项的统计独立性意味着误差项的算术和几何平均独立性，因此它是不存在偏差的充分条件。第三，说明统计独立性虽然是充分条件，但不是无偏的必要条件。第四，它证明了误差项的同方差性既不是没有偏差的必要条件也不是充分条件。第五，它表明在半对数规范中，对于在自变量（例如正态）的每个值处具有相同定性分布形状的对数误差项，算术平均独立，但异方差性，对对数形式的估计将给出有偏算术平均值参数的估计量（而具有同方差性，