英文文献：The log-linear return approximation, bubbles, and predictability-对数线性回报近似、泡沫和可预测性
英文文献作者：Tom Engsted,Thomas Q. Pedersen,Carsten Tanggaard
英文文献摘要：
We study in detail the log-linear return approximation introduced by Campbell and Shiller (1988a). First, we derive an upper bound for the mean approximation error, given stationarity of the log dividendprice ratio. Next, we simulate various rational bubbles which have explosive conditional expectation, and we investigate the magnitude of the approximation error in those cases. We find that surprisingly the Campbell-Shiller approximation is very accurate even in the presence of large explosive bubbles. Only in very large samples do we find evidence that bubbles generate large approximation errors. Finally, we show that a bubble model in which expected returns are constant can explain the predictability of stock returns from the dividend-price ratio that many previous studies have documented.

我们详细研究了坎贝尔和席勒(1988a)提出的对数线性收益近似。首先，给出了对数股息价格比的平稳性，给出了平均逼近误差的上界。接下来，我们模拟了各种具有爆炸性条件期望的理性泡沫，并研究了这些情况下近似误差的大小。我们惊奇地发现，即使存在巨大的爆炸性泡沫，坎贝尔-席勒近似也非常准确。只有在非常大的样本中，我们才能找到气泡产生巨大近似误差的证据。最后，我们证明了一个预期收益为常数的泡沫模型可以解释股利价格比率的股票收益的可预测性，许多先前的研究已经证明。