Journal of Financial and Quantitative Analysis

Research Articles

The Log-Linear Return Approximation, Bubbles, and Predictability

Tom Engsteda1, Thomas Q. Pedersena2 and Carsten Tanggaarda3



a3 Department of Economics and Business, Aarhus University, Fuglesangs allé 4, DK-8210 Aarhus V, Denmark.


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 dividend-price ratio. Next, we simulate various rational bubbles that 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.

(Online publication February 13 2012)


This paper is a substantially revised version of Sections 2 and 3 in an earlier working paper by the first and third authors, “The Log-Linear Return Approximation With and Without Bubbles” (2007). We thank an anonymous referee and seminar participants at CREATES and at Lund University for constructive comments. We also acknowledge support from CREATES funded by the Danish National Research Foundation.