State-space estimation of

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2  One possible exception to this problem may be the case of closed-end funds where the net asset value of the fund can be viewed in many cases as being the fundamental and thus sustained premia or discounts as representing positive or negative bubbles (Ahmed, et al, 1997).

3 For a review of models arising as these conditions are relaxed, see Rosser (1991, Chaps. 4, 5, 15).

4 For more on state-space models and their estimation using the Kalman filter, see Harvey (1989).

5 For more detailed discussion of financial, macroeconomic, and institutional changes occurring in the Japanese and German economies during this period, see Rosser and Rosser (1996, Chapters 6,10).

6 This premium exceeded 100% before the crash (Ahmed et al., 1997). Given that most closed-end funds usually exhibit discounts, it is very hard to explain this large premium as indicating anything other than exuberant speculative behavior.

7 Although the stochastic rational bubble model assumes an exponential rise followed by a sudden crash, many apparent historical bubbles, such as the U.S. stock markets in 1929 and 1987, reached peaks around two months before they experienced sharp crashes. Such phenomena can be explained by models in which there are heterogenous agents, with better informed agents selling out at the peak and the crash occurring later when the less informed agents panic (DeLong, et al, 1990).

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