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The existence of periodically collapsing bubbles in stock markets, applying the Enders–Siklos momentum threshold autoregressive (MTAR) model, is empirically investigated in this paper. Using this nonlinear time series technique, we are now able to analyse bubble-driven run-ups in stock prices followed by a crash in a cointegration framework with asymmetric adjustment. Therefore, applying this technique makes possible a deeper insight into the behavior of stock prices than was previously possible using conventional cointegration tests. Although the results from the subsample 1871–1995 cannot be interpreted in favor of the existence of periodically collapsing bubbles in the US stock market, the findings from the 1871–2001 sample period indicate their presence.

Since the 1980s, there has been continuous research interest into the phenomenon of speculative bubbles in share prices, motivated primarily by actual developments in stock markets and the increasing importance of stock markets for investors. The analyses of speculative bubbles are intimately related to time series analysis so that advances in econometrics enable an investigation of open questions in this field and their use provide the opportunity to obtain further insights into the characteristics of stock markets. One potential example is the class of periodically collapsing bubbles (Evans, 1991) and the development of econometric techniques designed to capture nonlinear adjustment mechanisms in a cointegration framework Enders & Granger, 1998 and Enders & Siklos, 2001. Studies investigating the consistency of dividend and stock price data with the market fundamental hypothesis Blanchard & Watson, 1982, Shiller, 1981 and West, 1987 are confronted with the difficulty that the contribution of hypothetical rational bubbles to stock prices are not directly distinguishable from the contribution of unobservable market fundamentals. As an alternative testing strategy, Diba & Grossman, 1984 and Diba & Grossman, 1988a proposed the use of standard unit root and cointegration tests Bhargava, 1986, Dickey & Fuller, 1981 and Engle & Granger, 1987 for stock prices and observable fundamentals to obtain evidence for the existence of explosive rational bubbles. This approach relies on the argument that if stock prices are not more explosive compared to dividends, then rational bubbles do not exist because they generate an explosive component into stock price time series. The empirical evidence reported in Diba and Grossman was not conducive to the conclusion that there are explosive bubbles in US stock prices.1 Evans (1991) argued that the test approaches put forward by Diba and Grossman are unable to detect an important class of rational bubbles, namely periodically collapsing bubbles. The application of standard unit root and cointegration techniques leads, with a high probability, to incorrect conclusions with respect to the presence of bubbles in stock prices. Evans' Monte Carlo simulations show that even in the presence of periodically collapsing bubbles, stock prices are not more explosive than dividends using standard unit root and cointegration tests (see also Charemza & Deadman, 1995). The explanation relies on the logic of standard unit root and cointegration tests; they assume a unit root as the null hypothesis and a linear autoregressive process under the alternative hypothesis. In the case of periodically collapsing bubbles, the bubble component is a nonlinear process, which falls outside the alternative hypothesis. Relying on simulated data with periodically collapsing bubbles present, the findings of standard unit root and cointegration tests reported in Evans' study incorrectly show the absence of bubbles in the majority of cases. Basing his central argument solely on Monte Carlo simulations and highlighting the power properties of standard unit root and cointegration tests, Evans was not able to provide empirical evidence as to whether periodically collapsing bubbles are actually present in US stock prices due to lack of techniques suitable to deal with nonlinear processes in a cointegration framework. Therefore, the presence of this class of rational bubbles in stock prices remained an open question. This paper tries to fill this gap using recent advances in the field of time series modelling. Applying the momentum threshold autoregressive (MTAR) model Enders & Granger, 1998 and Enders & Siklos, 2001 for stock prices and dividends makes possible the empirical investigation of the existence of periodically collapsing bubbles in stock prices in a cointegration framework.2 By taking into account asymmetries in departures from the long-run equilibrium relationship, the MTAR technique is designed to empirically capture the characteristics of periodically collapsing bubbles. We apply this approach to annual and monthly US time series for the period from 1871 to 2001 as well as for the 1871–1995 subsample on real stock prices and dividends. Furthermore, we provide Monte Carlo simulation findings as to whether the critical test for symmetry in the Enders–Siklos approach is sufficiently powerful to detect asymmetry when using the data generating process outlined by Evans. The paper proceeds as follows. Section 2 describes the theory behind periodically collapsing bubbles as outlined in Evans (1991). Section 3 presents the MTAR technique to capture the behavior of this class of rational bubbles in stock prices together with the findings of the Monte Carlo study. Section 4 provides the empirical evidence for the US stock market, and Section 5 concludes.

This article introduces the MTAR cointegration model proposed by Enders and Siklos (2001) to empirically investigate periodically collapsing bubbles (Evans, 1991) in annual and monthly US stock prices. Evans stressed that even in the presence of periodically collapsing bubbles, stock prices will appear to be integrated and cointegrated with their fundamentals so that standard integration and cointegration tests are not able to detect this class of rational bubbles. The Enders–Siklos MTAR model is a generalization of the Engle and Granger (1987) two-step procedure allowing a formal test of rational speculative bubbles, which eventually burst after reaching high levels. Technically, the bubble component can be taken into account as a nonlinear process in the alternative hypothesis. Our Monte Carlo simulation findings show that the MTAR approach provides a sufficiently powerful test to detect periodically collapsing bubble behavior when the actual data generating process is given by the bubble model put forward by Evans. Relying on the MTAR technique, the empirical findings contained in this paper are interpretable in favor of the absence of periodically collapsing bubbles in the US stock market over the subsample 1871–1995. Deviations from the long-run equilibrium do not seem to indicate an asymmetric adjustment to the long-run relationship. This finding supports the results contained in Taylor and Peel (1998) who investigate the bubbles hypothesis for the US stock market relying on a different testing approach and they were not able to confirm the existence of periodically collapsing bubbles. However, the evidence for the sample including the rapid share price increases since the middle of the 1990s is interpretable in favor of the existence of periodically collapsing bubbles in US stock prices. According to the results for the 1871–2001 sample, in the short-run US stock prices exhibit run-ups followed by crashes, while in the long-run US share prices adhere to fundamentals.