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Not all bull and bear markets are alike: insights from a five-state hidden semi-Markov model

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Abstract

This paper employs the hidden semi-Markov model and a novel model selection procedure to determine different regimes in the US stock market. The empirical results suggest that the US stock market is switching between five states that can be classified into three bull states and two bear states. The three bull states are categorized as a low-volatility bull market, a high-volatility bull market, and a stock market bubble. One of the bear states represents a regular bear market, while the other corresponds to either a stock market crash or a market correction. The paper demonstrates that the five-state model is consistent with a number of stylized facts and provides many valuable insights into the regime-switching dynamics of the US stock market and the risk-reward pattern of each regime. Besides, the paper demonstrates that the five-state model enables investors to make better asset allocation decisions. Specifically, in out-of-sample tests, the asset allocation strategy based on the five-state model achieves higher performance with lower risk than the strategy based on the two-state model and the buy-and-hold benchmark.

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Notes

  1. The duration time is also known as the dwell time, occupancy time, holding time, or sojourn time.

  2. We remind the reader that the best model is the model with the smallest value of AIC.

  3. A somewhat similar critique of using the standard model selection criteria to determine the number of states in the HMM is presented by Pohle et al. (2017). These authors emphasize, among other things, that in selecting the number of states, one should consider the realism of the fitted candidate models assessed using expert knowledge.

  4. In other words, N(J) is the number of transitions between the states.

  5. The estimated mean return in state j is computed as the mean return over all months in the data sample decoded as being state j.

  6. Note that \(r_t\) denotes the time t total return on the stocks that consists of the capital gain return \(x_t\) and the dividend yield. In this section, variables \(\mu\) and \(\sigma\) denote the mean total return and the standard deviation of the total returns.

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  1. School of Business and Law, University of Agder, Service Box 422, 4604, Kristiansand, Norway

    Valeriy Zakamulin

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  1. Valeriy Zakamulin
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Appendix

Appendix

See Table 7.

Table 7 Dating of decoded states in the 5-state HSMM for the S&P 500 index
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Zakamulin, V. Not all bull and bear markets are alike: insights from a five-state hidden semi-Markov model. Risk Manag 25, 5 (2023). https://doi.org/10.1057/s41283-022-00112-y

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