Since the classical work of Samuelson (1965) and Leroy (1973), the random walk and martingale models of stock prices have formed the cornerstone of modern finance. Hence it is not surprising that an extensive empirical literature has considered deviations from these benchmark models. Several authors, including Lo & MacKinlay (1988), Fama & french (1988), Poterba & Summers (1988), Richardson & Stock (1989) and Boudokh & Richardson (1994) study long-run serial correlations in stock returns. Although this literature finds indications of a slowly mean reverting component in stock prices, deviations from normally distributed returns, time-varying volatility and small sample sizes have plagued existing tests and made it difficult to conclusively reject the random walk model.
This paper proposes a new approach to modelling time series dependence in stock prices that allow bull and bear hazard rates, i.e. the probability that a bull or bear market terminates next period, to depend on the age of the market. Inspection of these hazard rates yields new insights into long-run dependencies and deviations from parametric models of asset prices proposed in the literature, including the simple random walk model with a constant drift and models that allow for volatility persistence. By explicitly focusing on duration dependence in stock prices, the proposed tests are very different from the tests based on autocorrelations previously adopted in the literature. Our approach does not require that stock prices follow a low-order Markov process although this is a special case of our model when termination probabilities are memoryless.
We formalize bull and bear states in terms of movements between local peaks and troughs. Earlier studies such as Fabozzi & Francis (1977), Kim & Zumwalt (1979) and Chen (1982) consider definitions of bull markets based simply on returns in a given month exceeding a certain threshold value. Such definitions do not reflect long-run dependencies in stock prices and ignore information about the trend in stock price levels. By our definition the stock market switches from a bull to a bear state if stock prices have declined by a certain percentage since their previous (local) peak within that bull state. Likewise, a switch from a bear to a bull state occurs if stock prices experience a similar percentage increase since their local minimum within that state. This definition does not rule out sequences of negative (positive) price movements in stock prices during a bull (bear) market as long as their cumulated value does not exceed a certain threshold. By abstracting from the small unsystematic price movements that dominate time series as noisy as daily price changes this definition is better suited to capture long-run dependencies in the underlying drift in stock prices.
Most closely related to the current study is Pagan & Sossounov (2000) who also consider a definition of bull and bear states based on cumulated changes. They use a pattern recognition dating algorithm for bull and bear states that filters monthly returns through a sequence of censoring operations. Their study is concerned with modeling overall characteristics such as average durations and amplitudes of bull and bear markets. Our study does not impose minimal duration constraints on bull and bear markets and jointly models the full duration distribution of both short and long bull and bear states.
We find evidence of distinctly different duration dependence in bull and bear states. The longer a bull market has lasted, the lower its hazard rate and hence the lower the probability that it terminates next period. While the hazard rate of bear markets also declines initially, it has a U-shaped pattern and thus increases at longer horizons. Interest rates are also found to have an important effect on hazard rates. Higher interest rates are associated with an increase in the bull market hazard rate and a decrease in the bear hazard rate. They are therefore associated with a higher probability of being in a bear market and a lower bull market probability.
The finding of a hazard function that depends on age suggests that stock prices do not follow a low order Markov process but that the drift and the effect of interest rates on stock prices is related to the market’s memory of the time spent in the current state. This means that the effect of an interest rate change on stock prices depends on the age and type of the state where the change occurs. In addition it is necessary to account for the entire sequence of hazard rates.
The plan of the paper is as follows. Section 2 presents our definition of bull and bear market states. Section 3 characterizes the unconditional distribution of the durations and returns in bull and bear markets using more than a century of daily stock prices from the US. Section 4 discusses estimation of bull and bear markets whose hazard rate may depend on age. Section 5 reports empirical results while Section 6 undertakes a scenario analysis to investigate the effect of an interest rate change on stock prices. Section 7 briefly discusses further economic interpretation of our findings.
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