Duration Dependence in Stock Prices: An Analysis of Bull and Bear Markets
Lunde, Asger, Timmermann, Allan, Journal of Business & Economic Statistics
This article studies time series dependence in the direction of stock prices by modeling the (instantaneous) probability that a bull or bear market terminates as a function of its age and a set of underlying state variables, such as interest rates. A random walk model is rejected both for bull and bear markets. Although it fits the data better, a generalized autoregressive conditional heteroscedasticity model is also found to be inconsistent with the very long bull markets observed in the data. The strongest effect of increasing interest rates is found to be a lower bear market hazard rate and hence a higher likelihood of continued declines in stock prices.
KEY WORDS: Hazard model; Interest rate effect; Survival rate.
Since the seminal work by 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 and MacKinlay (1988), Fama and French (1988), Poterba and Summers (1988), Richardson and Stock (1989), and Boudokh and Richardson (1994), have studied long-run serial correlations in stock returns. Although this literature reports 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 article proposes a new approach to modeling time series dependence in stock prices that allow bull and bear hazard rates--that is, the probability that a bull or bear market terminates in the 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. The tests are more closely related to the duration dependence measures first proposed in the context of regime switching models used to analyze gross domestic product growth (Durland and McCurdy 1994) or stock market prices (Maheu and McCurdy 2000a). Although the two approaches share some of the same objectives--to capture potential duration dependence--they differ both in terms of the definition of the underlying states and in terms of econometric methodologies.
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. Faust (1992) demonstrated that existing tests for autocorrelation based on variance ratios have optimal power in testing the random walk hypothesis against certain classes of stationary autoregressive moving average models. However, these models form only a small subset of the alternatives that are interesting from an economic standpoint, such as nonlinear speculative bubble processes and processes in which the drift depends on past cumulated returns within a state. There is no result for the power of autocorrelation tests against nonlinear alternatives or processes with long memory. This is important, because there is mounting evidence of nonlinearities such as a switching factor in the mean and volatility of stock returns (cf. Maheu and McCurdy 2000a; Perez-Quiros and Timmermann 2000).
We formalize bull and bear states in terms of a filter that tracks movements between local peaks and troughs. Earlier studies, such as those of Fabozzi and Francis (1977), Kim and Zumwalt (1979), and Chen (1982), considered definitions of bull markets based simply on returns in a given month exceeding a certain threshold value. …