Abstract
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
Bull and bear markets are a common way of describing cycles in equity prices. To fully describe such cycles one would need to know the data generating process (DGP) for equity prices. We begin with a definition of bull and bear markets and use an algorithm based on it to sort a given time series of equity prices into periods that can be designated as bull and bear markets. The rule to do this is then studied analytically and it is shown that bull and bear market characteristics depend upon the DGP for capital gains. By simulation methods we examine a number of DGPs that are known to fit the data quite well—random walks, GARCH models, and models with duration dependence. We find that a pure random walk provides as good an explanation of bull and bear markets as the more complex statistical models. In the final section of the paper we look at some asset pricing models that appear in the literature from the viewpoint of their success in producing bull and bear markets which resemble those in the data. Copyright © 2002 John Wiley & Sons, Ltd.
1 INTRODUCTION
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
In the past two decades a great deal of attention has been paid to the documentation of many features of returns in equity markets. It has been shown that equity returns exhibit a range of features such as an equity premium, volatility clustering and fattailed densities e.g. see the surveys by Campbell, Lo and MacKinlay (1997) and Pagan (1996) inter alia. But much less attention has been paid to the feature that probably attracts more commentary than anything else, namely that there are extensive periods of time when equity prices rise and fall. Colloquially these periods of time are referred to as bull and bear markets respectively. Because it is less studied, the objective of this paper is to provide a framework for thinking about and analysing such phases of the market.
Since the movement from a bull market to a bear market phase (and conversely) involves a turning point in the market, Section 2 proposes a definition of such an event. The idea we use is motivated by similar research that has been carried out when detecting turning points in the business cycle—in particular the seminal work of Bry and Boschan (1971) and its application in King and Plosser (1994), Watson (1994) and Harding and Pagan (2002). We develop an algorithm that seems to be quite successful in locating periods in time that have been thought of as bull and bear markets in US equity prices. Once the turning points are established characteristics of the phases can be identified. This exercise is performed on monthly data for the equivalent of the S&P500 for the USA over the years 1835/1–1997/5.1
Section 3 gives a formal statement of the criterion selected to determine a turning point and Section 4 canvasses equivalent forms that are sometimes more useful for analysis. It emerges that the nature of bull and bear markets will depend upon the type of data generating process (DGP) for capital gains in the market. For example, if equity prices follow a random walk with normally distributed increments, all the characteristics of bull and bear markets will depend solely upon the mean and volatility of capital gains. In Section 5 we investigate the type of bull and bear markets generated by a number of statistical models that have been proposed as candidates for the DGP of the capital gains—a random walk, GARCH and EGARCH models and a hidden layer Markov chain model.
Section 6 looks at the type of markets generated by DGPs that come from economic rather than purely statistical models. We provide a general discussion about this and then focus in some detail upon models by Gordon and St–Armour (2000) and Campbell and Cochrane (2000). We also look at some work by Donaldson and Kamstra (1996) which explains the bull and bear markets of the 1920s and we consider some aspects of their explanation. Since economic models are generally of the calibrated variety they are capable of being simulated, whereupon the output from them may be passed through the dating algorithm to establish whether the bull and bear markets they imply match up with what has been observed. In general we find that all models have difficulty in producing realistic bull and bear markets and we use our framework to provide a simple explanation of why this is so. Finally, one byproduct of the framework is that it facilitates the study of extreme events such as large increases in stock values during bull markets. We look at this use in the context of equity market behaviour in the 1920s. Experiments such as this may also be useful for Value at Risk work focusing on event risk.
2 DATING ALGORITHMS
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
In its earliest manifestation the definition of bull and bear (B&B) markets seems to correspond to that given in Chauvet and Potter (2000, p. 90, fn. 6): ‘In stock market terminology, bull (bear) market corresponds to periods of generally increasing (decreasing) market prices.’
Recent usage in the financial press, however, seems to have refined this to insist on the rise (fall) of the market being greater (less) than either 20% or 25% in order to qualify for these names. In many ways the more general definition offered in the quotation above would seem to be closer to that used to describe contractions and expansions in the business cycle literature while the second, by emphasizing extreme movements, would be analogous to ‘booms’ and ‘busts’ in the real economy. We will adopt the first definition, although it will become clear that the analysis could equally have been done with the second and we actually do measure this to some extent. Thus our definition implies that the stock market has gone from a bull to a bear state if prices have declined for a substantial period since their previous (local) peak. This definition does not rule out sequences of negative price movements in stock prices during a bull market or positive ones in bear markets, but we will have to provide some extra rules to restrict the extent of these movements.
Given this definition we need to be able to describe turning points in the series. In the business cycle literature an algorithm for doing this was developed Bry and Boschan (BB) (1971). It is important to recognize that the BB program is basically a patternrecognition program and it seeks to isolate the patterns using a sequence of rules. Broadly these are of two types. First, a criterion is needed for deciding on the location of potential peaks and troughs. This is done by finding points which are higher or lower than a window of surrounding points. Second, durations between these points are measured and a set of censoring rules is then adopted which restricts the minimal lengths of any phase as well as those of complete cycles. Because one is simply seeking patterns in the data, the philosophy underlying the BB program is relevant to any series, but the nature of asset prices is sufficiently different from real quantities as to suggest that some modification may be needed in the precise way that pattern recognition is performed. In particular, while the determination of a set of initial peaks and troughs of the business cycle is done by using (monthly) data that are smoothed, and from which ‘outliers’ have been removed, this is not as attractive with monthly asset price data. In fact, the process of eliminating ‘outliers’ may actually be suppressing some of the most important movements in the series. Considerations such as these lead us to make a number of modifications to the BB procedures.
Our first deviation from BB is not to smooth any of the series, while the second relates to the size of window used in making the initial location of turning points. In the BB program this is six. It is not entirely clear how to choose this parameter when dealing with asset prices. But given the lack of smoothing it seemed sensible to make this slightly longer, and we eventually settled on eight months as the appropriate length for asset prices.
Our second deviation relates to some rule for deciding on the minimum time one can spend in any phase. In business cycle dating this is six months. To try to determine something that would be appropriate for stock prices we consider some of the earliest formal literature that emphasizes the terms ‘bull and bear markets’. This literature, Dow Theory, was developed by Charles Dow at the turn of the century and popularized by W. P. Hamilton in editorials in the Wall Street Journal. Dow theory saw the stock market as composed of three distinct movements and distinguished between
The daily fluctuation…a briefer movement typified by the reaction in a bull market or the sharp recovery in a bear market which has been oversold…and the main movement. which decides the trend over a period of many months' (Hamilton, 1919).
Hamilton also regarded the main or primary trend as ‘The broad upward and downward movements known as bull and bear markets’ while the secondary reaction was ‘an important decline in a primary bull market or a rally in a primary bear market. These reactions usually last from three weeks to as many months.’
Since this paper shares with Dow theorists a fundamental interest in the primary movements, the quotations above point to a minimal length for a stock market phase of three months. We therefore set it at four months.2
Some minimum length to the complete cycle also needs to be prescribed. Dow Theory is somewhat vaguer about this. Dow defined a primary bull market as one with a broad upward movement, interrupted by secondary reactions, and averaging longer than two years. Furthermore, Hamilton says
There are the broad market movements; upwards or downwards, which may continue for years and are seldom shorter than a year at the least (Wall Street Journal, 26 February 1909).
Twentyfour months would therefore be one possibility but, with an eye on the hedging engaged in by Hamilton above, as well as the general recognition that bull and bear markets are unlikely to be equal in duration, the case for setting a complete cycle shorter than two years is strong. In business cycle dating the minimal cycle length is fifteen months, so we stay close to it at sixteen months. Moreover, this fits with the identification of original peaks and troughs as using a symmetric window of eight periods.
Finally, given the sharp movements that have been seen in stock prices it does seem as if some quantitative constraint needs to be appended to the rules above. Consider October 1987, for example. In terms of peaks and troughs the contraction lasted only for 3 months, after which a recovery occurred, so this would not be regarded as a bear market due to the duration of the price decline being too short. That seems unsatisfactory. Allowing bear markets to have less than a 3month minimum duration would, however, almost certainly produce many spurious cycles. Hence an extra constraint that the minimal length of a phase (four months) can be disregarded if the stock price falls by 20% in a single month was appended to the rules. Appendix B sets out the rules used to establish the turning points of the series studied in this paper.
There are other algorithms that we might use. Instead of describing how a turning point occurs, we might instead define a binary random variable S_{t} taking the value unity when a bull market occurs at time t and zero for a bear market, and then describe how one goes from the state S_{t} to S_{t+1}. Such an algorithm was followed by Lunde and Timmermann (2000). Since the probability at time t of a peak would be Pr(S_{t} = 1, S_{t+1} = 0), and the method used by Lunde and Timmermann to date a switch from a bull to bear market focuses upon Pr(S_{t+1} = 0S_{t} = 1), it is clear that the differences reside in the need to specify Pr(S_{t} = 1) in the Lunde–Timmermann case, i.e. in practice one needs to know the initial state S_{0} to perform their dating. Obviously a turningpoint approach does not require knowledge of S_{0}. Both methods have their attractions. The fact that one does not need to guess at S_{0} means that the turningpoint method appeals. However, describing what would cause a transition between states rather than a turning point may make it easier to take account of the magnitude of price changes as a determinant of changes in states. To do so via the dating method one needs to incorporate the magnitude restriction as a censoring operation, and this will be done in the later numerical work. Of course we might use a combination of the two.
3 SOME FACTS ON BULL AND BEAR MARKETS
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
Figure 1 plots the natural log of the monthly stock price index lnP_{t} for the USA over the period 1835/1–1997/5. The series is equivalent to the S&P500 and the data sources are given in Appendix A. It is clear that there are many expansions and contractions in the series, particularly the enormous decline in stock values during the Great Depression.
To summarize this history we apply the algorithm that incorporates the dating methods discussed above to the series. Once we establish where the turning points occur it is possible to summarize various characteristics of the movements between each of these points; such expansions and contractions are termed phases. We consider five such measures of the nature of these phases.
 (1)
The average duration of each phase, D.
 (2)
The average amplitude of each phase, A. For convenience, we define ‘amplitude’ as the difference in the logs of the stock price from one turning point to another. This does not yield an exact measure of the actual percentage change in the equity price over a phase owing to the fact that these movements are sometimes large and therefore the approximation ln(1 + x) = x breaks down.
 (3)
The average cumulated movements in lnP_{t} over each phase, C.
 (4)
The ‘shape’ of the phases as measured by their departure from being a triangle. The index used for this purpose is the ‘excess’ index in Harding and Pagan (
2002),
EX, which is the average over all phases of the value of (
C − 0.5
A − 0.5(
A ×
D))/
D for each phase.
 (5)
The fraction of expansions and contractions for which A ≥ 0.18 and A ≤ −0.22. These numbers translate into increases in the equity price of more than 20% and decreases of less than 20%. The motivation for considering such statistics is that some definitions of bull and bear markets require expansions and contractions that are of these magnitudes. We will refer to these as the B^{+} and B^{−} proportions.
After defining S_{t} as a binary random variable taking the value unity if a bull market exists at time t and zero if it is a bear market, we can estimate the quantities above in the following way. First, the total time spent in an expansion is and the number of peaks (hence expansions) is given by .3
Therefore the average duration of an expansion will be 4
The average amplitude of expansions will be
To obtain the cumulated change over any expansion we have to define Z_{t} = S_{t}Z_{t−1} + S. Then Z_{t} contains the running sum of ΔlnP_{t} provided S_{t} = 1, with the sum being automatically reset to zero whenever S_{t} = 0. Hence the total of cumulated changes over all expansions is
with the average being
All these quantities can be found up to a factor of proportionality from regressions, e.g. Ĉ comes from the regression of Z_{t} against unity. To get Ĉ one needs to adjust for an incorrect scaling factor, e.g. the regression coefficient in the regression just described would need to be multiplied by T/NTP. The estimated excess is also a multiple of (1/NTP) where the multiple is the sum of (Ĉ_{i} − 0.5Â_{i} − 0.5Â_{i}D̂_{i})/D̂_{i} over all the phases i = 1, …, NTP. Finally, since the series (1 − S_{t+1})S_{t} has unity at the peak of an expansion and zeros elsewhere, while Z_{t} contains the amplitude of each expansion at the point in time t, the amplitudes of expansions are the nonzero values of (1 − S_{t+1})S_{t}Z_{t}. Consequently,
where I[a] = 1 if a is true and zero otherwise. Bear market statistics are found in the same way by replacing S_{t} with 1 − S_{t}.
To compute standard errors for the estimators we resort to some asymptotic theory. Consider . We can write this as , where p̂ = NTP/T. Then, since
converges to E(S_{t}), the delta method says that the distribution of √T(D̂ − D) is asymptotically determined by that of [p^{−2}√T(p̂ − p)]E(S_{t}). Now p̂ is the regression coefficient of (1 − S_{t+1})S_{t} against unity and so we can find its standard deviation using HAC standard errors. We use the formula in Hamilton (1994, p. 283, eq. (10.5.21)) for this, and account for 20thorder serial correlation. It is clear that all the statistics above are asymptotically multiples of p̂^{−1} and so the same theory applies.
As one might expect from the derivation, it is not clear that asymptotic theory will be very effective here as it hinges on getting a good approximation to the distribution of p̂^{−1}. Indeed simulations of the D̂ estimators from a model in which the underlying DGP for equity prices was a random walk with drift showed that the distribution was highly nonnormal even when the sample size exceeds 1200. Thus the robust standard errors found by asymptotic analysis may be unreliable as guides to the dispersion of the estimators. Nevertheless, because we are generally interested in whether a particular model can generate the observable characteristics of bull and bear markets, it is possible to simulate sets of observations (of length equal to the sample size) from the DGP of that model, and thereby find pvalues of tests for the hypotheses that the model and sample characteristics are the same.
Table I provides the statistics just described for the USA for three sample sizes as well as asymptotic standard errors. As we have noted above, the latter may be an unreliable indicator of the precision of the estimators so we provide (in the footnote to the table) the standard errors that one would get if the DGP had been a random walk with drift whose parameters are estimated over the period 1889/1–1997/5. There are some obvious differences but, by and large, one would get much the same story from using either set.
Table I. Statistics on bull and bear markets in US stock price dataa,b  1835/1–1997/5  1889/1–1997/5  1945/1–1997/5 


Bear duration  15 (1.68)  14 (1.70)  12 (2.09) 
Bull duration  25 (2.70)  25 (3.17)  27 (4.74) 
Bear amplitude  0.31 (0.03)  0.31 (0.04)  0.23 (0.04) 
Bull amplitude  0.43 (0.05)  0.45 (0.06)  0.46 (0.09) 
Bear cumulated  2.67 (0.28)  2.59 (0.32)  1.52 (0.29) 
Bull cumulated  7.29 (0.79)  7.71 (0.96)  7.70 (1.40) 
Bear excess  0.024 (0.002)  0.021 (0.002)  0.014 (0.003) 
Bull excess  0.019 (0.004)  0.026 (0.004)  0.030 (0.006) 
B^{−}  0.60 (0.07)  0.52 (0.06)  0.38 (0.07) 
B^{+}  0.83 (0.09)  0.88 (0.10)  0.93 (0.16) 
The statistics of Table I are interesting. It is clear that bull markets tend to be longer than bear markets and the durations agree quite closely with those attributed to Hamilton in 1921 (Rhea, 1932, p. 37) who claimed that, over the preceding 25 years, bull markets had lasted an average of 25 months while bear markets had lasted 17 months. Over time it seems as if bear markets have become shorter and weaker while bull markets have become stronger. The US stock market also exhibits expansions and contractions that deviate quite a lot from a triangular shape and this tendency may have become more emphatic over time. The fact that there is a departure from a triangle in the evolution of the markets is also true for the US business cycle; see Sichel (1994) and Harding and Pagan (2002). Finally, it is clear that most bull markets rise more than 20% while a much smaller fraction of bear markets culminate in a fall of more than 20%.
As a check on the dating algorithm, Table II compares our results for the USA to some postwar stock market cycle dates quoted in Niemira and Klein (1994, Table 10.2, p. 431) that have been used by Chauvet and Potter (2000). Their results are in parentheses. The correspondence is quite good, except for an extra contraction from April 71 to November 71 (the Niermira/Klein dating stops before the last contraction identified). There was a 10% contraction over this period and it may be that the Niermira/Klein results incorporate some censoring based on the magnitude of movements in share prices.
Table II. Postwar US stock market cycles: two dating methods (Niermira/Klein in parentheses)Peak  Trough 

1946/5 (1946/4)  1948/2 (1948/2) 
1948/6 (1948/6)  1949/6 (1949/6) 
1952/12 (1953/1)  1953/8 (1953/9) 
1956/7 (1956/7)  1957/12 (1957/12) 
1959/7 (1959/7)  1960/10 (1960/10) 
1961/12 (1961/12)  1962/6 (1962/6) 
1966/1 (1966/1)  1966/9 (1966/10) 
1968/11 (1968/12)  1970/6 (1970/6) 
1971/4  1971/11 
1972/12 (1973/1)  1974/9 (1974/12) 
1976/12 (1976/9)  1978/2 (1978/3) 
1980/11 (1980/11)  1982/7 (1982/7) 
1983/6 (1983/10)  1984/5 (1984/7) 
1987/8 (1987/9)  1987/11 (1987/12) 
1990/5 (1990/6)  1990/10 (1990/10) 
1994/1  1994/6 
Finally, as mentioned in the Introduction, most attention in the literature has been paid to the mean and volatility of ΔlnP_{t}. Moreover, Slutsky (1937) and Fisher (1925) both emphasized that what seem to be regular ups and downs in a series can simply arise from stochastic variation. Fisher termed this phenomenon the ‘Monte Carlo cycle’. Malkiel (1973) noted that a random walk in stock prices would produce cycles. Hence, as there is clearly going to be some connection between B&B market phenomena and these two moments, we present values for the mean µ and standard deviation σ of ΔlnP_{t} for each of the three samples in Table I. These quantities are used in the simulations of the next section.
Table III shows that the mean capital gain has been increasing over time and the standard deviation has been falling. However, the decline in the latter is really quite small and statistically insignificant.
Table III. Mean and standard deviation of capital gains  1835/1–1997/5  1889/1–1997/5  1945/1–1997/5 

µ  0.0031  0.0039  0.0066 
σ  0.044  0.0448  0.0404 
µ/σ  0.07  0.09  0.16 
4 THE ANALYTICS OF BULL AND BEAR MARKETS
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
To gain some appreciation of how the type of DGP determines B&B markets, return to how initial turning points in a series were selected. A peak was taken to have occurred at time t if the event
occurs, where P_{t} is the level of the stock price. Thus the probability of the event PK occurring will depend upon the joint distribution of and, to determine that probability, one requires a specification of the DGP for ΔlnP_{t}. For example, if ΔlnP_{t} was N(µ, σ^{2}), then the Pr(PK) would be solely a function of µ/σ, since the turning points in lnP_{t} are identical to those in ln(P_{t}/σ). Consequently, it is likely that the probability will rise with µ (the mean capital gain) and decline with σ. Of course, there is more to the dating rules than that. After the initial turning points are found a set of censoring operations is applied that will change the probability of ‘final’ turning points. Unfortunately, it becomes very hard to assess the precise impact of those operations analytically and so we will be forced to resort to numerical simulation. Nevertheless, the insight obtained from looking at what determines the initial turning points is extremely valuable in analysing bull and bear markets. In particular, it is clear that, regardless of the model for ΔlnP_{t}, the ratio of µ to σ will be a key determinant of cycle characteristics. As an illustration of this point, note that the movements in this ratio in Table III are very suggestive about the actual changes over time in bull and bear market characteristics noted in Table I. Moreover, it is clear that any theoretical model which claims to provide an explanation of historical bull and bear markets will have to be capable of reproducing the historical values of µ and σ. Since µ is related to the equity premium one must therefore be able to replicate that as well as the volatility of capital gains. Whether this is sufficient is something that we investigate in the next section.
5 SOME STATISTICAL MODELS OF RETURNS
 Top of page
 Abstract
 1 INTRODUCTION
 2 DATING ALGORITHMS
 3 SOME FACTS ON BULL AND BEAR MARKETS
 4 THE ANALYTICS OF BULL AND BEAR MARKETS
 5 SOME STATISTICAL MODELS OF RETURNS
 6 SOME ECONOMIC MODELS
 7 CONCLUSION
 Acknowledgements
 REFERENCES
 APPENDIX A: STOCK MARKET DATA
 APPENDIX B: PROCEDURE FOR PROGRAMMED DETERMINATION OF TURNING POINTS
 Supporting Information
As the previous section showed, it is the DGP of ΔlnP_{t} that is the key to understanding bull and bear markets. To this end we might categorize the potential DGPs into those for which the capital gain is a martingale and those for which it is not. The simplest martingale model would just be the basic random walk with drift.
 (1)
where ϵ_{t} is n.i.d.(0,1).5 Columns two and three of Table IV provide a summary of the bull and bear markets that would be seen in realizations of the DGP (1) when viewed through the dating filter described earlier.6 In the simulations µ = 0.0039 and σ = 0.0448 are taken from the US data for 1889/1–1997/5; see Table III. Ten thousand realizations of the DGP of the chosen statistical model are used to provide the sampling statistics. In parentheses under each quantity is the standard error of the estimator of that quantity using a sample of observations of the same length as 1889/1–1997/5. However, since the estimators are nonnormal, pvalues were also computed. If there is a single asterisk then the pvalue is less than 0.05, while a double asterisk means it is less than 0.01. If there is no asterisk the pvalue is greater than 0.05.
Table IV. US bull and bear markets generated by various statistical modelsa,b  Data  RW µ = 0  RW µ ≠ 0  GARCH µ ≠ 0  EGARCH µ ≠ 0  DDMSDD 


Dur bear  14  20** (2.36)  16 (1.93)  16 (1.92)  16 (1.95)  16 (2.00) 
Dur bull  25  20** (2.35)  24 (3.08)  25 (3.26)  26 (3.41)  26 (3.48) 
Amp bear  −0.31  −0.33 (0.034)  −0.27 (0.03)  −0.26 (0.04)  −0.30 (0.04)  −0.28 (0.04) 
Amp bull  0.45  0.33** (0.034)  0.42 (0.05)  0.42 (0.06)  0.46 (0.05)  0.44 (0.06) 
Cum bear  −2.59  −4.31* (1.14)  −2.70 (0.68)  −2.63 (0.78)  −2.87 (0.88)  −2.78 (0.85) 
Cum bull  7.71  4.34** (1.09)  7.22 (2.04)  7.42 (2.16)  8.39 (2.41)  8.48 (2.61) 
Ex bear  0.020  −0.00** (0.006)  −0.00** (0.005)  −0.000** (0.006)  0.006* (0.007)  0.005* (0.008) 
Ex bull  0.026  0.00** (0.006)  0.00** (0.007)  0.000** (0.008)  0.011* (0.009)  0.007* (0.01) 
B^{−}  0.52  0.69* (0.08)  0.56 (0.09)  0.50 (0.10)  0.56 (0.1)  0.50 (0.09) 
B^{+}  0.88  0.80 (0.07)  0.88 (0.06)  0.85 (0.07)  0.89 (0.06)  0.84 (0.07) 
It is clear that the random walk with drift does quite well at replicating the bull and bear markets actually observed, but the pure random walk (µ = 0) fails quite badly. In fact, with the latter we would expect symmetry in the characteristics of the phases, since the probabilities of an initial peak and a trough occurring would be identical. The only exception to this comes with respect to the B^{+} and B^{−} statistics. There the different censoring thresholds (−0.22 and 0.18) utilized to ensure that B&B market movements produce a 20% movement in P_{t} are the sources of the asymmetries in these proportions. Given this result on the importance of µ it is clear that an explanation of its magnitude will be a key element in getting the nature of bull and bear markets right. A notable deficiency with the random walk model is its implication that phases should, on average, look like triangles, whereas this is clearly not so.
A possible extension to the basic random walk model is motivated by the fact that (1) implies that capital gains are normally distributed, while the sample excess kurtosis for the 1899–1997 period is 9.2. Consequently, one wants to adopt a DGP that produces realizations for ΔlnP_{t} from a nonnormal density. One response would be to change the density for ϵ_{t} to others with fatter tails, e.g. Student's t, but it is more interesting to generate the excess kurtosis ‘endogenously’. Some standard ways of doing that are to treat ΔlnP_{t} as being either a GARCH(1,1) or EGARCH(1,1) process, i.e. σ in (1) is replaced by σ_{t} which varies with the past history of returns. Accordingly, these models were fitted to US capital gains over 1889/1–1997/5 yielding:
In order to perform a valid comparison with the random walk model we also make the means of the GARCH and EGARCH capital gains identical to those in the data, i.e. µ = 0.0039.
Columns four and five of Table IV then process the simulated output from the GARCH and EGARCH processes. Given the symmetry of the GARCH process it is not surprising that it has little effect upon mean durations, but in general it seems that the GARCH model does not add much to the explanation of B&B markets over that provided by the random walk. Given that the GARCH specification produces fatter tails in the distribution of ΔlnP_{t}, it is a somewhat surprising outcome that bull and bear markets are slightly less extreme under it. Of course it has to be remembered that it is cumulated shocks that are important for bull and bear markets and the GARCH model is just as likely to produce a large positive shock as a negative one and these operate to offset one another. The EGARCH model tends to provide a better match to most of the phase characteristics than the GARCH model does. In particular, it is the only model that has the ability to produce shapes of phases that resemble the data, although even then the p values suggest that there is still some discrepancy between them.
The above models have ΔlnP_{t} being a martingale difference. It has long been observed that there is little linear dependence in ΔlnP_{t}, motivating the search for some nonlinear structure. One might fit some general nonlinear models, such as neural networks or threshold autoregressions, but in some ways it would be nicer to be able to produce the nonlinearity in a framework that preserves the flavor of the topic being examined. Because of the emphasis being laid upon the two types of markets, it is useful to try to obtain the requisite nonlinearity by utilizing the literature on hidden layer Markov chains. Hamilton's (1989) work is the bestknown example of this in econometrics, although in other fields there have been other versions.
In its simplest form Hamilton's model replaces (1) with
 (2)
where z_{t} is a random variable taking the values of zero and unity whose evolution is governed by a Markov chain with transition probabilities p_{00} = Pr(z_{t} = 0z_{t−1} = 0) and p_{11} = Pr(z_{t} = 1z_{t−1} = 1). Which state corresponds to which type of market is essentially arbitrary. For the purpose of presentation and comparison of results we identify the bull state as that which has a higher mean capital gain and label it the state corresponding to z_{t} = 1, although we stress again this is quite arbitrary. Many applications of Hamilton's model and its extensions have been made in the econometric literature. Pagan and Schwert (1990) applied the basic model to US stock returns from 1835 until 1925. Recently attempts have been made to generalize this model to allow for the transition probabilities to depend upon the length of time spent in a particular state, i.e. to produce duration dependence. Maheu and McCurdy (2000) is a good example. They fitted a model in which the transition probabilities had the same format as in Durland and McCurdy (1994), namely
 (3)
where d_{t}, the duration of time spent in the jth state in the current phase at time t, is constrained to not exceed 16. The model they preferred was called DDMSDD and we took the parameters from their Table V to simulate it. One problem is that they fit the model to monthly US returns rather than to capital gains. Whilst the volatility of the returns series is much the same as capital gains, since the dividend yield shows relatively small monthly variation, the mean of returns is higher than that of capital gains. Hence we adjusted the µ_{0} and µ_{1} parameter estimates in their model by a scaling factor of 1.7 so that the overall mean of the simulated data agreed with that in Table IV. By keeping the mean and variance of the simulated returns equal to that of the data we are therefore solely studying the effect of introducing duration dependence into the model. Data is simulated from this model in column 6 of Table IV. It provides no improvement on the results from the EGARCH model.7
Table V. US bull and bear markets generated by the Gordon–St Armour model, equivalent random walk and data (1960/1–1992/6)a,b  Data  GSA  RW 


Bear duration  11 (2.67)  10 (4.92)  15 (3.5) 
Bull duration  25 (5.48)  52* (21.30)  27* (7.2) 
Bear amplitude  −0.27 (0.059)  −0.12* (0.070)  −0.20 (0.04) 
Bull amplitude  0.44 (0.099)  0.41 (0.162)  0.39 (0.09) 
Bear cumulated  −1.49 (0.323)  −0.3 (1.88)  −1.76 (0.95) 
Bull cumulated  6.81 (1.52)  16.7 (16.3)  7.70 (4.69) 
Bear excess  0.029 (0.004)  0.000 (0.024)  0.00** (0.007) 
Bull excess  0.025 (0.008)  0.003 (0.026)  0.00** (0.013) 
B^{−}  0.5 (0.11)  0.22 (0.16)  0.33 (0.16) 
B^{+}  1.0 (0.22)  0.71 (0.21)  0.83 (0.14) 
In summary, one can conclude that the broad characteristics of B&B markets are a consequence of the random walk nature of the data but that the ‘shapes’ of the markets are not well accounted for by any of the statistical models studied here. It does seem however that whatever feature is needed to account for the shapes, it is likely that it will involve some asymmetry in the conditional density of the capital gains process.