312 Economic Cycles
with the inequality reversed for troughs. The algorithm ensures that peaks and
troughs alternate, so that an expansion is immediately followed by a contraction,
and vice versa. Finally, the algorithm considers combining phases, or creating new
phases, according to a set of predetermined rules. For instance, a censoring rule for
business cycles is that either a contraction or expansion must last a minimum of
two quarters and complete cycles must last a minimum of five quarters.
BBQ can also mark turning points for other types of series. For example, to
locate a potential peak for thegrowthcycle in GDP, replace the requirement that
yt−yt− 2 >0 with the requirement thatyt−yt− 2 >0, and so on. Pagan and
Sossounov (2003) modify BBQ to factor in themagnitudeof growth rates in financial
series. Other applications of nonparametric methods to mark the turning points
include Lunde and Timmermann (2004) and Ohn, Taylor and Pagan (2004), who
investigate bull and bear markets; Cashin, McDermott and Scott (2002), who inves-
tigate booms and slumps in commodity markets; Eichengreen, Rose and Wyplosz
(1995), who investigate exchange rate crises; and Ibbotson, Sindelar and Ritter
(1994), who examine hot and cold IPO markets.
7.2.2.2 Markov chain models
Hamilton’s (1989) innovative Markov chain switching-regime model can also be
employed to mark time. For the parametric switching-regime model, a latent ran-
dom variable,s∗t, governs the state orregimewith, say,s∗t =0 indicating low or
negative average growth, ands∗t=1 indicating high or positive average growth.
Two states, signifying negative and positive average growth rates, are adequate to
mark the turning points sinceyt<0 indicates a downswing andyt>0 indicates
an upswing in the level of GDP.
Consider a simple latent-structure model for GDP:
yt−μs∗t=φ(yt− 1 −μs∗t− 1 )+εt. (7.1)The mean of the growth-rate process switches between “low” and “high,” with
μ 0 <μ 1. For each datetin the sample, Hamilton shows how to obtain an estimate
ofP(s∗t= 0 |FT), whereFTcontains the past, or even thecomplete, sample history
of growth rates,{yt}t=1,...,T.
DefineS∗t= 1 [0.5−P(s∗t= 0 |FT)], so thatS∗t=1 during projected high-growth
phases, andS∗t=0 during low-growth phases. The observed binary series,{St∗}, can
be used in a survival, orduration, analysis to determine whether the probability
of remaining in a given phase, either contraction or expansion, depends on how
long one has been in it. An important consideration is that Hamilton’s (1989)
model is such thats∗tevolves with the probability of remaining in a given phase
independent of its duration, so that contractions and expansions are assumed to be
duration independent. Of course, this makes{S∗t}less than ideal to represent phases
that may actually beduration dependent.
Although Durland and McCurdy (1994), Filardo (1994), Diebold, Lee and
Weinbach (1994), Macheu and McCurdy (2000) and Jensen and Liu (2006) gener-
alize Hamilton’s assumptions in various directions, no parametric model matches