Joe Cardinale and Larry W. Taylor 311statistics that directly address the ability of a model to replicate business cycle
characteristics.
7.2.1 Reasons for marking time
Letyt=log(GDPt), and consider mapping{yt}to a binary time series,{St}, with
St=0 for recessions andSt=1 for expansions. Pagan (2004) and Harding and
Pagan (2007) offer the following rationale for the reader’s interest in{St}:
1.Stfrequently emphasizes features of yt that are not immediately obvious.Observing
the behavior ofytover different phases affords us a better understanding of its
features.
2.Stmay be more meaningful to decision makers than yt. Reaction to an economic
downturn is often strong among the electorate, and it is of interest to determine
whether the probability of exiting a downturn depends on how long one has
been in it and whether such exit probabilities, orhazards, have changed in
fundamental ways in recent history.
3.Stmay be the object of interest if questions are asked about the synchronization of
cycles across sectors or countries.IfStis derived from many underlying series, it is
often more convenient to compare the representativeStvalues than to compute
a large number of correlations from the underlying series.
4.Stis generally more robust than ytto relatively unimportant short-lived shocks.Such
shocks may substantially affect statistics based on GDP growth rates but have
little impact on overall trends. In contrast,Stemphasizes the qualitative trend,
up or down.
7.2.2 Techniques for marking time
In short, mapping{yt}to{St}yields fruitful insights about cycle asymmetries, per-
sistence, and synchronization. Some nonparametric rules for marking time are
exceptionally simple. For yearly aggregate data, Neftci (1984) and Cashin and
McDermott (2002) employ the calculus rule,St = 1 (yt > 0 ). For quarterly
data, one often-used rule in the popular press is the extended Okun rule. The
rule states that, for a recessionary phase, termination is signified by two suc-
cessive quarters of positive growth,(yt+ 1 >0,yt+ 2 > 0 ). Similarly, for an
expansionary phase, termination is signified by two successive quarters of nega-
tive growth,(yt+ 1 <0,yt+ 2 < 0 ). The simplicity of such rules is very attractive,
and any limitations of such rules are quickly revealed by visual inspection of the
observed series. In fact, regardless of the rule employed, the constructed turning
points should visually coincide with those apparent in a plot of the observed time
series,{yt}.
7.2.2.1 BBQ
To locate turning points in thelevelof GDP, the BBQ algorithm first determines a
potential set of local peaks and troughs. Timetis a local peak if:
(yt−yt− 2 >0,yt−yt− 1 >0,yt−yt+ 1 >0,yt−yt+ 2 > 0 ),