326 Economic Cycles
is negative duration dependence; if flat, there is no duration dependence. However,
the shape of the hazard function is but one example of the types of shapes associ-
ated with cycles. In particular, Harding and Pagan (2002) and Pagan and Sossounov
(2003) discuss the typical shapes of phases, either contractions or expansions. They
address the following issues:
- Amplitudes of phases
- Cumulative movements within phases
- Asymmetric behavior of phases.
After marking the turning points, the binary series,{St}, is employed along with
the observed underlying series,{yt}, to describe the shape of a phase. For example,
during economic expansions, GDP is observed to rise quickly at first and then
slows its ascent before finally reversing direction, thus marking the beginning of a
contraction.
In Figure 7.2, we present a stylized economic expansion. Theyaxis represents
log(GDP), oramplitude, and thexaxis represents time spent in an expansionary
phase, orduration. On the time axis, time A represents the first turning point,
the trough, and time B the second turning point, the peak. The amplitude of the
expansion is the vertical distance between points A and B, measuring the change in
GDP from trough to peak. In this instance the amplitude is log(GDPB)−log(GDPA).
The hypotenuse of the triangle is a benchmark representing a constant increase
in amplitude, with increases in amplitude proportional to the time spent in the
expansionary phase.
Descriptive measures of interest include the average duration and average ampli-
tude of the expansions in the sample, measures of the variability in durations and
amplitudes, and a measure to show how closely growth in GDP adheres to the
hypotenuse depicted in Figure 7.2. For our sample ofnexpansions, we observe the
durations{T 1 ,T 2 ,...,Tn}, and the amplitudes{A 1 ,A 2 ,...,An}.
Actual pathAMPLITUDEGDPlog(GDPA)log(GDPB)ABDurationFigure 7.2 Stylized expansion phase