Palgrave Handbook of Econometrics: Applied Econometrics

246 Investigating Economic Trends and Cycles

0

2.5ε + 4

5.0ε + 4

7.5ε + 4

1.0ε + 5

0 30 90 90 120

Figure 6.1 The functionY(t)=βexp{rt+γcos(ωt)}as a model of the business cycle. Observe
that, whenr>0, the duration of an expansion exceeds the duration of a contraction

9.5

10.0

10.5

11.0

11.5

0 30 90 90 120

Figure 6.2 The function ln{Y(t)}=ln{β}+rt+γcos(ωt)representing the logarithmic busi-
ness cycle data. The durations of the expansions and the contractions are not affected by
the transformation

9.7

9.8

9.9

10.0

10.1

10.2

0 30 90 90 120

Figure 6.3 The functionμ+γcos(ωt)representing the detrended business cycle. The
durations of the expansions and the contractions are equal

whereμ=ln{β}. Since logs effect a monotonic transformation, there is no displace-
ment of the local maxima and minima. However, the amplitude of the fluctuations
around the trend, which has become linear in the logs, is now constant.
The final step is to create a stationary function by eliminating the trend. There
are two equivalent ways of doing this in the context of the schematic model.