Palgrave Handbook of Econometrics: Applied Econometrics

Joe Cardinale and Larry W. Taylor 327

7.6.1 Durations

Assume that durations and amplitudes constitute random samples. If so, theTi′s

follow identical distributions,TiD(μT,σT^2 ), withTistatistically independent

fromTjfori=j. Further,T=^1 n

n



i= 1

Tiis a consistent estimator forμTandT

aN(μ
T,σ

2
T/n). This assumes, of course, that the time line has been successfully
segmented to ensure that each of theTi′sfollows the same distribution, that is,
with no mixing of distributions.
Small sample inference, however, is particularly problematic for durations. Con-
sider that theTi′srarely come from the normal distribution. In fact, for the
discrete case with constant hazards, theTi′sfollow the geometric distribution,Ti

GEOM(μT,μ^2 T−μT), withμT= 1 /p, wherepis the constant hazard. However,
since the geometric distribution is considerably right-skewed, for very small sam-
ples it is generally unwise to construct confidence intervals onμTby employing
thet-distribution, since normality of the durations is one of the assumptions sup-
porting its use. Simulations by Pagan and Sossounov (2003) also suggest thatTis
significantly skewed in small samples.

7.6.2 Amplitudes

Amplitudes are less problematic. For theith expansion, the amplitude is calculated

asAi=yTi−y 0 =

Ti



j= 1

yj, withy 1 =y 1 −y 0 ,y 2 =y 2 −y 1 , and so on. In Figure 7.2,

y 0 equals log(GDPA)andyTiequals log(GDPB). The amplitude of the expansion is
thus the sum of the growth rates from time A to B. Although Harding and Pagan
(2002) convincingly argue that GDP growth rates are not statistically independent,
their sum may be approximately normally distributed by either Gordin’s (1969) or
Hannan’s (1973) Central Limit Theorems (CLT) for stationary ergodic processes;
see also White (1984, Ch. V). Thus, approximate normality for the amplitudes,Ai,
is more plausible than normality for the durations,Ti, provided that expansions
are sufficiently long to allow the CLT to work for eachAi.

7.6.3 Cumulative gain

Another measure of interest is thecumulativegain from trough to peak, that is, the
area under the curve that describes the actual path of GDP. An approximation to
this gain is obtained by adding together the area in rectangles of unit length and
height equal toyj−y 0. The approximation, however, is too large since each such
rectangle overstates the actual area by approximately(yj−yj− 1 )/2. Correcting for
the overstatement, Harding and Pagan (2002) approximate the cumulative gain
for theith expansion by:

Fi=

∑Ti

j= 1

[

(yj−y 0 )−(yj−yj− 1 )/ 2

]

(7.18)

=

⎡

⎣

∑Ti

j= 1

(yj−y 0 )

⎤

⎦−Ai/2, (7.19)