Joe Cardinale and Larry W. Taylor 3197.4.2.1 The GMD test
Mudambi and Taylor (1995) devise a generalized method of moments (GMM) esti-
mator based on the moment condition,V(T)−[E(T)]^2 −E(T)−γ=0. For the
geometric density,γ=0, and they determine whether GMD =( 1 /n)
n
i= 1(Ti−T)^2 −T^2 −T is statistically significant from zero. Once normalized, GMD is asymptoti-
callyN(0, 1), and should be especially sensitive to IFRA and DFRA alternatives since
it is completely analogous to a continuous-time test based onV(T)−[E(T)]^2 = 0
that is designed for such alternatives. Because GMD is highly skewed in finite
samples, however, simulations are necessary to obtain finite-sample critical values.
7.4.2.2 The SB test
Closely related to the GMD test is thestate-basedSB test proposed by Pagan (1998),
which has a number of positive attributes:
- SB focuses directly on the conditional probabilities.
- SB involves a regression and so is easy to explain to a nonspecialist.
- SB can be used to examine prediction issues.
- SB can be used to study how the exit probabilities have changed over time since
the parameters can be recursively estimated. - SB can be easily modified to estimate discrete-time hazard functions, with or
without covariates.
Although SB can be applied to whole cycles, it is best to apply the test to the
separate half cycles, or phases. In other words, one should apply SB first to con-
tractions and then to expansions. In fact, Mudambi and Taylor (1991) show that
it is incompatible for expansions, contractions, and whole cyclesallto follow
a constant-hazard geometric distribution. So, if expansions and contractions are
duration independent, it is certain that whole cycles are duration dependent. Like-
wise, if whole cycles are duration independent, there is necessarily some form of
statistical dependence in the half-cycle components.
Consider a small sample ofcontractionsobserved at monthly intervals. In the
example below, there are gaps in the time line because most months of expansion
are excluded from the sample. In fact, the only included months of expansion are
those that correspond to a turning point.
Jan Feb Mar Apr ... Sep Oct Nov Dec Jan Feb ... Jun Jul
St 0 0 0 1 ... 0 0 0 0 0 1 ... 0 0
dt− 1 0 1 2 3 ... 0 1 2 3 4 5 ... 0 1St=0 indicates a month of economic contraction, andSt=1 indicates a month of
economic expansion. For instance, the first complete contraction began in January
and ended in March, with April as the turning point. The string ofStvalues rep-
resents two complete contractions and one incomplete, orcensored,contraction.
The durations of the contractions areT 1 =3,T 2 =5, andT 3 =2. Any censored
observations will invariably be at the beginnning and/or end of the string, and