Palgrave Handbook of Econometrics: Applied Econometrics

Joe Cardinale and Larry W. Taylor 321

squares estimator:

1

m

m



t= 1

(St−S)dt− 1 (7.9)

=

1

m

m



t= 1

Stdt− 1 −Sd (7.10)

=

n

m

T−(

n

m

)^2

1

n

n



i= 1

(Ti+ 1 )Ti

2

(7.11)

=S

1

2

{ 2 T−S[ ̃σ^2 +T

2

+T]}. (7.12)

Since, for the geometric distribution,plimT = 1 /pandplim ̃σ^2 =( 1 −p)/p^2 ,
it follows that:

plimS

1

2

{ 2 T−S[ ̃σ^2 +T

2

+T]} =p

1

2

{

2

1

p

−p

[

( 1 −p)

p^2

+

(

1 /p

) 2

+

1

p

]}

=0. (7.13)

An immediate implication is thatplim̂β 1 =0 for a constant-hazard function. On
the other hand, just as with GMD, the distribution of SB is skewed right in finite
samples, and thus it is necessary to use simulations to obtain finite-sample critical
values.
In spite of their skewed distributions, GMD and SB are asymptotically pivotal;
that is, asymptotically they do not depend on unknown parameters. For asymptot-
ically pivotal statistics, the bootstrapped critical values are generally more accurate
than those based on first-order asymptotic theory. Horowitz (2001) and Davidson
and MacKinnon (2006) explain why it is desirable to use pivotal statistics when
bootstrapping.

7.4.3 Strong-form tests

Diebold and Rudebusch (1991) and Ohn, Taylor and Pagan (2004) employ the chi-
square goodness-of-fit test to determine if durations follow the geometric density.

The test statistic isχ^2 =

K



j= 1

[(Oj−Ej)^2 /Ej], whereOjis the observed frequency in

thejth bin andEjis the expected frequency in thejth bin. The expected frequency
is derived under the null distribution, in this case the geometric density. A well-
known rule of thumb is that the expected frequency,Ej, should be at least 5 for
all bins (see Hoel, 1954). To be on the safe side, Ohn, Taylor and Pagan (2004)
use 6 instead of 5. If one adheres to this rule-of-thumb, long-term experience sug-

gests thatχ^2 approximately follows its asymptotic chi-square distribution withK-1
degrees of freedom. Nonetheless, both Diebold and Rudebusch (1991) and Ohn,
Taylor and Pagan (2004) employ simulation to obtain finite-sample critical values.

7.5 Modeling with covariates

Most researchers using aggregate-level data segment the time line to control
for heterogeneity of the exit probabilities. For example, Edwards, Biscarri and