The duration of fixed exchange rate regimes

by

λ(t) = ι_πn^pr(^t<^τ<^{t + d}t^lT ≥ ^t)                       (1)

dt→0             dt

= f (t)
S (t)

Clearly, both representations in terns of the hazard function and the probability density
function contain exactly the sane infornation. However, the hazard function is useful
in so far as its shape provides a definition of duration dependence. Positive duration
dependence exists at the point t* when dλ(t)∕dt > 0 at t = t*. The probability that a
regine will end increases as the regine increases in length of tine. Conversely, negative
duration dependence exists at the point t* when dλ(t)∕dt < 0 at t = t*. The condition
that dλ(t)∕dt = 0 for every t defines a so-called memoryless system. Clearly, it is possible
that the hazard function evolves with tine in a non-nonotonic fashion alternating between
positive and negative duration dependence.

There are different approaches to estimating the hazard function. The Kaplan-Meier
estimator is a non-parametric approach. The random spell is written as T^* in the absence
of censoring. The censoring time is C . Then, the observed random variable is T =
min(T^*, C). Suppose that there are k completed durations in our sample of size n, where
k < n since some observations are censored, and because two or more observations can
have the same duration. We define a variable δ that takes the value 1 if the observation
is censored, and zero otherwise. We assume that if T = t and δ = 1, censoring happens
immediately after time T . We can order the completed durations from smallest to largest,
t1 < t2 < ... < tk . We denote the number of durations that end at time ti by di , and the
number of durations censored between ti and ti+1 by mi . The risk set is the set of durations
that are eligible to end at time t_i and is defined as

k
ni = ∑(mj + dj )
j≥i

The scalar ni is really the number of durations neither completed nor censored before

<<   6   7   8   9   10   11   12   >>

More intriguing information