The duration of fixed exchange rate regimes

are conditional probabilities, φ(.) will be nonnegative and there is no necessary restriction
on the vector of coefficients β . Second, we can interpret the estimated coefficients as the
constant proportional effect of x on the conditional probability λ(.).

In many practical applications, the underlying distribution is unknown. However, it
is still possible to make no arbitrary assumption about the form of this distribution and
resort to semi-parametric estimation. For the sake of the exposition, suppose that all n
observations are uncensored. We can order observed durations from smallest to largest,
t₁ < ... < t_n . The conditional probability that the first observation concludes a spell at
time t1 , given that all of the n durations could have ended at time t1 , is

λ(t1,X1,β, λo)

∑n=1 λ(tι,Xi ,β,λo)

This quantity is the contribution of the first observation to partial likelihood. The
numerator is the hazard for the individual whose spell completes at time t1 , while the
denominator is the sum of the hazards for individuals whose spells could have ended at
time t₁. If we adopt the specification λ(t, x, β, λ_o) = λ_o(t)φ(x₁ , β) this ratio becomes

λo (t)φ(xι,β )    =   φ(xι,β)

λo(t) ∑n=ι φ(Xi,β) = ∑n=ι φ(Xi,β)

Only the order of completed durations provides information on the unknown coeffi-
cients. The baseline hazard λ_o cancels out and therefore, we do not have to make an
assumption on its underlying distribution. It is recovered from the partial likelihood esti-
mation. In general, the log-likelihood function is obtained as

nn

^ln L*(^β) = Σ ^lnφ(xj^,β) ^{- ln}Σ ^φ(^xi^,β)
j=1                     i=j

We shall proceed in two steps. Firstly, we estimate the hazard function using the
non-parametric estimator for the whole sample and sub-samples of the data. This graph-
ical evidence can illustrate differences in duration dependence across types of economies.
Secondly, we make use of a proportional hazard specification to account for time-varying
explanatory variables to assess how these affect the conditional probability that a given

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