Econometric Analysis of Cross Section and Panel Data - Free

Duration Analysis 707
with covariates. In addition to allowing us to treat grouped durations, the panel data
approach has at least two additional advantages. First, in a proportional hazard
specification, it leads to easy methods for estimating flexible hazard functions. Second,
because of the sequential nature of the data, time-varying covariates are easily
introduced.
We assume flow sampling so that we do not have to address the sample selection
problem that arises with stock sampling. We divide the time line into M þ 1 intervals,
½0; a 1 Þ; ½a 1 ; a 2 Þ; ...; ½a M 1 ; a M Þ; ½a M ; yÞ, where the a m are known constants. For
example, we might have a 1 ¼ 1; a 2 ¼ 2; a 3 ¼ 3, and so on, but unequally spaced
intervals are allowed. The last interval, ½a M ; yÞ, is chosen so that any duration falling
into it is censored at a M : no observed durations are greater than a M . For a random
draw from the population, let c m be a binary censoring indicator equal to unity
if the duration is censored in interval m, and zero otherwise. Notice that c m ¼ 1
implies c mþ1 ¼ 1: if the duration was censored in interval m, it is still censored in interval
m þ 1. Because durations lasting into the last interval are censored, c Mþ1 1 1.
Similarly, y m is a binary indicator equal to unity if the duration ends in the mth interval
and zero otherwise. Thus, y mþ1 ¼ 1ify m ¼ 1. If the duration is censored in
interval m ðc m ¼ 1Þ, we set y m 1 1 by convention.
As in Section 20.3, we allow individuals to enter the initial state at di¤erent calendar
times. In order to keep the notation simple, we do not explicitly show the conditioning
on these starting times, as the starting times play no role under flow
sampling when we assume that, conditional on the covariates, the starting times are
independent of the duration and any unobserved heterogeneity. If necessary, startingtime
dummies can be included in the covariates.
For each person i, we observe ðy i1 ; c i1 Þ; ...; ðy iM ; c iM Þ, which is a balanced panel
data set. To avoid confusion with our notation for a duration (T for the random
variable, t for a particular outcome on T ), we use m to index the time intervals. The
string of binary indicators for any individual is not unrestricted: we must observe a
string of zeros followed by a string of ones. The important information is the interval
in which y im becomes unity for the first time, and whether that represents a true exit
from the initial state or censoring.
20.4.1 Time-Invariant Covariates
With time-invariant covariates, each random draw from the population consists of
information on fðy 1 ; c 1 Þ; ...; ðy M ; c M Þ; xg. We assume that a parametric hazard
function is specified as lðt; x; yÞ, where y is the vector of unknown parameters. Let T
denote the time until exit from the initial state. While we do not fully observe T,
either we know which interval it falls into, or we know whether it was censored in a