Seminar on Longitudinal Analysis James Heckman University of - - PowerPoint PPT Presentation

Intro Review Models Covariates Diagnostic Check Duration Model Tests

Seminar on Longitudinal Analysis

James Heckman University of Chicago This draft, May 20, 2007

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Intro Review Models Covariates Diagnostic Check Duration Model Tests

Review of Data Collection Techniques Before addressing issues regarding the modeling and estimation of individual histories, we first review the type of data available on processes such as fertility, employment and unemployment, and

• ther state transition processes.

For illustration, suppose we examine the characterization of a two-state unemployment/employment process over continuous time with asynchronous switching. In the next figure, let t = 0 be the starting date and T0 be the termination date. Individual I experiences three transitions in this time period, while II makes only two

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Two-State Unemployment/Employment

1 t=0 T0 I II

0 = unemployed 1 = employed

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Event history data

Event history data provides the analyst with information on:

1

all of the switches of each individual

2

length of time spent in each state

This data is extremely rare. A few examples of such data sets:

Fertility studies of the Bicol region of the Philippines Another from Denmark Seattle employment study

These studies are available only for a finite length of time. Thus, the ending date of the survey censors the final observation: the duration of the individual’s stay in the last state is unknown.

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CPS data collection

The CPS data collection technique actually provides exceedingly little information for longitudinal analysis despite its breadth. The CPS survey follows a group of individuals in the following fashion:

1

At time t = 0 an initial survey is performed.

2

After 9 months, a second survey is made.

3

After 11 months, a followup is performed.

The following four “case histories,” each represented as a single broken line (line for unemployment and break for employment), illustrate how choppy the CPS data is.

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CPS Survey

t=0 2 months T=9 T=11 A B C D t

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The questioning about past history of unemployment spells is such that:

A is recorded as one uninterrupted spell of unemployment B is also recorded as an uninterrupted spell C is a spell of known duration and ending date D is known only as lasting between 2 and 9 months and is right-censored.

Such data doesn’t even allow computation of an unbiased mean duration of unemployment spells, let alone allowing us to answer questions regarding the dependence of spell length on past history

• f the individual or an calendar time.

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Prospective Point Sampling

A good deal of occupational information is collected on this basis, sometimes (misleadingly) called continuous history data. Compared with the CPS framework, this scheme is very simple. At chosen sampling intervals, the individual reports the state he or she currently occupies. The detail of such surveys depends greatly on the chosen sample interval and the characteristics of the transition process.

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Prospective Point Sampling

One-year intervals may be sufficiently narrow for a fertility series, but too wide for employment transitions. Multiple transitions may occur within the interval. Unless retrospective information is gathered at the sampling dates to fill in such gaps, this method may convey too little information regarding the time dynamics of the underlying process.

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Event Count Data

Such surveys record the number of switches between states only. There is no information on duration, incidence rates or spacing.

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Partial observability

This is a situation arising when the degree of variability recorded is not the same across variables. In some surveys, earnings is recorded on a yearly basis while unemployment is followed at three-month intervals. Methods to study correlation between disparate series will be addressed.

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Partial observability

The moral of this review to those involved in data collection design is

1

Collect all retrospective data

2

the wider the chosen sampling intervals, the noisier the recording

• f the underlying process — although the surveying may be less

costly

3

the more characteristics regarding each individual are collected, the easier to control for individual errors in reporting.

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Duration Models Standard hazard specifications A duration model characterizes the probability of an event

• ccurring as a function of elapsed time, when the waiting time is

governed by some random distribution. Let T be the waiting time until the occurrence of an event. S(t) = exp

• −

t h(u) du

• =

the probability that T exceeds some time span t h(t) = d dt [− log S(t)] = g(t) S(t), where g(t) is the density associated with G(t) = 1 − S(t).

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S(t) is called the survivor function and h(t) the hazard function, and G(t) is the distribution of spell lengths. Note that when discussing a given population of individuals and a single event which each individual experiences only once, the functions have the following interpretations: S(t) = the fraction of people who have not yet experienced the event f (t) = rate of occurrence per unit time h(t) = rate of occurrence per individual still to experience the event, or “at risk”

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A crude approximation to the survivor function S(t) is a step function giving the number surviving as a percent of the total starting population. Each time an event occurs, the number of survivors falls in a discrete fashion.

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Survivor Step Function

1 t S(t)

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An interesting transformation of the S(t) function is − log S(t) = − t h(u) du. the curvature of this integrated hazard may be read from the next graph as either

A the rate of occurrence is slowing down with time and the number

• f individuals who leave,

B the rate of occurrence is speeding up, or C the rate of occurrence changes with time and the number of individuals who leave.

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Curvature of Integrated Hazard

• log S(t)

t B C A

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Giving a functional form to the specification of the hazard is a problem of balancing off flexibility to incorporate all of cases A, B and C against conveyance: finding a parametrization which is easy to work with. Six types of hazard function are given below:

1

The simplest is of course a constant hazard over time: h(t) = C for all t > 0. Somewhat restrictive.

2

The Weibull family of distributions offers simplicity and more flexibility: h(t) = αtα−1, α > 0. α = 1 then this yields a specific constant hazard. 0 < α < 1 then h(t) is decreasing in t. α > 1 then h(t) is increasing in t.

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The Gompertz distributions also offer flexibility and convenience, with a problem when the parameter γ is negative. h(t) = exp(γt).

For γ > 0, h(t) is increasing. For γ < 0, h(t) is decreasing, but the distribution function is

• defective. The probability P(T > t) = S(t) does not go to zero

as t goes to infinity:

• h(u) du

=

• exp(γu) du = 1

γ [exp(γt) − 1] for γ < 0 S(t) = exp

• −1

γ (exp(γt) − 1)

• lim

t→∞ S(t)

= exp 1 γ

• = 0

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The Box-Cox hazard is a commingling of the preceding two: h(t) = exp γtλ − 1 λ

• lim

λ→0 h(t)

∼ Weibull λ = 1 ∼ Gompertz γ = 0 ∼ constant hazard

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The unimodal parametrization allows h(t) to be initially increasing and subsequently decreasing. The inflection point of curve C is the mode of the hazard function. h(t) = λα(λt)α−1 1 + (λt)α 0 < α < 1 then h(t) is decreasing from +∞ α = 1 then h(t) is decreasing from +1 α > 1 then h(t) is unimodal with mode

√α−1 λ

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Finally, a more complex hazard function with a concrete basis for the parametrization is the following: Let t = elapsed calendar time x = age of an individual female A hazard specification for the fertility of women who have had a child might be as follows: h(t, x) = ρη(x)(η(x)t)α−1 exp [−η(x)t]

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Notice the relation between this function and the unimodal hazard

• f §5.

There the parameter which governs the mode is η(x). By specifying that η(x) declines with the age of the female, the declining fertility of older women is incorporated into the hazard. An example of η(x) would be η(x) =        if x < 14 1 if 14 ≤ x < 25

• 1 −

x−25

20

τ2 if 25 ≤ x < 45 if x ≥ 45

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1 14 25 45 0(x) x

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Mixtures of constant hazard rates Even in the simple model in which all individuals have some constant hazard rate, θ, and these rates are distributed across the population according to some mixing density, m(θ), problems of identification arise. First, we show that such a mixing model always generates a declining proportional hazard rate, and then discuss distinguishing such a model from one in which each individual has a decreasing hazard rate.

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This form of heterogeneity, a mixture of constant hazard rates is common in the duration analysis literature. S(t) = ∞ exp(−θt) dµ(θ) where dµ(θ) is equivalent to m(θ) dθ, with m(θ) = mixing density of θ in the population θ = unobserved heterogeneous hazard rates

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For a population with such a combination of individuals, the proportional, or population hazard rate, is always decreasing over time: h(t) = d dt [− ln S(t)] = ∞

0 θe−θt dµ(θ)

∞

0 e−θt dµ(θ)

d dt h(t) = ∞

0 θe−θt dµ(θ)

2 − ∞

0 e−θt dµ(θ)

∞

0 θ2e−θt dµ(θ)

∞

0 e−θt dµ(θ)

2 By the Cauchy-Schwartz inequality for L2, the sign of

d dth(t) is

negative.1

1

x(s) y(s) ds 2 ≤

• x2(s) ds
• y 2(s) ds.

Let

• x2(s) ds =

∞ e−θt dµ(θ) and

• y 2(s) ds =

∞ θ2e−θt dµ(θ)

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This result is quite general but the converse is not true. One restriction that helps but does not completely resolve the difficulty of distinguishing between mixtures of constant hazards and mixtures of decreasing hazards is that of complete monotonicity. That is, mixtures of constant hazards must have derivatives which alternate in sign: (−1)n dnS(t) dtn ≥ 0 for all t ≥ 0, n ∈ I+ If data is fine enough to support differentiation, then a simple sufficient condition (see Feller 1971, Vol. II) for this property to be violated is −h′′(t0) + 3h(t0)h′(t0) − h3(t0) > 0.

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However, it may be that an alternative specification with decreasing hazards, S(t) = ∞ e−θtα dµ(θ), 0 < α < 1 (1) is identical to a constant hazard model ∞ e−θt dµ∗(θ), (2) where the new mixture is properly defined. Both (1) and (2) are completely monotone, and of course “observationally equivalent.”

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This underidentification can be overcome only by further a priori assumptions. If only densities with finite means are allowed then (2) may be ruled out. For example, if dµ(θ) is a gamma density, then dµ∗(θ) must have a “fat” tail. Alternatively, if there is a theoretical restriction on the time dependence, which is presumed the same for all individuals, i.e., α is known, then the mixture dµ(θ) is identified.

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Actuarial Estimators Non-parametric estimates of S(t) and t

0 h(u) du will involve

maximum likelihood estimators over a function space. No distributional assumptions regarding m(θ) are made. The first example of such a likelihood is an actuarial estimator on a single-state process.

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The actuarial estimator is based on the following observation: if failure times across a population are governed by the same distribution but the information on the process gives only the number of survivors at arbitrary time intervals,

I1 I2 I3 t1 t2 tk t3

. . .

then the survivor function evaluated at each one of the time points S(tk) = P(T > tk) = P(T > t1, T > t2, . . . , T > tk) = P(T > tk | T > t1, T > t2, . . . , T > tk−1) × P(T > tk−1 | T > t1, T > t2, . . . , T > tk−2) . . . × P(T > t1) = P(T > tk | T > tk−1) P(T > tk−1 | T > tk−2) · · · P(T > t1)

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Notice that if T > tk−1, then T > tj, for j < k − 1. If there are η1 individuals at risk at the beginning of time interval I1, then an estimated probability of surviving each interval is

• P(T > tj) = 1 − 1

η1 . The estimated survivor function is

• S(tk) =

k

• j=1
• 1 − 1

ηj

• .

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If the number of failures within an interval is greater than one, then the atuarial estimation may be modified: Let di be the number of terminations in interval Ii.

• S(tk) =

k

• j=1
• 1 − dj

ηj

• .

The Kaplan-Meier non-parametric maximum likelihood estimator is an extension of this actuarial construct.

• S(t) =
• ti<t
• 1 − dj

ηj

• .

Here t1 < t2 < · · · < t are the actual times at which individuals experience the event: di = number of individuals exiting at the ith event time ηi = number of individuals at risk at the ith time

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Given the non-parametric likelihood, in order to perform hypothesis testing, we need a standard deviation as a function of time. Greenwood’s formula gives

• γ(t) =

S(t)

• kt
• j=1

d0 (η − j)(η − j + 1) kt = value of k such that t ∈

• t(k), t(k+1)
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The Aalen estimator of integrated hazard is a good descriptive device which uses the same technique used in procedures for estimating multi-state transition rates which may involve complicated time dependence. Where the survivor function is S(t) = exp

• −

t h(u) du

• .

A procedure to estimate the integrated hazard sums the ratio of exiting individuals to the number remaining at risk at each event time:

• t

h(u) du =

• i=ti<t

di ηi

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Let S(t) be the Kaplan-Meier estimator. The relation of the Aalen estimate to S(t) may be seen by the following transformation: − ln

• 1 − di

ηi

• =
• i=ti<t

ln

• 1 − di

ηi

• =
• t

h(u) du

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Each term of this Kaplan-Meier integrated hazard, when written as a series expansion is − ln

• 1 − di

ηi

• = di

ηi − d2

i

2η2

i

+ d3

i

6η3

i

− · · · . The Aalen estimator ignores the higher order terms of this expansion. If the number at risk is large, most of the weight in this series does indeed fall on the first term alone. The Aalen estimator also tends to correct the bias introduced by the nonlinear transformation − ln(S(t)).

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In failure time models, once an individual experiences the event, he is out of the pool of individuals at risk for the rest of the survey. In the following example taken from economics, transitions between unemployment, employment and being out of the labor force may be repeated any number of times. The Aalen estimator may be used in this context as well in testing a hypothesis regarding the transition rates between states.

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Given three employment states, there are six possible transition rates.

U E O

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Define the following event rates: rU,E(t) = expected number of unemployment to employment transitions per unit time, per individual at risk. rO,E(t) = expected number of transitions from out of the labor force to employment Flinn and Heckman pose the question of whether unemployment and out of the labor force should be designated as separate classifications.

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To test this, they examine the hypothesis H0 : rU,E(t) = rO,E(t). First, the assumption that transitions depend on past history is made. Suppose data available gives a counting process over the entire population. The six cumulative counts are N

e (t) = (NU,E(t), NN,E(t), NE,U(t), . . .)

Note that individuals may appear in these counts more than once.

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The Aalen estimator for the integrated transition rate is t rU,E(u) du =

• k:0≤tk<t

dk YU(tU,E

(k) )

where YU = the number of individuals in state U at time t. Although YU need not decline monotonically with multistate flows, it is still the number of individuals at risk in state U.

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The test relies on the query, “how frequently do events occur per time period per individual?” The event times of the two transition patterns do not have to match, but having complete event history data is crucial.

O, E s1 s2 s3 sk . . . U, E t1 t2 t3 tk . . .

The test asks whether t rU,E(u) du =

• k:0≤tk<t

dk YU(tU,E

(k) )

is equal to t rO,E(u) du =

• k:0≤tk<t

dk YO(tO,E

(k) )

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If event history were not available, and instead prospective point data were, then multiple intermediate transitions would be unobservable. To infer what jumps occurred between observed points, on might try to fit a Markov or semi-Markov process.

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Estimation of Separable Hazard Models In this section, the problem of estimating the form of time dependence is addressed. Specifically, the sensitivity of the time dependence estimate to the form of the distribution of unobserved heterogeneity assumed and to the parametrization of the time dependency is examined. Finally, the nonparametric method of the EM algorithm is presented as an alternative to the standard maximum likelihood methods.

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If the survivor function is of the separable form S(t | x) = [S0(t)]

exp X

fβ

then partial likelihood estimation will yield a ˆ β estimate, and parametric specifications of the time dependence, such as h0(t) = αtα−1 or h0(t) = eαt, where S0(t) = exp

• −

t h0(u) du

• may be estimated with standard techniques.

Given the variety of functional forms that might be chosen for the time dependence, on would perhaps like to data to speak for itself, in the absence of any theory on time dependence.

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A nonparametric approach would proceed in the following fashion: Choose a priori a set of time intervals that need not correspond to jump times associated with occurrences of transitions. They must satisfy the condition that they are long enough to be non-empty of events. h0(t) =          eα1 for 0 ≤ t < t1 eα2 for t1 ≤ t < t2 . . . eαk for tk−1 ≤ t < tk

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This places no restrictions of monotonicity on the number of modes for the hazard function. The integrated hazard will be t h0(u) du =

k

• j=1

eαj (tj − tj−1) for a stepwise hazard, as in the graph below:

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Stepwise Hazard Function

1 2 3 4 5 6

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The estimated hazards are αk = ln dk = ln

• i∈Rk

δi exp

• X

e iβ e

• where

dk = number of individuals who experience the event

• f interest in the half open interval [tk−1, tk)

Rk = set of individuals at risk in the interval [tk−1, tk) δi =1 if the individual is uncensored. Again, δi points out that non-censored individuals provide information regarding time dependence.

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If the estimated ˆ h0(t) is of the form in the graph (A), which appears unimodal, this would support a specification of h0(t) = λα (λt)α−1 1 + (λt)α for α > 1.

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(A) Estimated ˆ h0(t)

h(t) t

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(B) Estimated ˆ h0(t)

t

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Survivor Analysis and GMLE A brief look at the theory underlying generalized maximum likelihood follows. (See Kiefer and Wolfowitz.) Define dµ(x) to be a dominating measure if for dP0(x) = f (x) dµ(x), µ(A) = 0 ⇒ implies that P0(A) = 0. Let P be the measure of all probability measures. For every pair of probability measures P1 and P2 in the class P, define f (x; P1, P2) = dP1(x) d(P1 + P2)(x)

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Radon-Nikodym derivative Here, f (x; P1, P2) plays the role of the likelihood. The measure ˆ P is a generalized maximum likelihood estimator if f (x, ˆ P, P) ≥ f (x, P, ˆ P)

• r

d ˆ P(x) d(ˆ P + P)(x) ≥ dP(x) d(ˆ P + P)(x) .

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Let T1, T2, . . . , Tn be the times an event of interest occurs in a population surveyed. To extend the analysis, we now allow a censoring of the data in a random manner. Let C1, C2, . . . , Cn be the censoring times: this is equivalent to an individual dropping out of a sample population before experiencing the event in question.

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The observable is Yi = min(Ti, Ci). At each time Yi, an individual either makes the transition out of the state having experienced the event or drops out from the sample’s observed population “prematurely.” Let the δi variable indicate whether yi is censored or not: δi = if the individual is censored 1 if not Consider the probability distributions on x = ((y1, δ1), (y2, δ2), . . . , (yn, δn)) .

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Finding a generalized maximum likelihood estimator of the hazard rate is the same as finding a P(x) such that the observed events are given the maximum probability: P(x) =

n

• i=1
• Pr(T = y(i))

δi {Pr(T > yi)}1−δi . The first bracketed term is the probability of an event occurring at exactly time T. It is given weight only if the event is uncensored, i.e., when δi = 1. The second term is the probability that the event occurs any time after the time T, which is the most information that a censoring at T yields. It receives weight only if δi = 0.

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More succinctly, the problem is to max

n

• i=1

pi ·

• n
• j=1

pj

• for p1, p2, . . . , pn ≥ 0.

When the number of individuals is finite, the sum of the number

• f events and censoring must also be finite. (The two are equal.)

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When the time partitioning is fine enough so that no two events

• ccur exactly simultaneously, then the solution to the maximum

problem is exactly the Kaplan-Meier estimate: ˆ Pi = δ η − i + 1

i−1

• j=1
• 1 −

δj η − j + 1

• .

A comparison of time dependent rate estimates depends on the assumption of a homogeneous population. Finding differences across covariates requires further investigation.

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Duration Models with Covariates Complicating the underlying process by postulating that it is affected by some covariates (which we assume are observable, for now) leads to gains in estimation efficiency to some specification

• f the form of duration dependence.

The issue which arises, on which statisticians and social scientists are divided, is whether to model the durations (i.e., waiting times) themselves, or to model the rates of exit (the hazard function). Application of a standard regression framework to durations imply complex hazard specifications. Statisticians tend to favor modeling curves to fit the hazard itself with more convenient parameterizations.

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Regression Framework In order to apply linear regression to waiting times, a standard technique is to take a log transformation, mapping non-negative waiting times onto the entire real line. Then a symmetric disturbance term εi may be applied to a regression of log durations: ln ti = β0 +

k

• j=1

βjxji + εi. xji = the value of the jth covariate for individual i. εi = iid, ∼ Φ(0, δ2) where Φ is the normal cdf.

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The theorizing here is at the level of linking the covariates to the expected value of log duration. The hazard is subsumed in this specification, and is worth examining. Although the waiting times are straightforward, the hazard is complex (read “crazy”).

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Let ln T = β0 + X

• ′β
• + ε

T = exp

• β0 + X
• ′β
• + ε
• .

The conditional survivor function is P(T > t | X

• ) = P
• exp
• β0 + X
• ′β
• exp (ε) > t
• = P
• exp (ε) > t exp
• −β0 − X
• ′β
• = P
• ε > ln t − β0 − X
• ′β
• = 1 − Φ
• ln t − β0 − X
• ′β
• .

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The survivor function is S(t) = 1 − Φ(T) = exp

• −

t h(u) du

• .

The hazard my be retrieved as well by differentiation. t h(u) du = − ln

• 1 − Φ
• ln t − β0 − X
• ′β
• h(t) =

1 t φ(ln t − β0 − X

• ′β
• )

1 − Φ(ln t − β0 − X

• ′β
• ).

Another drawback of this method is having to know the completed waiting times ti. Traditionally, one may know only transition times censored in some fashion.

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Consider a More General Approach This approach postulates a general separable hazard specification where the time dependent portion is multiplicatively separable from the portion which varies with X

• , some vector of covariates, i.e.,

h(t | X

• ) = ψ(t)u(x),

where ψ(t), u(x) are non-negative valued functions.

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An example of such a hazard is the Cox specification: h(t | X

• ) = ψ(t) exp(X
• ′β
• )
• r,

exp

• −

t h(u | X

• ) du
• = [S0(t)]

exp(X

• ′β
• ) ,

where S0(t) = exp

• −

t φ(u | X

• ) du
• .

A full maximum likelihood estimator of the hazard and the coefficients on the covariates is constructed as follows.

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Define Di = indicator for individuals who experience an event of interest at time i. Ci = indicator for censored individuals: those who drop out of the sample before they experience an event. By looking at the conditional survivor function, we find the contribution to the likelihood for individuals experiencing the event is

• S0(t(i))

exp(X

• β
• ) −
• S0(t+

(i))

exp(X

• 1β
• )

where t+

(i) denotes the time just after t(i), i.e., t(i) + ∆.

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For a censored individual, the contribution is

• S0(t+

(i))

exp(X

• β
• )

. The likelihood is therefore L =

• ℓ∈Di
• S0(t(i))

exp(X

• β
• )

−

• S0(t+

(i) + ∆)

exp(X

• ℓβ
• )

×

• ℓ∈Ci
• S0(t+

(i))

exp(X

• ℓβ
• )

.

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Following a method suggested by Cox, if one is interested in the covariates, then one might maximize the partial likelihood. This ignores the time-dependent part of the hazard, separating out the changes which enter through the covariates. Given an additional requirement that the covariates X

• be

time-invariant, (see B. Efron). The Cox likelihood proposal is max

β n

• i=1

exp(X

• iβ
• )
• ℓ∈R(t(i)) exp(X
• ℓβ
• )

where R(t(i)) is the number of individuals at risk at time t(i).

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Heuristically, this term may be related to the full maximum likelihood by the following argument. The conditional probability that person (i) experiences an event at t(i) given that R(t(i)) individuals are at risk and that exactly on event occurs at time t(i) is h(t | Xi) = P(t < T < t + δ | T > t) ∼ φ(t) exp(X

• ′

iβ).

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By the definition of conditional probability, this is φ(t) exp(X

• (i))β
• ℓ∈R(t+) φ(t) exp(X
• ℓβ) =
• the fraction with covariates

X

• i who exit in interval (t, t + z)
• total population at risk

as of time t

• Making the assumption that the nature of time dependence is the

same for all individuals, φ(t) cancels out, yielding one element of the Cox partial likelihood objective.

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A Diagnostic Check for Specific Hazards Given the expense of generalized maximum likelihood estimation, it is useful to have simple diagnostic tests of the specification of the time dependence in a duration model. A test of proportional hazards under Weibull time dependence can be performed by a graphical technique: Let S(t | x

• ) = P(T > t | X
• = x
• ) = [S0(t)]

exp(X

• β
• )

Taking a log transformation twice over, we have − ln S(t | x) = − ln S0(t) exp(X

• β
• )

ln(− ln S(t | x)) = ln(− ln S0(t)) + X

• β
• .

(3)

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When the time dependence part of the survivor function is of the Weibull family, then S0(t) = exp(−tα) and the first term of the equation (3) is ln(− ln S0(t)) = α ln t and ln(− ln S(t | x)) = α ln t + X

• β
• .

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Graphing ln(− ln S(t | x)), a “double” log transformation of the survivor function against ln(t) should produce a family of straight lines of the same slope, α. The intercepts should vary according to X

• β
• under this separable

specification of S(t | x). If the graph does not conform then this convenient specification does not apply.

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Double log Transformation of Survivor Function Against ln(t)

ln( -ln( S(t) ) ) ln( t ) slope = V

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More generally, unless parallel curves are generated by plots of ln[− ln S(t | x

• )] against ln t for various chosen values of the

covariates x

• , then the assumption of separable cannot hold.

Example, see J. Menken, J. Trussel, D. Stampel, O. Babokol, Demography, Vol. 18, 1981 pp 181-200 (on marital dissolution)

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A Duration Model from Economic Theory In a search model of unemployment, the Poisson arrival of new job

• ffers and a reservation wage strategy of income maximizing

workers generates observed unemployment spells which have fundamentally non-separable hazards. Heterogeneity across workers is likely to exist either in costs of search or wage offer distributions. This model is outlined here as an example of a duration model based on optimizing behavior by economic agents. A thorough presentation may be found in Lippman & MaCall, (1976).

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Let λ= Poisson encounter rate with new job offers V = value of search rV = reservation wage c = instantaneous cost of search r = instantaneous interest rate F(w) = distribution of wage offers, assumed to have finite mean.

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Agents maximize income subject to the following scheme: If search cost C is incurred, job offers arrive at rate λ independent

• f c.

Wage offers are independently drawn without recall from distribution F(w). Agents are infinite-lived and jobs last forever, having present discounted value w

r .

The value of search is V =

• −c∆t

1+r∆t + 1−λ∆t 1+r∆t V + λ∆t 1+r∆tEmax

w

r , V

• + O(∆t) if V > 0

0 otherwise.

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Passing to the limit, we have c + rV = λ r ∞

rV

(w − rV ) dF(w) and reservation strategy: d =

• 1

if w > rV , job offer accepted if w ≤ rV , job offer rejected. Then the probability that an unemployment spell exceeds duration tu, given hazard rate of exit (acceptance of a job) hu = λ(1 − F(rV )), is Pr(Tu > tu) = exp(−λ(1 − F(rV ))tu).

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Returning to the entire population, with some observed characteristics x and unobserved characteristics θ, we have Pr(Tu > tu | x

• ) =
• Θ

exp [−λ(x, θ)(1 − F(rV (x, θ)))tu] dµ(θ) where Θ =

• θ | 0 < λ

r ∞

rV

(w − rV (θ, x)) dF(w) − c(x, θ) ≤ ω

• Obviously, there is no general separable hazard specification that

emerges. Instead, the hazard and covariates are linked by the solution to the Bellman equation, in which the reservation wage rV is implicitly defined. See Heckman and Singer (1984).

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Duration Models with Unobservables In most applications, the analyst has data on some process, along with some observable characteristics regarding individuals included in the survey. In addition to these, there may be other characteristics of these individuals which are factors affecting the process but which are unmeasured. For example, in the Stanford study of heart transplant recipients, it is likely that each patient differed in some dimension, call it “frailty”, which affected their survival times after receiving transplants. Thus, the survivor function includes an unobservable θ: P(T > t | x

• , θ) = exp(−H(t)U(x
• )V (θ))

= exp(−H(t) exp(x

• ′β
• + θ))

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The statistician has the following issues to address: what estimation strategy to use to obtain ˆ β in the presence of θ, and what estimation can reveal regarding the distribution of θ itself. Even when this distribution of θ is of no interest to the analyst, its presence will affect the consistency of estimation of β. Assuming some form of the mixing distribution of θ, dµ(θ), the data may be confronted with the integrated survivor function, where θ has been integrated out: P(T > t | X

• ) =
• exp(−H(t) exp(X
• ′β
• + θ))dµ(θ)

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Commonly used functional forms for dµ(θ) are gamma and normal distributions Gamma: dµ(θ) =

baθa−1 exp(−bθ) Γ(a)

dθ Normal: dµ(θ) =

exp(−(θ−a)2/2b) √ 2πb

dθ. Both of these offer a flexible, analytically tractable and computationally convenient family of distributions. Both are fully described by the specification of two parameters a, b above.

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For all the claimed convenience of these specifications, along with lognormal variation, they do not all yield the same qualitative estimates for the coefficients on time dependence and observed covariates. Sensitivity of these estimators is apparent in one study, the Heckman and Singer analysis of labor earnings data, but not in the Manton, Stallard and Vaupel study of mortality risks among the aged.

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Mortality Risks Among the Aged

• I. Weibull Parameter Estimates: φ(t) = αtα−1

Age Gamma Heterogeneity Inverse Gaussian No Heterogeneity 65+ 5.89 (.05) 5.88 (.08) 5.44 (.04) 67+ 6.00 (.06) 5.98 (.09) 5.46 (.04) 70+ 6.39 (.07) 6.35 (.11) 5.69 (.05)

• II. Gompertz Parameter Estimates: φ(t) = eγt

Age Gamma Heterogeneity Inverse Gaussian No Heterogeneity 65+ 7.48 · 10−2 (.09 · 10−2) 7.72 · 10−2 (.21 · 10−2) 6.26 · 10−2 (.06 · 10−2) 67+ 7.56 · 10−2 (.10 · 10−2) 7.77 · 10−2 (.23 · 10−2) 6.11 · 10−2 (.06 · 10−2) 70+ 7.97 · 10−2 (.12 · 10−2) 8.13 · 10−2 (.25 · 10−2) 6.18 · 10−2 (.05 · 10−2)

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Although the labor economics study shows qualitative differences in the model and in the duration dependence, the mortality study shows duration dependence that is robust to alternative specifications of the mixing distribution. It has yet to be shown what characterizes a data set which will be robust to alternative heterogeneity specifications.

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Alternatively, in a study of child mortality, the specification of time dependence is shown to be sensitive to the inclusion of unobservables in the estimation strategy. Nonparametric maximum likelihood estimation has been shown to give unbiased estimates of covariate coefficients provided one has chosen a particular form of time dependence in Monte Carlo studies. This is where theory must play an important role.

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0.2 0.2 0.4 0.4 0.6 0.6 w/ unobservables

• est. by NPMLE

unobservables ignored NPMLE unobservables ignored

Weibull Gompertz

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Without strong enough theory to suggest the “time” underlying functional form of time dependence, these studies suggest, the effect of covariates and time dependence cannot be distinguished, even with the use of a nonparametric estimation strategy.

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The following digression on an area of statistics on the extraction

• f “true test scores” focuses on the issues that underlie the

reasons why duration models with underlying heterogeneity need strong restrictions a priori from theory to obtain identification. The demands on the data from models of this sort are far greater than those of regression models.

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True test scores are discussed in Lord and Novick, Statistical Theory of Mental Test Scores (1968). Let X = observed test score ξ = errors, with density hξ u = unobserved true test score, with density gu X = ξ + u f (X) = ∞

−∞

hξ(X − t) gu(t) dt

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Notice that a survivor function of form S(t) =

• k(t | θ) dµ(θ)

is a general form of hξ(X − θ). The density of true test scores gu(t) is the analog of the mixing distribution. The question of both problems is fundamentally the same: When can an observed histogram be decomposed and purged of some noise? J.W. Tukey does so with strong assumptions regarding the densities hξ and gu in “Named and Faceless Values,” Sankhya 1974.

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Heuristically, you want to decompose observed from true test scores, something that can be accomplished only with some already known characteristics of the errors:

• θi = Xi + σ2

Z

σ2

X

(X − Xi) In this linear model, an empirical Bayes estimator θ uses a variance

• btained from a previous study by ETS on errors in test scores to

extract an estimate of u.

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For the survivor analysis, we need a nonlinear version of this. For each individual, identify a normal density with mean θi and variance σ2

i where σ2 i is the variance of θi.

The estimated time score distribution must be extracted from an

• bserved histogram, where each portion of that histogram is one
• bservation on an underlying distribution for that one individual.

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21 2n 22

. . .

0(2 , F )

^ i i 2

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2 2

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The estimated time score distribution is “reflated” by the normally distributed errors: gu(θ) = 1 n

n

• η(

θi, σ2

i ).

Again, the analytical device most useful depends on the question at hand. If each individual realization is important, the histogram is useful. If the population distribution is under discussion, the histogram must be reflated by the errors.

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Nonparametric estimation of the mixing distribution Two approaches to extracting estimates of µ(θ), α and β with a nonparametric approach to µ(θ) are:

1

to use a maximum amount of information to obtain the largest number of jump points. If η is the number of points of increase used then N − η is the number of remaining θ’s available to calculate the probabilities along those discrete levels of dµ(θ).

2

to use a coarse distribution, say with 2 or 3 jumps and recompute the maximum likelihood for additional jump points until the likelihood no longer increases.

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The model, including some simplifying notation, for each individual ℓ yields a probability of experiencing the event at time tℓ is f (tℓ, θ | X

• ℓ) = f (tℓ; λj,ℓ) = αtα−1 λj,ℓ exp(−tα λj,ℓ)

f (tℓ, θ | X

• ) = f (tℓ; λj,ℓ) =

J

• j=1

pj αtα−1 λj,ℓ exp(−tα λj,ℓ) =

J

• j=1

pj f (tℓ; λj,ℓ) λj,ℓ = exp(X

• β
• + θj)

J = number of points of increase in mixing distribution pj = probability of experiencing an event

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For censored individuals S(tℓ | λ

• ℓ) =

J

• pj exp(−tα λj,ℓ)

tℓ = censoring time Since θ is unobserved, the pj’s and λ’s can’t be separated, and the previous method of proportional hazards is invalid.

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Direct Estimation of Time-Dependent Models with Unobserved Heterogeneity The problems of estimating such a model with no a priori restrictions on heterogeneity, such as whether dµ(θ) is unimodal, has finite mean or any other moments, is apparent by examining the absolute simplest model conceivable. Imagine there are only two types of individuals. Their values of the unobservable are θ1 and θ2. Their proportions in the sample population are p and 1 − p.

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The data may then be confronted with an integrated survivor function S(t | X

• ) = p exp
• −H(t) exp(X
• ′β
• + θ1)

+(1 − p) exp(−H(t) exp(X

• ′β
• + θ2))
• There are three parameters to be assigned values relating to the

heterogeneity: θ1, θ2, p. These are in addition to the vector β

• and whatever α is implicit in

the time dependence specifications H(t).

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2 2

1 2

1-p p

2

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For three support points, we have S(t | X

• ) =

3

• k=1

pk exp(−H(t) exp(X

• ′β
• + θ)).

For any n support points, 2n − 1 free parameters are introduced.

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It is then no surprise that in order to distinguish between survivor functions, which are all monotone decreasing functions of time, across separate possible combinations of θ mixtures, Monte Carlo studies indicate that the size of the sample must be on the order

• f 20,000 observations.

One Monte Carlo study used a sample created using a gamma distributed unobservable. The nonparametric maximum likelihood technique was poor at reaching any discrete mixture approaching the shape of the continuous gamma function. For example, the next figure was not forthcoming.

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2 p

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However, the data did yield estimates of the covariates β which were consistent with the actual ones used to create the sample.

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The E. M. (Expectation - Maximization) algorithm involves two separate steps. E-step: p(m+1)

j

= N

ℓ=1 p(m) j f (tℓ;λ(m)

j,ℓ )

PJ

j=1 pjf (tℓ;λ(m) j,ℓ ) = N

ℓ=1 p(m) j

φ(m+1)

j,ℓ

M-step: ˜ L = N

n=1

J

j=1 ln f (tℓ; λj,ℓ)

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The procedure is as follows:

1

Select starting value: (p(0)

1 . . . p(0) J ), (θ(0) 1 . . . θ(0) J ), α(0), β(0)

• .

2

Form φ(1)

j,ℓ = f (tℓ;λ(0)

j,ℓ)

PJ p(0)

j

f (tℓ;λ(0)

j,ℓ).

Substitute into L(1) = ln f (tℓ; λj,ℓ)φ(1)

j,ℓ

and maximize L to obtain α(1), β(1), θ(1).

3

Calculate p(1)

j , a weighted average of p(0) j .

4

Repeat from step 2, with φ(2)

j,ℓ , and L(2) to convergence.

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Demster, Laird and Rubin show this procedure does indeed converge under exceedingly weak assumptions on f . Problems do arise with this technique: the likelihood function tends to have bumps - the convergence may yield a local rather than a global maximum. This technique can be very slow to converge over “flat” portions

• f the likelihood.

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In practice, it makes sense to combine this method with a steepest ascent route. Also, this is an unbalanced problem in that some part (time dependency and observed covariates) is parametric and the other part (unobservables) is non parametric. Consistency of an estimate of m(θ) is available only for huge data sets (10k), whereas convergence is quicker for α and β.

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The recoverability of m(θ) is difficult for even the simplest form of the survivor function. Take S(t | θ) = exp(−θt). Observable data is on S(t) = ∞ exp(−θt) dµ(θ).

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A Laplace transform of the probability distribution µ(θ) is invertible if S(t) is known for continuous time. Very small changes in S(t) will yield large differences in dµ(θ). The analytical problem yields only approximations of S(t) for some time t.

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Using the EM algorithm often yields a mixing distribution with only 3-8 points. A spiked estimated mixing distribution is likely an approximation to a true distribution with more variance:

2 21 22 23 24

the data not being rich enough to distinguish between all such grouped peaks.

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Given a discrete mixing distribution, there is thus a finite numbers of durations: g(t | θ) = θ exp(−θt) (exponential) g(t | θ) = αtα−1 exp(θ) exp(−tα exp θ) (Weibull) ¯ g(t)

(observed)

= ∞

−∞

g(t, θ) dµ(θ) (mixed density)

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An interesting mathematical result on variation diminishing transformations allows the analyst to infer information regarding the mixing distribution from the histogram of the durations. If g(t | θ) exhibits sign regularity, then the number of times that the histogram ¯ g crosses an arbitrary constant c must exceed the number of times that the mixing distribution crosses the same constant function c.

g t c c 2 m(2)

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Formally, sign regularity requires that for all values t1 < t2, θ1 < θ2, either det g(t1 | θ1) g(t1 | θ2) g(t2 | θ1) g(t2 | θ2)

• ≥ 0

Or that the same determinant is less than zero for all t1 < t2, θ1 < θ2. Every member of the exponential family of functions exhibits this property. The variation diminishing property is { number of sign changes in m(θ) − c for θ ∈ R} ≥ { number of sign changes in g −(t) − c for t ∈ R}

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This not only determines the minimum number of modes that m(θ) may have, but gives local information on amplitude, since c may be chosen arbitrarily.

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Tests which are insensitive to the specification of heterogeneity An important question given these results which show sensitivity

• f estimators to heterogeneity specification is what simple tests

are reliable indicators of the basic structural form of the time dependence in a duration model.

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For example, one study by Chahnazarian, Menken and Choe shows that a commonly used practice of picking an arbitrary time segmentation on a duration model in order to run logistic regressions on transitions at say 6 month intervals is quite sensitive to that arbitrary choice. They show that covariate coefficients vary qualitatively depending

• n the choice of time segmentation.

The lesson, one might say, is to devote some effort to modeling the underlying dynamic process before seeking to obtain results on what variables affect that process!

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Thus, some preliminary tests are in order to investigate what forms of switching processes are consistent with the data. First, discuss properties of Markov switching processes in discrete time. Following this, I present a test which can reject a broad class of models all based on the notion of partial exchangeability.

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Finally, this is extended to a continuous time model of a particular form

• f separability, namely where switching times are given by a Poisson

process (i.e. exponential waiting times) but where the switches across states depend only on the current state (Markovian).

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Discrete time switching models The following discussion of Markov processes leads to an indication of the assumptions that must be made in order to test duration models when only prospective point sampling data is available, not event history data. Let the following diagram represent transitions between two states, represented by the two points 0 and 1. The dashed red line is one individual’s history, the solid line is the other’s.

t 1

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The definition of the Markov property is P{X(k∆) = ik | X(k − 1)∆ = ik−1, X(k − 2)∆ = ik−2, . . . , X(0) = i0} = P {X(k∆) = ik | X(k − 1)∆ = ik−1} , for k = 1, 2, . . . k, where k = 0 is the first survey. Time homogeneity requires that P{X(k∆) = 1 | X(k − 1)∆ = ik−1} is the same for all sampling times k.

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Suppose that one only has two surveys. All that may be estimated is P(X(∆) = i1 | X(0) = i0) = mi0,i1. A 2 × 2 transition matrix which summarizes probabilities assigned to the possible transitions (0, 0), (0, 1), (1, 0), (1, 1) at the survey times k = 0, k = ∆: P(0, ∆) = M =

• a

1 − a 1 − b b

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To examine the properties of such a Markov process, assume that we have a discrete time event model where the transition times are identified with the sampling times. Assume that switches occur “just before” the sampling times. Let the transition matrix have the actual values M = 1/4 3/4 5/8 3/8

• .

If switches occurred twice as often, say at times 0, ∆/2, and ∆, then in order for the property of time stationarity to hold, there must be a new “double-time” transition matrix M0 such that M0M0 = M

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Similarly, if time stationarity is to hold with sampling and switching times at 0, ∆/3, 2∆/3, ∆, then M0M0M0 = M For the above value of M, the first condition does not hold true, but the second does. As the number of matrices changes, the existence of the roots changes.

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A Continuous Time Switching Process Assume that two rates describe waiting times for the transitions from state 0 to state 1 and vice versa, and that transition probabilities are independent across spells, P(T0 > s) = exp(−r0s) P(T1 > s) = exp(−r1s).

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A basic test of this specification, as before, is a graph of the log of present of the population surviving, against time: one should observe linear functions.

• ln S(t)
• ln S(t)

r t r t

1

Duration of spell in state 0 Duration of spell in state 1

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The transition rates may be summarized by the matrix R: R = −r0 r0 r1 −r1

• Reducing R to eigenvalue form:

R = H    λ1 · · · ... · · · λs    H−1 exp(∆R) = H    exp(λ1) · · · ... · · · exp(λs)    H−1 Here ∆ = length of time between observations.

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Taking logs, ∆R = ln M

• r

1 ∆ ln M = R. If this data allows an R matrix with real roots, −r0 r0 r1 −r1

• (4)

then this model has probabilistic interpretation M = B    λ1 · · · ... · · · λs    B−1

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In general, ln M = B(ln | λ1 | +i(arg(λj) + 2πk))B−1 where the right hand term is the polar decomposition, and where M =

• a

1 − a 1 − b b

• 1

∆ ln M = ln(a + b − 1) a + b − 2 a − 1 1 − a 1 − b b − 1

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If a + b > 1, then the observations are compatible with a unique continuous time transition model with transition rates r0 and r1 and r0 = (1 − a) ∆ ln(a + b − 1) a + b − 2 r1 = (1 − b) ∆ ln(a + b − 1) a + b − 2 If a + b < 1, then ln(a + b − 1) is complex and there is no probabilistic interpretation of the R matrix.

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Suppose that the transition rates are dependent on some

• bservables x
• :

r0 = c0 exp(x

• β)

r1 = c1 exp(x

• β)

What happens when this form is used on discrete sampling data? P(0, ∆) = exp(∆R) exp

• ∆

−c0 exp(x β) c0 exp(x β) c1 exp(x β) −c1 exp(x β)

• =

m00 m01 m10 m11

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The likelihood is then

N

• i=1

m

δi

00

00 m δi

01

01 m δi

10

10 m δi

11

11

=

N

• i=1

(exp(∆R(x

• β)))δ00

00 (exp(∆R(x

• β)))δ01

01

(exp(∆R(x

• β)))δ10

10 (exp(∆R(x

• β)))δ11

11

Reference: J. Cohen and B. Singer “Malaria in Nigeria: Constrained continuous time Markov models for discrete-time Longitudinal Data” Edited by S. Levin. “Lectures in Mathematics in Life Sciences.” American Mathematics Society pp69-133, 1979.

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Nonstationary Continuous Time Models and Discrete Time Data Here we ask when transition matrix M =

• a

1 − a 1 − b b

• ,

based on observations in two states, is consistent with a non-stationary continuous time model. 1) ∂P(s, t) ∂t = P(s, t)R(t) 2) ∂P(s, t) ∂s = −R(s)P(s, t) P(t, t) = I Pij(s, t) = P(x(t) = j | X(s) = i) for s < t R(t) = −r0(t) r0(t) r1(t) −r1(t)

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Nonstationary Continuous Time Models and Discrete Time Data In the R matrix, r(t) can be any non-negative function of t. Solutions to the differential equations (1) and (2) are conditional probabilities. There exists a P(0, ∆) iff (a + b) > 1. Given this condition holds, a problem with identification still needs to be solved. Assumptions or indications fro theory need to be applied to parametrization.

• Ref. GS Goodman 1970, “An Intrinsic Time Model for Nonstationary

Markov Chains”, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie.

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In the case where continuous event history data is available, the nonstationary Markov property can be tested and the nonstationary may be characterized. First using the language of a counting process. N

• (t) = (N01(t), N10(t))

=    cumulative transitions cumulative transitions from 0 → 1in time ; from 1 → 0 in time 0 ≤ s < t 0 ≤ s < t   

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If the Markov property holds up, then by examining the 0 → 1 transitions, t

• r0(u) du =
• k:tk≤t

1 Y0(t01

k ),

where Y0(t) is the number of individuals at risk in state 0 at time t.

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For separate time periods t0, t1, · · · , this gives rise to an estimated curve as below,

• Z t

r0(u) du

t0 t1 t2 t t

"

which conforms to a Weibull function.

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If t r0(u) du = tα then r0(t) = αtα−1. Notice that the estimation above applies the same Aalen’s theory

• f counting processes presented at the beginning of these notes.

The variance is Var t r(u) du

• =
• k:t0,1

k≤t

• 1

Y0(t(0,1)

k

) 2 . This technique cannot be applied with point sampling data.

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An example of the theoretical restriction that must be imposed with point sampling data follows. It can be of a relatively simple form, as in the study of malaria: the climate’s effect on the mosquito population led to the restrictive presumption that the winter incidence rates of malaria would be distinctly different than the summer’s. R(t) =

• R1

for 0 < t < ∆/2 R2 for ∆/2 < t < ∆ P(0, ∆/2) = exp(∆/2R1) P(∆/2, ∆) = exp(∆R2) M =

• a

1 − a 1 − a a

• =
• α

1 − α 1 − α α β 1 − β 1 − β β

• = M1M2

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In order for the transitions over a year to have a Markovian probabilistic interpretation, a > 1/2, and for the 6 month seasons, a > 1/2, β > 1/2. From the condition that M = M1M2, it follows that a = αβ + (1 − α)(1 − β). Graphically, this restriction requires (α, β) be of a lows of points for each possible a. With data from point sampling available at six month intervals, the estimated ˆ a, ˆ α, and ˆ β can be used to test this restriction.

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Time Aggregation There are dangers in using comparisons between estimated discrete time Markov transitions and continuous time switching probabilities to draw conclusions regarding the dynamics of the process. For example, comparing the sum of the diagonal elements of the matrix Mk to the sum of the diagonals of P(0, k∆) may be interpreted as an examination of those who stay in the originating state over all time periods k and those who begin and end in the same state, but who may make transitions in and out in the intervening periods.

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Since the second condition seems a weaker condition, one might expect the following regularity: tr(Mk) < trˆ P(0, k∆). This cannot be claimed as certainty, since the Markov model applied to discrete sampling points would not capture asynchronous switching times. Suppose that population were not homogeneous, but a simple mixture of “movers” and “stayers”. Stayers making up {s : 0 ≤ s < 1} proportion of the population.

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Then the one period transition probability is P(0, ∆) = SI + (1 − s)M. Over k periods, P(0, k∆) = SI + (1 − s)Mk. In this case, a similar diagnostic check would be whether tr[SI + (1 − s)M]k < tr[SI + (1 − s)Mk] (5)

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Unfortunately a reversal of this inequality can follow simply by changing the time interval ∆ or the size of s. For analytical convenience, consider the 3 × 3 matrix in two variables a and b. M =   a 1 − a b 1 − b a 1 − a   . Here, condition (Eq. 5) is met for k = 3 only if a, b satisfy {(a, b) : a2 − 2ab + b > 1/3}

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In addition, in this range, the reversal occurs when the fraction of stayers, s, exceeds 1/2. Although this is a specific case, such reversals are common. Changing the time scale from, say, six months to a year can also cause a reversal. In continuous time, these reversals do not occur lim

h→0

E(Nij(t + h) − Nij(t) | Ft) h = Λij(t, ω) Λij(t) = rij(t)Yi(t, ω).

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The Markov assumption in continuous time is exp

• −

s+t

s

r1(u) du

• = P(T1 > t | X(s) = 1)

Reference: Singer and Spilerman, “The Representation of Social Processes by Markov Methods”, Am. Journal of Sociology, 82 (1976) 1-54. Hansen and Sargent, “The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities”, Econometrica 1983 (March).

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Testing homogeneity Suppose that in a multi-wave panel study, certain individual histories are available only for a subset of the panels. For example, the Nigerian data on malaria had full dynamics for a half of the total population. An hypothesis for testing would naturally be: “do the individuals who are missing from some panels have the same underlying transition probabilities?”

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Prospective point sampling over two periods gives an estimator ˆ P(0, ∆) =

• a

1 − a 1 − b b

• =
• ηij

ηi+

• ,

where ηij : number of individuals in state i at time 0 and in state j at time ∆. ηi+ : number of individuals in state i at time 0 The criterion for a matrix of transition probabilities to be consistent with the data is a + b > 1. Thus the test of H0 : a + b > 1 is a test that mixtures of Markov models with heterogeneity holds over the two waves of panel data. Let η00 η0+ = ˆ a; η11 η1+ = ˆ b

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Introduce the decision rule for testing H0:

1

if ˆ a + ˆ b > 1 + δ1, then retain H0 as a working hypothesis;

2

If 1 − δ2 ≤ ˆ a + ˆ b ≤ 1 + δ1, then no decision;

3

If ˆ a + ˆ b < 1 − δ2, then reject H0.

The cutoff points are determined by choosing some power of the test α1, α2. The theory for finding δ1, δ2 | α1, α2 is based on the binomial distribution. With a cell size of at least 12, the normal distribution may be used for the binomial one: α1 = Φ   −2δ2

• 1

η0+ + 1 η1+

  Φ ∼ N

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Therefore, δ = −1 2Φ−1(α1)

• 1

η0+ + 1 η1+ . The data is more informative as the sample size, η0+ and η1+ get large. ˆ r0 = ln(ˆ a + ˆ b − 1) ˆ a + ˆ b − 2 (1 − ˆ a) ˆ r1 = ln(ˆ a + ˆ b − 1) ˆ a + ˆ b − 2 (1 − ˆ b) Because of reduced computer costs, re-sampling plans to test an estimate of ˆ r are becoming feasible.

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References

• P. Diaconis and B. Efron, “Computer Intensive Methods in

Statistics”, Scientific American 1983

• B. Efron, “The Jackknife, the Bootstrap & other Resampling

Plans”, Society of Industrial & App. Math. Monograph, 1982

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Test of Exchangeability Given a model with discrete time periods and an index function approach to dealing with unobservables affecting decisions, “exchangeability” tests the compatibility of such a model to event history data on an individual’s sequence of choices. Specifically, Let d(t, i) =

• 1, if Y (t, i) ≥ 0

0, if Y (t, i) < 0 Y (t, i) = β0 + β1z(t, i) + γd(t − 1, i) + ε(t, i) (6)

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The coefficient γ measures the effect of a post decision on the succeeding period’s choice. ε(t, i) = V (i) + u(t, i) where u(t) for t = 0, 1, 2, · · · are iid with a common symmetric distribution F. Given event history data, each individual record is a sequence {i0, i1, i2, · · · ik}, where ij = 0 or 1.

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One hypothesis is that the probability of a run of events is a mixture of Bernoulli trials. This would rule out a nonzero γ in the model (Eq. 6). That is, Y (t) would depend only on current observables. Pi0,i1,··· ,ik = π

Pk

i=0 ij(1 − π)

Pk

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Here, µ(π) is a mixture of probabilities, π = prob(d(t) = 1) for t = 0, 1, 2, · · · . To write this in notation compatible with Eq. 6,

• γ

dm(v(i))

• z

F(−(β0 + β1ξℓ + v))

Pk(1−ij)

(1 − F(−(β0 + β1ξℓ + v)))

Pk ij dµ(ξ)

where ξℓ = value of observed covariate z.

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For a collection of sequences of equal number of switches to be exchangeable, all sequences with the same number of 1’s occur with the same frequency. Obviously, sequences of Bernoulli trials have this property: PA(1 − P)1−A depends only on the count of events, and not upon the

• rder.

Definetti’s sequences are exchangeable, then the mixing distribution in a heterogeneous Bernoulli trial model is uniquely identified. Definetti’s theorem: If sequences of length k = 2, 3, · · · are exchangeable then there is a unique mixing distribution µ(π) such that pi1,i2,i3,··· ,iJ =

• π

PJ ij(1 − π) PJ(1−ij) dµ(π).

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To actually apply this theorem to a model in which the probabilities π depended upon covariates z,

• π(z)A(1 − π(z))1−A dµ(π)

requires taking the integral transform to extract µ(π). This notion of exchangeability also requires arbitrarily long strings

• f sequences.

Thus, a more practical notion is that of partial exchangeability.

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Partial exchangeability is a weaker notion, and requires only that all sequences which belong to the same equivalence class occur with the same frequency. Two sequences of 0’s and 1’s belong to the same equivalence class if they begin and end in the same slate and if they have the same number of each type of intermediate transitions. The order of these intermediate transitions may vary. Thus, {1, 0, 1, 1} and {1, 1, 0, 1} belong to the same equivalence class, and {0, 1, 0, 0} and {0, 0, 1, 0} belong to a separate equivalence class. Partial exchangeability requires P{1, 0, 1, 1} = P{1, 1, 0, 1}

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Definetti’s theorem for Markov Chains For sequence frequences of length k = 2, 3, 4, · · · which are partially exchangeable, there exists a unique Markov chain that defines the data. P{i0, i1, · · · , ik} =

• M

¯ Pµ

i0mi0,i1mi1,i2mi2,i3 · · · mii−1,ik dµ(M)

This integration takes place over the space M of all stochastic matrices with dimension determined by the number of possible states. The data being finite, this theorem is useful only for rejecting the hypothesis that a Markov process generates the data. Constituting a test for 4 × 4 stochastic matrices requires breaking the data up into equivalence classes sequences of 0’s, 1’s, 2’s, 3’s and 4’s.

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Instead of testing each equivalence class separately, linear combinations of the conditions may be constructed. Any violation among any of the equivalence is sufficient grounds for rejecting the Markov property. Note that this theorem does not hold for anything but discrete data over regular intervals. If the time between panels varies, this theorem does not apply. However, with event history data, in continuous time, mi0,i1 has an analogue in exp(∆R). Reference:

• P. Diacouis and D. Freedman, “Definetti’s Theorem for Markov

Chains”, Annals of Probability”, 1981 Bishop, Feenberg and Holland, “Discrete Multivariate Analysis”

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The strategy for testing partial exchangeability is as follows. From the chi-square statistic

CN

• k=1

|F|k,n

• ℓ=1

{(Nℓ,k − N+,k/ | F |k,n)}2 N+,k/ | F |k,n ∼ χ2

2n−Cn−d

where N+,k =

|F|k,nNℓ,k

• l=1

, |F|k,n =number of types of sequences in the equivalence class Nℓ,k =number of observed occurrences of the ℓth type in the kth equivalence class N+,k =the total number of sequences observed from the kth equivalence class Cn =the total number of equivalence classes d =number of degenerate equivalence classes with only 1 member

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Obviously, the best way to proceed is to test partial exchangeability for sequences of shortest length first. Rejection at length k is the same as rejection for all k+j, j > 0. A problem in practice may arise for longer sequences, in that some types of sequences may not be recorded, perhaps due to sampling. If one is not going to reject on the basis of this sampling zero, there are methods of smoothing the data to obtain nonzero counts for every cell. See Bishop, Feinberg and Holland, “Discrete Multivariate Analysis” for their discussion of smoothed contingency tables.

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Stochastic Nonstationarity: A Special Case There are some economic models with fundamental non-stationarity which is an integral part of the theory. For example, in models of consumer response to advertising of a particular product brand, most people switch early in an ad campaign. In a model where use of brand X is defined as state X and several brands are available, there is a way to test whether one kind of time dependence exists. The time process hypothesized is of a special form.

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Let X(k) be a Markov process which, as before, is time homogeneous. This governs movements between states. Let us introduce a separate clock N(t) which counts switching times, where the switching rate is governed by a Poisson process (exponential waiting times) Y (t) = X(N(t)) P(N(t) = k) = λ(t)k k exp(−λ(t)), λ(t) = t h(µ)dµ

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Let M be the transition matrix underlying X(k). We can disentangle the discrete switching process from the time inhomogeneous Poisson process if all the heterogeneity in the population is in transition probabilities of M. The Poisson switching rate, however, must be the same for the entire population and be invariant to the state one is.

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Event History Data Given this fundamental non-stationarity, one cannot hope that picturing an arbitrary interval ∆ at which to examine the data will display the Markov properties examined by the partial exchangeability test.

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Instead, the non-stationarity can be addressed in the following way. Look at the number of transitions in the entire population from 0 to t, H(t) = t h(u) du where P(N(t) = k) = H(t)k

k!

exp(−H(t)). H(t) = E{N(t)} = the expectation of the number of jumps in the non-stationary Poisson process up until time t, the integrated hazard.

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A plot of the estimate of H(t) looks perhaps like the graph below, where the transition rate is higher at the start and levels off slowly:

H(t) ^ t

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The probability of an entire history of states (i0, i1, · · · , in), with transitions which occurred at times (0, t1, · · · , tn) Prob(X(0) = i0, X(∆1) = i1, · · · , X(∆n) = in) = ρi0 exp[(t1 − t0)(M − I)]i0,i1 exp[(t2 − t1)(M − I)]i1,i2 · · · exp[(tn − tn−1)(M − I)]in−1,in

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In order to address the data, however the times cannot be chosen arbitrarily. Instead, a simple inversion redefines the measure of the “clock” to shorten intervals where many switches occur and lengthen them when few occur. Taking ˆ H(t) as the guide, define sk = H−1(tk) H−1 being a monotone increasing function. Thus, over the new segmentation sk, the expected number of jumps is constant:

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This transformation returns the data to a “stationary” environment, and the previous discussion of partial exchangeability applies. For panel data, a similar procedure can be used, however, it requires at the very minimum, four panels of data. A detection scheme for this separable Poisson non-stationarity would be as follows. Where the panel data is available at intervals ∆, estimate H(K) for all available points.

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Notice we can write ˆ H(k∆) = − ln det ˆ P(0, k∆) P(0, k∆) = exp(λ(k∆)(M − I)) Since the transition probability matrix P is exactly the probability

• f exiting any given starting state over the interval (0, k∆).

For the model to be validated by the data, the following equalities must be satisfied up to sampling variation. for ˆ P(0, k∆) = exp(ˆ λ(k∆)(M − I)) 1) H−1(∆) ln ˆ P(0, ∆) + I ≈ M 2) H−1(2∆) ln ˆ P(0, 2∆) + I ≈ M . . . k) H−1(k∆) ln ˆ P(0, k∆) + I ≈ M

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More general non-stationary specifications have yet to be studies. Surveys of panel data will, in general, not be sufficient to handle mixtures of non-stationary Markov processes, e.g., Y (t) = X(Nα(t)) where α is distributed in the population with density dµ(α). P(0, k∆) =

• M

exp (λα(k∆)(M − I)) dµ(α). This heterogeneity in the population requires that a different rate

• f occurrence λα must be extracted for every member of the

population. see Julian Kielson, “Markov Chain Models-SPELLCHECK and Exponentiality”, Springer-Verlag, New York, 1979 and Kielson’s (1979) discussion of uniformization.

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A duration model generated by a diffusion process is a second form

• f stochastic dynamics which is relatively easy to link to data.

Asset price paths are one example where diffusions have been used to describe observed data. Although analytic solutions to such a stochastic difference equation can be quite difficult to obtain, where exit times from an initial slate are theoretically generated by the first passage of such a price path over a given value, say a reservation price, the distribution of waiting times is straightforward. The solution to the diffusion equation in its entirety is unnecessary for the analysis of the resulting observed waiting times.

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First Passage Times-An Application of Stochastic Dynamics Suppose that an unemployment spell ends at the first time that wage time series exceeds a fixed reservation wage. The fixed reservation wage is an assumption that is made for simplifying the presentation. The duration of an unemployment spell is Tx,θ = min(t : X(t) = θ) where x = wage, θ = reservation wage.

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X(t) T 2

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P(Tx,θ > t) depends on the movement of the offered wage process X(t), since θ is fixed for any particular individual. If m(θ) is the population mixing distribution, ∞ P(Tx,θ > t)m(θ) dθ = P(Tx > t) X(t) is assumed to be a diffusion process where realizations are solutions of a stochastic differential equation: dX(t) = σ(X, t) dW + b(X(t)) dt X(t) = x0 + t σ(X(u)) dW (u) + t b(X(u)) du

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The second component of the right hand side describes Brownian motion, the third term a “drift” over time. Brownian motion requires that increments W (tk) − W (tk−1) be independent of each other, with normality.

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Distribution theory offers useful results for integrated normal distributions. Time dependence introduced as a “drift” coefficient does not disturb these results. P {W (t1) ∈ [A1, B1], W (t2) ∈ [A2, B2], · · · , W (tk) ∈ [Ak, Bk]} = Bk

Ak

Bk−1

Ak−1

· · · B1

A1

exp(y 2

1/2σt1)

√2πσ2t1 · exp((y2 − y1)2/2σ(t2 − t1))

• 2πσ2(t2 − t1)

· · · · dy1 dy2 · · · dyk

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σ2(x′) = lim

h→0

E((x(t + h) − x(t))2 | x(t) = x′) h = the conditional “local” variance b(x′) = lim

h→0

E(x(t + h) − x(t) | x(t) = x′) h = the conditional mean, interpreted as “drift” Introduce u = u(x, t), a function of spatial position depending on x and t. du dt = σ2(x) d2u dx2 + b(x) du dx u(x, t) =

• p(t, x, y)f (y) dy

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Green’s function, P(X(t) = y | (X(0) = x0)) = exp((y − x)2/2tσ) √ 2πσ2 gives the probability of seeing X(t) = y, given an initial starting value of the process X. With Brownian motion, this is simply the normal density.

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A test which rejects all diffusion processes as underlying the given data is the property total positivity. That is, every diffusion process may be written as an integral P(Tx,θ > t) = ∞

t

q(s, x, θ) ds where q has the following property. For any two ordered time sequence s1 < s2 < s3 · · · < sk and t1 < t2 < t3 · · · < tk, the condition (−1)

m(m−1) 2

det    q(s1 + t1 | x, θ) · · · q(s1 + tk | x, θ) . . . . . . q(sk + t1 | x, θ) · · · q(sk + tk | x, θ)    ≥ 0 must hold for all s, t sequences.

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To apply this notion, let g(t) = − dP

dt (Tk > t) be the density of

• bserved durations.

Then g must satisfy sign regularity conditions if it is a duration density for first passage times of a diffusion with positive drift. det g(s1 + t1) g(s1 + t2) g(s2 + t1) g(s2 + t2)

• for ordered time intervals t1 < t2, s1 < s2, or the same determinant

must always be negative. Using a histogram of observed durations, ˆ g(t), to estimate g(t),

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All possible combinations may be computed quite simply without ever having to manipulate the stochastic differential equation itself. In fact, an equivalent form of the condition is for intervals I1, I 2, I3, and I4 is ˆ g(I1)ˆ g(I4) ˆ g(I2)ˆ g(I3) < 1 These are the odds ratio from a contingency table. Confidence intervals from the standard normal distribution apply.

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