Bayesian Estimation of Discrete Duration Models

Michele Campolieti

A thesis subrnitted in conforrnity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Economics University of Toronto

O Copyright by Michele Campolieti 1997

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Abstract

Bayesian Estimation of Dimete Duration Models Doctor of Phiiosophy Michele Campoliai

Department of Economics University of Toronto

1997

This thesis is comprised of three chapters which discuss Bayesian estimation of discrete

tirne duration models.

n i e fist chapter presents the multiperiod probit model and discusses estimation

with a Gibbs sampler with data augmentation. As an empirical illustration, the multiperiod

probit model is used to estimate a duration model using employrnent duration data for

New Brunswick, Canada. The results from Bayesian estimation are compared with

maximum likelihood estimation of a logit hazard model. Bayesian estimation of a model

with unobserved heterogeneity is s h o w to be a simple extension of a estimation of a

mode1 with no unobserved heterogeneity.

The second chapter discusses parametric and nonparametric specifications of

duration dependence. 1 then propose an alternative to these specifications that can capture

the features of both specifications. 1 employ a Shiller smoothness prior to restrict the

curvature of the parameters in the nonparametric duration dependence specification and

hence put restrictions on the shape of the badine hazard. The smoothness pnor is used to

help 'filter' the 'noise' that comrnonly appears in estimates of the baseiine hazard. The

methods are illustratecl in an empiricai exercise with employrnent duration data from the

Canadian province of New Brunswick. The estimates with the smoothness prior are

compared with the competing alternatives to modehg duration dependence.

In the third chapter a Bayesian estirnator for a discrete time duration model is

proposed which incorporates a nonparametric specification of the unobserved

heterogeneity distribution through the use of a Dirichlet process prior. This estimator

offen distinct advantages over the nonparametric maximum iikelihood estimator

(NPMLE) of this model. First it allows for exact final sarnple inference. Second, it is easily

estimated and mixed with nonparametric specifications of the baseline hazard. An

application of the mode1 to employment duration data fiom the Canadian province of New

Brunswick is provided.

iii

1 mua fkst thank my parents Guiseppe and Rosaria for dowing me the

opportunity to concentrate on my shidies.

I owe my next debt to my Ph.D. cornmittee. In particular, 1 thank Professor Gary

Koop for his constant and u n w a v e ~ g support and encouragement. 1 also owe a great

debt to Professor Dale Poirier for agreeing to take me on at a late stage of my work.

My thesis was begun and completed at the Institute for Policy Analysis. The

Institute provided me with the resources and faciiities most graduate students just dream

about. The faculty and staffat the Institute made coming in to work fun and productive. In

particular 1 thank the two directors of the Institute during my stay, Professors James

Pesando and Frank Mathewson, and Sharon Bolt.

1 would like to thank my fiiends in the department: Dajiang Guo, Walid Hejazi,

Jack Parkinson and William Ma; and al1 my fiends back home: Frank and Tony Nicodemi,

Tony Martino, Tony Verduci, Gianni, Ivano, Lemy, Rob, Rico and Vito, for their support

and fnendship over the last several years.

TabIe of Contents

List of Tables

List o f Figures

Chapter 1

Bayesian Estimation of Duration Models: An Application of the

Introduction

The Likelihood Function and Maximum Likelihood Estimation

Bayesian Estimation

1 -3.1 Gibbs Sarnpling

1 -3 -2 Bayesian Estimation Without Unobserved Heterogeneity

1.3.3 Bayesian Estimation Wit h Unobsewed Heterogeneity

1.3 -4 Evaiuation o f Numerid Accuracy

The Data

Empirical Resul t s

Concluding Remarks

Bibliogr aphy

Chapter 2

Flexible Duration Dependence Modelling in Hazard Models

Introduction

Shiller's Smoothness Prior

Application to Modelling Duration Dependence

Empirical Illustration

2.4.1 Data

2.4.2 Empincal Results

Concluding Remarks

Bibliography

Chapter 3

Bayesian Semiparametric Estimation of Discrete Duration Models: An

Application of the Dirichlet Process Pnor

3.1 Introduction

3.2 The Dirichlet Process Prior

3.3 The Mode1

3 -3.1 The Gibbs Sampler With a Multivariate Normal Prior

3 -3.2 The Gibbs Sampler With the Dirichlet Process Prior

3.4 Empincal Illustration

3 -4.1 The Data I l l

3.4.2 EmpiridResdts

3.5 Concluding Remarb

Bibüograp hy

vii

List of Tables

Chapter 1

Table 1 : Description of Variables Table 2a: Multiperiod Probit Specification 1, Duration Approach,

No Random Effects, Parametric Duration Dependence Table 2b: Multipenod Probit Specification 1, Insured Weeks Approach,

No Random Effects, Pararnetric Duration Dependence Table 3a: Multipenod Probit Specification 2, Duration Approach,

No Random Effects, Pararnetric Duration Dependence Table 3b: Multiperiod Probit Specincation 2, Insured Weeks Approach,

No Random Effects, Parametric Duration Dependence Table 4a: Maximum Likelihood Estimates for Logit Model,

Specification 1, Duration Approach Table 4b: Maximum Likelihood Estimates for Logit Model,

Specification 1, Insured Weeks Approach Table 5a: Maximum Likeliiood Estimates for Logit Model,

Specification 2, Duration Approach Table 5b: Maximum Likelihood Estimates for Logit Model,

Specification 2, Insured Weeks Approach Table 6a: Multiperiod Probit Spdca t ion 1, Duration Approach,

Random Effects, Parametric hiration Dependence Table 6b: Multiperiod Probit Specification 1, Insured Weeks Approach,

Random Effects, Parametnc Duration Dependence Table 7a: M-ultiperiod Probit Specitication 2, Duration Approach,

Random Effects, Parametnc Duration Dependence Table ïb: Multiperiod Probit Specification 2, Insured Weeks Approach,

Random Effects, Parametric Duration Dependence Table 8a: Multiperiod Probit Specification 1, Duration Approach,

No Random Effectg Nonparametric Duration Dependence Table 8b: Multipenod Probit Specification 1, Insured Weeks Approach,

No Random Effects, Nonparametnc Duration Dependence Table 9a: Multiperiod Probit Specification 2, Duration Approach,

No Random Effects, Nonparametric Duration Dependence Table 9b: Multiperiod Probit Specification 2, Insured Weeks Approach,

No Random Effects, Nonparametric Duration Dependence Table 10a: Multiperiod Probit Specification 1, Duration Approach,

Randorn Effects Nonparametric Duration Dependence Table lob: Multiperiod Probit Specification 1, insured Weeks Approach,

Random Effects, Nonparametric Duration Dependence Table 1 la: Multiperiod Probit Specification 2, Duration Approach,

Random Effects, Nonparametric Duration Dependence

viii

Table 1 1 b: Multiperiod Probit SpeCincation 2, Insured Weeks Approach, Random Effects, Nonparametric Duration Dependence 40

Table 12: Hazsrd Esthates, Duration Approach, No Unobserveci Heterogeneity 41

Table 13: Hazard Estimates, Insured Weeks Approach, No Unobserved Heterogeneity 42

Table 14: Hazard Estimates, Duration Approach, Unobserveci Heterogeneity 43

Table 15: Hazard Estimates, Insured Weeks Approach, Unobsewed Heterogeneity 44

Table 1 6 : Characteristics of Individuals 45 Table 1 7: Predicted Probabilities for Representative

Individuals at Selected Durations 46

Chapter 2

Table 1 : Description of Variables Table 2: Descriptive Statistics for the Data and Durations Table 3: Posterior Means of Eligibility Variables for

Alternative Duration Dependence Specifications Table 4: Posterior Means of Eligibility Variables with

the Shiller Smoothness Prior

Chapter 3

Table Table Tabf e

Table

Table

Table

Table Table

1 : Description of Variables 2: Descriptive Statistics for the Data and Durations

3 a: S pecification 1, Multivariate Normal Pior on Heterogeneity Parameters, Duration Approach, Parametric Duration Dependence

3b: Specfication 1, Multivariate Normal Prior on Heterogeneity Parameters, Insured Weeks Approach, Parametric Duration Dependence

4a: S pecification 2, Multivariate Normal Prior on Heterogeneity Parameters, Duration Approach, Parametric Duration Dependence

4b: Specification 2, Multivariate Normal Pior on Heterogeneity Parameters, Insured Weeks Approach, Parametric Duration Dependence

5 : Posterior Variance of Random Effects 6a: Specification 1, Multivariate Normal Prior on Heterogeneity

Paramet ers, Duration Ap proach, Nonpararnetric Duration Dependence

Table 6b: SpeCincation 1, Mdtivariate NoRnal Prior on Heterogeneity Parameters, Insured Weeks Approach, Nonparametric Duration Dependence

Table 7a: Spedication 2, Multivariate Normal Prior on Heterogeneity Panuneters, Duration Approach, Nonparametric Duration Dependence

Table 7b: Specification 2, Multivariate Normal Prior on Heterogeneity Parameters, Insureci Weeks Approach, Nonpararnetric hiration Dependence

Table 8a: Specification 1, Dirichlet Process Prior on Heterogeneity Pararneters, Duration Approach, Parametric Duration Dependence

Table 8b: Specification 1, Dirichlet Process Prior on Heterogeneity Parameters, Insued Weeks Approach, Parametric Duration Dependence

Table 9a: Specification 2, Dirichlet Process Prior on Heterogeneity Parameters, Duration Approach, Parametric Duration Dependence

Table 9b: Specification 2, Dirichlet Process Prior on Heterogeneity Parameters, Insured Weeks Approach, Parametnc Duration Dependence

Table 10a: Specification 1, Dirichlet Process Prior on Heterogeneity Pararneters, Duration Approach, Nonparametnc Duration Dependence

Table 1 Ob: Specification 1, Dirichlet Process Prior on Heterogeneity Parameters, Insured Weeks Approach, Nonpararnetnc Duration Dependence

Table 1 la: Specification 2, Dirichlet Process Prior on Heterogeneity Parameters, Duration Approach, Nonparametnc Duration De pendence

Table I 1 b: Specification 2, Dirichlet Process Prior on Heterogeneity Pararneters, Insured Weeks Approach, Nonparametnc Duration Dependence

Table 12: Postenor Variance of Random Effect, Pararnetric Duration Dependence

Table 13: Posterior Variance of Random Effect, Nonparametnc Duration Dependence

Table 14: Hazard Estimates, Duration Approach, Multivariate Normal Prior on Heterogeneity Parameters

Table 15: Hazard Estimates, Insured Weeks Approach, Multivariate Normal Pno r on Het erogeneity Paramet ers

Table 16: Hazard Estimates, Duration Approach, Dirichlet Process Pnor on Heterogeneity Parameters

Table 17: Hazard Estimates, Insured Weeks Approach, Dirichlet Process Prior on Heterogeneity Parameters

Table 18: Estimates of a Table 19: Estimates of a for sensitivity analysis

List of Figurer

Chapter 1

Figure 1 - 1 : Plot of the Empirical Hazard Function for the Employment Duration Data

Figure 1-2: Plot of the Empirical Survivor Function for the Employment Duration Data

Figure 1-3: Plot of the Empirical Hazard Function for the Employment Duration Data by Year

Figure 1 -4: Predicted Hazard, Representative Individual 1, Parametric Duration Dependence

Figure 1-5: Predicted Hauvd, Representative Individual 2, Parametric Duration Dependence

Figure 1-6: Predicted Hazard, Representative Individuai 3, Pararnetric Duration Dependence

Figure 1-7: Predicted Hauud, Representative Individuai 4, Parametnc Duration Dependence

Figure 1-8: Predicted Hazard, Representative Individual 1, Nonparametric Duration Dependence

Figure 1-9: Predicted Hazard, Representative Individual 2, Nonparametric Duration Dependence

Figure 1 - 10: Predicted Hazard, Representative Individuai 3, Nonpararnetric Duration Dependence

Figure 1 - 1 1 : Predicted Hazard, Representative Individual 4, Nonpararnetric Duration Dependence

Chapter 2

Figure 2- 1 : Plot of the Empirical Hazard Figure 2-2: Estimated Hazard, Step Specification 1 Figure 2-3 : Estimated Hazard, Step Specification 2 Figure 2-4: Estimated Hazard, Step Specification 3 Figure 2-5: Estimated Hazard, Step Specification 4 Figure 2-61 Estimated Hazard, 4th Order Time Polynornial Figure 2-7: Estirnated Hazard, Smoothness Prior d=l Figure 2-8: Estimated Hazard, Smoothness Prior d=4 Figure 2-9: Estimated Hazard, Smoothness Pnor d=7

Chapter 3

Figure 3- 1 : Plot of the Empirical Hazard by Year

Figure 3-2: Plot of the Estimated Hazard, Specification 2, Eligibility Determineci by lnsured Weeks

Figure 3-3: Plot of the Estimated Hazard, Specification 2, Eligibitity Detennind by hiration

Figure 3-4: Plot of the Estimated Hazard, Specincation 2, Eligibility Determined by Insured Weeks

Figure 3-5: Plot of the Estirnated Hazard, Specification 2, ELigiility Determin4 by Dwation

Figure 3-6: Postenor of Alpha, Parametric BaseIine Hazard, Insured Weeks Approach

Figure 3-7: Postenor of Alpha, Parametric Baseline Hazard, Duration Approach

Figure 3-8: Posterior of Alpha, Nonparametric Baseline Hazard, Insured Weeks Approach

Figure 3 -9: Posterior of Alpha, Nonparametric Baseline Hazard, Duration Approach

Chapter

Bayesian Estimation of Duration

Models: An

Multiperiod

Application of the

Introduction

Duration analysis is concerned with the study of transition between states. Examples

include the duration of employment or unemployment spells, time between trades in

hancial markets and the duration of wars. Interest centers on the conditional probability

of exiting a state; for exarnple, what is the probability that you will be employed on

the 10th day given that you have already been unemployed 9 days. Both continuous

and discrete time models are used to estimate duration models.' In this paper the

transition between employment and unemployment will be modeled as a discrete process.

A cornrnon choice for the functional form of the exit probability is the logit cumulative

density function (CDF), primarily because maximum likelihood estimation is straight

forward because the loglikelihood function is globally concave and because analytical

'For a discussion of continuous versus discrete time modelling see Heckman and Singer (L984a) and Lancaster (l9W).

expressions are available for the probabilities. This paper uses an alternative choice

for the functional form of the exit probability, the cumulative normal distribution, and

estimation is done with Bayesian methods.

The posterior distribution for this choice of continuation probability, the multipenod

probit model, cannot be analyzed directly. However, when the posterior distribution

is augmented with latent data it is relatively eary to sample the conditional posterior

distributions. A Gibbs sampling algorithm with data augmentation can then be applied

to generate s series of draws from the conditional posterior distributions that will converge

to draws from the joint postenor distribution. The Gibbs sampling approach has the

advantage that it allows for exact small sample inference and results can be obtained to

any desired degree of accuracy by varying the number of draws taken.

In any study of duration data an important issue to address is unobserved hetero-

geneity. Unobserved heterogeneity refers to the differences in the distributions of the

dependent variables that remain even after controlling for the effect of observable vari-

ables. This heterogeneity can arise from two sources; the mispecification of the functional

form and unobservable variables such as motivation. For example, if you were studying

the duration of unemployment spells, more motivated individuals may be more likely to

exit unemployment more quiddy because they put a lot more effort into the search for a

new job. Unobserved heterogeneity is an issue that is important to econometricians and

applied economists because it can lead to biased and inconsistent estimates (see Neu-

mann (1995)). Thus ignoring unobserved heterogeneity can lead to spurious inference for

the controls for observable heterogeneity that are present in the model.

In the multiperiod probit framework presented in this chapter unobserved hetero-

geneity can be modelled by one of two methods. Either including individual specific

random effects on some subset of the covariates (the method implemented in this paper)

or by allowing ali the parameters that are estimated to Vary across al1 households. The

marginal posterior distribution of the random effects (or the random coefficients) for each

individual can then be exarnined to determine the extent of the heterogeneity present.

Another advantage of using a Gibbs sampler is that the move from modeling without

u n o b s e d heterogeneity to including unobserved heterogeneity oniy requires that more

distributions be added to the Gibbs sampling algorithm.

Some studies applying Gibbs sampling methods to the analysis of discrete economic

choices over many time penods have already begun to appear. Geweke, Keane and

R u d e (1994) present a simulation study of the multinomial multiperiod probit. Rossi,

Meculloch and Allenby (1993) use the multinomial probit to study the direct target

marketing. McCdoch and Rossi (1994) app1y the multinomial probit model to brand

choice as well as briefly discussing the multiperiod probit.

This chapter discusses Bayesian es tirnation of a parametric unobserved heterogenei ty

distribution. Unobserved heterogeneity is modelled by induding a vector of Gaussian

random effects. Whiie the likelihood function for a model with random effects can be

evaluated numerically with quadrature. The implementation of the model becomes more

dificult as the number of random effects in the model is increased. By increasing the

number of random effects that are included the order of the integration that is necessary to

evaluate the likelihood function increases and so makes quadrature less feasible. However,

for the Bayesian increasing the number of random effects that are present only means

that the random effects will have to be sampled from a multivariate distribution rather

than a univariate distribution.

The chapter proceeds as follows. Section 2 discusses choices of iùnctionsl form for

the continuation probability and maximum likelihood estimation with and without un-

observed heterogeneity present. Section 3.1 discusses Gibbs sampiing and data augmen-

tation. Section 3.2 presents the Gibbs sampling algorithm for the multiperiod probit

without trying to account for the effect of unobserved heterogeneity. Section 3.3 presents

the Gibbs sampling algorithm for the multiperiod probit with random effects. Section

3.4 discusses some of the diagnostics that have been developed to measure numerical ac-

curacy of Gibbs sampling estimates. The multiperiod probit mode1 and the logit model

are both applied to study employrnent duration data from the Canadian province of New

Brunswick. Section 4 contains a short discussion of the data used and Section 5 presents

the results kom mmcimum likelihood estimation of the logit model and from the Gibbs

sarnpler algorithm for the multiperiod probit. Section 6 contains concluding remarks.

The Likelihood and Maximum Likeli-

hood Estimation

In discrete tirne duration models before one can construct the likelihood function a func-

tional form for the continuation probability At must be selected. One of the most common

choices in the empincal literature is the logit cumulative distri bution f~nct ion:~

Another possible choice is the cumulative normal distribution function:

Once a specification for the continuation probability has been selected the likelihood

function can be constructed. The contribution of each household or spell to the likelihood

for a completed speil is given by

where Th is the length of the spell for individual h. This is the product of the continu-

ation probabilities for the Th - 1 periods the individual survives rnultiplied by the exit

-

2A Bayesian anabis of the logit model specification is possible but numerical techniques must be used. Either the Tierney-Ksdane approximation or Monte Cerlo Inkgration with Importance sampling, with a rnultivariate t-distribution as an importance fuction, are feasible for this model. T h e numerical methods have been suassfully applied by Koop and Poirier (1993) to the multinomial logit model.

probability for the last period. For a censored or uncompleted spell the contribution to

the likelihood function is given by

where Tc the censoring point. Here the contribution to the likelihood function is just the

product of the continuation probabilities for the Tc periods the individual survives. The

loglikelihood function for the sample can be written as

where C is the set of completed spells and h indexes households. Because the loglikelihood

function for the logit specification is globally concave maximum likelihood estimation is

straightforward.

Unobserved heterogeneity is introduced in the logit specification (see Nickel (1979))

by including random intercepts el, ..., ON. which are an i.i.d. draw from a discrete

distribution with Ne points of support and associated probabilities pl, ..., p ~ ~ , where

Ne Ci=, pi = 1. The 9, can be distributed independently across speils or distributed in-

dependently across individu& but is fixed across spells for a given individud3 The

contribution to the likelihood for an individual with a completed spell will be

For an uncompleted spell the contribution to the likelihood will be given by

When multiple spells are availible for each household then it is more intereting to have the hetr* geneity distribution vary acroes households rather than spells.

with continuation probability

Unlike the logit specification without random intercepts, this specification is not globally

concave and so maximization can introduce considerable computational difficulties, see

Meyer(l990) or Baker and Rea (1993). This specification can be estirnated using the

methods of Ham and Rea (1987) or the H e h a n and Singer (1984b) dgonthm.

1.3 Bayesian Estimation

1.3.1 Gibbs Sampling

By Bayes d e

the posterior distribution p(8ly) will be proportionai to the likelihood l(y;O) times

the pnor distribution p(O). The posterior distribution summarizes all the available

information about O conditional on the observecl data. However, most researchers are

aiso in interested in E( g(0) 1 y ) = j g(B)p(81 y)dO , where g(0) is some function of interest.

Comrnon choices for g(0) are Bi for the posterior mean of Bi and 0; for the second

moment of Bi. If the posterior distribution is tractable then expressions for the posterior

mean and variance can be obtained analytically. When the posterior distribution is not

tractable numerical techniques have been developed to facilitate analysis (see Koop (1994)

for a survey). These indude simple Monte Carlo integration, Monte Carlo integration

with importance sampling (Geweke (1989)), the Laplacian approximations of Tiemey

and Kadane (1986) and most recently Markov Chain Monte Carlo methods such as

the Gibbs sampler (Gelfand and Smith (1990)) and the Metropolis-Hastings aigorithm

(Tierney (MM)).

The Gibbs sampler is useful in problems where it is not possible to sample from the

joint postenor directly but it is possible to sample from the conditional posterior distri-

butions. For example, suppose the joint postenor distribution is f (01 y) = f ($1, &, O, 1 y)

and this distribution has conditional distributions that c m be sarnpled from at relatively

low cost then a Gibbs sarnpling algorithm can be implemented. Start with initial values

for the Bi ,( Op), OP), Op)) and then draw from the conditional distributions

(1) (1) (11, ..., Q ~ W , ehM), &Ml) The above cycle is then iterated M times producing a sample (8, ,O2 , O,

. After an initial transient phase, the draws kom the Gibbs sampler will converge to

draws from the joint pos terior dis tribut ion under fairly weak regularity conditions (see

Tiemey (1994)) . Unlike Monte Carlo integration methods the draws fiom the Gibbs

Sampling algorithm will not be independent but correlated because of the conditioning

on the previous draw.

In certain problems the conditional distributions used in the Gibbs sampling algo-

rithm may not be easy to sarnple but become so when the joint posterior distribution

is augmented with some latent data. This is the data augmentation method of Tanner

and Wong (1987) . For exarnple, suppose the conditional distributions of the Bi in the

previous example are not tractable but become so when augmented by some latent data

z. The Gibbs sampling algorithm with data augmentation draws kom the following

condi tional distributions

Gibbs sampling with data augmentation is applied kequently in Bayesian analyses of

limited dependant variables for example binary and polychotomous response models,

Albert and Chib (1993a), and the Tobit model, Chib (1989).

1 A.2 Bayesian Estimation Without Unobserved Heterogeneity

Using the cumulative normal density function as the continuation probability the poste-

rior distribution for the hazard model will be given by

where

/ 1 if the spell is cornpleted Dh = 1

1 O O t herwise

and p(P) is the prior distribution. A duration can be viewed as a finite chain of discrete

choices made by an individual over a period of time. For example, the individual may

decide if he wishes to be unemployed or to find a job. This sequence of discrete choices

will produce a duration of a given length.

Consider the latent variable regression4

41dentification in the Zt iper iod pmbit rnodel is achieved by setting the variance of the latent utility to unity.

where e u is IIDN(0,I) and Xht is vector of o b s e d characteristics for individual h

at thne t and Uht is the unobserved latent data. If Uht > O the individual survives

otherwise he exits. By introducing the latent variables Uht it is possible to apply data

augmentation in the Gibbs sarnpler. The introduction of the latent variables makes the

condi t ional distributions of the augmented posterior distribution like those of the normal

linear regression model and hence makes the Gibbs sampler feasible.

Since Uhl is un~bsemble the econometrician observes only the durnmy variable dht

where

1 if survives dht =

O otherwise.

The econometncian will then have a sequence dh = (dhi , dh2, ..., dhTh) on each individual

where Th is the period in which individual h exits or is the censoring point. A cornpleted

spell of Th periods will look like (1,1, ... ,1,0) while a censored spell of Th penods will

look like (1,1, ..., 1,1).~ If the latent data are not used to augment the postenor, the

posterior distribution for the multiperiod probit, i.e. p(Ply), will be

where the dht are defined above. Note that this is just the posterior distribution for the

hazard model with normal CDF as the continuation probability.

To construct the joint posterior of ,O and the latent data, let P(dh) denote the prob-

'The Framework is flexible enough to handh multiple spells, for wample consider the seqenœ of dummy variables (1,1,0,1,l,l,l,l). This sequence denotes two spells, spell 1 is a a~rnpleted spell that I a s t s 3 periods while spell2 is œnsored at 5 periods.

ability that individual h's sequence of d's is di,

The probability density kernel can be written as

The posterior distribution for the multiperiod probit mode1 p(P, UI data), where U is a

vector of latent variables, will be given by the product of the probability density kernel,

the prior distribution of 0, an integrating constant and a term for the consistency of the

or derings

where 1(X E A) is the indicator function that is qua1 to 1 if X is contained in the set A

and takes the value zero otherwise. While this posterior distribution does not d o w for

easy analysis, the conditional posterior distributions are relatively easy to sample because

the latent variables Uht are used in a data augmentation step in the Gibbs sampler.

The Gibbs sampler for the multiperiod probit ssmples from the following two condi-

tional distributions6

(i) Draw the latent data

-- -

'While 1 have assumed I.I.D. disturbances the Gibbs sampling algorithmn can accomodate alterna- tive error structures. For example, autoregressive disturbances can be stimated by adding two more conditional distributions to the sampler . One to sample the autoregressive parameters and anot her to sample the initial disturbances.

tmcated at the left by O if survive Uhtldata, f l - N(XLtfl, 1) t

truncated at the right by O if exit

for h = 1 ,..., N, t = 1 ,...,Th,

and

(ii) Draw ,O

where Nk denotes the k dimensional multivariate normal density.

With a uniform prior on B, i.e. p(P) aconstant, the conditional posterior mean and

variance are given by:

With a multivariate normal prior on P , P - N ( b , ) , the posterior conditional postenor - -B

mean and variance are given by:

and

where ~ - ' i s the inverse of the prior covariance matrix and ,B is - B

1.3.3 Bayesian

In hazard models the

Estimation Wit h Unobserved

issue of unobserved heterogeneity m u t

the pnor rnt~an.~

Het erogeneity

be addressed. The ex-

planatory variables in a hazard model are included to control for the heterogeneity in a

sample. Heterogeneity arises when different individuals in the sample have different dis-

tributions of the dependent variables. Problems can arise when interpreting the results if

the attempt to control for heterogeneity is incomplete and some heterogeneity remains . For example, unobserved heterogeneity can lead to misleading inferences in interpreting

duration dependence or the effects of the included explanatory variables. This hetero-

geneity can arise from two sources; the mispecification of the functional form and unob-

servable variables. With the multiperiod probit specification, unobsented heterogenei~

can be modeled by allowing individuals to have some individual specific pararneters on

some subset of the covariates or by allowing each individual to have their own vector of

parameters? The marginal posterior distributions of these individual specific parameters

can then be used to infer the degree of the heterogeneity present in the sample.

The multiperiod probit framework presented here wili incorporate unobservable het-

7 ~ h e Gibbs sampler is d e d in FORTRAN ïï using the truncated univariate normal random num- ber generator in Geweke (1991) and the rnultivariate normal random number generator in the IMSL Mat h/Stat Library.

B~cCul loch and Rossi (1994) modelled heterogeneity coefficients on ail the mvariates in the model.

in the multinomial probit by inctuding random

erogeneity by allowing for some of the covariates to have randorn effects associated with

them. The continuation probability with random effects will be O(XhP + &ah), where

Xnr is a vector of covariates, ,6 is a vector of h e d regression parameters, w~ is a sub-

set of & and bh is a vector of individual specific parameters which are distnbuted as

ap - &(O, D) .

The contribution to the likelihood huiction for an individual for this specification wiIi

be

a q dimensional integral that is intractable. However, Albert and Chib (1993b) show

that if the posterior distribution is augmented with latent data, as in the binary probit

model, a Gibbs sampler can be implemented. The Gibbs sarnpler requires sampling the

latent data and the parameter vector 0, as in the Gibbs sampler presented in Section

3.2, the random coefficients & and the covariance matrix of the random effects D.

The sarnpler is then:

(i) Draw the latent data

truncated at the left by O if survive UnLldat~, p, 6hl D N(Xhtp + w;,6h9 1) 1

truncated at the right by O if exit

for h=1, ..., N and t=l, ..., Th,

(ii) Draw the 6h

for h=1, ..., N and t=l , ..., T h ,

where bh = K1wL(& - XhP) and ~l = (WiWh + D-') Wh is a Th x q matrix containing the covariates for the random effects

Xi, is a Th x k matrix containing the X's for the fixed regression parameters

& is a T h x 1 vector containing the latent data.

The prior on bh is bhlD - N,(O, D ) and the prior on D-' is D-' - W,(po,&).

(iii) Draw ,O

for h=1, ..., N and t=l , ..., Th,

where = BL' Xi(& - Whbh)) and Bi = &Xh) if the non-informative

p ior p(P) occonstant is used. If the multivariate pnor on p is used the posterior mean

and variance wiu be given by B = B~'(c- ' P + ~ h = ~ Xh(& - Whbh)) and BI = ( C-' - B - B

+ CE, Xkxh).

(iv) Draw the inverse of the covariance matrix of the random effects D-'

for h=I, ..., N.

1.3.4 Evaluat ion of Numerical Accuracy

Recently, convergence diagnostics and measures of accessing numerical accuracy similar

to those available for simple Monte Carlo integration and Monte Carlo integration with

importance sampling have b e n developed for Gibbs sampling methods, Cowles and

Carlin (1995) offer a comparative review of these diagnostics. Geweke (1992) shows that

the Gibbs sampling estimate of E( g(0) 1 y), 3 = & XE, g(@, where M is the number of

induded draws, will hava the asymptotie distribution N( E( g(B)I y), F) , where S(0)

is the spectral density of g(@) evaluated at fkequency zero. Geweke suggests using 4% as a Numerical Standard Error (NSE) for 3 .

1 estimate the spectral density with the Newey-West (1987) covax-iance matrix esti-

where

and

1 used 3 lags for every 100 included draws when constructing the covariance matrix

estirnat~r.~

1.4 The Data

To illustrate the usefulness of the multiperiod probit model I present an illustrative

example. Both the logit specification for the continuation probability and the multiperiod

probit model are applied to employrnent duration data for the province of New Brunswick,

Canada. The data are taken from the Canadian Labour Market Activity Survey (LMAS).

This is a weekly panel data set covenng a probability sample of individuals over the penod

1988 through 1990. In the initial year of the swey , information is collected through a

' ~ h e estirnates did not seem to be sensitive to the number of lags, 1 also t ried 1 and 2 lags for every 100 included draws.

supplement to the monthly Labour Force S w e y (LW), which is wry much like the

U.S. Curent Population Survey. In subsequent years respondents are re-contacted and

interviewecl about their labour market adivities over the intemning penod of t h e .

This survey provides weekly information on respondents' penodr of employment. For

New Brunswick there are 1518 employment spells which range in length from 1 to 96

weeks available for the 999 individuals in the sarnple.l0 Of these spells 384 are cemred

and the average duration is 25.51 weeks.

Baker and Rea (1993) use these data to study the effect of the Canadian unemploy-

ment insurance (UI) program on employment durations. In 1989, individuals had to

accumulate between 10 and 14 'insured weeks7 of employment in the year preceding a

UI daim to qualify for benefits. The precise number of weeks, the Variable Entrance

Requirement (VER), varied acroas the 48 'economic regions' in Canada according to the

local unemployment rate. This feature of the Canadian UI system came up for periodic

renewal. At the end of 1989, a dispute between the House of Commons and the Senate

delayed the passage of a bill that would have renewed the VERS for the following year.

As a result, the VERS were not renewed for 1990, and in the first II months of that

year the entrance requirement was set to 14 weeks in all the economic regions of Canada,

regardless of economic conditions. l l Baker and Rea use the 'experiment ' represented by

this change in the VER'S to examine the effects of the UI eligibility d e s on employment

duration. In particular, they look for spikes in the employment hazard in the week that

individuals qualik for UI benefits.

For preliminary data analysis the empirical hazard and survivor were cakulated. The

empirical hazard is calculatecl as

l0I3aker and F h (1993) use this same LMAS data to examine the effects of UI eligibility requiremeds on the employment hazard. 1 ch- to study the data for New Bmnswick because these authors find evidence of u n o k e d heterogeneity when conducting their analysis of this province. They calculate an information matrix test and cannot reject the nul1 hypothesis of no unobserved heterogeneity. "See Baker and Rea (1993) for more details.

where hj is the number of exïts at duration t j divided by the number continuing at

duration t j for the completed spells in the sample. The empirical survivor function is the

Kaplan-Meier estimator

The estimates of the empirical hazard and survivor for the first 24 weeks of employment

are plotted in Figures 1 and 2. In Figure 3 the empirical hazard is plotted for the

spells in 1989 and 1990 separately. In 1989 the VER for all the 'economic regions' in

New Brunswick was set at 10 weeks. However, not aU individuals faced this entrance

requirement because some were repeaters (individuals who had previous UI claims) facing

longer eligibility requirements. The empirical hazard may exhibit spikes at other weeks

if there are a lot of repeaters in each region. In Figure 3 we see that there is a large

decrease in the hazard at 10 weeks between 1989 and 1990. This cross-year variation

in this spike suggests some sort of UI effect; i.e. 10 week spells were less likely in 1990

when the VER was set at 14 weeks in all regions of the province. Another one of the

features readily apparent from the plots of the empirical hazard are the regular spikes that

appear approximately every two weeks. Baker and Rea (1993) note that these spikes can

appear for a variety of reasons; digit preferences (the tendency of individuals to report

the length of their employment speIls rounded off to the nearest even number or multiple

of one month), calendar effects (the tendency of spells to begin or end at the beginning or

end of a month) or local employrnent initiatives which provide a relatively large number

of jobs of fked duration.

Baker and Rea construct dummy variables to capture the effect of UI provisions on

employrnent duration. The dummy variables are defined for the points in time, or periods,

that an individual: 1) initidy qualifies for UI, 2) qualifies for UI but is stiil accumulating

benefit entitlement and 3) qualifies for UI at the m&um entitlement. They take two

different approaches to identifjr these periods. The first, the 'Duration' approach, assumes

that an individual enters an employment spell with no accumulated insured weeks from

previous employment spells. The second, the 'hsured Weeks' approach, counts insured

weeks fiom any employment spell between the last benefit claim and the start of the

current employment spell. The insured weeks from previous spells count towards the

entrance requirement but are hard to identify, i.e. there is a problem identifyuig UI

receipt. The insured weeks approach tries to assign UI receipt the duration approach

doesn't.

Using the 'Duration' approach, EL1 takes the value 1 in the week that the individ-

ual's current employment duration satisfies the local Unemployment Insurance eligibility

requirement and a value of O in all other weeks. This variable captures spikes in the

employment hazard which are correlated with the week in which individuals initially

qualify to make Unemployment Insurance claims. The variable EL2 takes the value 1

in the penod in which the individual's current duration f d s between the initial week of

eligibility and the week in which the maximum benefit entitlement is reached. EL2 is

used to capture the effect of additional entitlement on the hazard. Finally, EL3 takes

the value 1 in the weeks in which the individual has qualified for UI at the maximum

benefit entitlement for his region. EL3 captures the more permanent effects of eligibility

on employment duration. The other set of eligibility variables calleci ILI, IL2 and IL3

are constructed in the same fashion as EL1, EL2 and EL3 but using an estimate of

insured weeks instead of the current employment duration to determine an individual's

UI eligibility. Further details of these two approaches, as well as a discussion of their

relative merits, can be found in Baker and Rea (1993).

1.5 Empirical Results

Two specification. of the duration model are estimated. Specification 1 includes the

constant, a fourth order polynomial in duration, a time (year) effect and controls for age,

educat ion, real hourly earnings , the provincial unemployment rate, gender , marriage,

schwl attendance, previous receipt of unemployment insurance benefits as well as the

eligibility variables. A full description of the construction of these variables is reported

in Table 1. Specification 2 indudes the constant, a fourth order polynomial in duration,

the yeas dummy and the eligibility dummies. These two specifications were estimated

with maximum likelihood for the logit functional form for the continuation probability

and with the Gibbs sampler for the multiperiod probit.12

While some advocate the use of multiple chains (Rubin and Gehnan (1992)) to im-

prove and help monitor the convergence of the Markov chain. 1 have found that a single

long diain with a suitable number of burn in draws is adequate for the multiperiod probit

Gibbs sampler. McCdoch and Rossi (1994) have shown that Gibbs sampler for the bi-

nomial probit converges. My experience with simulated data indicates that not only does

the Gibbs sampler for the multiperiod probi t converge but convergence is very l3

In addition to the fast convergence the Gibbs sampler for the multipenod probit is not

sensitive to the initial starting values. l4

The ML estimates and the posterior means for specifications with no unobserveci

heterogeneity are presented in Tables 2a to 5b. The prior on ,û was made quite diffuse

and centered at zero, p - N(O,~OOI#~ Negative parameter estimates in both the logit

and probit specification indicate that the continuation probability decreases with the

l * ~ h e Gibbs sampler results were obtained with 2200 ciraws, with the first 200 draws discardeci. 13~lbert and Chib (199%) make a similar observation for the binary probit model. 141 experimented with different starting values and found that bad starting values affect the early

draws. However, with a suitable number of initial ciraws d i ï e d the posterior mean from the Gibbs sampler with bad starting values will not be very d8erent than the posterior mean obtained from a Gibbs Sarnpler with good starting values.

151 also estimated the mode1 with the prior on beta proportional to a constant. The posterior means were almost identical to those obtained with the diffuse multivariate normai prior.

covariate and therefore the hszard increases. The fbst thing to note is that the parameter

estimates for all the covariates have the same sign in the logit and multiperiod probit

results. For specification 1 (the specification with controls), the estimate of the EL1

parameter indicates an increase in the employment hazard in the initial week of UI

eligibility, in both the probit and logit models. This is similar to the result in Baker

and Rea. Using the 'Insured Wseks' approach, the estirnate of the parameter on the

IL1 variable still indicates an increase on the employment hazard in both models. So

both the estimates of EL1 and IL1 suggest that there is an initial eligibility effect on the

employment hazard. The estimates of EL2 and IL2 indicate a decrease in the hazard in

both the logit and probit models. Finally, the estimate of EL3 indicates an increase in

the hazard in both specifications, while the estimate of IL3 indicates a decrease. Also

note that the posterior means of the eligibility variables from specification 2 are very

similar to those obtained in specification 1.

The multiperiod probit specification was also estimated using the random effects

mode1 presented in section 3.3. The pnor on bh is - &(O, D), the prior on D-'

is D-' Wq(pO,&) and the prior distribution for ,8 is, 0 - N(0, 100Ik). The prier

distribution for the precision matrix of the random effects is picked to be diffuse & =

101, and po = 44 + q + N. The covariates are the sarne as those in specifications 1 and

2 with the random effects entering through the eligibility variables. l6 These results are

presented in tables 6a, 6b, 7a and 7b. Using the 'Duration' approach and specification 1,

the postenor means of EL1 and EL3 are about half the values obtained without random

effects. On the other hand, the posterior mean of EL2 is slightly larger. In the 'Insured

Weeks' approach IL1 is about 50% smaller than the value obtained without random

effects, while the pasterior means of IL2 and IL3 are about twice as large. Similar

patterns are observed in the posterior means of the eligibility variables in specification

2 . The numerical standard errors (NSE) for the specifications estimated with random

1 6 ~ h e r e are random e f k t s associated with ELi, i=L,2,3. So each individual has a vector of 3 random effets.

effects also tend to be slightly larger than those obtained without random effects.17 These

results still indicate that there is an increase in the employment hazard in the week an

individual becornes eligible for UT.

As an alternative method of modeling duration dependence the time polynomial in

duration was replaced wit h a step function. A series of dummy vari ables were constmcted

to take single values for weeks 2 to 14 and then groupings for weeks 1516, 17-18, 19-

20, 21-25, 26-30, 31-40, 41-50, 51-60 and for spells longer than 61 weeks. A 4 t h order

time polynomial is a restrictive specification of duration dependence because it imposes

a great deal of smoothness on the baseline hazard. A step function will allow a more

flexible estirnate of the baseline hazard because it will reflect the spikes that appear in

the empincal hazard. For example, the single week dumrny variable groupings in weeks

2-14 will allow the step specification 1 have selected to capture the spikes that appear in

the empirical hazard. Baker and Rea (1993) were primarily interested in capturing the

spikes that appear in the employment hazard, because of this the choice nonparametric

specification of duration dependence, Le. the step function, is an important issue.

Results for a mode1 induding the step function, but no random effects are presented

in Tables 8a, 8b, 9a and Sb. The posterior means for the covariates used to control for

the observable heterogeneity are similar to those in the time polynomial specification.

The estirnates of the eligibility variables constructed using the 'Duration' approach, i.e.

the ELi (i=l, 2, 3) variables, all have negative posterior means and so they al1 indicate

increases in the employrnent hazard. The posterior means of the eligibility variables

constructed using the 'Insured Weeks' approach, the Li (i=1,2,3) variables, are similar

to those obtained from the specification using the t h e polynomial, but the posterior

standard errors are mudi larger. The NSEs are all in the same range as the values

ob t ained wi th the time polynomial.

Tables 10a, lob, 1 l a and l l b present the postenor means and variances for the spec-

-

1 7 ~ h e specifications with and without random eff& were estimated with 2200 draws from the Gibbs sampler with the first 200 draws discadecl to remove the effects of the initial conditions.

ifications 1 and 2 with both random effects and the step function. These results appear

to be more sensitive to the addition of the random effects. Estimation of specification

1 using the 'Duration' approach produces only 1 eligibility variable which increases the

employment hazard, EL3. The posterior means of EL1 and EL2 are both positive (they

both decrease the employment hazard) and they have fairly large posterior standard er-

rom. Estimation using the 'Insured Weeks' approach is more "successful" , the posterior

means of the ILi variables have the sarne signs as those without random effects, but

the posterior standard errors are quite large. In specification 2, the posterior means of

EL1 and EL3 are negative, so both EL1 and EL3 increase the employment hazard. The

posterior means of the L i variables are all positive. Therefore, the effect of UI eligi-

bility on the employment hazard becomes more difficult to interpret with the addition

of the random effects to these specifications. The week of eligibility has a positive, but

very s m d effect on the employment hazard in specification 1, using the 'Iwured Weeks'

approach and in specification 2 using the 'Duration' approach. Estimation of the other

two specifications leads to results that indicate that the employment hazard would f d

in the initial week of eligibility. This decrease in the employment hazard is quite s m d

using the 'Insured Weeks' approach and specification 2, but fairly large for specification

1 using the 'Duration' approach.

To better illustrate the effect of the week of eligibility on the hazard across different

specifications, 1 present the value of the hazard at weeks O and at the week of eligibility,

14 weeks for the 'Duration' approach and 10 weeks for the 'Insured Weeks' approach, at

the sample means of the covariates. These results are presented in Tables 12 to 15. For

the specifications without any unobserved heterogeneity we see that with specification

1 the nonpararnetnc specification of duration dependence produces a larger increase in

the hazard when the insured weeks approach is used. However, the results in Tables

12 and 13 indicate that there will be a larger increase in the hazard for specification 2

(both 'Insured Weeks' and 'Duration' Approach) when a parametric duration dependence

specification was used. There was also a larger increase in the hazard for specification 1

with the ' hu red W h ' approach when a parametnc duration dependence specification

was used. For the specifications with unobserved heterogeneity (Tables 14 and 15) the

parametnc duration dependence specification produced larger increases in the week of the

week of eligibility for the specification 1 with the 'Duration' approadi, and specification

2 using both the 'Duration and ' hu red Weeks' approach. The nonparametnc duration

dependence specification was only able to produce a larger increase in the hazard for week

of eligibility in specification 1 when the 'Insured W h ' approach was used to construct

the eligi bili ty variables.

As a predictive exercise 1 have also computed the conditional probabiiity of ending

an employment spell given so many weeks of employment for certain representative indi-

viduals for both the probit and logit specifications. I consider four 'types' of individuals.

Individual 1 is a married male who is older than 44 years of age and has not completed

hi& school. Individual 2 is a university educated mamed female who is older than 44

pars of age. Individual 3 is a university educated single male between 24 and 44 years

of age. Individual 4 is a male who is older than 44 years of age, married and has a trade

certificate. The individuals and their characteristics are presented in more detail in Table

16.

Table 17 presents the probability that an individual will exit an employment spell

for speils that are 5, 10, 14, 20 and 25 weeks conditional on being employed 4, 9, 13,

19 and 24 weeks respectively. The predicted exit probabilities were similar across the

two continuation probability specifications. The probit exit probabilities are larger for

individuals 1 and 2 but smaller for individuals 3 and 4. The predicted exits for the

probit mode1 are slightly smaller when the step function is used instead of the time

polynornid. The predicted exit probabilities for both specifications are also increasing

with the length of the speil, as they almost doubled kom 5 to 25 weeks. Figures 4 to 7

are plots of the hazard rates for each of the representative individuals for the multiperiod

probit specification with a time polynomial, while Figures 8 to 11 are plots of the hszards

for the representative individuals when the step function was used to mode1 duration

dependence. The plot of the employment hazard for 'representative' individual 1, whose

speil occurred in 1990, reveals a large spike in the employment hazard at 14 weeks. The

other 'representative' individuals have employment spells in 1989 and so they have a

spike in their employment hazards at 10 weeks.

1.6 Concluding Remarks

The multiperiod probit model is used to estimate a discrete time duration model using

a Gibbs sampler with data augmentation. The results from Bayesian estimation of the

multiperiod probit are compared with those from maximum likelihood estimation of a

logit hazard model. The results show that there is not much difierence in the specifi-

cations; covariates which decrease the hazard do so in both the multipenod probit and

logit specifications of the continuation probabili ty. Estimation of the multiperiod probi t

with random effects does not change the posterior means of most of the covariates by

very much. However, the posterior means of the variables constructed to capture the

effects of UI eligibility on the employment hazard are more sensitive to the introduction

of random effects and the step function. The results from the empirical illustration indi-

cate that there is an increase in the employment hazard in the week that an individual

qualifies for UI, in most of the specifications. However, this increase is not apparent in

the specifications with the random effects and a step function specification of duration

dependence. Some of the results from this specification indicate that there would be

a decline in the employment hazard in the week an individual qualifies for UI. More

importantly, perhaps, the estimation of the specification with unobserved heterogeneity

and the step function is a straightforward extension of the model with no unobserved

heterogeneity: only two more conditional distributions have to be added to the Gibbs

ssmpling algorithm. None of the numerical problems referred to by Baker and Rea (1993)

or Meyer (1990) when trying to estimate a specification with heterogeneity and a step

function were encountered.

'Iàble 1: Description of Variables

unemployment rate: the monthly unemployment rate

hourly earnings : Average hourly eaniings, all values converted to 1989 dollars

age 16-24: 1 for those 1424 years of age in 1988, O otherwise

age 25-44: 1 for those 25-44 years of age in 1988, O otherwise

high school: 1 if high school graduate, O otherwise

post secondary: 1 if post secondary education, O otherwise

universiSr: 1 if university degree, O otherwise

trade certificate: 1 if trade certificate or diplorna, O otherwise

past UI receipt: 1 if the individual received UI income in the previous year

marital status: 1 if the individual is married

school attendance: 1 if the individual attended school in the year of the current week,

O otherwise

sex: 1 if the individual is fernale, O othemise

year: 1 if a week during the year 1990, O otherwise

EL1 or ILI: 1 in the week individual satisfies the local UI eligibility requirement

EL2 or IL2: takes the value 1 in the weeks that the individual satisfies the local UI

eligibility requirement, but has not yet achieved the maximum benefit entitlement, O

otherwise

EL3 or IL3: takes the value 1 in the weeks that the individual has satisfied the local UI

eligibility requirement and has reached the maximum benefit entitlement , O otherwise

Note: For age the excluded group is age 45-64 years of age, for education the excluded

group are individu& that have not completed high school.

'Igble 2a: Multipenod Probit Specification 1, Duration Approach, No

Random Effécts, Parametric Duration Dependence

Variable

constant

unemployment rate

hourly eaniings

age: 16-24

age: 25-44

hi& school

post secondary

university

trade certi ficate

past Ul receipt

marital status

school attendance

sex

Far

EL1

EL2

EL3

Posterior Mean

1 .SM48

0.0298

0.0002

-0.0400

0.0022

0.0890

O. 1405

O. 1198

0.141 1

-0.1665

0.0639

-0.2274

-0.0992

0.1046

-0.3557

0.0471

-0.1375

Posterior Variance NSE

0.0323 0.0052

0.0002 0.0004

1.2E5 0.0001

0.0027 0.0018

0.0016 0.0012

0.0013 0.0011

0.0013 0.0010

0.0062 0.0020

0.0049 0.0019

0.0009 0.0010

0.0014 0.0012

0.0016 0.0011

0.0008 0.0008

0.0008 0.0008

0.0044 0.0017

0.003 1 0.0017

0.0053 0.0021

Table 2b: Muitiperiod Probit Specification 1, Insureci Weeks Approach, No

Random Effects, Parametric Duration Dependence

Variable

constant

unemployment rate

hourly earnings

age: 1624

age: 25-44

high school

post secondary

trade certificate

past UI receipt

marital status

school attendance

sex

Pos terior Mean

1.9869

0.0275

-3.6E5

-0.0368

0.0097

0.0848

0.1388

0.1233

O .O997

-0.1568

0.0632

-0.2340

-0.0932

0.1207

-0.2121

0.0939

0.0428

Posterior Variance

0.0374

0.0002

l . l E 5

0.0027

0.0015

0.0012

0.0012

0.0054

0.0069

0.0010

0.001 1

0.0017

0.0008

0.0007

O .O043

0.0021

0.0025

NSE

0.0061

0.0004

0.0001

0.0012

0.0010

0.0010

0 .O009

0 -0027

0.0028

0.001 1

0.0008

0.0010

0.0009

O ,0008

0.0015

0.0013

O .O0 14

Table 3a: Muitiperiod Probit Specification 2, Duration Approach, No

Random Effeds, Parametric Duration Dependence

Vari able Post erior Mean Pos terior Variance NS E

constant 2.2286 0.0058 0.0019

Yem O. 1009 0.0008 0.0007

EL1 -0.3629 0.0048 0.0018

EL2 0.0061 O .O030 0.0017

EL3 -0.1504 0.0051 0.0017

Table 3b: Multiperiod Probit Spedication 2, Insured Weeks Approach, No

Ftandom Effects, Parametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 2.2245 0.0053 0.0020

Far 0.1209 0.0007 0.0007

IL1 -0.2303 0.0049 0.0019

IL2 0.0794 0.0021 0.0012

IL3 0.0379 0.0021 0.0014 Note: For Tables 2a - 3b the NSE is reported for the mean

a b l e rda: Maximum Likelihood Estimates for the Logit Mode1 Specification

1, Duration Approach

Variable

con& ant

unemployment rate

hourly eamings

age: 16-24

age: 25-44

high school

post secondary

univer si ty

trade certificate

past UI receipt

marital statu

school attendance

sex

Far

EL1

EL2

EL3

Posterior Mean

3.7655

0.0627

0.0292

-0.0957

0.0238

0.1846

0.3301

0.3152

0.2255

-0.3808

0.1530

-0.5300

-0.2259

0.2205

-0.8000

0.0197

-0.3089

Standard E m r

O .47W

0.0332

0.0077

0.1117

0.0885

0 -0856

0.0856

OS750

O. 1883

0.0745

0.0833

0.0929

0.0702

0.0679

0.1415

0.1288

0.1608

Table 4b: Maximum Likelihood Eatimates for the Logit Mode1 Specification

2, Insured Weeb Approach

Variable

constant

memployment rate

hourl y earnings

age: 16-24

age: 25-44

high school

post secondary

universi ty

trade certificate

past UI receipt

marital status

school attendance

sex

Y'==

EL1

EL2

EL3

MLE

3.7260

0 .O693

0.0002

-0.1021

0.0204

O. 1908

0.3268

0.3095

0.2116

-0.3622

0.1519

-0.5362

-0.2259

0.2681

-0 -4586

0.2279

0.1 143

Standard Error

0.4586

0.0334

0.0077

0.1143

0.091 1

0.0823

0.0819

0.1759

O. 1817

0.0753

0.0831

0.0958

0.0682

0.0680

O. 1497

O. 1094

0.1148

a b l e 5 a MLE for Logit Model Specification 2, Duration Approach

Variable Name MLE Standard Error

constant 4.6379 0.0924

Table 5b: MLE for Logit Model Specification 2, Insured Weeks Approach

Variable Narne MLE Standard Error

constant 4.5806 0.1477

Year 0.2608 O .O643

IL 1 -0.478 1 0.1526

IL2 0.1407 0.0967

IL3 0.0599 0.0944

mble 6a: Multiperiod Probit Specification 1, Duration Approach, Randorn

Effects, Paramettic Duration Dependence

Variable

constant

unernployment rate

hourly earnings

age: 16-24

age: 25-44

high school

post secondary

university

trade certificate

past UI receipt

marital status

school attendance

sex

Year

EL1

EL2

EL3

Pos t erior Mean

2.1556

0.0156

-0.0016

-0.0565

-0.0018

0.0993

O. 1 707

0.1600

0.1382

-0.1653

0.0657

-0.2461

-0.1 140

0.1 192

-0.1961

0.1327

-0.0515

Posterior Variance

0.4776

0.0203

1.9E-5

0.005 1

0.0025

0.0021

0.0309

O. 0074

0.0088

0.0013

0.0018

0.0028

0.0018

0.0015

O. 1074

0.0658

0.0360

NSE

0.M7

0.0030

2.2E5

0.0036

0.0022

0.0017

0.0030

0.0035

O. O038

0.0014

0.0015

0.0027

0.0023

0.0019

0.0217

0.0169

0.0119

'IIible 6b: Multiperiod Probit Specification 1, Insured Weeks Approach,

Random Effwts, Parametric Duration Dependence

Variable

constant

unemployment rate

hourly eaniings

age: 16-24

age: 2544

hi& school

post secondary

University

trade certificate

past UI receipt

marital status

school attendance

swc

year

EL1

EL2

EL3

Pos terior Mean

2.1999

0.0149

-0.0012

-0.0615

-0.0022

0.1013

0.1592

0.1393

0.1 159

-0.1529

0.0647

-0.2523

-0.1 173

0.1361

-0.0577

0.1827

0 .O994

Posterior Variance

0.0340

0.0002

1 . G E X

0.0025

0.0018

0.0019

0.0023

0.0070

0.0083

0.0012

0.0016

0.0029

0.0019

0.0024

0.1223

0.0446

0.0176

NSE

0.0081

0.0006

0.0002

0.0014

0.0011

0.0020

0.0023

0.0028

0.0037

0.0014

0.0016

0.0025

0.0023

O .O028

0.0228

0.0139

0.0085

Table ?a: Mdtiperiod Probit Specification 2, Duration Approach, Random

Effects, Parametric Duration Dependence

Variable Posterior Mean Pos terior Variance NSE

constant 2.1139 0.0064 0.0022

Far O. 1244 0-0023 0.0019

EL1 -0.2594 0.0698 0.0115

EL2 0.0529 0.0392 0.0079

EL3 -0.1143 0.0406 0.0088

Table 7b: Multiperiod Probit Specification 2, Insured Weeks Approach,

Random Effects, Parametric Duration Dependence

Variable Post erior Mean Pos terior Vari ance NS E

constant 2.1774 0.0017 0.001 8

Far O. 1506 0.0027 0.0019

IL1 -0.1299 0.0795 0.0 108

IL2 0.1121 0.051 1 0.0064

IL3 0.0519 0.0416 0.0048

'lhble 8a: Mdtiperiod Probit Specification 1, Duration Approach, No

Random Effects, Nonparamattic Duration Dependence

Variable

constant

unempIoyment rate

hourly earnings

age: 16-24

age: 25-44

hi& school

post secondary

universi ty

trade certificate

past UT receipt

marital status

school attendance

sex

Y==-

EL1

EL2

EL3

Posterior Mean

1 .SI392

0.0348

3.33-5

-0.0410

0.0059

0.0862

O. 1434

0.1433

O .O977

-0.1761

0.0627

-0.2305

-0.0960

0.0976

-0.2005

-0.1017

-0.4008

Post erior Variance

0.0456

0.0002

9.6E6

0.0027

0.0017

0.0017

0.0014

0.0058

0.0066

0.001 1

0.0013

0.0018

0.0008

0.001 1

0.0065

0.0058

0.0109

NSE

0.0064

0.0004

8.8E5

0.0017

0.0013

0.0008

0.001 1

0.0027

O. 0024

0.0010

0.001 1

0.0009

0.0008

0.0010

0.0019

0.0023

O. 0033

Table 8b: Multiperiod Probit Specification 1, Insured Weeks Approach, No

Random Effects, Nonparametrk Duration Dependence

Variable

constant

unemployment rate

hourly earninp

age: 16-24

age: 25-44

high school

post secondary

University

trade certificate

past UI receipt

marital status

school at tendance

sex

Posterior Mean

1.9341

O .O354

-0.0002

-0.0398

0.0027

0.0959

0.1475

0.1 209

0.0958

-0.1710

0.0629

-0.2376

-0.095 1

0.1284

-0.0296

0.0085

0.0014

Posterior Variance

0.0403

0.0002

5.5M

0.0022

0.0014

0.0013

0.0011

0.0052

0.0056

0.0009

0.0013

0.0018

0.0009

0.0008

0.0065

0.0025

0.0027

NSE

O.ûû66

0.0004

8.5E5

0.0014

0.0010

0.0012

0.0010

0.0022

0.0022

0.0008

0.0012

0.0013

0.0010

0 .O009

0.0023

0.0013

0.0016

'hble 9a: Multiperiod Probit Specification 2, Duration Appraach, No

Random Effects, Nonparametric Duration Dependence

Variable Posterior Mean Pos terior Variance NSE

constant 2.2568 0.0111 0.0035

F a r 0.0885 0.0008 0.0007

EL1 -0.1946 0.0070 0.0022

EL2 -0.0895 0.0049 0.0020

EL3 -0.3884 0.0077 0.0019

Table 9b: Multiperiod Probit Specification 2, Insured Weeks Approach, No

Random Effects, Nonparametric Duration Dependence

Vari able Posterior Mean Pos terior Variance NS E

constant 2.2400 0.0124 0.0036

Table lûa: Multiperîod Probit Specification 1, Duration Approach, Random

Effects, Nonparametric Duration Dependence

Viuiable

constant

unemployment rate

hourly earnings

age: 1624

age: 25-44

high school

post secondary

universi ty

trade certificate

past UI receipt

marital statu

school attendance

sex

F a r

EL1

EL2

EL3

Pos t erior Mean

1.9269

0.0342

-9.8E-4

-0.0489

0.0014

O. 1109

0.1769

0.1824

0.1107

-0.1625

0.0728

-0.2333

-O. 1 147

O. 1243

0.2178

O. 1138

-0.2257

Posterior Variance

0.0068

0.0002

1.3E5

0.0041

0.0025

0.0033

0.0042

0.0014

0.0082

0.0018

0.0022

0.0024

0.0021

0.0025

0.3954

0.0981

0.0636

NSE

0.0014

0.0007

1.4W

0.0025

0.0020

0.003 1

0.0036

0.0065

0.0031

0.0018

0.0018

0.0017

0.0025

0.0277

0.0043

O. 0209

0.0160

Table lob: Multiperiod Probit Specifieation 1, Insured Weeks Approach,

Randorn Efkts , Nonpatametric Duration Dependence

Variable

constant

unemployment rate

hourly earnings

age: 1624

age: 25-44

high school

post secondary

universi ty

trade certificate

past UI receipt

marital status

school attendance

sex

Pos t erior Mean

2.0138

0.0317

-0.0011

-0.0671

-0.0111

O. 1086

0.1618

0.1630

0.1176

-0.1567

0.0669

-0.2543

-0.1132

0.1461

-0.0029

0.1890

0.0732

Posterior Variance

0.5122

0.0016

2.53-5

0.0063

0.0030

0.0019

0.0026

0.0088

0.0079

0.0014

0.0014

0.0048

0.0025

NSE

0.0466

0.0026

2.5E-4

0.0043

0.0026

0.0019

0.0025

Table lla: Multiperiod Probit Specification 2, Duration Approadi, Random

Effects, Nonparametric Duration Dependence

Variable Posterior Mean Pos terior Variance NS E

constant 2.0854 0.0079 0.0048

Far 0.1151 0.0021 0.0025

EL1 -0.0506 0.0124 0.0046

EL2 0.0214 0.0132 0.0047

EL3 -0.0813 0.0139 0.0048

Table Ilb: Multiperiod Probit Specification 2, Insured Weeks Approach,

Random Effects, Nonparametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 2.3113 0.0357 0.0119

F a r O. 1235 0.0020 0.0025

IL1 0.0869 0.1112 0.0214

IL2 0.1567 0.0426 0.0 134

IL3 0.0681 0.0191 0.0089 Note: For Tables 6a to l l b the reported NSE is for the mean

Table 12: Hazard Estimates, Duration Approach, No Unobserved

Heterogeneity

PARAMETRIC D URATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

1 0.03 116 0.07144 6.39

2 0.01019 O .O7677 6.97 NONPARAMETRIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00951 0.07336 7.71

Tabie 13: Hazard Estimates, Insured Weeks Approach, No Unobserved

Heterogeneity

PARAMETRIC D URATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

1 0.01040 0.03913 3.26

2 0.01012 0.05893 5.82 NONPARAMETRIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00969 0.02845 2.94

2 0.01054 0.03174 3.01

Table 14: Hazard Estirnates, Duration Approach, Unobserved Heterogeneity

PARAMETRIC DURATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

2 0.01459 0.04999 3.43 NONPARAMETRIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

Table 15: Hazard Estimates Insured Weeks Approach, Unobserved

Heterogeneity

PARAMETRIC DURATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00981 0.03336 3.39

2 0.01349 0.04131 3 .O6 NONPARAMETFLIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00868 0.04458 5.14

2 0.01461 0.04163 2 -85

H d a t 14 W& Note: For Tables 12 to 15 A =

Ttible 16: Characteristics of Individiirils

Characteristics

constant

Unemployment Rate

Hourly Wage

Age 1624

Age 2 5 4 4

High School

Post Secondary

University

Trade Certificate

Past UI Receipt

Mancied

School At tendance

Sex

Year

Table 17: Predicted Probabilities for Representative Individuals at Selected

Durations

LOGIT SPECIFICATION Individual 5 Weks 10 Weeks 14 Weks 20 Weeh 25 Weks

MULTIPERIOD PROBIT SPECIFICATION, PARAMETRIC DURATION DEPENDENCE Individual 5 Weks 10 Weks 14 Wireks 20 Weekç 25 Weeks

MULTIPERIOD PROBIT SPECIFICATION, NONPARAMETRIC D URATION DEPENDENCE Individual 5 Weeks 10 Wieeks 14 Weeks 20 Weeks 25 Wéeks

1 0.01798 0.04549 0.07133 0.03529 0.03913

2 0.01352 0.05443 0.04592 0.02713 0.03016

3 0.00673 0.03019 0.02571 0.01438 0.01614

4 0.00759 0.03442 0.02848 0.01612 0.01806

I 1 I I

O 20 40 60

Ernployment Duration in Week

Figure 1-1: Plot of the Empirical Hazard Function for the Ernployment Duration Data

Figure 1-2: Plot of the Empirical Survivor Function for the Employment Duration Data

1 3 5 7 9 11 13 15 17 19 21 23 DURATION IN WEEKS

1 - year 1989 +year 1990 1

Figure 1-3: Plot of the Empirical Hazard for Employrnent Duration Data by Year

Figure 1-4: Predicted Hazard, Representative Individual 1, Parametric Duration Depen- dence

Figure 1-5: Predicted Hazard, Representative Individual 2, Parametric Duration Depen- dence

1 L 1 I 1

O 20 40 60

Employment Ouration in Weeks

Figure 1-6: Predicted Hazard, Representative Individual 3, Parametric Duration Depen- dence

r I I I

O 20 40 60

Ernployment Duration in Weeks

Figure 1-7: Predicted Hazard, Representative Individual 4, Parametric Duration Depen- dence

Figure 1-8: Predicted Hazard, Representative Individual 1, Nonparametnc Duration Dependence

Figure 1-9: Predicted Hazard, Representative Individual 2, Nonparametric Duration Dependence

Figure 1- 10: Predicted Hazard, Representative Individual 3, Nonparametric Duration Dependence

Figure 1-11: Predicted Hazard, Representative Individual 4, Nonparametric Duration Dependence

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Chapter 2

Flexible Durat ion Dependence

Modelling in Hazard Models

2.1 Introduction

In a hazard mode1 duration dependence occurs when the transition probability for a

state is affected by the length of time in a state. Let X ( t ) denote the hazard rate (exit dA( t ) probability). Positive duration dependence at the point t* occurs when dt Ir=r* > 0.

Positive duration dependence occurs when the exit probability for a state increases as the

time in the state increases. Negative duration dependence at the point t' occurs when

Tlkto < 0, and refers to an exit probability that decreases as the length of time in s

state increases. For example, for unemployment duration data the exit probability from

unemployment may fa11 as the length of the spell lengthens.

Duration dependence in discrete tirne hazard models is usually modelled in two fash-

ions. The first, parametric duration dependence, is typicaily modelled using a low order

time polynomial, for example, a cubic or fourth order polynomial in duration. As an

alternative to time po1ynomia.l~ Ham and Rea (1987) and Meyer (1990) suggested using

a more flexible nonparametnc specification. Both the pararnetric and nonparametnc

specifications have disadvantages and advantages associated with their use. In this pa-

per I propose an alternative specification of duration dependence that can lie in between

these two competing specifications and can also nest the two competing specifications

as special cases. A smoothness prior (Shiller (1973)) is imposed on the step function

coefficients to restrict the curvature of the badine hazard and ailow for a more flexible

range of shapes for the estimate of the baseline hazard.

The use of tirne polynornials, or parametnc specifications of duration dependence, is

often criticized because a time polynomial imposes a strong pararnetnc restriction on the

shape of the badine hazard. This is especiaiiy true for low order time polynomials such

as a linear trend or quadratic specification. Typically higher order time polynomials are

used to be more 'flexible' about the duration dependence specification, for example 6th

order polynomials in Ham and Rea (1987) and 5th order polynomials in Gunderson and

Melino (1990). However, as the order of the polynomial increases numerical problems are

often encountered when atternpting to maximize the likelihood function. Eberwein, et.

al. (1995) argue in favor of time polynornials for several reasons. First, the addition of

all the time dummies to the quasi-likelihood function increases the number of parameters

to be estimated and can make maximization more difficult, particularly with a nonpara-

metric unobserved heterogeneity distribution. Secondly, the use of a time polynomial

allows the data to determine the order of the polynomial, reducing the need to make

arbitrary decisions on how the time durnmies should be grouped. And thirdly, a time

polynomial with suficiently hi& order can mimic a step function, hence a higher order

time polynomial is also a flexible specification of duration dependence.

The nonparametric specification of duration dependence is considered to be a more

flexible form for the duration dependence. With discrete data, duration dependence

can be modelled nonparametrically with a step function as suggested by Ham and Rea

(1987) and Han and Hausman (1990). A step function is a series of time vsrying dummy

variables specified for points in time or intervals of tirne, for example

For example, if you had a spell of 5 periods a possible step h c t i o n specification

be (O, 1, 0, 0, O), (0, 0, 1, 0, O), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1). A step fimction is said

to have 'dense bins' if there are just a few groupings for the time intervals, conversely

'thin bins' occur when there are many time intervals. hstead of restricting the duration

dependence, and hence the shape of the baseline hazard, to take a very strong parametric

form a nonparametric specification whidi may not place any restrictions on the shape of

the baseline hazard can be used.

Han and Hausman (1990) and Meyer (1995) argue in favor of step functions because

the shape of the hazard is often irregular and unlikely to approlcirnated by a simple

parametric form. The use of a nonparametric baseline hazard is a benefit because it

puts no restrictions on the shape of the underlying hazard in discrete data. The step

function is becorning a more common method of modelling duration dependence in recent

years; for example, Meyer (1990), Han and Hausman (1990), Baker and Rea (1993),

Narendranathan and Stewart (1993) and Meyer (1995) all use step functions. However,

the use of the step function is not completely free of problems. From a computational

point of view, Baker and Rea (1993) and Meyer (1990) have reported difficulties in

estimating a duration model with a quasi-likelihood that contains both the step function

and nonparametric heter~geneit~.' There is also the issue of how to group observations

when constnicting the time dummies. There are some suggestions in the literature but

there is no universally accepted method of specibng a step fhction. Ham and Rea

(1987) specified a step function so that there are at least 3% of the observations in each

bin and the last bin contained the residual 2% of the observations. Han and Hausman

-- - - - ---

lMeyer (1990) was able to estimate the model by assuming a parametric form for the heterogeneity distribution, the gamma distribution.

(1990) estimated a step function with 40 steps and they also remarked that they would

have no problems estimating a step function with up to 50 steps ( redy thin bins). At

the other extreme Narendranathan and Stewart (1993) pick a step h c t i o n specification

with relatively few parsmeters ( redy dense bins). Because there is no consensus on how

to select how dense or thin the bins will be in the step specification, researchers tend to

experiment with specifications until they find one that fits their needs.

In this chapter 1 present a duration dependence specification that can lie between

these two competing specifications and can also nest the two competing specifications

as special cases. A srnoothness prior (Shiller (1973)) is irnposed on the step h c t i o n

coefficients to restrict the 'smoothness' of the step coefficients and hence restrict the

curvature of the underlying baseline hazard. Shiller (1973) proposed the smoothness

pnor as an alternative to restricting the coefficients in a distributed lag model to come

from some parametric family or to lie on sorne pararnetric function, for example a low

order polynomial. The srnoothness prior reflects and imposes a lag curve that is smooth,

but without restricting the coeflicients in the distnbuted lag to come from or be restricted

to lie on a particular hinctional form which is chosen arbitrarily.

Placing a smoothness pnor on the duration dependence coefficients will restrict the

'smoothess' of the coefficients of the step function, and hence place restrictions on the

curvature of the baseline hazard. By varying the degree and the variance of smoothness

pnor 1 can obtain very smooth shapes for the baseline hazard, like that produced by a

parametric duration dependence specification. Alternatively, 1 can d o w for more jagged

shapes in the baseline hazard, like that produced by a thin bin specification or allow

the shape of the hazard to lie somewhere between these two extrema. This means that

the smoothness pnor can be used as a filter to remove spurious noise from the baseline

hazard while employing a thin bin step function specification.

The chapter proceeds as follows. Section 2 contains an introductory discussion of the

smoothness pnor and i ts use in distributed lag models. Section 3 presents the multiperiod

probit model which is used to model the duration data and discusses how the smooth-

ness pnor can be incorporated in a hazard model. This section also presents the Gibbs

sampling algorithm that is used to estimate the model. Section 4 presents an empincal

application of the model to employment duration data from the Canadian province of

New Brunswick. This section shows that the smoothnsss prior c m be used to generate

a wide range of shapes for the baseline hazard which includes the parametric and non-

parametric baseline hazard rnodels as special cases. Section 5 contains some concluding

remarks.

2.2 Shiller's Smoothness Prior

Shiller (1973) proposed using the smoothness prior as an alternative to restricting the

coefficients in a distributed lag model to come hom some special distribution or restricting

the distributed lag coefficients to lie on a low order polynomial. For example, the Almon

lag restricts the coefficients in the distributed lag to lie on a low order polynomial, another

alternative is t O assume the lag coefficients are proportional to a geometric distribut ion

with an unknown decay rate. The smoothness pnor reflets and imposes 'smoothness' in

the lag cuve but without restricting the lag coefficients to lie on or come from a s m d

elementary class of functions. While an Almon lag would impose exact restrictions on the

coefficients in a distnbuted lag model, Taylor (1974) showed that Shiller's smoothness

pnor is equivalent to imposing stochastic restrictions rather than exact restrictions.*

So rather than make the implausible assumption that the lag coefficients lie exactly on a

polynomial of given degree, Shiller assumes that the lag coefficients should lie reasonably

close to such a polynomial.

Consider a tirne senes y, which follows a linear distnbuted lag on a time series xt

*Corradi (1977) shows that under certain conditions Shiiler's smoothness prior can be shown to be obtained t~ a special case of a smoothing spline. These srnoothing splines are used extensively in nonpararnetric estimations (see Hiirdle (l994)). k n t I y , Tsionas (1996) has also discusseù t h e use of smoothing splines in a Bayesian framework.

3~oirier (1975) a h discusser the use of splines in distributed lag models as an alternative to Almon lag specifications.

where X is the lag length (a known constant), the Pi are the unknown coefficients and is

a stationary stochastic process with zero mean. Shiller defines the dth degree smoothness

pnor by first defining the d+l order clifFerences of the Pi (denoted u)

where & is a (A - d - 1 x A) matrix of d+l differences with rank equal to X - d - 1 and

,B = (Pi , A, ... , PX)' The matrix % contains the restrictions on the lag coefficients. For

example, if d=2 and X = 6,4

The pnor distribution of u is assumed to be normal with prior variance c2 and a pior

mean equal to zero, so u - N (O, c21).

Shiller explains the interpretation of the smoothness prior with an illustrative exam-

ple. "If d=l and E is smd, then the priors attach hi& probability to all shapes in which

the dope changes slowly, highest of ail to a straight line, and s m d probability to jagged

shapes. A jagged shape which lies within a narrow band around a straight line is still

accorded low probability, while a shape which deviates markedly from a straight line but

does so gradually is given high probability. If d=O, then the pnors assert that the Iag is

unlikely to involve large jumps; it will probably proceed in small s t e ~ s . " ~

4 ~ f exact restrictions were king i m w , this example would constrain the distributeci lag coefficients to lie dong a quadratic polynomial.

'Shiller (1973), p. m.

Combining the prior distribution for the d+l order differences and the likelihood

function, by Bayes law, Shiller obtains the posterior distribution for

The posterior diskibution of f l is multivariate normal with posterior mean b and posterior N N

variance 02(X'X)-'.

2.3 Application t O Modelling Durat ion Dependence

The transition between states is modelled as a discrete process. To construct the likeli-

hood function for the hazard mode1 the standard normal cumulative distribution function,

iP(XhP), is selected to be the functional form for the continuation probability, where Xht

contains controis for observable heterogeneity and the duration dependence specifica-

tion. The contribution of each household or individual to the likelihood function for a

completed spell is given by

where T h is the length of the speil for individual h. This is the product of the continuation

probabilities for the Th - 1 periods the individual survives multiplied by the probability

the individual exit8 in penod Th. For a censored or uncompleted spell the contribution

to the likelihood function is giwn by

where T, is the censonng point. Here the contribution to the likelihood function is just

the product of the continuation probabilities for the Tc periods the individual survives.

The likelihood function for all the individuals in the sarnple can be written as

where

1 if individual h survives dht =

otherwise

Combining the Likelihood function with the prior distribution on P we obtain, by Bayes

d e , the joint posterior distribution for the hazard mode1

where p(B) is the prior distribution for and dht is defined abow.

While this posterior distribution does not provide closed form solutions for posterior

functions of interest, Albert and Chib (1993) have shown that a Gibbs sampler with a

data augmentation step can be used to obtain Monte Carlo estimates of the posterior

functions of interest; for exarnple the posterior mean and variance. The Gibbs sampler

is a Markov Chain Monte Carlo method that requires sampling from the conditional

posterior distributions of the joint posterior distribution. Under fairly weak regularity

conditions the draws from the conditional posterior distribution wili converge to draws

from the joint posterior distribution.

Consider the following quivalent specification for the continuation probability

where Xht is a vector of controls for observable heterogeneity for individual h at tirne t

and Dat is the control for duration dependence at time t for individual h and ,O = (3 and X i t = ( 3, DM ). DN can take one of two forms:

i) a time polynomial specification, for example a cubic Dit = (t, t2, t3), or

ii) the step function specification where Dkt wili be a vector that will take the following

fom

where the t subscript denotes the t-th element of DL. The dumrny variables in the

vector will take the value 1 in the inteml [&,q and the value O at other points in time.

The parameters for the duration dependence specification can be estimated dong with

the parameters controlling for the observable heterogeneity as one block in the Gibbs

sampler. However, if a nonpararnetric specification of duration dependence is selected

and the smoothness pnor is used on the step function, then the parameter vector can be

broken up into two blocks; the component corresponding to the observable heterogeneity

6 and the component corresponding to the duration dependence 7. These two blocks

are then sampled fkom their respective conditionai posterior distributions in the Gibbs

sampler.

Introduce the unobserved latent data (Albert and Chib(1993)) and consider the latent

variable regression

where ~ h t Niid N (O , 1) . If Uht > O the individual survives, otherwise he Bots. D e h e the

1 i fUu>O binary indicator variable du = . The Unt are unknown, however, the

O otherwise distribution of the Uht conditional on dht is truncated univariate normal. By introducing

the latent data UN it is possible to apply a data augmentation step (Tanner and Wong

(1987)) in the Gibbs sampler. The introduction of the latent data makes the conditional

distributions of the data augmented posterior like those of the normal linear regression

mode1 and hence makes the Gibbs sampler feasible.

The Gibbs sampler, with the smoothness pnor irnposed on the step function, will

requiring sampling kom the following conditional distributions:

(i) Sample the latent data Uht

tnuicated at the left by O if survive Uht16,7, data, dht - N (Th$ + D&y, 1

truncated at the right by O if exit

for h=l, ..., N and t=l, ..., Th.

(ii) Sample the parameter wctor 6

where Cs= (6-' + xLi xzi XhtXht)-'=d 6 =Cs &-'&+ xh=i 121 X&ht - Dht7))

(iii) Sample the parameter vector y

716, Uht, data - N (7, -7) ,

C Y N rCI- - I . - where = D'D and 7 = (DILI)-' DtU, D is the rnatrix [ y ] where

with Üht = Uht - Tht6 and k = i and % is the matrix of differences.

The smoothness prior is used to put restrictions the duration dependence specification

and hence on the baseline hazard. The amount of smoothness that occurs will depend on

value of d as well as the variance parameter (. For example, srnall values of d, e-g. d=l , will lead to very smooth shapes for the baseline hazard. Larger values of d can allow

for more jagged shapes in the baseline hazard but how jagged the hazard appears will

depend on the size of the variance (. Smder values of , other things being qua1 wiU

allow for smoother shapes in the hazard. Larger values of E , will produce a more jagged

shape for the hazard.

If the smoothness prior is not imposed on the step function, or the parametric spec-

ification of duration dependence is used, the Gibbs sarnpler requires sarnpling from the

following conditional distributions:

(i) Sarnple the latent data Uiu

truncated at the left by O if survive UhtlP, dat%dht N (&P7 1) 1

truncated at the right by O if exit

for h=l, ..., N and t=l, ..., Th-

(ii) Sample the parameter vector p

2.4 Empirical Illustration

2.4.1 Data

As an empirical illustration the smoothness prior is used to estimate the hazard for em-

ployment duration data from the Canadian province of New Brunswick. This duration

data was first used by Baker and Rea (1993) to study the effect of the Canadian Unem-

ployment Insurance (UI) program on employment duration. The data are taken hom the

Canadian Labour Market Activie Survey (LMAS). This is a weekly panel data set cov-

ering a probabiliw sample of individuals from the province of New Brunswick, Canada,

over the period 1988 to 1990. In the initial year of the sunrey, information is collected

through a supplement to the monthly Labour Force S w e y (LFS), which is very much

like the U.S. current population survey. In subsequent years respondents are re-contacted

and interviewed about their labour force activities over the intervening penod of time.

This survey provides weekly information on respondents' pen0d.s of employment. There

are 1518 employment spells which range in duration from 1 to 96 weeks for the 999

individuals in the sample.

Baker and Rea (1993) use this data to study the effect of Canadian UI program

provisions on employment durations. In 1989, individuals had to accumulate between

10 and 14 'insured weeks' of employment in the year preceding a UI claim to qualify for

benefits. The precise number of weeks the Variable Entrance Requirement (VER), varied

across the 48 'economic' regions of Canada according to local unemployrnent rate. This

feature of the Canadian UI system came up for periodic renewal. At the end of 1989,

a dispute between the House of Commons and the Senate delayed passage of the Bill

that wodd have renewed the VERs for the following year. As a result, the VERs were

not renewed for 1990, and in the first 11 months of that year the entrance requirement

was set to 14 weeks in dl the economic regions of the country regardless of economic

conditions. Baker and Rea use the 'experiment' represented by this change in the VERS

to examine the eEects of UI eligibility d e s on employment duration. In particular they

look for spikes in the employment hazard in the week individuals qualify for UI benefits.

Baker and &a construct dummy variables to capture the efTect of UI provisions on

employment duration. The dummy variables are defined for the points in time, or periods,

that an individual: 1) initially qualifies for UI, 2) qualifies for UI but is still accumulating

benefit entitlement and 3) qualifies for UI at the maximum entitlement. They take two

different approaches to identiS these periods. The first, the 'Duration' approach, assumes

that an individual enters an employment spell with no accumulated insured weeks from

previous employment spells. The second, the 'Insured Weeks' approach, counts insured

weeks from any employment spell between the last benefit claim and the start of the

current employment speil.

Using the 'Duration' approach, EL1 takes the value 1 in the week that the individu-

als's current employment duration satisfis the local Unemployment Insurance eligibility

requirement and a value O in all other weeks. This variable captures spikes in the em-

ployment hazard which are correlated with the week in which individuals initially qualify

for UI daims. The variable EL2 takes the value 1 in the period in which the individ-

ual's current duration falls between the initial week of eligibility and the week in which

maximum benefit entitlement is reached. EL2 is used to capture the effect of additional

entitlement on the hazard. Finally, EL3 takes the value 1 in the weeks in which the

individual has qualified for UI at the maximum benefit entitlement for his region. EL3

captures the more permanent features of eligibility on employment duration. The other

set of eligibility variables called ILI, IL2 and IL3 are constructed in the same fashion

as ELl, EL2 and EL3 but using an estimate of insured weeks instead of the current

employment duration to determine an individual's UI eligibility. W h e r details of these

two approaches, as well as a discussion of their relative merits, can be found in Baker

and Rea (1993).

For preliminary data analysis the hazard rate, the empirical hazard, was estimated

with no controls for either observable or unobservable heterogeneity. The empincal haz-

ard is calculated as X(tj) = 2 , where hj is the number of e i t s at duration t j divided

by the number continuing at duration t j for the completed spells in the sample. The

empirical hazard is plotted for the first 75 weeks in Figure 1. One of the features readily

apparent from the plot of the empincal hazard are the regular spikes in the empirical haz-

ard that appear apprcximately every two weeks. Baker and Rea (1993) note that these

spikes c m appear for a variety of reasons; digit preferences (the tendency of individuals

to report the lengths of their employment spells rounded off to the nearest even number,

or to a multiple of a month), calendar effects (the tendency for spells to begin or end

at the beginning or end of a calendar month) or local unemployrnent initiatives which

provide a relatively large number of jobs of fixed duration. Baker and Rea note in their

discussion of the empirical hazard that the probably cause of these spikes is response

error, as individuals round off their employment duration to the nearest even number.

They also note that the spikes at approximate 'month's ends' appear to be slightly larger,

suggesting an additional source of aggregation.

The specification of the duration mode1 includes an intercept, controls for duration de-

pendence, a time (year) effect and controls for age, education, real hourly earnings, the

provincial unemployrnent rate, gender, marriage, school attendance, previous receipt of

unemployment insurance as well as the eligibility variabld A full description of the con-

struction of these variables is provided in Table 1 and sampie statistics are presented in

= ~ h e specifkation is a only estirnated using the mi, i=1,2,3 eligibility variables, i.e. using the 'Duration' approach to identify the week of eligibility.

Table 2. 1 only present covariate estimates for the eligibility variables to conserve space.

1 also present plots of the estimated hazard rates.? 1 choose this method of presenting

my results because 1 believe that the effect of the smoothness prior is best illustrated by

presenting the plots of the estimated hazard rather than presenting posterior meana in

tables.

Baker and Rea (1993) were primarily interested in h d i n g spikes in the employment

hazard in the week individuals qualify for UI benefits. For spells occurring in 1989 this

spike appears in the tenth week for most individuals in New Brunswick because the VER

was set at 10 weeks for all the 'economic regions' of New Brunswick. Howewr, some

individuals are repeaters, i.e. individuals not making their f b t claim, who face longer

eligibility requirements. This means that if there are large nurnbers of repeaters in these

'economic' regions then there will be other spikes in the hazard besides the large one

at 10 weeks. For speb occurring in 1990, because of the temporary legislation enacted,

this spike (where individuals satisfy the UI eligibility requirement) is more pronounced in

week 14. The presence of these two spikes can be clearly seen in the plot of the empincal

hazard in Figure 1. As discussed in the previous section another feature of the empirical

hazard, that is also readily apparent in Figure 1, are the regular spikes that appear

approximately every two weeks. Since these spikes are for the most part just 'noise' in

the data, particularly after 30 weeks, it is a worthwhile exercise to 'filter' the data and

reduce the amount of noise in the data. 1 'filter' the noise using the smoothness pnor to

smooth the spikes in the hazard.

As a benchmark I estimated the hazard mode1 with the following specifications

of duration dependence. A 4th order time polynornial was used as the parametrie

specification! Four nonparametric specifications for duration dependence were also used:

'The estimated hazard rate is computed by computing the hazard rate for each individual in the sample, at each iteration in the Gibbs sampler, and averaging the hazard rates acrcss the individuals. The average hazard rate from each iteration in the Gibùe sampler is then taken and averaged across iterations.

8 ~ h e 4th order time polynomia1 was selected because it was the highest order that was supported by the data.

(1) Step specification 1, week specific durnmy variables for weeks 2-14 and groups for

weeks 15-16, 17-18, 19-20, 21-25, 26-30, 31-40, 41-50, 51-60 and another group for weeks

after week 61.

(2) Step specification 2, week specific dummy variables for weeks 2-35 and weeks 36-

40,41-50, 51-60 and 6 1 f are grouped together.

(3) Step specification 3, week specific dummy variables for weeks 2-42 and weeks 43-45,

46-48,4450, 51-60 and 61+ are grouped together.

(4) Step specification 4, has no week specific dumrny variables, weeks 2-10, 11-20, 21-30,

31-40, 41-50, 51-60 and 61+ are grouped together.

I used a proper but very diffuse multivariate normal pnor to estimate the models, ,O - N(O,100&).

The hazard estimates from these five specifications are plotted in Figures 2-7. The

features of the hazard estimates vary across specifications. For the 4th order time poly-

nomial there is a very smooth shape to the hazard with s m d spikes occurring in weeks

10 and 14. Step specification 4 is the dense bin specification. The wide groupings for

the time intemls elirninates most of the spikes that are present in the empirical hazard

and also gives the predicted hazard a steplike shape. Step specification 1 is the not so

dense bin specification. The specification of dummy variables for single periods d o w s

the hazard to capture the spikes that appear in the first 14 weeks. After week 14, time

intervals are grouped so the estimated hazard function isn't able to capture the spikes

that are appear in the empirical hazard. Step specification 2 is a thin bin specification

that does not begin to group observations until week 35. This specification captures the

spikes that appear up to 35 weeks. But as with the other two specifications the hazard

takes on a steplike shape after the time intervals are grouped. Step specification 3 is

an even thinner bin specification with single penods up to week 42. After week 42 the

hazard begins to take on a steplike shape.

Table 3 presents the posterior means of the eligibility variables (ELi, i=l, 2, 3) for

various duration dependence specifications. The posterior mean of EL1 from the more

restrictive duration dependence specifications (the t h e polynomial and step specification

4) are much closer to the d u e hom the specification with no duration dependence speci-

fication than the more flexible duration dependence specifications. For EL2 the posterior

from the step specifications are all very close to postenor mean fiom the specification

with no controls for duration dependence. However, all the posterior means have fairly

large posterior standard errors. For EL3, the postenor from the spedcation with no

controls for duration dependence and the 4th order time polynomial are fairly sirnilar.

T h e posterior means from the step specifications range from 3 to 5 times larger than the

postenor mean from the specification with no duration dependence controls.

The two cnticd parameters in the smoothness prior are the order of differencing d and

the pnor variance of the smoothness prior c. Smaller values of d indicate smoother shapes

for the hazard. Smder values of also imply that the hazard will be much smoother.

A s m d value of < indicates that there will be small deviations from the smooth shape,

larger values of E allow for larger deviations from the pararnetric form assumed and hence

the results should not differ greatly from the results produced when a smoothness prior

was not imposed. The smoothness prior was used on step specification 2 for difference

orders d=1,4,7 and for values of 5 = 100,1,0.01. The restriction matrices (R, , i=1,4,7)

will be of dimension (NDUM-d-1 x NDUM), where NDUM is the number of bins in the

step function, and the restriction matrices will take the following forme

' ~ h e nonzero entries in the rstridion matrices can be obtained from Pascal's triangle.

and

Step specification 2 was selected because it captures the eligibility spikes at 10 and

14 weeks and because it is also capable of capturing the spikes between 15 and 35 weeks.

A dense bin step specification with wider time intervals will produce an estimate of

the hazard that has a more pronounced step shape. A thin bin specification with b e r

tirne intervals will produce an estimate of the hazard that is more jagged. Baker and Rea

(1993) were interested in capturing the spikes in the employrnent hazard that were c a w d

by changes in the eligibility requirements for the unemployment insurance prograrn. Be-

cause of the nature of the question they were interested in a dense bin specification wiil

not be appropriate because it does not capture most of the spikes that are present in the

empirical hazard. On the other hand, picking a step specification with really thin bins

may result in ovefitting and capturing more of the noise that is present in the hazard. 1

have attempted to select a step specification that lies in between these two extremes.

The estimates of the hazard are plotted in Figures 7-9. Each figure contains 4 panels:

panel (a) is the estimated hazard with no smoothness restrictions, panel (b) displays the

estimate of the hazard for = 0.01, panel (c) presents the hazard estimate when < = 100

and panel (c) presents the hazard estimate when = 1.0.

The results for d=l are presented in Figure 7. Selecting d=l imposes a great deal of

smoothness, so the estimate of the hazard should be much smoother than the estirnate

produced without the smoothness prior. In Figure 7 panel (c), with = 100 although

the coefficients in the step function are constrained to have a lot of smoothness they are

allowed to deviate from the smooth shape in a substantial manner. Because of this there

are only s m d differences between the hazard estimate with the smoothness prior and

the hazard estimate without the smoothness prior. The only dinesence between the two

hazard estimates is that the spikes in the estimate with the smoothness prior are slightly

smaller. As is decreased to 1 the hazard estimate becomes slightly smoother than the

hazard estimate in panel (c). However, as < falls to 0.01 there is a very substantial change

in the shape of the hazard estimate. Most of the spikes in the hazard, except for the

spikes at 10 and 14 weeks, are smoothed. There is also an increase in the hazard at about

week 18, after which the hazard begins to decline srnoothly. The hazard estimate looks

very similar to the hazard produced when a 4th order time polynomial was used as the

duration dependence specification.

In Figure 8 the difference order d was set to 4 and the hazard was estimated with the

three values of c. In panel (c), for 5 =IO0 there is not much difference in the hazard esti-

mate with the smoothness prior and without the smoothness prior. Decreasing the value

of E to 1, in panel (d), produced a smoother estimate of the hazard in the sense that the

spikes becarne less pronounced. This differs from the results when the differencing order

was set at 1, which showed that there was not a lot of smoothing occurring. Decreasing

the value of to 0.01 also produced a much smoother hazard then with larger values of

c. However, the hazard estimate was not as smooth as that produced with the difference

order d set at 1. The hazard estimate's spikes at IO and 14 w& are more pronounced

then the spikes with d=l. In addition to this, while the hazard estimate with d=l looks

like it increases at 20 weeks and then declines slowly with d=4 the jump at 20 weeks

looks more like a spike. After 20 weks the hazard is fairly flat until30 weeks and then

drops to 35 weeks where the steplike shape that appears when observations have been

grouped is more pronounced then with d=l.

In Figure 9 the differencing order was set at 7. In panel (b) the hazard estimate with

5 = 0.01 was not as smooth as the estimates with a lower order of differencing. The

spikes at 10 and 14 weeks are of simiiar magnitude as those in the hazard estirnate with

d=4. Most of the differences between the estimated hazard lies between weeks 18 and

32. In the hazard with d=4 the increase in the hazard at 20 weeks begins to look like

a spike, with d=7 the spike shape becomes ewn more apparent. The hazard also does

not have the plateau like qualities as the hazard estimate with d=4. In addition to this,

with d=7 a spike at 32 weeks also begins to form.

Table 4 presents the posterior means of the eligibility variables for the smoothness

pnor specifications. For t=.01 and d=1,4,7 the posterior means are all fairly similar

and quite close to the posterior mean of EL1 when step specification 4 was used. For

= 100, the posterior mean of EL1 for the different smoothness pior specifications are

all very similar but they are about 50% larger than the value of the postenor mean from

Table 2 for step specification 2. For E=l, the posterior means are about 50-100% larger

than the values in Table 2 for step specification 2. However, the posterior means from

the specifications with d=4 and 7 are close to the posterior mean from the specification

estirnated with a 4th order time polynomial. For EL2 the posteriors means from the

specifications estimsted with the smoothness pnor were all at least 50% larger than the

values from Table 3. For a given value of the posterior means with the specifications

order of d=4 and 7 were quite similar. The specification estimated with = 100 produced

posterior means that were similar across all the differencing orders. The posterior means

for EL3 with the smoothness pior are also 50-100% larger than the values in Table 2.

As, before for a given value of k the posterior means with a difference order of 4 and 7

are fairly close. Also for < = 100 the posterior means from all three smoothness prior

specifications are fairly similarly

The data set used in the empiricd example contains almost 39000 bùiary outcornes,

so there is substantial sample information. For a given value of the differencing order

a large value of E indicates a very diffuse or vague wioothness prior so the resdts will

be very much like the estimates produced without the smoothness prior. As -, O the

nonsample information becomes more 'precise' and we see a much greater impact on the

smoothness in the hazard estimates. Rom Tables 3 and 4 the posterior means from

the various smoothness pnor specifications indicate that as the order of the diEerencing

increases the smoothing effect of C decreases and that the results for higher order of

differencing with a given value of E are similar.

2.5 Concluding Remarks

Shiller 's smoothness pior is applied to a nonpararnetric duration dependence specifica-

tion. The smoothness prior is used as a method of smoothing the spikes that often appear

in the estimates of the hazard functions. It is shown that by varying the hyperparame-

ters of the smoothness pnor it is possible to obtain very smooth shapes for the baseline

hazard, like those produced by a parametric specification of duration dependence, or to

allow a more jagged baseline hazard with irregdar spikes and troughs, like those that

are often present in the empirical hazard function. The choice of 5 appears to be the

most important prior hyperpararneter used to calibrate the amount of smoothness. Small

values of E wili produce much smoother hazard estimates, larger values of E will allow

much more jagged hazard estimates and hence indicate much less smoothing.

Table 1: Description of Variables

unemployment rate: the monthly unemployment rate

hourly earnings : Average hourly earnings, all values converted to 1989 dollars

age 1624: 1 for those 14-24 years of age in 1988, O otherwise

age 25-44: 1 for those 2 5 4 4 years of age in 1988, O otherwise

high school: 1 if high school graduate, O otherwise

post secondary: 1 if post secondary education, O otherwise

university: 1 if university de-, O otherwise

trade certificate: 1 if trade certifiate or diplorna, O otherwise

past UI receipt: 1 if the individuai received UI income in the previous year

marital status: 1 if the individual is married

school attendance: 1 if the individual attended school in the year of the current week,

O otherwise

sex: 1 if the individual is female, O otherwise

year: 1 if a week during the year 1990, O otherwise

EL1 or ILI: 1 in the week individual satisfies the local UI eligibility requirement

EL2 or IL2: takes the value 1 in the weeks that the individual satisfies the local UI

eligibility requirement, but has not yet achieved the maximum benefit entitlement, O

otherwise

EL3 or IL3: tcrkes the value 1 in the weeks that the individual has satisfied the local UI

eligibility requirement and has reached the maximum benefit entitlement, O otherwise

Note: For age the excludeà group is age 4564 pars of age, for education the excluded

group are individuals that have not completed high school.

Table 2: Descriptive Statistics for the Data and Durations

Variable

Unemployment Rate

Hourly Earnings

age: 16-24

age: 25-44

High School

Post Secondary

Trade Certificate

University

Past UI receipt

Marital Status

School Attendance

Sex

Sample Mean

SAMPLE STATISTICS FOR DURATTONS

Standard Error

1.13

4.57

O .48

0.50

0.41

0.44

0.19

0.18

0.49

0.48

0.39

0.48

Sample Mean Standard Error Minimum Maximum % Censored

25.51 20.96 1.0 96.0 25.3

Table 3: Posterior Means of Eligibility Variables for Alternative Duration

Dependence Specifications

Specificat ion EL1

No Duration Dependence -0.4834

(0.055)

4th Order Time Polynomial -0.3688

(0.063)

Step Specification 1 -0.2128

(0.070)

Step Specification 2 -0.1994

(o. 077)

Step Specification 3 -0.1935

(0.092)

Step Specification 4 -0.4576

(0.064)

Note: Posterior Standard Enors in parentheses

'Ilable 4: Posterior Megns of Eligibility Variables with the Shiller

Smoothness Prior

Value of d and

d=l J = 0.01

Notes:

Posterior Standard Errors in Parentheses

- Value of d refers to the ordering of differencing

EL2

-0.3079

(0.045)

-0.1878

(O. 063)

-0.2923

(0.065)

-0.3197

(0.047)

-0.2635

(O. 047)

-0.2428

(o. 047) -0.3158

(0.051)

-0.2456

(0.056)

-0.2334

(0.061)

k=j where is the prier variance of the smoothness prior

i 1 I 1

O 20 40 60

Ernplayment Ouration in Weeks

Figure 2-1: Plot of the Empirical Hazard

Figure 2-2: Estimated Hazard, Step Specification 1

Figure 2-3: Estimated Hazard, Step Specification 2

Employment Durauan in Weeks

Figure 2-4: Estimated Hazard, Step S~ecification 3

L I I 1

O 20 40 60

Employment Ouration in Weeks

Figure 2-5: Estimated Hazard, Step Specification 4

Ernplayment Ouration in Weeks

Figure 2-6: Estimated Hazard, 4th Order Time Polynornial

Emplayment Duration in Weeks ( a l

Emplayment Duranon in Weeks ( c 1

Emplayment Duraüon ai Weeks ( b 1

Employment Ouration in Weeks ( d l

Figure 2-7: Estimated Hazard, Smoothness Prior d=l

Notes:

(a) Step Specification 2 with no Smoothness Prior

(b) Step Specification 2, Smoothness Prier d=l and = 0.01

( c ) Step Specification 2, Smoothness Prior d=l and E = 100

(d) Step Specification 2, Smoothness Prior d=l and < = 1.0

Employment Duration in Weeks ( a 1

Emplcynent Ouration in Weeks ( b )

Employment Duration in Weeks ( c 1

Employment Ouration in Weeks ( d l

Figure 2-8: Estimated Hazard, Smoothness Pnor d=4

Notes:

(a) Step Specification 2 with no Smoothness Pnor

(b) Step Specification 2, Smoothness Prior d=4 and < = 0.01

(c) Step Specification 2, Smoothness Prior d=4 and < = 100

(d) Step Specification 2, Smoothness Prior d=4 and ( = 1.0

Emplayment Ouration in Weeks ( a l

Ernployment Ouration in Weeks ( c 1

Emplcynent Dumion in Weeks ( b )

Ernployment Duration in Weeks ( d l

Figure 2-9: Estimated Hazard, Smoothness Pnor d=7

Notes:

(a) Step Specification 2 with no Smoothness Prior

(b) Step Specification 2, Smoothness P i o r d=7 and E = 0.01

( c ) Step Specification 2, Smoothness Pnor d=7 and E = 100

(d) Step Specification 2, Srnoothness Prior d=7 and ( = 1.0

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Chapter 3

Bayesian Semiparametric Estimation

of Discrete Duration Models: An

Application of the Dirichlet Process

Prior

Introduction

In any study of duration data it is important to address the issue of unobserved het-

erogeneity. Unobserved heterogeneity refers to any differences in the distributions of

the dependent variables that remain even after controlling for the efTect of observable

variables. It typically has two sources: mispecification of the functional form of the

econometnc mode1 and omission of important but perhaps unobservable variables from

the conàitioning set. As an exarnple of the latter, more motivated individuals may tend

to exit unemployrnent more quiddy because they put more effort into the search for a

new job.'

'Keifer (1988) discusses some of the consequencxs of ignoring unobserveci heterogeneity. En prticular he notes that mispecification of the hazard leads quite generally to a downward bias in the estimateci

Increasingly retmxchers have adopted nonparametric specifications of the distribu-

tion of unobserved heterogeneity, due to the biases that can resdt when an inappre

priate parametric assumption is made (Heckman and Singer (1984)). For the classical

econometrician there are two obstacles to foIlowing this strategy. First , we do not have

an asymptotic distribution theory for the nonparametric maximum likelihood estimator

(NPMLE)2, although it is consistent (Heckman and Singer (1984)). Second, as a prac-

tical mat ter, nonpararnetnc estimation of the unobserved heterogeneity distribution can

present severe computational difficulties, especially when it is mixed with a nonparamet-

ric specification of the baseline hazard? While some sort of parametric assumption on

the baseline hazard c m lessen these problems, this in itself may not be an innocuous

assumption.

In this chapter, 1 propose a Bayesian estimator for discrete duration models, which

incorporates nonparametnc unobse~ed heterogeneity and avoids the problems of the

NPMLE. First , i t dows for exact finite sarnple inference. Second, it is easily estimated

and mixed with a nonparametric baseline hazard. The hazard model is specified as a

multiperiod probit model, which is estimated using a Gibbs sampler with data augmenta-

tion. Unobserved heterogeneity is introduced through individual specific random effects.

Finally, 1 use a Dirichlet process pnor on the random effects, which allows for nonpara-

metric analysis of this part of the model in the sense that the likelihood function does not

impose restrictions on the distnbution of the data (Chamberlain and Imbens (1995)).

The Dirichlet process pnor (a pnor on the space of distribution functions) allows

the support of the prior distribution to include al1 distributions on the real line. This

impact of duration dependence on the probability of completing a speii. Keifer (1988) also discusse the direction of the bias for the coefficient estirnates for the contmls for observable heterogeneity.

'van der Vaart (1996) is able to prove the asymptotic normaiity of the NPMLE for certain special cases. However, he does not offer a general proof. In related work, Bearse, et. al. (1996) propose a kemel estimator as an alternative to the NPMLE. They provide proofs of consistency and qrnptotic normality for their estimator.

31n fact many authors have found it impossible to estimate both the unotxerved heterogeneity dis- tribution and duration dependence specification nonparametrically because of numerical problems. For one of the more successful attempts s e Narendrathan and Stewart (1993).

increase in the support will in turn d o w a much wider range of shapes for the posterior

distribution. Hence, the Dirichlet process pnor will permit a more 'flexible' estimator

of the unobsend heterogeneity distribution4 by allowing the possibility of multimodal,

skewed and fat-tailed distributions (see Escobar and West (1995) and (1992) for exam-

ples). In the current application, the posterior distribution of the random effects will be

a mixture of normal densities and point densities. Escobar (1994) also notes that the

Dirichlet process prior indudes a smoothing parameter that dows it to use the data

in a local fashion like a nonparametnc estimator, such as the NPMLE, or a parametric

estimator that uses the data in a global fashion. Therefore this model also provides a

theoretical basis and modelling framework to address practical problems with nonpara-

metric methods; for example, local versus global smoothing and smoothing parameter

estimation.'

To estimate the model, all that is required is the ability to sample from the conditional

postenor distributions, which is considerably easier to irnplement than NPMLE. Findy,

the Dirichlet process prior allows the econometrician to nest the parametric heterogeneiw

distribution (the Gaussian random effects), or estimate a mixture of normal and point

mass distributions as an alternative. How much the estimator departs from the baseiine

parametric case is determinecl by the data rather than a priori assumptions.

The next section contains an introduction to the Diridilet process prior. Section 3.1

presents the Gibbs sampler for the multiperiod probit rnodel with a multivariate normal

prior on the random effects, while Section 3.2 discusses the changes that must be made

to the Gibbs sampler to incorporate the Dirichlet process prior on the random effects.

4McCulloch and h i (1994) propose a more flexible unobserved heterogeneity distribution in their random coefficient multinornial probit model by specifying a finite mixture of normal distributions as the heterogeneity distribution. Finite mixtures of normal distributions are often p r o p d as an alternative to nonparamet ric modelling.

' ~ h e current analysis draws on recent advances in Bayesian simulation techniques, namely the Gibbs sampler. The use of Gibbs sampling has solved some of the computational problems that were previously encountered when using the Dirichlet p m prior and have made implementation much easier. Recent work in the statistics literature includes papers by Eacobar and West (1995) (density estimation) and West, Müller and k b a r (1994) (estimating regpission functions and densit ies nonparametrically ) .

Section 4 contains an empirical application of the model to employment duration data

from the Canadian province of New Brunswick. Baker and Rea (1993) used this data to

study the impact of changes in the Canadian unemployment insurance program on em-

ployment durations (see also Chnstofedes and McKenna (1995) and Green and RiddeU

(1993)). To anticipate the main hdings of this exercise, 1 find no substantial deviations

fiom the pararnetric heterogeneity mode1 when 1 impose a parametric specification of the

baseline hazard. On the other hand, slightly more deviations from the baseline paramet-

ric heterogeneity model arose when the baseline hazard was specified nonparametrically.

However, the use of a nonparametric specification of the baseline hazard makes i t difficult

to distinguish the specification wi th the parametnc heterogeneity fiom the specification

with nonparametnc heterogeneity. It is important to note that the extent of the devia-

tions from the badine parametric model were determined by the data and not by apriori

assurnptions. Finally, Section 5 contains some concluding remarks.

3.2 The Dirichlet Process Prior

The Dirichlet process pnor has recently been the subject of renewed interest in the statis-

tics li terature and the econometrics li terature. For example, Chamberlain and Imbens

(1995) discuss the application of the Dirichlet process pnor to instrumental variable es-

timation and quantile regression. Ruggeiro (1994) diwusses Bayesian semiparametric

estimation of proportional hazards models. In the statistics literature the Dirichlet pro-

cess prior is being used to estimate normal means models, some of the serninal papers

include Bush and MacEachren (1995), Escobar (1994), Escobar and West (1992, 1995)

and West, Muer and Escobar (1994). Recent advances in Bayesian computation, narnely

the Gibbs sampler (Gelfand and Smith (1990)), have solved some of the computational

problems that arise when using the Diridilet process prior and have made implernentation

much easier.

The canonical framework for the Dirichlet process mode1 for a normal means problem

is given by the likelihwd y,,,, pe, (y) i=l , ..., n and the prior Bi - G with additional

uncertainty about the prier distribution G:

where G l a - D(aG0) rneans that G is random distribution generated by a Dirichlet

process with base measure aGo.

Consider the following normal means problem from Escobar and West (1995).' Sup

pose that the data YI, . .. , Y, are conditionally independent and normally distributed

Yi Ilri .- N ( b , CF?) with mean p; and variance CT,~ and let Ti = ( p i , CF:). Suppose the

rj come from some pnor distribution G(*) on 92 x W+. If G(*) is uncertain and modeled

as a Dirichlet process then the data will come from a Dirichlet mixture of normals. In

particular suppose that G- D(aGO), a Diridilet process dehed by a, where a is a posi-

tive scalar. Go ( 0 ) is a specified bivariate distribution function over 92 x 92+. Go ( a ) is the

prior expectation of G(-) and a is the precision parameter, determining the concentration

of the prior for G(*) about Go(-) . The precision parameter a represents the weight of our

belief thst G(-) is the distribution of Go ( O ) . The parameter a allows us to determine

how large the deviations from the baseline parametric Go ( 0 ) case are. Large values of

a imply that there are not large deviations from the baseline parametric model. Con-

versely, small values of a imply that there are large deviations from the parametic case

and Bi is estimated by pooling the other values of Bi together. The parameter cr adjusts

the estimator to behave either like a parametric estimator that uses the data in a global

manner or a nonparametric estimator that uses the data in a local fashion.

A key feature of the Dirichlet process model is the discreteness of G (=) . This means

' ~ h e following discussion follows Escobar and West (1995) very closely.

102

that in any sample of size n from G ( e ) there is a positive probability of there being

identical values, i.e. with positive probability the 3 reduce to k<n distinct values. This

is best illustrated by examining the conditionai prior of the m. Let a = (xl, ..., n,). For any i= 1, . . . , n, let di) be x without q , so di) = (nl , . . . , T,- 1 ,9+ 1, . . . , xn ) . Then the

conditional pnor for ai1x") is

l for positive integers where b(ni) denotes a unit mass point at x = y and &-1 =

r. This means that ri is drawn from Go ( 0 ) with probability A or it is taken from the

existing values of ni with probability &.

The next step is to specify the prior mean Go(*) of G(0). For example, the conjugate

pnor for the normal means problem, the normal-inverse-gamma prior, could be used.

Assume that oc2 - G(4, f) with shape 1 and scale and - N(m, ) for some

mean m and scale factor T > 0.

For each i the conditionai posterior for is given by

where D, = (yl, ..., yn) is the observed data. Gi(-) is the normal-inverse-gamma posterior

distribution whose cornponents are cr2 - G ( 2 , 9 ) with Si = b + w, and - N(xi, Xa:) with X=& and = . The weights Q are dehed by

and

1 QJ a ) , for j=1, ..., n and j # i,

( 2 q 2 ) 4

r(&) subject to qo + ... + qi-l+ qi+i + ... + Q,, = 1, "th M=(l+r)g and c(a)=,W&.

Gi(*) is the posterior distribution of q under the prior Go (a) . The weight qo is pro-

portional to a times the marginal density of Yi evaluated at the data yi with Go (*) as

the prior for q. The weight Q is proportional to the likelihood of data yi being a sample

from a normal distribution Yilq or just the density function of N( b, O?) at the point

Yi . The conditional posterior distribution niIn('), D,, is a weighted mixture of the best

guess for the prior Go ( 0 ) with single atom distributions on the other values on which we

condi t ioned.

This conditional posterior distribution can be sampled using two methods. The k t

method is described in Escobar and West (1995). Sample a value of ri from Go (-) with

probability qo or else use an existing value of ri with probability Q. The second method

was proposed by MacEachren (1994) and Bush and MacEachren (1995). MacEachrenls

method is a two step procedure that is equivalent to sampling from the pwterior distri-

bution directly, as Escobar and West (1995) do. In the first step a cluster structure for

K is generated. The cluster structure partitions nl , . . . , ?r, into k sets using a multinomial

distribution. All of the r,s in a particular set are identical, but those in different sets

will differ. Call the k sets of the TS nf , ... , iri and let denote the common value of

the T* in the set i. In the second step the I$ are sampled from Go (=) . The problem with

the algorithm used by Escobar and West (1995) is that sometimes not enough new draws

are taken from Go ( 0 ) and so the bulk of the ri are taken previous values. This means

that the Gibbs sampler will take much longer to mix owr the posterior distribution and

lead to poor estimates of posterior quantities. MacEadiren's algorithm doesn't run into

this problem and so has better convergence properties, which he illustrates in his paper

analytically as well as with some simulated data examples.

3.3 The model

3.3.1 The Gibbs Sampler with a Multivariate Normal P ~ o r

In this papa the transition between states is modelled as a discrete process. To construct

the likelihood function for the hazard model the standard normal cumulative distribution

function, 8 ( X h P ) , is selected to be the functional form for the continuation probabil-

ily. The contribution of each household or individual to the likelihood function for a

completed speli is given by

where Th is the length of the speLi for individual h. This is the product of the continuation

probabilities for the Th - 1 penods the individual sunrives multiplied by the probability

the individual exits in period Th. For a censored or uncompleted spell the contribution

to the likelihood hinction is given by

where Tc is the censoring point. Here the contribution to the likelihood function is just

the product of the continuation probabilities for the T, penods the individual survives.

The likelihood function for all the individuals in the sample can be written as

where

1 if individual h survives at time t dht = { O O therwise

Combining the likelihood hinction with the prior distribution on B we obtain, by Bayes

105

d e , the joint postenor distribution for the hazard model

where p(p) is the prior distribution for P. In this section, we consider the case where p(P) is multivariate normal. Even with

the multivariate normal pnor this posterior distribution does not provide closed form so-

lutions for posterior quantities of interest, for example the postenor rnean and variance.

However, Monte Carlo methods, such as the Gibbs sarnpler, can be used to sample from

the joint postenor distribution. Once we have the draws from the posterior distribution,

we in effect "know" everything because the posterior distribution summarizes all the

available information about the parameters of interest conditional on the observed data

and pnor information. However, the draws from the posterior distribution can also be

used to obtain summary measures of the posterior distribution such as the pasterior mo-

ments and quantiles of the distribution. Directly drawing from joint posterior distribution

is, dortunately, a formidable task. However, a Gibbs sampler with data augmentation

can be implemented for this model to obtain draws fiom the joint posterior distribution.

The Gibbs sarnpler is a Markov Chain Monte Carlo method that requires sampling from

the condi tional postenor distributions, which in this case are more tract able. Under

fairly weak regularity conditions the draws from the condi tional posterior distributions

wiIl converge to draws from the joint posterior distri b ~ t i o n . ~

To implement the Gibbs sampler for the multiperiod probit model introduce the latent

data such that8

'~cCul loch and Roasi (1994) prove that the Gibbs sampler for a binomial probit model converges under fairly weak regularity conditions.

' ~ h e identification of the model is achieved by assuming the variance of the btent utility equals one.

where Xht is a vector of observable characteristics for individual h at time t and Uht

is the unobserved latent data for individual h at time t. If UN > O the individual

survives otherwise he exits. The UM are unobservable, however, the distribution of the

Uht conditional on du (the binary indicator variables) is truncated univariate normal.

By introducing the latent data Uht it is pmible to apply a data augmentation step

(Tanner and Wong (1987)) in the Gibbs sampler. Data augmentation is hequently used

in Bayesian analysis of limited dependent variable models; for example, the probit model

(Chib and Albert (1993a)) and multinomial probit model (McCdoch and Rossi (1994)

and Geweke, Keane and Runkle (1994)). The introduction of the latent variables makes

the conditional distributions of the augmented postenor distribution like those of the

normal linear regression model and hence makes the Gibbs sampler feasible.

The multipenod probit framework presented here WU incorporate unobsenred hetero-

geneity by allowing some of the covariates to have random effects associated with them.

The continuation probabiliw with random effects therefore is given by O(XhtP + wLtOh ),

where Xht is a vector of covariates, /3 is a vector of regression parameten, wht is a subset

of Xht and Oh is a vector of individual specific rsndom effects whjch are distnbuted as

Bh(D - N(0, D). The contribution to the likelihood function for this specification is

a q-dimensional integral that is often intractable. However, Albert and Chib (1993b) show

that if this posterior is augmented with latent data, as in the binary probit model, a Gibbs

sampler can be implemented. So rather than integrating out the random effects from

likelihood function, 1 include them as parameters to be sampled in the Gibbs sampler.

The Gibbs sampler requires sampling the latent data, the parameter vector ,O, the random

effects Bi, and the covariance matrix of the random effects D. In the empirical work

that follows the random effect wiil not be a vector but will enter ody through the

intercePt The marginal posterior distribution of the random effect can then be examined

to determine the extent of the heterogeneity present (Allenby and Rossi (1993)).

The Gibbs sampler requires sampling from the following conditional distributions:"

(i) Sample the latent data UN

truncated at the left by O if survive U,l p,oh,a2,data - N ( X 3 + ~ & 6 h ? 1 ) 1

truncated at the right by O if exit

for h = 1, ..., N and t = 1 , ..., Th.

(ii) Sample the random efFect Oh

P, vu, 2, data - N(bh, KI), h = 1, ..., N ,

where bh = V ~ ' W L ( G - XhP), V i l = (WAWh + ü2), where Wh is a T h x 1 matrix

containing the covariates for the random effects, Xh is Th x k matrix containing the

covariates for individual h and is a Th x 1 vector containing the latent data for

individual h. The pnor on the random effect is Oh Io2 - N(0, 02).

(iii) Sample the parameter vector p

- where f l = B;'(&'P + XEi X;(G - Wh&) and Bi = (&' + c;..=, xi&) if the

mdtivariate pnor on 0 is used, i.e. - Nk(& &)-

(iv) Sample the variance of the random effect c2

gAn alternative specifcstion for allowing unobeerved heterogeneity to enter is to allow for a11 the parameters in the mode1 to be random.

1°T'he Gibbs sampler is d e d in FORTRAN 77. The truncated univariate normal random numbers are generated using the random number generator in Geweke (1991). Al1 the other random number generators are either subroutines in the IMSL Math/Stat Library or some transformation of an IMSL random number generator.

where IG (a,b) denotes an inverse gamma distribution with shape parameter a and sade

parameter b. The pnor on u2 is IG(u, 6).

3.3.2 The Gibbs Sampler with the Dirichlet Process Prim

With the addition of the Dirichlet process pnor to the random effects Oh there is a slight

modification to the Gibbs sarnpIer presented in the previous section. Step (ii) in the

Gibbs sampler is replaced by:

(iia) Sample the random effect Oh h m the Dirichlet process conditional posterior

where bh and Vc 'are defined as they were in (ii) , O(") = (O1, .. . , Oh- 1, Oh+.

and

for k=l, ..., N and k# h, subject to qo + ... + qh-1 + qh+i + ... + q~ = 1 and 6(Ok)

denotes a mass point at 8 = Bk and kh is Th x 1 vector of 1s.

To implement MacEachren's two step algorithm defbe the latent configuration indi-

cator Sh = k if observation h is an element of cluster k. Then the Oh will be assigned to

a duster according to P (sh = klO(h), S(h), k) = qLk where

and nk is the number of observations in ciuster k. If Sh = O then draw a new value of

Oh from the nonnal postenor distribution. Assign each observation to a cluster and then

draw the clustered values of Bi, 8,'. So we c m rewrite (iia) as :

(iia') Sample the la tent indicator variables fiom the mult inornial distribution

if h=O 7

i f h > O

(iib) Sample the distinct ciuster value 84 from

where bh = ChESh I / ~ ' W L ( ~ - Xh@, Vil = (ChEW WLWh + nhü2) , where Wh is a

Th x 1 matrix containing the covariates for the random effects, Xh is a Th x k mat&

containing the covariates for the parameter vector ,û and & is a Th x 1 vector containing

the latent data. The sum is over all the observations that Lie in the duster denoted by

the latent indicator Sh.

The precision parameter cr for the Dirichlet process prior can also be sampled dong

with the other parameters in the Gibbs sampler. Escobar and West (1995) show that a

can be sampled by first sampling a latent variable 17 from a Beta distribution and then

sampling the value of a, conditional on the value of r ) , from a mixture of two Gamma

distributions. The Gibbs sampler will then be given by (i), (iia'), (iib), (iii) , (iv) and

(v) Sample the latent variable g

q(a, k , data - Beta(ai + 1, N ) ,

where k is the distinct number of clusters and N is the sample size.

(vi) Sample the smoothing parameter a

al?, k, data - r,G(a + k, b - logr)) + (1 - ?r,)G(a + k - 1, b - log q ) ,

where a and b are pnor hyperparameters kom the pnor distribution of a, which is

assumed to be G(a,b), a gamma distribution with shape parameter a and scale parameter

b, and the mixture weight ~r, is given by = Nyi!&.

3.4 Empirical Illustration

3.4.1 The Data

The data are taken from the Canadian Labour Market Activity Survey (LMAS). This is

a weekly panel data set cowring a probability sample of individuals from the province

of New Brunswick, Canada, over the penod 1988 to 1990. In the initial year of the sur-

vey, information is collected through a supplement to the monthly Labour Force Sunrey

(LFS), which is very much like the U.S. Current Population survey. In subsequent years

respondents are re-contacted and intervieweci about their labour market activities over

the intemning period of time. This survey provides weekly information on respondents'

periods of employment. For New Brunswick there are 1518 employment spells which

range in length from 1 to 96 weeks for the 999 individuals in the sample. Of these spells

384 are censored and the average duration is 25.5 weeks.

Baker and Rea (1993) use these data to study the effect of the Canadian unemploy-

ment insurance (UI) program on employment durations. In 1989, individuals had to

accumulate between 10 and 14 'insured weeks' of employment in the yesr proceeding a

UI daim to qualify for benefits. The precise number of weeks, the Variable Entrance

Requirement (VER), varied across the 48 'econornic regions' in Canada according to the

local unemployment rate. This feature of the Canadian UI system came up for periodic

renewal. At the end of 1989, a dispute between the House of Commom and the Senate

delayed passage of a bill that would have renewed the VERS for the following year. As

a result, the VERS were not renewed for 1990, and in the first 11 months of that year

the entrance requirement was set to 14 weeks in all the economic regions of Canada, re-

gardless of economic conditions. Baker and Rea use the 'experiment' represented by this

change in the VERS to examine the effects of UI eligibility d e s on employment dura-

tion. In particular they look for spikes in the employment hazard in the week individuals

qualify for UI benefits.

For preliminary data analysis the empirical hazard is calculated. The empirical hazard

is calculated as I ( t j ) = 9 , where hj is the number of exits at duration t j divided by the "r number continuing at duration t j for the completed spells in the sample. The estimate

of the empirical hazard for the fist 24 weeks of employment is plotted in Figure 1 for the

spells in 1989 and 1990 separately. In Figure 1 we see that there is a large decrease in the

hazard at 10 weeks between 1989 and 1990. This cross year variation in this spike suggests

some sort of UI effect; i.e. 10 week spells were less likely in 1990 when the VER was set

at 14 weeks in all regions of the province." Another one of the features readily apparent

from the plots of the ernpiricai hazard are the regular spikes that appear approximately

every two weeks. Baker and Rea (1993) note that these spikes can appear for a d e t y

of reasons; digit preferences (the tendency of individuals to report the length of their

employment spells rounded off to the nearest even number or multiple of one month),

calendar effects (the tendency of spells to begin or end at the beginning or end of a

month) or local employment initiatives which provide a relatively large number of jobs

of fixed duration.

Baker and Rea constmct dummy variables to capture the effect of UI provisions on

employment duration. The dumrny variables are dehed for the p0int.s in time, or periods,

l1The VER was set at 10 weeks in al1 the 'economic' regions of New Brunswick in 1989. However, not al1 individuals faced this requirement. lndividuals who were repeaters, i.e. individuals who have made previous UI claims, faced longer eligibility requirements. If tliere are large numbers of repeaters this will cause spikes in the employment hazard, in 1989, other than the main spike at 10 w d c s .

that an individual: 1) initially qualifies for UI, 2) qualifies for UI but is still accumulating

benefit entitlement and 3) qualifies for UI at the rnmimum entitlement. They take

taro different approaches to identify these periods. The k t , the 'Duration' approadi,

assumes that an individual enters an employment spell with no accumulated insured

weeks from previous employment spells. The second, the 'Insured Weeks' approach,

counts insured weeks from any employment spell between the last benefit claim and the

start of the current employment spell. Using the 'Duration' approach, EL1 takes the

value 1 in the week thst the individuals's current employment duration satisfies the local

Unemployment Insurance eligibility requirement and a value O in all other weeks. This

variable captures spikes in the employrnent hazard which are conelated with the week

in which individuals initially quaiify to UI claims. The variable EL2 takes the value 1

in the period in which the individual's current duration falls between the initial week

of eligibility and the week in which maximum benefit entitlement is reached. EL2 is

used to capture the effect of additional entitlement on the hazard. Finally, EL3 takes

the value 1 in the weeks in which the indîvidual has qualified for UI at the maximum

benefit entitlement for his region. EL3 captures the more permanent features of eligibility

on employment duration. The other set of eligibility variables called ILI, IL2 and IL3

are constructed in the same fashion as EL1, EL2 and EL3 but using an estimate of

insured weeks instead of the current employment duration to determine an individual's

UI eligibility. Further details of these two approaches, as well as a discussion of their

relative merits, can be found in Baker and Rea (1993).

Two specifications of the duration mode1 with random effects are estimated. Specifi-

cation 1 includes the constant, controls for duration dependence, a time (year) effect

and controls for age, education, real hourly eanüngs, the provincial unemployment rate,

gender, marriage, schml attendance, previous receipt of unemployment insurance as well

as the eligibility variables. Specification 2 includes the constant, controls for duration

dependence , the yesr dumniy and the eligibility dummy variables. A full description

of the construction of these variables is provided in Table 1 and sample statistics are

presented in Table 2.

The baseline hazard is modeiled with two aiternative specifications. The first is to

use a parametnc specification, for example a low order polynomial in duration. A 4th

order polynomial was selected because that was the highest order that was supported by

the data. The alternative to this parametnc specification of the baseline hazard is to use

a nonparametric specification that is often referred to as a step function (Ham and Rea

(1987) and Meyer (1990))'' and allow a more flexible form for the duration dependence.

The step function models the baseline hazard by constructing a series of time varying

dummy variables. After some experimentation the following specification for the step

function was selected. The dummy variables will take single values for weeks 2-14 and

then groupings for weeks 15-16, 17-18, 19-20, 21-25, 26-30, 31-40, 41-50, 51-60 and for

spells longer than 61 weeks.13 l4

The results for the muitiperiod probit with a multivariate normal pnor for the hetero-

geneity parameters are presented in Tables 2a-5b. The Gibbs sampler was run for 1200

iterations with the first 200 iterations discarded to d u c e the effect of initial conditions.

The pnors for the mode1 were chosen to be proper but very diffuse. The pnor on the

parameter vector f l was N(0, 100Is). The @or on the variance of the random effect a2

was IG(u, 6) where v = 44 and 6 = 12. For the parameter a, the smoothing parameter

12With discrete data, duration dependence can be modelled nonparametrically with a step fundion as suggested by Han and Hausman ( 1990).

130ther spec ih t ions of thestep fundion are also passible. A specification that has wider groupings for the time intervals will produce a hazard estirnate that has a more p r o n o u n d step shape. A specification t bat has finer time intervals will produœ an estimate of the hazard that is more "jagged" . I have chosen a specification that lies in between these two extremes.

14The choiœ between parametric and nonparametric specifications of duration dependence is of in- terest in Baker and Rea's study because they were primarily interestexi in capturing the spikes in the ernployment hazard. A parametric specification of duration dependenœ will produce a much smoother estirnate of the hazard, which will smooth many of the spikm that appear in the estimate of t he empir- ical hazard. The nonparametric d a t i o n of duration dependence will capture most of spikes in the hazard, if there are enough single week dummy variables included in the step specification, i.e- if the time intevals are very fine.

for the Dirichiet process, the prior was G(a,b), where a=l and b=2. Negative parame-

ter estimates in the multiperiod probit mode1 indicate that the continuation probability

deceases with the covariate and hence the hazard increases with the covsriate. For spec-

ification 1 (the specification with controls for observable heterogeneity), the estimate of

the EL1 parameter indicates an increase in the employment hazard in the initial week of

UI eligibility. This is similar to the result in Baker and Rea. Using the 'Insured Weeks'

approach, the estimate of the parameter on the IL1 variable also indicates an increase

in the employment hazard in both models. The estimates of EL2 and IL2 indicate a

decrease in the employment hazard. Fially, the estimate of EL3 indicates an incresse

in the hazard in both specifications, while the estimate of IL3 indicates a decrease. The

variance of the random effect is fairly s m d and is identical to 4 decimal place for ail the

specifications that were estimated.

Tables 6a, 6b, 7a and 7b present the results for the specifications estimsted with

the nonpararnetnc specification of the baseline hazard. These results appear to be very

sensitive to this specification of the duration dependence. There are some changes in the

sign of the postenor means for the covariates controlling for the observable heterogeneity.

In Table 6a the posterior mean of EL1 remains negative but is alrnost twice as large as

the value obtained from the specification with the time polynomial. The EL2 variable

is now found to increase the hazard and EL3 has a mu& larger postenor mean. For

the ILi variables, IL1 and IL3 have a positive effect on the hazard. For specification 2

the ELi variables all have the same sign and are sirnilar to the values in Table 6a for

specification 1. The results for the ILi variables are much different than those obtained

for specification 1. The ILi variables are all negative and slightly smaller than the values

obtained using the 'Duration' approach. Also note that the postenor variances for the

eligibility variables are all much smaller. As with the time polynomial specification aiI

the estimates of the variance of the random effects are identical to 4 decimal places.

Tables 8a, ab, Sa and 9b present the results for the specifications using a parametnc

specification of the baseline hazard and the Dirichlet process prior. Tables 8a and Sb

contain the estimates for specification 1 and Tables 9a and Sb contain the estimates for

Specification 2. These results are very similar to those in Tables 3a, 3b, 4a and 4b (same

specifications but with the multivariate normal prior on the heterogeneity parameters).

The posterior means of the year effect, EL1 and EL3 are all slightly hrger while the

posterior mean of EL2 is slightly smaller when the Dirichlet process pnor is used. The

posterior variance of the random effect ( s e Table 12) is much also smaller when the

Dirichlet process prior is used.

Tables 10a, lob, I l a and l l b present the posterior means and variances for the

specifications with the step function and the Dirichlet process prior. These results differ

much more from those obtained with the multivariate normal prior on the heterogeneity

parameters. The posterior mean of the control for the year effect is about 5 smailer

than that obtained with the Dirichlet process prior. The eligibility variables alI have

the same signs as the specification with the multivariate normal prier but the posterior

means tended to be much smder. The posterior mean of EL1 was almost 50% smaller

than that obtained with the multivariate normal pnor. The posterior means of EL2

and EL3 were much more smaller than those obtained with the multivariate normal

prior. The results for the 'Insured Weeks' approach also show the same patterns as the

posterior means obtained with the 'Duration' approach and the Dirichlet process prier.

The posterior rnean of the control for the year is about j smaller than that obtained with

the multivariate normal pnor on the heterogeneity parameters. The posterior means of

the eligibility variables al l tended to be smaller than those obtained with the multivariate

normal pnor on the heterogeneity parameten. The posterior variance of the random

effect (see Table 13) were smaller than those for the specifications estimated with the

multivariate normal prior.

To help illustrate the effect of the week of eligibility on the hazard across the specifi-

cations in the hazard I present the hazard at weeks O and the week of eligibility, 14 weeks

for the 'Duration' approach and 10 weeks for the 'Insured weeks' approach, at the sarnple

means of the covariates. These results are presented in Tables 14 to 17. For the speci-

fications with the multivariate normal pnor on the heterogeneity parameters (Tables 14

and 15) the parametric duration dependence produces a larger increase in the hazard be-

tween week O and the week of eligibility for all specifications. When the Dirichlet process

prior is used on the random effect the nonpararnetric duration dependence specification

produces a larger increase in the hazard (when both the 'Duration' and 'Insured Weeks'

approaches are used to constmct the eligibility variables) when specification 1 is used.

With specification 2 the pararnetric duration dependence specification produces a larger

increase in the hazard for both the 'Duration' and the 'Insured Weeks' approach.

As an alternative method of comparing results, plots of the estimated hazard function

are also presented. The hazard is estimated by computing the exit probabilities for al1 the

individuals in the sample and awraging across all the individuals at each iteration in the

Gibbs sampler. These exit probabilities are then averaged over all the iterations, after

discarding the first 200 iterations, to produce a Monte Carlo estimate of the hazard.

The plots of the estimated hazard functions are presented in Figures 2-5. The first

panel in each figure contains a plot of the estimate of the hazard from the specification

estimated with the multivariate normal prior on the heterogeneity parameters, while the

second panel contains the estimate of the hazard from the specification estimated with

the Dirichlet process pnor.

Figures 2 and 3 contain the hazard estimates for the specifications which use the

parametric specification of the baseline hazard. In Figure 2 there is not much of a

difference between the estimates of the hazard with the Dirichlet process prior and with

the mdtivariate normal pior on the heterogeneity parameters. However, in Figure 3 we

see that the estimate of the hazard from the specification which uses the Dirichlet process

prior specification is not as smooth as the estimate which uses the multivariate normal

prior on the heterogeneity parameters. In particular, the spikes at 10 and 14 weeks are

more pronounced when the Dirichlet process pnor is used to estimate the model. Figures

4 and 5 present the hazard estimates for the specifications which use the nonpararnetric

specification of the baseline hazard. In Figures 4 and 5 we see that the two estimates of the

hazard are not very different both are still quite jagged, aithough the estimate from the

specifications with the multivariate nomal pnor on the heterogeneity parameters appear

to be slightly wioother. This result is similar to the finding in Han and Hausman (1990)

and Manton, Stallard and Vaupel (1986) that the nonparametric specification of the

baseline hazard reduces the sensitivity of the estimates to the parametric heterogeneity

sssumpt ion.

The estimates of the smoothing parameter for the Dirichlet process pnor are pre-

sented in Table 18 (Figures 6-9 are plots of the posterior of a). The values of a, the

smoothing parameter for the Dirichlet process, were smaller for the specifications that

used the nonparametric specification of the baseline hazard. The smder vaiues of a for

specification 2 with the nonparametric specification of the baseline hazard indicate that

there was more clustering, i.e. a smaller number of distinct clusters, than the specifi-

cations estimated with the parametric specification of the baseline hazard. Hence there

were slightly more deviations from the baseline parametnc model when a nonparametric

specification of the baseline hszard is used. However, the estimates of a are still very

large for all the specifications that are estimated. This means that while there is some

clustering in the random effects, the large value of a indicates that there are not large

deviations from the baseline parametric model. To check the sensitivity of the results to

the parameter a the prior hyperpararneters for a were picked to be a=0.001 and b=0.001.

The selection of these s m d values of a are motivated by the fact that the prior for log a

will be uniform as a 3 O and b -, O. The new estimates of a are in Table 19. These

results indicate that while the values of a are smaller than those in Table 18 they are

still quite large ranging from 345.60 to 424.91.15

''The posterior means from specifications estimated with the prior hyperpanimeters a==.ûûl and b=û.001 did not d8er greatly fiom the resuIts presented in Tables 8a, 8b, 9a, 9b, lûa, lob, 1 la and 1 lb. There were also no substantial changea in the estirnate of the hazard function.

3.5 Concluding Remarks

Bayesian estimation of a discrete time duration model with a normal cumulative density

function as the functional form for the continuation probability was presented. Unob-

senred heterogeneity was allowed to enter through individual specific randorn effects.

This heterogeneity distribution was estimated in two fashions. The first approach was to

use a multivariate normal prior on the random effects and estimate the heterogeneity dis-

tribution parametrically. The second approach was to allow a more flexible form for the

heterogeneity distribution by introducing a Dirichiet process pnor on the random effect

and effectively allow for Bayesian nonparametnc estimation of this part of the model.

The method was applied to a set of employment duration data from New Brunswick,

Canada.

The results from the empirical section indicate that there are not substantial devia-

tions from the baseline parametric heterogeneity specification when a pararnetric spec-

i fication of the baseline hazard is employed. However , when a nonparametric speci fica-

tion of the baseline hazard is used the estimates of a, the smoothing parameter for the

Dirichlet process pnor, indicate that there are slightly more deviations from the baseline

parametric heterogeneity model. The large estimates of a indicate that there is not a

great deal of clustering occurring with the random effects. For there to be considerable

clustering much smaller values of a, would have to be obtained (e.g. single digit values

of a).

The estimates from the specification with a nonparametric baseline hazard also indi-

cate that once a nonparametnc specification of the baseline hazard is used the estimate

of the hazard will be less sensitive to the specification of the unobserved heterogeneity

distribution. These results indicate that the parametric model would be s&cient to

model the unobserved heterogeneity when a nonparametnc specification of the baseline

hazard is used. The specification of a parametric baseline hazard makes the estimate of

the hazard slightly more sensitive to the specification of the unobserved heterogeneity

distribution.

'Igble 1: Description of Variables

unemployment rate: the monthly unemployment rate

hourly earning : Average hourly earnings, ail values convertcd to 1989 dollars

age 1624: 1 for those 14-24 years of age in 1988, O otherwise

age 25-44: 1 for those 2544 years of age in 1988, O otherwise

high school: 1 if hi& school graduate, O otherwise

post secondary: 1 if post secondary education, O otherwise

university: 1 if university degree, O otherwise

trade certificate: 1 if trade certificate or diploma, O othefFKise

pst UI receipt: 1 if the individual received U? income in the previous year

marital status: 1 if the individual is married

school attendance: 1 if the individual attended school in the year of the current week,

O otherwise

sex: 1 if the individual is female, O otherwise

year: 1 if a week during the year 1990, O otherwise

EL1 or ILI: 1 in the week individual satisfies the local UI eligibility requirement

EL2 or IL2: takes the value 1 in the weeks that the individual satisfies the local UI

eligibility requirement, but has not yet achieved the maximum benefit entitlement, O

ot herwise

EL3 or IL3: takes the value 1 in the weeks that the individual has satisfied the local UI

eligibility requirernent and has reached the maximum benefit entitlement , O otherwise

Note: For age the excluded group is age 45-64 years of age, for education the exduded

group are individuals that have not completed high school.

Tàble 2: Descriptive Statistics for the Data and Durations

Variable

Unemployment Rate

Hourly Earnings

age: 16-24

age: 25-44

High School

Post Secondary

Trade Certificate

University

Past UT receipt

Marital Status

School At tendance

Sex

Sample Mean

11-89

9 .O3

0.37

0.47

0.22

0.28

0.04

SAMPLE STATISTICS FOR D URATIONS

Standard Error

1.13

4.57

0.48

0.50

0.41

0.44

0.19

0.18

0.49

0.48

0.39

0.48

Sample Mean Standard Error Minimum Maximum % Censored

25 -5 1 20.96 1.0 96.0 25 -3

a b l e 3a: Specification 1, Multivariate N o r d Prior on Heterogeneity

Parameters, Duration Approach, Paramettic Duration Dependence

Variable

constant

unemployment rate

hourly earnings

age: 16-24

age: 25-44

high school

post secondary

uni ver si ty

trade certificate

past UI receipt

marital status

school attendance

sex

Far

EL1

EL2

EL3

Pos t erior Mean

2.0215

0.0263

-0.0002

-0.0425

0.0159

0.0862

0.1417

0.1323

0.1085

-0.1646

0.0662

-0.2326

-0.0946

0.1046

-0.3630

0.0139

-0.1451

Posterior Variance

0.0377

0.0002

9.4E6

0.0021

0.0017

0.0018

0.0016

0.0054

0.0075

0.0010

0.0013

0.0018

0.0008

0.0009

0.0043

0.0033

0.0049

NSE

O .O074

0 -0006

0.0001

0.0018

0.0017

0.0018

0.0016

O .O027

O .O038

0.0014

0.0015

0.0016

0.0009

0.0011

0.0023

O. 0024

0.0025

a b l a 3b: Specification 1, Multivariate Normal Prior on Heterogeneity

Parameters, Insured Weeks Approach, Parametric Duration Dependence

Variable

constant

unemployment rate

hourly esrnings

age: 16-24

age: 25-44

high school

post secondary

university

trade certificate

past UI receip t

marital status

school a t t endance

sex

Far

ILI

IL2

IL3

Post erior Mean

2.0335

0.0252

0.0004

-0.0524

-0.0009

0.0096

O. 1405

0.1309

0.1167

-0.1617

0.0630

-0.2309

-0.0942

0.1255

-0.2309

0.0977

0.0427

Post erior Variance

0.041 1

0.0002

1.2E5

0.0020

0.0016

0.001 2

0*0012

0.0070

0.0065

0.0013

0.0013

0.0015

0.0009

0.0007

0.0015

0.0025

0.0029

NSE

0.0082

0.0005

0.0001

0.0016

0.0015

0.0012

0.0014

0.0038

0.0033

0.0016

0.0014

0.0014

0.0012

0 .O009

0.0014

0.0019

0.0022

a b l e Ba: Specification 2, Multivariate Normal Prier on Heterogeneity

Parameters, Durat ion Approach, Parametric Durat ion Dependence

Variable Posterior Mean Posterior Variance NSE

constant 2.2299 0.0106 0.0027

F a r O. 1089 0.0008 0.0013

EL1 -0.3670 0.0050 0.0020

EL2 0.0084 0.0027 0.0021

EL3 -0.1445 O. 0046 0.0024

Table 4b: Specification 2, Multivariate Normal Prior on Heterogeneity

Parameters, Insured Weeks Approach, Parametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 2.2509 0.0102 0.0028

Year O. 1231 0.0008 0.0012

IL1 -0.2290 0.0054 0.0024

IL2 0.0805 0.0023 0.0018

IL3 0.0414 0.0023 0.0018

Table 5: Posterior Variance of the Randorn Mect

Specification Duration Approach Irisured Weeks Approach

1 0.00194 0.00194

2 0.00194 0.00193

'hble 6% Specification 1, Multivariate Normal Prior on Heterogeneity

Parameters, Duration Approach, Nonparametrie Duration Dependence

Variable

constant

unempioyment rate

hourly eamings

age: 16-24

age: 25-44

hi& school

post secondary

universi ty

trade certificate

past UI receipt

marital status

school at tendance

sex

year

EL1

EL2

EL3

Post erior Variance

0.4033

0.0004

7.631-5

0.0049

0.066 1

0.0218

0.0046

0.0241

0.1339

0.0068

0.0012

0.0013

0.0209

0.0023

0.0099

0.0039

0.0043

NSE

0.0421

0.001 1

0.0005

0.0015

0.0174

0.0098

0.0147

O . O O M

0.0249

0.0052

O .O069

0.0069

0.0098

0.0244

0.0041

0.0032

0.0033

'Iàble 6b: Specification 1, Multivariate Normal Prior on Heterogeneity

Parameters, Insureci Weaks Approach, Nonparametric Duration Dependence

Variable

constant

unemployment rate

hourly eamings

age: 16-24

age: 25-44

high school

post secondary

trade certificate

past UI receipt

marital status

school attendance

Pos t erior Mean

3.6332

0.5085

-0.0471

-0.0154

-0.3825

-0.0898

0.0419

O -421 1

0.3497

0.2902

-0.0794

0.1161

-0.2118

-0.2873

-0.4082

-0.2910

-0.6512

Post erior Variance

0.1349

0.0025

4.0EX

9.53-5

0.0326

0.0236

0.0878

0.0099

0.0986

0.0503

0.0046

0.0101

0.0090

0.0227

0.0077

0.0036

0.0027

NSE

0.0215

O. 0027

0.0011

6.4E4

0.0118

0.0103

0. OOS?

0.0062

0.0212

0.0145

O. 0039

O. O063

0.0058

0.0101

0.0037

0.0026

0.0026

Tàble 7a: Specification 2, MultiMnate Normal Prior on Heterogeneity

Parameters, Duration Approach, Nonparametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 0.7702 0.0148 0.0065

Far 0.4225 O. O073 0-0027

EL1 -0.7346 0.0102 0.0029

EL2 -0.5875 0.0061 0.0027

EL3 -0.9802 0.0107 0.0027

Table 7b: Specification 2, Multivariate Normal Prior on Heterogeneity

Parameters, Insured Weeks Approadi, Nonparametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 0.5671 0.0203 0.0092

Far 0.5461 0.0242 0.0027

IL1 -0.4021 0.0106 0.0047

IL2 -0.3291 0.0046 0.0035

IL3 -0.7741 0.0042 0.0036

'Iàble &: Specification 1, Dirichlet Procees Prior on Heterogeneity

Parameters, Duration A p p r d , Parametric Duration Dependence

Vanable

constant

unemployment rate

hourly eamings

age: 16-24

age: 25-44

hi& school

post secondary

universi ty

trade certificate

past UI receipt

marital status

school attendance

sex

Year

EL1

EL2

EL3

Posterior Mean

2.0389

0.0184

-0.0001

-0.0385

-0.0025

0.0867

0.1366

O. 1074

0.1280

-0.1584

0.0600

-0.2127

-0.0791

0.1111

-0.3587

0.0014

-0.1 544

Pœterior Variance

0.0467

0.0002

l.lE-5

0.0019

0.0018

0.0012

0.0015

0.0057

0.0062

0.0013

0.0012

0.0025

0.0012

0.001 1

0.0058

0.0029

0.0048

NSE

0.0089

O .Oûû7

0.0001

0.0016

0.0019

0.0013

0.0016

0.0032

0.0034

0.0015

0.0014

0.0021

0.0017

0.0013

0.0022

0.0018

0.0028

a b l e 8b: Spacification 1, Dirichiet Pro- Prior on Heterogeneity

Parameters, Inilured Weelcs Approadi, Parametrie Duration Dependence

Variable

constant

unemployment rate

hourly eamings

age: 16-24

age: 2544

hi& school

post secondary

University

trade certificate

past UI receipt

marital statu

school at tendance

sex

year

IL1

IL2

IL3

Pos t erior Mean

1 .W86

0.0279

O .O00 1

-0.0525

0.0019

O .O844

O. 1438

0.1193

0.1226

-0.1624

0.0633

-0.2302

-0.1002

0.1215

-0.2089

0.0929

0.0392

Poetenor Variance

0.0384

0.0002

1 .OE5

0.0022

0.0016

0.0016

0.0013

0.0055

0.0058

0.001 1

0.0015

0.0017

0.0009

0.0008

0.0044

0.0021

0.0019

NSE

0.0076

0.005

0.0001

0.0018

0.0016

0.0016

0.0012

0.0033

0.0027

0.0013

0.0016

0.0014

0.0009

0.001 1

O .O022

0.0018

0.0018

'Iàble 9a: Specification 2, Dirichlet Process Prior on Heterogeneity

Paramet ers, Durat ion Approach, Parametric Duration Dependence

Variable Pos terior Mean Pos t erior Variance NSE

constant 2,0964 0.0134 0.0063

Far 0.1231 0.0009 0.0012

EL1 -0.3679 0.0052 0.0024

EL2 0.0005 0.0028 0.0019

EL3 -0.1752 0.0052 0.0050

Table 9b: Specification 2, Dirichlet Process Prior on Heterogeneity

Parameters, Insured Weeks Approach, Parametric Duration Dependence

Variable Posterior Mean Posterior Variance NSE

constant 2.2595 0.0079 0.0025

Far O. 1221 0.0009 0.001 1

IL1 -0.2232 0.0045 0.0016

IL2 0.0818 0.0020 0.0020

IL3 0.0424 0.0020 0.0020

Table 10a: Specification 1, Dirichlet Process Prior on Heterogeneity

Parameters, Duration Approadi, Nonparametric Duration Dependence

Variable

constant

unempioyment rate

hourly earnings

age: 1624

age: 25-44

high school

post secondary

University

trade certificate

past UI receipt

marital statu

school attendance

sac

Far

EL1

EL2

EL3

Poe t erior Mean

1.5363

0.0519

-0.0014

-0.0373

0.0022

0.0922

0.1350

0.1122

0.1615

-0.2109

0.0603

-0.2278

-0.0087

O. 2059

-0.4888

-0.1277

-0.1406

Posterior Variance

O. 1647

9.23-4

1.1E-5

0.0024

0.0023

0.0017

0.0032

0.0061

0.0145

0.0155

0.0018

0.0124

0.0019

0.0122

0.0874

0.0082

0.0107

NSE

0.0094

6.7M

0.0001

0.0019

0.0017

0.0010

0.0013

0.0026

0.0036

0.0026

0.0013

0.0022

9.83-4

0.0029

0.004 1

0.0019

0.0021

'Iàble lob: Specification 1, Dirichlet Process Prior on Heterogeneity

Parameters, Insureci Weeks Approach, Nonparametric Duration Dependenœ

Variable

constant

unemployment rate

hourly eaniings

age: 16-24

age: 25-44

high school

post secondary

trade certificate

past UI receipt

marital status

school attendance

sex

Poe teri or Mean

1 A888

0.0547

-0.0015

-0.0400

0.0013

0.0869

0.1332

0.1071

0.1405

-0.2061

0.0536

-0.2165

-0.0843

0.2365

-0.2304

-0.0390

-0.0516

Posterior Variance

O. 1883

0.001 1

l.lE.5

0.0028

0.0020

0.0022

0.0034

0.0065

0.0156

0.0168

0.0210

0.0098

0.0025

0.0132

0.0276

0.0053

0.0069

NSE

0.0094

6.3E-4

1.2u

0.0018

0.0014

0.0014

0.0023

0.0028

0.0028

0.0026

0.0015

0.0019

0.0012

0.0025

0.0034

0.0023

0.0024

'Iàble lla: Specification 2, Dirichlet Procesa Prior on Heterogeneity

Parameters, Duration Approach, Nonparametric Duration Dependenœ

Variable Posterior Mean Posterior Variance NS E

constant 2.0617 0.0051 0.0032

Fm O. 1567 0.0011 0.0014

EL1 -0.4289 0.0053 0.0022

EL2 -0.0914 0.0021 0.0016

EL3 -0.0627 0.0015 0.001 5

Table l lb: Specification 2, Dirichlet Process Prior on Heterogeneity

Parameters, Insured Weeks A p p r d , Nonparametric Duration Dependence

Variable Postenor Mean Posterior Variance NSE

constant 2.0324 0.0042 0.0025

Far 0.1766 0.0009 0.001 1

IL1 -0.2554 0 -0057 0.0023

IL2 -G.0353 0.0020 0.0018

IL3 0.0092 0.0015 0.0015

Note: In Tables 3a to l l b the reported NSE is for the posterior mean

Table 12: Posterior Variance of the Random Effect, Parametric Duration

Dependence

Specification Duration Approach In sud Weeks Approach

1 0.00194 0.00193

2 0.00194 0.00193

Tàble 13: Posterior Variance of the Random Effect, Nonparametric

Duration Dependence

Specification Duration Approach Insured Weeks Approach

1 0.00006 0.00014

2 0.00194 0.00193

'Iàble 14: Hazard Estirnates, Duration Approach, Multivariate Normal Prior

on Unobserved Heterogeneity Parameters

PARAMETRIC DURATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00339 0.06372 18.8

2 0.01033 0.07139 6.91 NOWARAMETRIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00035 O .O4869 13.7

2 0.01 125 O -0384 1 3.41

Table 15: Hazard Estimates, Insurecl Weeks Approadi, Multivariate N o r d

Prior on Unobserveci Heterogeneity Parameters

PARAMETRIC D URATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

2 0.00141 0.00583 4.14 NONPARAMETRIC DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

1 0.00673 0 .O3899 5.79

Table 16: Hazard Estimates, Duration Approach, Diridilet Process Prior on

Unobsewed Heterogeneity Parameters

PARAMETRIC DURATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

2 0.01353 0.08624 6.37 N O N P A R A M ~ C DURATION DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

Table 17: Hazard Estimates, Insured W-ks Approach, Dirichlet Process

Prior on Unobserved Heterogeneity Paramet ers

PARAMETRIC DURATION DEPENDENCE Specification Hazard at O weeks Hazard at 14 weeks A

2 0,01002 0.05834 5.82 NONPARAMETRIC DUFUT~ON DEPENDENCE

Specification Hazard at O weeks Hazard at 14 weeks A

HaPrrd at 14 Weeks Note: For Tables 14 to 17 A = -, ,,

Table 18: Estimates of a

Baseline Hazard Specification Duration A p p r d I n s d Weeks Approach

Parametric 682.12 567.92

Nonparametnc 526.93 521.36

Table 19: Estimates of a for Sensitivity Analysis

B~seline Hazard Specification Duration Approach Insured Weeks Approach

Parame t ric 424.91 372.73

Nonparamet ric 345.60 347.57

DURATION IN WEEKS

Figure 3-1: Plot of the Empirical Hazard by Year

Figure 3-2:

Emplayment Ouration in Weeks ( 2 1

Plot of the Estimated Hazard, Specification 2, Eligibility Detefznined by Lnsured Weeks

Figure 3-3:

EmpIayment Ouraiion in Weeks ( 1 1

Employment Ouration in Weeks ( 2 1

Plot of Estimated Hazard, Specification 2, Eligibility Determined by Duration

Ernployment Duman in Weeks ( 2 )

Figure 3-4: Plot of the Estimated Hazard, Specification 2, Eligibility Determined by Insured Weeks

Emplayment Duranan in Weeks ( 2 )

Figure 3-5: Plot of the Estimate Hazard, Specification 2, Eligibility Determined by Du- ration

Notes for Figures 2 to 5:

1 denotes the specification estimated with a multivariate normal p io r on the hetero-

genei ty parameters

2 denotes the specification estimsted with the Dirichlet process prior on the heterogene-

ity parameters

Figure 3-6: Posterior of Alpha, Parametric Baseline Hazard, Insured Weeks Approach

200 400 600 oao 1 000

ALPHA

Figure 3-7: Posterior of Alpha, Parametic Baseline Hazard, Duration Approach

Figure 3-8: Postenor of Alpha, Nonparametric Baseline Hazard, hsured Weeks Approach

ALPHA

Figure 3-9: Posterior of Alpha, Nonparametric Baseline Hazard, Duration Approach

Notes for Figures 6-9:

The plots of the postenor of a are for specification 2 and the prior on a is G(1,2)

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