Chapter 55
DURATION MODELS: SPECIFICATION, IDENTIFICATION AND
MULTIPLE DURATIONS
GERARD J VAN DEN BERG
Department of Economics, Free University Amsterdam**, De Boelelaan 1105, 1081 HV Amsterdam,
The Netherlands; CEPR; and Tinbergen Institute
Contents
Abstract 3383
Keywords 3383
1 Introduction 3384
2 Basic concepts and notation 3387
3 Some structural models of durations 3389
3.1 Standard search model 3389
3.1 1 Stationarity 3389
3.1 2 Nonstationarity without anticipation 3391
3.1 3 Nonstationarity with anticipation 3391
3.2 Repeated-search model 3393
4 The Mixed Proportional Hazard model 3394
4.1 Definition 3394
4.2 Time-varying explanatory variables 3397
4.3 Theoretical justification 3400
5 Identification of the MPH model with single-spell data 3405
5.1 Some implications of the MPH model specification 3405
5.2 Identification results 3408
5.3 Interaction between duration and explanatory variables in the observed hazard 3412
5.4 The sign of the interaction 3414
5.5 Specification of the unobserved heterogeneity distribution 3418
5.5 1 Discrete distributions 3418
5.5 2 Gamma distributions 3419
5.5 3 Suggestions from economic theory 3420
5.5 3 1 Suggestions from equilibrium search models 3420
5.5 3 2 Suggestions from on-the-job search models 3421
5.6 Effects of misspecification of functional forms 3422
6 The MPH model with multi-spell data 3426
6.1 Multi-spell data 3426
6.2 Identification results 3427
7 An informal classification of reduced-form multiple-duration models 3431
8 The Multivariate Mixed Proportional Hazard model 3437
8.1 Definition 3437
8.2 Identification results 3438
8.2 1 Competing risks 3439
8.2 2 Successive durations 3441
8.3 Specification of the bivariate unobserved heterogeneity distribution 3442
8.3 1 Dimensionality 3442
8.3 2 The dependence between the durations 3444
9 Causal duration effects and selectivity 3447
9.1 Lagged endogenous durations 3447
9.2 Endogenous shocks 3448
10 Conclusions and recommendations 3449
References 3453