Testing for unobserved heterogeneity in exponential and Weibull duration models

We examine the use of the likelihood ratio (LR) statistic to test for unobserved heterogeneity in duration models, based on mixtures of exponential or Weibull distributions. We consider both the uncensored and censored duration cases. The asymptotic null distribution of the LR test statistic is not the standard chi-square, as the standard regularity conditions do not hold. Instead, there is a nuisance parameter identified only under the alternative, and a null parameter value on the boundary of the parameter space, as in Cho and White (2007a). We accommodate these and provide methods delivering consistent asymptotic critical values. We conduct a number of Monte Carlo simulations, comparing the level and power of the LR test statistic to an information matrix (IM) test due to Chesher (1984) and Lagrange multiplier (LM) tests of Kiefer (1985) and Sharma (1987). Our simulations show that the LR test statistic generally outperforms the IM and LM tests. We also revisit the work of van den Berg and Ridder (1998) on unemployment durations and of Ghysels et al. (2004) on interarrival times between stock trades, and, as it turns out, affirm their original informal inferences.     

Isotonic estimators of monotone densities and distribution functions: basic facts

We present short proofs of some basic results from isotonic regression theory. A straightforward argument is given to show that the left continuous version of the concave majorant of the empirical distribution function maximizes the likelihood function f↦f (X,)… f (X _n ) within the class of non-increasing densities. Similarly, it is shown that the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of interval censored data has an interpretation in terms of the left derivative of a convex minor ant. Finally, a short proof is given to show that the number of vertices of the concave major ant of the uniform empirical distribution function is asymptotically normal with asymptotic mean and variance both equal to log n .     

Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1

We give a new proof of the asymptotic normality of a class of linear functionals of the nonparametric maximum likelihood estimator (NPMLE) of a distribution function with "case 1" interval censored data. In particular our proof simplifies the proof of asymptotic normality of the mean given in Groeneboom and Wellner (1992). The proof relies strongly on a rate of convergence result due to van de Geer (1993), and methods from empirical process theory.     

Identification of first-price auctions with non-separable unobserved heterogeneity

We propose a novel methodology for identification of first-price auctions, when bidders’ private valuations are independent conditional on one-dimensional unobserved heterogeneity. We extend the existing literature ( and ) by allowing the unobserved heterogeneity to be non-separable from bidders’ valuations. Our central identifying assumption is that the distribution of bidder values is increasing in the state. When the state-space is finite, such monotonicity implies the full-rank condition needed for identification. Further, we extend our approach to the conditionally independent private values model of Li et al. (2000), as well as to unobserved heterogeneity settings in which the implicit reserve price or the cost of bidding varies across auctions.     

The role of observed and unobserved heterogeneity in the duration of unemployment

This paper studies the degree to which observable and unobservable worker characteristics account for the variation in the aggregate duration of unemployment. I model the distribution of unobserved worker heterogeneity as time varying to capture the interaction of latent attributes with changes in labor-market conditions. Unobserved heterogeneity is the main explanation for the duration dependence of unemployment hazards. Both cyclical and low-frequency variations in the mean duration of unemployment are mainly driven by one subgroup: workers who, for unobserved reasons, stay unemployed for a long time. In contrast, changes in the composition of observable characteristics of workers have negligible effects.     

Efficiency Considerations in the Additive Hazards Model with Current Status Data

For current status data, L IN , O AKES and Y ing (1998) proposed a procedure for estimation of the regression parameters in the additive hazards model that makes clever use of martingale theory. However, one of the outstanding problems posed in the paper was the issue of efficient estimation, as their estimators do not attain the semiparametric information bound. In this paper, we explore this issue and provide a characterization of the NPMLE. We conduct efficiency comparisons between the NPMLE and the procedure of L IN et al . (1998) analytically and numerically through analysis of a dataset from a tumorigenicity experiment.     

The (mis)specification of discrete duration models with unobserved heterogeneity: A Monte Carlo study

Empirical researchers usually prefer statistical models that can be easily estimated with the help of commonly available software packages. Sequential binary models with or without normal random effects are an example of such models that can be adopted to estimate discrete duration models with unobserved heterogeneity. But an easy-to-implement estimation may incur a cost. In this paper we conduct a Monte Carlo simulation to evaluate the consequences of omitting or misspecifying the unobserved heterogeneity distribution in single-spell discrete duration models.     

Testing for time-invariant unobserved heterogeneity in generalized linear models for panel data

Recent literature on panel data emphasizes the importance of accounting for time-varying unobservable individual effects, which may stem from either omitted individual characteristics or macro-level shocks that affect each individual unit differently. In this paper, we propose a simple specification test of the null hypothesis that the individual effects are time-invariant against the alternative that they are time-varying. Our test is an application of Hausman (1978) testing procedure and can be used for any generalized linear model for panel data that admits a sufficient statistic for the individual effect. This is a wide class of models which includes the Gaussian linear model and a variety of nonlinear models typically employed for discrete or categorical outcomes. The basic idea of the test is to compare two alternative estimators of the model parameters based on two different formulations of the conditional maximum likelihood method. Our approach does not require assumptions on the distribution of unobserved heterogeneity, nor it requires the latter to be independent of the regressors in the model. We investigate the finite sample properties of the test through a set of Monte Carlo experiments. Our results show that the test performs well, with small size distortions and good power properties. We use a health economics example based on data from the Health and Retirement Study to illustrate the proposed test.     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Bias corrections for two-step fixed effects panel data estimators

This paper introduces large-T bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These models include systems of equations with limited dependent variables and unobserved individual effects, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time varying heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control variable from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control variable accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two-step nature of the procedure. As these two-step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large-T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.     

Partial rank estimation of duration models with general forms of censoring

In this paper we propose estimators for the regression coefficients in censored duration models which are distribution free, impose no parametric specification on the baseline hazard function, and can accommodate general forms of censoring. The estimators are shown to have desirable asymptotic properties and Monte Carlo simulations demonstrate good finite sample performance. Among the data features the new estimators can accommodate are covariate-dependent censoring, double censoring, and fixed (individual or group specific) effects. We also examine the behavior of the estimator in an empirical illustration.     

Mixture models of choice under risk

This paper is concerned with estimating preference functionals for choice under risk from the choice behaviour of individuals. We note that there is heterogeneity in behaviour between individuals and within individuals. By ‘heterogeneity between individuals’ we mean that people are different, in terms of both their preference functionals and their parameters for these functionals. By ‘heterogeneity within individuals’ we mean that the behaviour may be different even by the same individual for the same choice problem. We propose methods of taking into account all forms of heterogeneity, concentrating particularly on using a Mixture Model to capture the heterogeneity of preference functionals.     

Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

Unemployment Duration: An Analysis of Incomplete,Completed, and Multiple Spells in Emilia-Romagna

Sets of incomplete and completed spells of unemployment wereobtained from the Italian Quarterly Labour Force Survey,carried out by ISTAT in Emilia-Romagna (1993:1–1995:1).The data were analysed through a proportional hazards modelwith a Weibull specification of the baseline hazard, includingboth unobserved heterogeneity applied to the scale parameterof the duration distribution, and telescoping effect to accountfor spikes in the distribution of unemployment spells.They were compared formally also with a non-proportional hazardmodel based on a log-logistic distribution of duration.The estimation of the parameters was carried out separately on bothcompleted spells and quarterly incomplete spells to ascertaindifferences and to envisage the potential seasonal effect.The results showed that the shape parameters changed over time.The parameters of each covariate proved to be statisticallystable over time and were also equal to the parameters ofthe completed spells model.Therefore, the analysis of incomplete spells is fairly feasiblewhen an even (closed form) baseline hazard function is suitablefor data.     

Unemployment insurance and subsequent job duration: job matching versus unobserved heterogeneity

The relationship between Unemployment Insurance (UI) benefit duration, unemployment duration and subsequent job duration is investigated using a multi‐state duration model with state specific unobserved heterogeneity. I examine two potential explanations for the negative correlation between unemployment and job spell durations; UI benefits increase job matching quality (the ‘Matching’ effect) versus unobserved heterogeneity (‘Adverse Selection’). The Matching effect is found to be weak. Although new jobs accepted within 5 weeks of benefit termination seem to have a higher dissolution rate, the negative correlation between unemployment and job duration is mostly explained by unobserved heterogeneity. Various simulations indicate that increasing the maximum benefit duration by one week will raise expected unemployment duration by 1.0 to 1.5 days but will raise expected job duration by 0.5 to 0.8 day only. Copyright © 2001 John Wiley & Sons, Ltd.     

The duration of contract strikes in U.S. manufacturing

This paper develops new evidence on the hazard function for strike duration, and on cyclical changes in this function, using data on contract strikes in U.S. manufacturing industries. A flexible duration model is estimated, and it is found that the hazard rate is generally a U-shaped function of strike age. The level of industrial production is found to have a significant positive effect on the hazard rate: strike duration is countercyclical. A convenient parametric model of heterogeneity and duration dependence is introduced, in which the logit of the hazard rate is a polynomial function of strike age, up to a random individual effect drawn from a beta distribution. Estimates of this ‘beta-logit’ model indicate that it is difficult to detect the influence of unobserved heterogeneity on the aggregate hazard function for strike duration.     

Quasi-experimental evidence on the effect of aircraft noise on apartment rents

Inferring the implicit price of an environmental good hinges on ceteris paribus conditions that are often hard to justify. This paper uses an unexpected change in flight regulations as source of exogenous variation and identifies aircraft noise effects from price adjustments in the market for rental apartments. Controlling for spatial and apartment heterogeneity, we find that aircraft noise reduces apartment rents by about 0.5% per decibel. Our results indicate (i) that noise discounts are overestimated in cross-sectional studies because aircraft noise tends to be negatively correlated with omitted neighborhood and housing amenities and (ii) that noise effects are unlikely to be constant over the entire noise range.     

New Insights into Unemployment Duration and Post Unemployment Earnings in Germany*

Building on non‐stationary search theory ( Mortensen, 1977 ; Van den Berg, 1990 ), this article estimates the effects of UB on unemployment durations and future earnings using unique administrative data in Germany. We apply censored Box–Cox quantile regression. Our results imply that the length of entitlement shows only a weak effect on unemployment duration for entitlement lengths up to 12 months and no effect on post unemployment earnings. There are noticeable effects on exits from unemployment for entitlement lengths above 12 months. A high wage replacement rate for low‐wage earners is associated with a longer duration of unemployment and higher post unemployment earnings.     

Analysis of Labour Participation Behaviour of Korean Women with Dynamic Probit and Conditional Logit*

We analyse the dynamic labour participation behaviour of Korean women. State dependence under unobserved heterogeneity is considered, where the heterogeneity may be unrelated, pseudo‐related, or arbitrarily related to regressors. Three minor methodological contributions are made: interaction terms with lagged response are allowed in dynamic conditional logit; a three‐stage algorithm for dynamic probit is proposed; and treating the initial response as fixed is shown to be ill‐advised. The state dependence is about 0.6 × SD(error), higher for the married or junior college‐educated, and lower for women in their twenties and thirties. While education increases participation, college education has negative effects for women in their forties or above. Marriage has a high negative short‐term effect but a positive long‐term effect.     

The effects of dynamic feedbacks on LS and MM estimator accuracy in panel data models

The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined.