Unobserved heterogeneity in duration models
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Issues of heterogeneity in duration models can take on different forms. On the one hand, unobserved heterogeneity can play a crucial role when it comes to different sampling methods, such as
stock In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...
or flow sampling. On the other hand, duration models have also been extended to allow for different subpopulations, with a strong link to mixture models. Many of these models impose the assumptions that heterogeneity is independent of the observed covariates, it has a
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
that depends on a finite number of parameters only, and it enters the hazard function multiplicatively. One can define the conditional hazard as the hazard function conditional on the observed covariates and the unobserved heterogeneity. In the general case, the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
of ''ti*'' associated with the conditional hazard is given by ''F(t, xi , vi ; θ)''. Under the first assumption above, the unobserved component can be integrated out and we obtain the cumulative distribution on the observed covariates only, i.e. ''G(t ∨ xi ; θ , ρ) = ∫ F (t ∨ xi, ν ; θ ) h ( ν ; ρ ) dν'' where the additional parameter ρ parameterizes the density of the unobserved component ''v''. Now, the different estimation methods for stock or flow sampling data are available to estimate the relevant parameters. A specific example is described by Lancaster. Assume that the conditional hazard is given by ''λ(t ; xi , vi ) = vi exp (x β) α t α-1'' where ''x'' is a vector of observed characteristics, ''v'' is the unobserved heterogeneity part, and a
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
(often E 'vi''= 1) needs to be imposed. It then follows that the average hazard is given by exp(x'β) αtα-1. More generally, it can be shown that as long as the hazard function exhibits proportional properties of the form ''λ ( t ; xi, vi ) = vi κ (xi ) λ0 (t)'', one can identify both the covariate function ''κ''(.) and the hazard function ''λ''(.). Recent examples provide a nonparametric approaches to estimating the baseline hazard and the distribution of the unobserved heterogeneity under fairly weak assumptions. In
grouped data Grouped data are data formed by aggregating individual observations of a variable into groups, so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data. There are two major types of grouping ...
, the strict exogeneity assumptions for time-varying covariates are hard to relax. Parametric forms can be imposed for the distribution of the unobserved heterogeneity, even though
semiparametric In statistics, a semiparametric model is a statistical model that has Parametric statistics, parametric and nonparametric components. A statistical model is a parameterized family of distributions: \ indexed by a statistical parameter, parameter \t ...
methods that do not specify such parametric forms for the unobserved heterogeneity are available. Heckman, J. J. and B. Singer (1984): A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data. Econometrica, 52, pp. 271-320


References

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