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 or flow sampling. [1] 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 that depends on a finite number of parameters only, and it enters the hazard function multiplicatively. [2]

Stock financial instrument

The stock of a corporation is all of the shares into which ownership of the corporation is divided. In American English, the shares are commonly known as "stocks." A single share of the stock represents fractional ownership of the corporation in proportion to the total number of shares. This typically entitles the stockholder to that fraction of the company's earnings, proceeds from liquidation of assets, or voting power, often dividing these up in proportion to the amount of money each stockholder has invested. Not all stock is necessarily equal, as certain classes of stock may be issued for example without voting rights, with enhanced voting rights, or with a certain priority to receive profits or liquidation proceeds before or after other classes of shareholders.

Flow sampling, in contrast to stock sampling, means we collect observations that enter the particular state of interest during a particular interval. When dealing with duration data, the data sampling method has a direct impact on subsequent analyses and inference. An example in demography would be sampling the number of people that die within a given time frame ; a popular example in economics would be the number of people leaving unemployment within a given time frame. Researchers imposing similar assumptions but using different sampling methods, can reach fundamentally different conclusions if the joint distribution across the flow and stock samples differ.

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

One can define the conditional hazard as the hazard function conditional on the observed covariates and the unobserved heterogeneity. [3] In the general case, the cumulative distribution function 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.

Cumulative distribution function probability that random variable X is less than or equal to x.

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .

G(t ∨ xi ; θ , ρ) = ∫ F (t ∨ xi, ν ; θ ) h ( ν ; ρ ) dν [4]

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 [5] β) α t α-1

where x is a vector of observed characteristics, v is the unobserved heterogeneity part, and a normalization (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(.). [6]

In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. In the case of normalization of scores in educational assessment, there may be an intention to align distributions to a normal distribution. A different approach to normalization of probability distributions is quantile normalization, where the quantiles of the different measures are brought into alignment.

Recent examples provide a nonparametric approaches to estimating the baseline hazard and the distribution of the unobserved heterogeneity under fairly weak assumptions. [7] In grouped data, the strict exogeneity assumptions for time-varying covariates are hard to relax. Parametric forms can be imposed for the distribution of the unobserved heterogeneity, [8] even though semiparametric methods that do not specify such parametric forms for the unobserved heterogeneity are available. [9]

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.

Parametric equation parametric system of coordinates

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

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Survival analysis is a branch of statistics for analyzing the expected duration of time until one or more events happen, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer questions such as: what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival?

Poisson regression Statistical model for count data

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.

In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.

Semiparametric regression

In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. Semiparametric regression models are a particular type of semiparametric modelling and, since semiparametric models contain a parametric component, they rely on parametric assumptions and so may be misspecified, just like a fully parametric model.

Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing the material from which a manufactured component is constructed may double its hazard rate for failure. Other types of survival models such as accelerated failure time models do not exhibit proportional hazards. The accelerated failure time model describes a situation where the biological or mechanical life history of an event is accelerated.

In the statistical area of survival analysis, an accelerated failure time model is a parametric model that provides an alternative to the commonly used proportional hazards models. Whereas a proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant. This is especially appealing in a technical context where the 'disease' is a result of some mechanical process with a known sequence of intermediary stages.

The Heckman correction is any of a number of related statistical methods developed by James Heckman at the University of Chicago from 1976 to 1979 which allow the researcher to correct for selection bias. Selection bias problems are endemic to applied econometric problems, which make Heckman’s original technique, and subsequent refinements by both himself and others, indispensable to applied econometricians. Heckman received the Economics Nobel Prize in 2000 for his work in this field.

Partial (pooled) likelihood estimation for panel data assumes that density of yit given xit is correctly specified for each time period but it allows for misspecification in the conditional density of yi≔(yi1,…,yiT) given xi≔(xi1,…,xiT).

In a linear panel data setting, it can be desirable to estimate the magnitude of the fixed effects, as they provide measures of the unobserved components. For instance, in wage equation regressions, Fixed Effects capture ability measures that are constant over time, such as motivation. Chamberlain's approach to unobserved effects models is a way of estimating the linear unobserved effects, under Fixed Effect assumptions, in the following unobserved effects model

Control functions are statistical methods to correct for endogeneity problems by modelling the endogeneity in the error term. The approach thereby differs in important ways from other models that try to account for the same econometric problem. Instrumental variables, for example, attempt to model the endogenous variable X as an often invertible model with respect to a relevant and exogenous instrument Z. Panel data use special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time.

Hausman, Hall, and Griliches pioneered work in fixed-effect Poisson models for static panel data in the mid 1980s. Their outcome of interest was the number of patents filed by firms, where they wanted to develop methods to control for the firm fixed effects. Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the incidental parameter problem. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically

Fractional models are, to some extent, related to binary response models. However, instead of estimating the probability of being in one bin of a dichotomous variable, the fractional model typically deals with variables that take on all possible values in the unit interval. One can easily generalize this model to take on values on any other interval by appropriate transformations. Examples range from participation rates in 401(k) plans to television ratings of NBA games.

Given a probit model y=1[y* > 0] where y* = x1 β + zδ + u, and u ~ N(0,1), without losing generality, z can be represented as z = x1 θ1 + x2 θ2 + v. When u is correlated with v, there will be an issue of endogeneity. This can be caused by omitted variables and measurement errors. There are also many cases where z is partially determined by y and endogeneity issue arises. For instance, in a model evaluating the effect of different patient features on their choice of whether going to hospital, y is the choice and z is the amount of the medicine a respondent took, then it is very intuitive that more often the respondent goes to hospital, it is more likely that she took more medicine, hence endogeneity issue arises. When there are endogenous explanatory variables, the estimator generated by usual estimation procedure will be inconsistent, then the corresponding estimated Average Partial Effect (APE) will be inconsistent, too.

A dynamic unobserved effects model is a statistical model used in econometrics. It is characterized by the influence of previous values of the dependent variable on its present value, and by the presence of unobservable explanatory variables.

In many cases, there is an unobservable heterogeneity in the probit model. For instance, when modelling the consumption choice of a certain brand, consumers’ personal preference is unobserved but needs to be considered in the model. Owing to omitted variable or measurement error, endogeneity issue also could arise. A probit model including both of these two issues can be represented as:

Grouped duration data are widespread in many applications. Unemployment durations are typically measured over weeks or months and these time intervals may be considered too large for continuous approximations to hold. In this case, we will typically have grouping points , where . Models allow for time-invariant and time-variant covariates, but the latter require stronger assumptions in terms of exogeneity. The discrete-time hazard function can be written as:

In statistics as applied to econometrics, exponential regression models constitute a very large and popular class of regression models. Standard econometric concerns such as endogeneity or omitted variables can be accounted for in a more general framework. Wooldridge and Terza provide a methodology to both deal with and test for endogeneity within the exponential regression framework, which the following discussion follows closely. While the example focuses on a Poisson regression model, it is possible to generalize the test to other exponential regression models, although this may come at the cost of additional assumptions.

In statistics, in particular in the design of experiments, a multi-valued treatment is a treatment that can take on more than two values. It is related to the dose-response model in the medical literature.

Stock sampling

Stock sampling is sampling people in a certain state at the time of the survey. This is in contrast to flow sampling, where the relationship of interest deals with duration or survival analysis. In stock sampling, rather than focusing on transitions within a certain time interval, we only have observations at a certain point in time. This can lead to both left and right censoring. Imposing the same model on data that have been generated under the two different sampling regimes can lead to research reaching fundamentally different conclusions if the joint distribution across the flow and stock samples differ sufficiently.

References

  1. ↑ Salant, S. W. (1977): Search Theory and Duration Data: A Theory of Sorts. The Quarterly Journal of Economics, 91(1), pp. 39-57
  2. ↑ Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
  3. ↑ Lancaster, T. (1990): The Econometric Analysis of Transition Data. Cambridge University Press, Cambridge.
  4. ↑ Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
  5. ↑ i
  6. ↑ Lancaster, T. (1990): The Econometric Analysis of Transition Data. Cambridge University Press, Cambridge.
  7. ↑ Horowitz, J. L. (1999): Semiparametric and Nonparametric Estimation of Quantal Response Models. Handbook of Statistics, Vol. 11, ed. by G. S. Maddala, C. R. Rao, and H. D. Vinod. North Holland, Amsterdam.
  8. ↑ McCall, B. P. (1994): Testing the Proportional Hazards Assumption in the Presence of Unmeasured Heterogeneity. Journal of Applied Econometrics, 9, pp. 321-334
  9. ↑ 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
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