Elsevier

Journal of Econometrics

Volume 152, Issue 1, September 2009, Pages 37-45
Journal of Econometrics

Excess heterogeneity, endogeneity and index restrictions

https://doi.org/10.1016/j.jeconom.2009.02.004Get rights and content

Abstract

A discrete or continuous outcome is determined by a structural function in which the effect of some variables of interest is transmitted through a scalar index. Multiple sources of stochastic variation can appear as arguments of the structural function, but not in the index. There may be endogeneity, that is observable and unobservable variables may not be independently distributed. Conditions are provided under which there is local identification of measures of the relative sensitivity of the index to variations in pairs of its possibly endogenous arguments, namely ratios of partial derivatives of the index.

Introduction

Many questions arising in microeconometric practice lead to the use of models which include more unobservable latent variables than there are observable stochastic outcomes, that is excess heterogeneity. The latent variables often represent unobserved characteristics of individuals and of the environment in which they make decisions. The inclusion of such variables is common in, for example, models of durations (see van den Berg (2001)) such as mixed proportional hazard models, in discrete choice models, see for example Brownstone and Train (1998), Chesher and Santos Silva (2002), McFadden and Train (2000), and in count data models, see Cameron and Trivedi (1998). There is a large econometric literature concerned with random coefficients models which permit this sort of excess heterogeneity. Excess heterogeneity also arises in other cases, for example when there is measurement error and in panel data models.

It is common to find strong restrictions imposed on models that admit excess heterogeneity. Frequently the specification is fully parametric as in the mixed multinomial logit models of Brownstone and Train (1998). When parametric restrictions are not imposed there are usually strong semiparametric restrictions. For example: most of the single spell duration models used in practice that permit excess heterogeneity require there to be a single latent variate that acts multiplicatively on the hazard function; measurement error and “individual effects” in panel data models are usually required to be additive.

The aim of this paper is to explore the extent to which strong restrictions such as these can be relaxed, while preserving a model with the power to identify interesting structural features.

When there is excess heterogeneity the probability distributions of observable variables are relatively low-dimensional reductions of the distributions of structural variables, obtained by taking expectations over the distributions of supernumerary latent variates. Information about fundamental structural features may not survive the averaging process. In the face of this difficulty one possibility is to focus on the identification of averages of structural features, as in for example Imbens and Newey (in press). Sometimes knowledge of such averages is not what is required to understand the impacts of policy changes. It is interesting to explore alternatives. Another approach is to impose restrictions which shield certain structural objects from the effects of averaging. This is the approach studied here.

In the models explored here excess heterogeneity can arise from any finite number of sources and there is an index restriction. The index restriction requires the effect on an outcome of certain variables of interest to pass entirely through a scalar function of those variables, an index, and that this index be free of latent variates. Continuously distributed variables that appear in the index are permitted to be endogenous in the sense that they may covary with any or all of the latent variates that appear in the model.

The structural features whose identifiability is studied in this paper are ratios of derivatives of the index at some specified values of the variables that appear in the index. This is therefore a study of local identification. These ratios are referred to as index relative sensitivity (IRS) measures because they measure the relative sensitivity of the index, and therefore of the outcome, to variation in a pair of its arguments. When the index is linear the ratios do not depend on the values of the arguments of the index. Then conditions sufficient to achieve local identification of the value of an IRS measure achieve global identification of the ratio of coefficients of the linear index.

With more sources of stochastic variation than there are outcomes structural functions necessarily involve non-additive latent variates, as noted in Hurwicz (1950). The identifying model employed here admits non-additive latent variates and embodies triangularity restrictions as in Chesher, 2003, Chesher, 2005, Chesher, 2007a and Imbens and Newey (in press).

IRS measures are often of interest in models for binary outcomes. For example in discrete choice models of travel demand there is interest in the “value of travel time” defined as the ratio of coefficients on travel time and travel cost. There are other contexts in which the relative sensitivity of an index to variation in its arguments is of interest. For example in models of intrahousehold allocation there is interest in the relative sensitivity of expenditures to variations in the incomes of two partners; in models for the duration of unemployment there is interest in the relative sensitivity of unemployment duration to variations in unemployment benefits and other household income or the wage prior to unemployment. In all these cases one or more of the arguments of the index could be endogenous although this is a possibility frequently ignored, perhaps because it is not understood how to deal with endogeneity in this situation. It is this which motivates this study which mainly focuses on identification issues.

The remainder of the paper is organised as follows. The structural equation, index restriction and IRS measures are defined in Section 2 and examples of microeconometric models accommodated within the framework employed here are given in Section 3. The identification strategy, based on a “control function” argument, is introduced in Section 4. Related literature is briefly reviewed in Section 5. Identification theorems are given in Section 6 and estimation is briefly considered in Section 7. The main results associate IRS measures with functions of derivatives of various distribution functions involving observable variables. These apply when W, the outcome of interest, is discrete or continuous. When the outcome is continuous the IRS measures can be associated with functions of derivatives of conditional quantile functions and the expressions are given in Section 8. Section 9 concludes.

Section snippets

The structural equation and the IRS measures

In the models considered here the outcome of interest is a random variable W determined by a structural equation of the following form. W=h0(θ(Y1,,YM,Z1,,ZK),Z1,,ZL,U1,,UN). Scalar W may be discrete or continuous, U{Un}n=1N are latent variates, Y{Ym}m=1M are observable continuously distributed endogenous random variables which may covary with U, and Z{Zk}k=1K are observable continuously varying covariates whose covariation with U is limited to some degree to be specified. θ is the

Examples

This section gives examples of microeconometric models in which a structural equation of the form (1) arises.

Example 1 — Mixed hazard duration models

Consider hazard functions for a continuously distributed duration (e.g. of unemployment) W conditional on observable Y=y, Z=z, Z=z and on unobservable, possibly vector, E=e of the form: λ(w|θ(y,z),z,e) where θ is a scalar valued function. The conditional distribution function of W given Y, Z, Z and E is FW|YZZE(w|y,z,z,e)=1exp(Λ(w|θ(y,z),z,

Identification

The strategy employed in developing identification conditions for IRS measures is now outlined. This is done for the case in which the covariates Z, which appear in the structural function (1) but not in the index θ, are not present. Their presence would not change the argument below except in inessential details.2

Related literature

The basic idea employed in this paper dates back as far as Tinbergen (1930) in which the problem of identification in linear simultaneous equations systems was attacked by developing conditions under which values of structural form parameters could be deduced from values of parameters of regression functions — the reduced form equations of the linear simultaneous system.

The conditional distribution functions FW|ZY and FYm|Z, m{1,,M} are regression functions, namely of 1[Ww] on Z and Y, and

Identification of index derivatives

This section introduces four restrictions and then gives a theorem which states that a model embodying these restrictions identifies index derivatives up to a common factor of proportionality. Some remarks on the assumptions are provided as they are introduced. The theorem is proved in an Appendix.

To simplify the notation the covariates Z which appear in the structural equation (1) and in the examples of Section 3 are assumed absent. Their inclusion requires minor changes to the assumptions

Estimation

Theorem 1 and its two Corollaries point to estimation procedures. For example, with nonparametric estimates of the conditional distribution function derivatives, Rˆy, Rˆz, Sˆy and Sˆz, estimates, Φˆ and ϕˆ, of Φ and ϕ, can be assembled incorporating the restrictions to hand, and a minimum distance estimator ψˆ=argminψ(Φˆψϕˆ)Ω(Φˆψϕˆ) can be calculated using a suitable positive definite matrix Ω. 7

Identification via conditional quantile functions

So far the outcome, W, has not been required to be continuously distributed. Now suppose that it is, at least conditional on Y and Z lying in a neighbourhood of (ȳ,z̄). In this case the matrices of conditional distribution function derivatives that appear in Theorem 1 and Corollary 1 can be re-expressed in terms of derivatives of conditional quantile functions.

This is so because for a random variable A, continuously distributed conditional on B lying in a neighbourhood of b, bFA|B(a|b)=bQA|B

Concluding remarks

This paper has developed an identifying model for problems in which structural functions involve multiple latent variables and endogenous observed arguments. Examples of microeconometric models in which these features arise include count and duration data models admitting across individual heterogeneity and models in which household outcomes are determined by characteristics and experiences of individual household members.

The identifying models use an index restriction that shields certain

Acknowledgements

I thank Whitney Newey for remarks on an earlier paper (Chesher, 2002) which stimulated this work and to Lars Nesheim for helpful discussions. I thank the Leverhulme Trust for support through grants to the Centre for Microdata Methods and Practice (CeMMAP) and the research project “Evidence, Inference and Enquiry” and the Economic and Social Research Council for support of CeMMAP since July 1st, 2007 through grant RES-589-28-0001.

References (27)

  • Chesher, Andrew D., 2007b. Endogeneity and discrete outcomes. Centre for Microdata Methods and Practice Working Paper...
  • Andrew D. Chesher et al.

    Taste variation in discrete choice models

    Review of Economic Studies

    (2002)
  • Honoré, Bo E., Hu, Luojia, 2002. Estimation of cross sectional and panel data censored regression models with...
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