Top

Health Services and Outcomes Research Methodology

Published in:

01-12-2006

Estimation of policy effects using parametric nonlinear models: a contextual critique of the generalized method of moments

Author: Joseph V. Terza

Published in: Health Services and Outcomes Research Methodology | Issue 3-4/2006

Abstract

The typical empirical study in health services and outcomes research is aimed at estimating the causal effect that an exogenously imposed condition (e.g. a policy mandate) will have (or has had) on a specified outcome of interest. Controlling for unobservable confounding influences is of primary importance in such analyses. The instrumental variables (IV) method has been widely used for this purpose in the linear regression context. The present paper examines the pros and cons of alternative versions of the generalized method of moments (GMM) [of which the IV estimator is a special case] for the estimation of policy effects when endogeneity is present in a nonlinear regression setting. We show that conventional GMM is difficult to implement for policy analysis because it does not typically accommodate symmetry—similar treatment of both observable and unobservable confounders in the regression specification. Although, simple additive (nonsymmetric) regression specifications afford practical GMM estimators, they are difficult to defend from both intuitive and conceptual standpoints. Moreover, as we show via simulation, if symmetry is ignored and conventional GMM is applied based on an incorrectly specified non-symmetric model, then policy analytic estimates can be seriously biased. As a result, prospects for the development and application of intuitive consistent GMM-based policy effect estimators are dim. The problem stems from the reasonable desire on the part of the researcher to derive GMM estimators in the nonlinear framework that are based solely upon the conventional minimalist linear IV assumptions. We show, in the context of our formulation of a simple but consistent alternative to GMM in the probit case, that intuitively appealing additional assumptions about the data generating process of the policy variable will often be sufficient for the development of desirable alternatives to the GMM.

Available only for authorised users

For examples of extant methods that are less robust than GMM (e.g. maximum likelihood), see the references cited in Sects. 4 and 5 of this paper. The appeal of GMM (or the IV method in the linear case) stems from the fact that it can be based on modeling assumptions that are much less restrictive than those required by other parametric methods.

See Hansen (1982), or any modern graduate econometrics text for a detailed discussion of the GMM.

Mullahy (1997) places particular emphasis on this point in his discussion of endogeneity in nonlinear models. Gail et al. (1984) focus on unobservables as omitted regressors in nonlinear models that do not involve endogeneity.

Note that the policy variable x _p may be continuous or discrete and, therefore, x _p1 and x _p2 are not necessarily the only values in the support of x _p.

In general, M() must satisfy the premises of the dominated convergence theorem (see Bierens 1994, p. 25), for the interchange of the expectation and partial derivative to be legitimate. The specific forms of M() considered here satisfy these premises.

In their models, unlike those discussed here, the omitted variables are assumed to be independent of the treatment variable.

See that last full paragraph on p. 586 of Mullahy (1997) and footnote 4.

We refer to (18) as a “pseudo” regression model because the nonobservability of x _u precludes estimation via conventional nonlinear methods.

For detailed discussions of the “weak instruments” issue in the linear context, and methods for testing instrument weakness, see Bound et al. (1995), Staiger and Stock (1997), and Hahn and Hausman (2002). Methods have not yet been developed for evaluating instrument weakness in nonlinear models. The current state of the art is to rely on the significance of the joint Wald test of the instruments in an auxiliary (reduced form) regression of the endogenous variable x _p on w, assuming that a valid reduced form model exists.

Recall the third IV condition says \({E[y \vert x_{\rm p} , w, x_{\rm u} ] = E[y \vert x_{\rm p} ,x_{\rm o},x_{\rm u}]}\) . Therefore (16) implies that,\({E[y \vert x_{\rm p} , w, x_{\rm u}]_{{x}_{\rm p} = {x}_{\rm p}^\ast} = E[y_{ {x}_{\rm p}^\ast} \vert w, x_{\rm u} ]}\) , because if [x _o x _u] is comprehensive, then so is [w x _u] . From this it follows that \({E[y_{{x}_{\rm p}^\ast} \vert w, x_{\rm u} ] = M_{\rm S} (x_{\rm p}^\ast \beta_{\rm p} + x_{\rm o} \beta_{\rm o} + x_{\rm u}\beta_{\rm u} )}.\)

See the discussion surrounding equation (30).

The reference to Ashford and Sowden (1970) encompasses the use of the bivariate probit model as a means of consistently estimating a probit model with an endogenous treatment effect.

The generic 2SRI estimator comprises the following steps: (1) regress the endogenous variable on the instrumental variables using the appropriate nonlinear method; (2) regress the outcome on the endogenous variable, the observable confounders, and the residual from the first stage regression. This differs from the linear two-stage least squares estimator where, in the second-stage regression, the true value of the endogenous variable is replaced by its first-stage predicted value; and the first-stage residual is not included in the second-stage regression. This method requires parametric specifications for both first and second stage regressions.

Alexndre, P.K., French, M.T.: Labor supply of poor residents in metropolitan Miami, Florida: the role of depression and the co-morbid effects of substance abuse. J. Mental Health Policy Econ. 4, 161–173 (2001)

Alvarez, R.M., Glasgow, G.: Two-stage estimation of nonrecursive choice models. Polit. Anal. 8, 147–165 (1999)

Amemiya, T.: The nonlinear two-stage least-squares estimator. J. Economet. 2, 105–110 (1974)CrossRef

Amemiya, T.: Advanced Econometrics. Cambridge, Harvard University Press (1985)

Ashford, J., Sowden, R. Multivariate probit analysis. Biometrics 26, 535–546 (1970)PubMedCrossRef

Averett, S., Duncan, B., Rees, D., Argys, L.: Race, ethnicity, and gender differences in the relationship between substance use and adolescent sexual behavior. Topics Econ. Anal. Policy 4(1), 1–32 (2004)

Basur, O., Bradley, C.J., Gardiner, J.C., Given, C.: Testing and correcting for non-random selection bias due to censoring: an application to medical costs. Health Services Outcomes Res. Methodol. 4, 93–107 (2004)CrossRef

Bierens, H.J.: Topics in Advanced Econometrics: Estimation, Testing, and Specification of Cross-section and Time Series Models. New York, Cambridge University Press (1994)

Blundell, R.W., Smith, R.J.: Estimation in a class of simultaneous equation limited dependent variable models. Rev. Econ. Stat. 56, 37–58 (1989)CrossRef

Blundell, R.W., Smith, R.J.: Simultaneous microeconometric models with censored or qualitative dependent variables. In: Maddala, G.S., Rao, C.R., Vinod, H.D. (eds.), Handbook of Statistics, vol 2. Amsterdam, North Holland Publishers, pp. 1117–1143

Bollen, K.A., Guilkey, D.K., Mroz, T.A.: Binary outcomes and endogenous explanatory variables: tests and solutions with an application to the demand for contraceptive use in Tunisia. Demography 32, 11–131 (1995)

Bound, J., Jaeger, D.A., Baker, R.M. (1995). Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak. J. Am. Stat. Assoc. 90, 443–450 (1995)CrossRef

Burnett, N.J.: Gender economics courses in liberal arts colleges. J. Econ. Education 28, 369–377 (1997)CrossRef

Cameron A.C., Trivedi P.K.: Regression analysis of count data. Cambridge, UK, Cambridge University Press (1998)

Dagenais, M.G.: Inconsistency of a proposed nonlinear instrumental variables estimator for probit and logit models with endogenous regressors. Econ. Lett. 63(1), 19–21 (1999)CrossRef

DeSimone, J.: Illegal drug use and employment. J. Labor Econ. 20, 952–977 (2002)CrossRef

Dickie, M.: Parental behavior and the value of children’s health: a health production approach. Southern Econ. J. 71, 855–872 (2005)

Dhrymes, P.: Mathematics for Econometrics. New York, Springer-Verlag (2000)

Gail, M.H., Wiend, S., Piantadosi, S.: Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 71, 431–444 (1984)CrossRef

Gallant, R.: Nonlinear Statistical Models. New York, Wiley (1987)

Greene, W.H.: LIMDEP Version 8.0 Econometric Modeling Guide. Castle Hill, Australia, Econometric Software Inc. (2002)

Grogger, J.: A simple test for exogeneity in probit, logit, and poisson regression models. Econ Lett. 33, 329–332 (1990)CrossRef

Hahn, J., Hausman, J.: A new specification test for the validity of instrumental variables. Econometrica 70(1), 163–189 (2002)CrossRef

Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054 (1982)CrossRef

Koc, C.: Health-specific moral hazard effects. Southern Econ. J. 72, 98–118 (2005)

Lee, L.: Identification and estimation in binary choice models with limited ‘censored’ dependent variables. Econometrica 47, 977–995 (1979)CrossRef

Lee, L.: Simultaneous equations models with discrete and censored dependent variables, Chapter 9. In: Manski, C., McFadden, D. (eds.) Structural Analysis of Discrete Data with Econometric Applications. MIT Press, Cambridge, MA, pp. 346–364 (1981)

McGarrity, J.P., Sutter, D.: A test of the structure of pac contracts: an analysis of house gun control votes in the 1980s. Southern Econ. J. 677, 41–63 (2000).CrossRef

Mroz, T.A., Bollen, K.A., Speizer, I.S., Mancini, D.J.: Quality, accessibility, and contraceptive use in rural Tanzania. Demography 36, 23–40 (1999)PubMed

Mullahy, J.: Instrumental-variable estimation of count data models: applications to models of cigarette smoking behavior. Rev. Econ. Stat. 79, 586–593 (1997)CrossRef

Neslusan, C.A., Hylan, T.R., Dunn, R.L., Donoghue, J.: Controlling for systematic selection in retrospective analyses: an application to fluoxetine and sertraline prescribing in the United Kingdom. Value Health 2, 435–445 (1999)PubMedCrossRef

Newey, W.K.: Efficient estimation of limited dependent variable models with endogenous explanatory variables. J. Economet. 36, 231–250 (1987)CrossRef

Norton, E.C., Lindrooth, R.C., Ennett, S.T.: Controlling for the endogeneity of peer substance use on adolescent alcohol and tobacco use. Health Econ. 7, 439–453 (1998)PubMedCrossRef

Norton, E.C., Van Houtven, C.H.: Intervivos transfers and exchange. Southern Econ. J. 73, 157–172 (2006a)CrossRef

Norton, E.C., Van Houtven, C.H.: Informal care and medicare expenditures: testing for heterogeneous treatment effects. Working Paper-UNC-Chapel Hill and the Center for Health Services Research in Primary Care (2006b)

Rees, D., Argys, L., Averett, S.: New evidence on the relationship between substance use and adolescent sexual behavior. J. Health Econ. 20(5), 835–845 (2001)CrossRef

Ribar, D.C.: Teenage fertility and high school completion. Rev. Econ. Stat. 76, 413–424 (1994)CrossRef

Rivers, D., Vuong, Q.H.: Limited information estimators and exogeneity tests for simultaneous probit models. J. Economet. 39, 347–366 (1988)CrossRef

Sen, B.: Does alcohol-use increase the risk of sexual intercourse among adolescents? Evidence from the NLSY97. J. Health Econ. 21, 1085–1093 (2002)PubMedCrossRef

Smith, R.J., Blundell, R.W.: An exogeneity test for a simultaneous equation tobit model with an application to labor supply. Econometrica 54, 679–685 (1986)CrossRef

Staiger, D., Stock, J.H.: Instrumental variables regression with weak instruments. Econometrica 65, 557–586 (1997)CrossRef

Terza, J.V.: Estimating count data models with endogenous switching and sample selection. Working Paper, Department of Economics, Penn State University (1995)

Terza, J.V.: Estimating count data models with endogenous switching: sample selection and endogenous treatment effects. J. Economet. 84, 129–154 (1998)CrossRef

Terza, J.V.: Endogeneity in nonlinear parametric models: a guide for applied researchers in health economics. Working Paper, Department of Epidemiology and Health Policy Research, University of Florida (2006)

Treglia, M., Neslusan, C.A., Dunn, R.L.: Fluoxetine and dothiepin therapy in primary care and health resource utilization: evidence from the United Kingdom. Int. J. Psychiat. Clin. Practice 3, 23–30 (1999)CrossRef

Windmeijer, F.A.G., Santos Silva, J.M.C.: Endogeneity in count data models: an application to demand for health care. J. Appl. Economet. 12, 281–294 (1997)CrossRef

Wooldridge, J.M.: Quasi-likelihood methods for count data. In: Pesaran, M., Schmidt, P. (eds.) Handbook of Applied Econometrics, Volume II: Microeconometrics. Blackwell Publishers, Ltd., Malden, MA (1997)

Wooldridge, J.M.: Econometric Analysis of Cross Section and Panel Data. MIT Press, Cambridge MA (2003)

Title: Estimation of policy effects using parametric nonlinear models: a contextual critique of the generalized method of moments
Author: Joseph V. Terza
Publication date: 01-12-2006
Publisher: Kluwer Academic Publishers-Plenum Publishers
Published in: Health Services and Outcomes Research Methodology / Issue 3-4/2006
Print ISSN: 1387-3741
Electronic ISSN: 1572-9400
DOI: https://doi.org/10.1007/s10742-006-0013-0

2024 ASCO Annual Meeting

Springer Medicine

Estimation of policy effects using parametric nonlinear models: a contextual critique of the generalized method of moments

Abstract

2024 ASCO Annual Meeting

Springer Medicine

Abstract

Please log in to get access to this content

Other articles of this Issue 3-4/2006

Computerized tracking and follow-up techniques in longitudinal research with drug users

Predicting the length of stay of patients admitted for intensive care using a first step analysis

A test for proportional hazards assumption within the class of exponential conditional mean models

Methods for studying adverse events on surgical wait lists

Does the concordance between medical records and patient self-report vary with patient characteristics?

The impact of survey attrition on health insurance coverage estimates in a National Longitudinal Health Care Survey