A General Approach to Incorporating Selectivity in a Model
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Irregular Identification, Support Conditions and Inverse Weight Estimation
Irregular Identi¯cation, Support Conditions and Inverse Weight Estimation¤ Shakeeb Khany and Elie Tamerz August 2007 ABSTRACT. Inverse weighted estimators are commonly used in econometrics (and statistics). Some examples include: 1) The binary response model under mean restriction introduced by Lew- bel(1997) and further generalized to cover endogeneity and selection. The estimator in this class of models is weighted by the density of a special regressor. 2) The censored regression model un- der mean restrictions where the estimator is inversely weighted by the censoring probability (Koul, Susarla and Van Ryzin (1981)). 3) The treatment e®ect model under exogenous selection where the resulting estimator is one that is weighted by a variant of the propensity score. We show that point identi¯cation of the parameters of interest in these models often requires support conditions on the observed covariates that essentially guarantee \enough variation." Under general conditions, these support restrictions, necessary for point identi¯cation, drive the weights to take arbitrary large values which we show creates di±culties for regular estimation of the parameters of interest. Generically, these models, similar to well known \identi¯ed at in¯nity" models, lead to estimators that converge at slower than the parametric rate, since essentially, to ensure point identi¯cation, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the examples above, we derive rates of convergence under di®erent tail conditions, analogous to Andrews and Schafgans(1998) illustrating the link between these rates and support conditions. JEL Classi¯cation: C14, C25, C13. Key Words: Irregular Identi¯cation, thin sets, rates of convergence. -
MNP: R Package for Fitting the Multinomial Probit Model
JSS Journal of Statistical Software May 2005, Volume 14, Issue 3. http://www.jstatsoft.org/ MNP: R Package for Fitting the Multinomial Probit Model Kosuke Imai David A. van Dyk Princeton University University of California, Irvine Abstract MNP is a publicly available R package that fits the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. Examples where the multi- nomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each in- dividual, and complete or partial individual choice orderings of the available alternatives from the choice set. The estimation is based on the efficient marginal data augmentation algorithm that is developed by Imai and van Dyk (2005). Keywords: data augmentation, discrete choice models, Markov chain Monte Carlo, preference data. 1. Introduction This paper illustrates how to use MNP, a publicly available R (R Development Core Team 2005) package, in order to fit the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. -
Chapter 3: Discrete Choice
Chapter 3: Discrete Choice Joan Llull Microeconometrics IDEA PhD Program I. Binary Outcome Models A. Introduction In this chapter we analyze several models to deal with discrete outcome vari- ables. These models that predict in which of m mutually exclusive categories the outcome of interest falls. In this section m = 2, this is, we consider binary or dichotomic variables (e.g. whether to participate or not in the labor market, whether to have a child or not, whether to buy a good or not, whether to take our kid to a private or to a public school,...). In the next section we generalize the results to multiple outcomes. It is convenient to assume that the outcome y takes the value of 1 for cat- egory A, and 0 for category B. This is very convenient because, as a result, −1 PN N i=1 yi = Pr[c A is selected]. As a consequence of this property, the coeffi- cients of a linear regression model can be interpreted as marginal effects of the regressors on the probability of choosing alternative A. And, in the non-linear models, it allows us to write the likelihood function in a very compact way. B. The Linear Probability Model A simple approach to estimate the effect of x on the probability of choosing alternative A is the linear regression model. The OLS regression of y on x provides consistent estimates of the sample-average marginal effects of regressors x on the probability of choosing alternative A. As a result, the linear model is very useful for exploratory purposes. -
Lecture 6 Multiple Choice Models Part II – MN Probit, Ordered Choice
RS – Lecture 17 Lecture 6 Multiple Choice Models Part II – MN Probit, Ordered Choice 1 DCM: Different Models • Popular Models: 1. Probit Model 2. Binary Logit Model 3. Multinomial Logit Model 4. Nested Logit model 5. Ordered Logit Model • Relevant literature: - Train (2003): Discrete Choice Methods with Simulation - Franses and Paap (2001): Quantitative Models in Market Research - Hensher, Rose and Greene (2005): Applied Choice Analysis 1 RS – Lecture 17 Model – IIA: Alternative Models • In the MNL model we assumed independent nj with extreme value distributions. This essentially created the IIA property. • This is the main weakness of the MNL model. • The solution to the IIA problem is to relax the independence between the unobserved components of the latent utility, εi. • Solutions to IIA – Nested Logit Model, allowing correlation between some choices. – Models allowing correlation among the εi’s, such as MP Models. – Mixed or random coefficients models, where the marginal utilities associated with choice characteristics vary between individuals. Multinomial Probit Model • Changing the distribution of the error term in the RUM equation leads to alternative models. • A popular alternative: The εij’s follow an independent standard normal distributions for all i,j. • We retain independence across subjects but we allow dependence across alternatives, assuming that the vector εi = (εi1,εi2, , εiJ) follows a multivariate normal distribution, but with arbitrary covariance matrix Ω. 2 RS – Lecture 17 Multinomial Probit Model • The vector εi = (εi1,εi2, , εiJ) follows a multivariate normal distribution, but with arbitrary covariance matrix Ω. • The model is called the Multinomial probit model. It produces results similar results to the MNL model after standardization. -
Bias Corrections for Two-Step Fixed Effects Panel Data Estimators
IZA DP No. 2690 Bias Corrections for Two-Step Fixed Effects Panel Data Estimators Iván Fernández-Val Francis Vella DISCUSSION PAPER SERIES DISCUSSION PAPER March 2007 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Bias Corrections for Two-Step Fixed Effects Panel Data Estimators Iván Fernández-Val Boston University Francis Vella Georgetown University and IZA Discussion Paper No. 2690 March 2007 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of the institute. Research disseminated by IZA may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit company supported by Deutsche Post World Net. The center is associated with the University of Bonn and offers a stimulating research environment through its research networks, research support, and visitors and doctoral programs. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 2690 March 2007 ABSTRACT Bias Corrections for Two-Step Fixed Effects Panel Data Estimators* This paper introduces bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. -
A Multinomial Probit Model with Latent Factors: Identification and Interpretation Without a Measurement System
DISCUSSION PAPER SERIES IZA DP No. 11042 A Multinomial Probit Model with Latent Factors: Identification and Interpretation without a Measurement System Rémi Piatek Miriam Gensowski SEPTEMBER 2017 DISCUSSION PAPER SERIES IZA DP No. 11042 A Multinomial Probit Model with Latent Factors: Identification and Interpretation without a Measurement System Rémi Piatek University of Copenhagen Miriam Gensowski University of Copenhagen and IZA SEPTEMBER 2017 Any opinions expressed in this paper are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but IZA takes no institutional policy positions. The IZA research network is committed to the IZA Guiding Principles of Research Integrity. The IZA Institute of Labor Economics is an independent economic research institute that conducts research in labor economics and offers evidence-based policy advice on labor market issues. Supported by the Deutsche Post Foundation, IZA runs the world’s largest network of economists, whose research aims to provide answers to the global labor market challenges of our time. Our key objective is to build bridges between academic research, policymakers and society. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA – Institute of Labor Economics Schaumburg-Lippe-Straße 5–9 Phone: +49-228-3894-0 53113 Bonn, Germany Email: [email protected] www.iza.org IZA DP No. 11042 SEPTEMBER 2017 ABSTRACT A Multinomial Probit Model with Latent Factors: Identification and Interpretation without a Measurement System* We develop a parametrization of the multinomial probit model that yields greater insight into the underlying decision-making process, by decomposing the error terms of the utilities into latent factors and noise. -
Nlogit — Nested Logit Regression
Title stata.com nlogit — Nested logit regression Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description nlogit performs full information maximum-likelihood estimation for nested logit models. These models relax the assumption of independently distributed errors and the independence of irrelevant alternatives inherent in conditional and multinomial logit models by clustering similar alternatives into nests. By default, nlogit uses a parameterization that is consistent with a random utility model (RUM). Before version 10 of Stata, a nonnormalized version of the nested logit model was fit, which you can request by specifying the nonnormalized option. You must use nlogitgen to generate a new categorical variable to specify the branches of the nested logit tree before calling nlogit. Quick start Create variables identifying alternatives at higher levels For a three-level nesting structure, create l2alt identifying four alternatives at level two based on the variable balt, which identifies eight bottom-level alternatives nlogitgen l2alt = balt(l2alt1: balt1 | balt2, l2alt2: balt3 | balt4, /// l2alt3: balt5 | balt6, l2alt4: balt7 | balt8) Same as above, defining l2alt from values of balt rather than from value labels nlogitgen l2alt = balt(l2alt1: 1|2, l2alt2: 3|4, l2alt3: 5|6, /// l2alt4: 7|8) Create talt identifying top-level alternatives based on the four alternatives in l2alt nlogitgen talt = l2alt(talt1: l2alt1 | l2alt2, talt2: l2alt3 | l2alt4) Examine tree structure -
Simulation from the Tail of the Univariate and Multivariate Normal Distribution
Noname manuscript No. (will be inserted by the editor) Simulation from the Tail of the Univariate and Multivariate Normal Distribution Zdravko Botev · Pierre L'Ecuyer Received: date / Accepted: date Abstract We study and compare various methods to generate a random vari- ate or vector from the univariate or multivariate normal distribution truncated to some finite or semi-infinite region, with special attention to the situation where the regions are far in the tail. This is required in particular for certain applications in Bayesian statistics, such as to perform exact posterior simula- tions for parameter inference, but could have many other applications as well. We distinguish the case in which inversion is warranted, and that in which rejection methods are preferred. Keywords truncated · tail · normal · Gaussian · simulation · multivariate 1 Introduction We consider the problem of simulating a standard normal random variable X, conditional on a ≤ X ≤ b, where a < b are real numbers, and at least one of them is finite. We are particularly interested in the situation where the interval (a; b) is far in one of the tails, that is, a 0 or b 0. The stan- dard methods developed for the non-truncated case do not always work well in this case. Moreover, if we insist on using inversion, the standard inversion methods break down when we are far in the tail. Inversion is preferable to a rejection method (in general) in various simulation applications, for example to maintain synchronization and monotonicity when comparing systems with Zdravko Botev UNSW Sydney, Australia Tel.: +(61) (02) 9385 7475 E-mail: [email protected] Pierre L'Ecuyer Universit´ede Montr´eal,Canada and Inria{Rennes, France Tel.: +(514) 343-2143 E-mail: [email protected] 2 Zdravko Botev, Pierre L'Ecuyer common random numbers, for derivative estimation and optimization, when using quasi-Monte Carlo methods, etc. -
Simulating Generalized Linear Models
6 Simulating Generalized Linear Models 6.1 INTRODUCTION In the previous chapter, we dug much deeper into simulations, choosing to focus on the standard linear model for all the reasons we discussed. However, most social scientists study processes that do not conform to the assumptions of OLS. Thus, most social scientists must use a wide array of other models in their research. Still, they could benefit greatly from conducting simulations with them. In this chapter, we discuss several models that are extensions of the linear model called generalized linear models (GLMs). We focus on how to simulate their DGPs and evaluate the performance of estimators designed to recover those DGPs. Specifically, we examine binary, ordered, unordered, count, and duration GLMs. These models are not quite as straightforward as OLS because the depen- dent variables are not continuous and unbounded. As we show below, this means we need to use a slightly different strategy to combine our systematic and sto- chastic components when simulating the data. None of these examples are meant to be complete accounts of the models; readers should have some familiarity beforehand.1 After showing several GLM examples, we close this chapter with a brief discus- sion of general computational issues that arise when conducting simulations. You will find that several of the simulations we illustrate in this chapter take much longer to run than did those from the previous chapter. As the DGPs get more demanding to simulate, the statistical estimators become more computationally demanding, and the simulation studies we want to conduct increase in complexity, we can quickly find ourselves doing projects that require additional resources. -
Estimation of Logit and Probit Models Using Best, Worst and Best-Worst Choices Paolo Delle Site, Karim Kilani, Valerio Gatta, Edoardo Marcucci, André De Palma
Estimation of Logit and Probit models using best, worst and best-worst choices Paolo Delle Site, Karim Kilani, Valerio Gatta, Edoardo Marcucci, André de Palma To cite this version: Paolo Delle Site, Karim Kilani, Valerio Gatta, Edoardo Marcucci, André de Palma. Estimation of Logit and Probit models using best, worst and best-worst choices. 2018. hal-01953581 HAL Id: hal-01953581 https://hal.archives-ouvertes.fr/hal-01953581 Preprint submitted on 13 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Estimation of Logit and Probit models using best, worst and best-worst choices Paolo Delle Sitea,∗, Karim Kilanib, Valerio Gattac, Edoardo Marcuccid, André de Palmae aUniversity Niccolò Cusano, Rome, Italy bCNAM LIRSA, Paris, France cUniversity Roma Tre, Rome, Italy dMolde University College, Molde, Norway eENS Paris Saclay, France Abstract The paper considers models for best, worst and best-worst choice probabilities, that use a single common set of random utilities. Choice probabilities are de- rived for two distributions of the random terms: i.i.d. extreme value, i.e. Logit, and multivariate normal, i.e. Probit. In Logit, best, worst and best-worst choice probabilities have a closed form. -
HURDLE and “SELECTION” MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009
HURDLE AND “SELECTION” MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Introduction 2. A General Formulation 3. Truncated Normal Hurdle Model 4. Lognormal Hurdle Model 5. Exponential Type II Tobit Model 1 1. Introduction ∙ Consider the case with a corner at zero and a continuous distribution for strictly positive values. ∙ Note that there is only one response here, y, and it is always observed. The value zero is not arbitrary; it is observed data (labor supply, charitable contributions). ∙ Often see discussions of the “selection” problem with corner solution outcomes, but this is usually not appropriate. 2 ∙ Example: Family charitable contributions. A zero is a zero, and then we see a range of positive values. We want to choose sensible, flexible models for such a variable. ∙ Often define w as the binary variable equal to one if y 0, zero if y 0. w is a deterministic function of y. We cannot think of a counterfactual for y in the two different states. (“How much would the family contribute to charity if it contributes nothing to charity?” “How much would a woman work if she is out of the workforce?”) 3 ∙ Contrast previous examples with sound counterfactuals: How much would a worker in if he/she participates in job training versus when he/she does not? What is a student’s test score if he/she attends Catholic school compared with if he/she does not? ∙ The statistical structure of a particular two-part model, and the Heckman selection model, are very similar. 4 ∙ Why should we move beyond Tobit? It can be too restrictive because a single mechanism governs the “participation decision” (y 0 versus y 0) and the “amount decision” (how much y is if it is positive). -
Using Heteroskedastic Ordered Probit Models to Recover Moments of Continuous Test Score Distributions from Coarsened Data
Article Journal of Educational and Behavioral Statistics 2017, Vol. 42, No. 1, pp. 3–45 DOI: 10.3102/1076998616666279 # 2016 AERA. http://jebs.aera.net Using Heteroskedastic Ordered Probit Models to Recover Moments of Continuous Test Score Distributions From Coarsened Data Sean F. Reardon Benjamin R. Shear Stanford University Katherine E. Castellano Educational Testing Service Andrew D. Ho Harvard Graduate School of Education Test score distributions of schools or demographic groups are often summar- ized by frequencies of students scoring in a small number of ordered proficiency categories. We show that heteroskedastic ordered probit (HETOP) models can be used to estimate means and standard deviations of multiple groups’ test score distributions from such data. Because the scale of HETOP estimates is indeterminate up to a linear transformation, we develop formulas for converting the HETOP parameter estimates and their standard errors to a scale in which the population distribution of scores is standardized. We demonstrate and evaluate this novel application of the HETOP model with a simulation study and using real test score data from two sources. We find that the HETOP model produces unbiased estimates of group means and standard deviations, except when group sample sizes are small. In such cases, we demonstrate that a ‘‘partially heteroskedastic’’ ordered probit (PHOP) model can produce esti- mates with a smaller root mean squared error than the fully heteroskedastic model. Keywords: heteroskedastic ordered probit model; test score distributions; coarsened data The widespread availability of aggregate student achievement data provides a potentially valuable resource for researchers and policy makers alike. Often, however, these data are only publicly available in ‘‘coarsened’’ form in which students are classified into one or more ordered ‘‘proficiency’’ categories (e.g., ‘‘basic,’’ ‘‘proficient,’’ ‘‘advanced’’).