Simulating Generalized Linear Models
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An Instrumental Variables Probit Estimator Using Gretl
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics An Instrumental Variables Probit Estimator using gretl Lee C. Adkins Professor of Economics, Oklahoma State University, Stillwater, OK 74078 [email protected] Abstract. The most widely used commercial software to estimate endogenous probit models offers two choices: a computationally simple generalized least squares estimator and a maximum likelihood estimator. Adkins [1, -. com%ares these estimators to several others in a Monte Carlo study and finds that the 1LS estimator performs reasonably well in some circumstances. In this paper the small sample properties of the various estimators are reviewed and a simple routine us- ing the gretl software is given that yields identical results to those produced by Stata 10.1. The paper includes an example estimated using data on bank holding companies. 1 Introduction 4atchew and Griliches +,5. analyze the effects of various kinds of misspecifi- cation on the probit model. Among the problems explored was that of errors- in-variables. In linear re$ression, a regressor measured with error causes least squares to be inconsistent and 4atchew and Griliches find similar results for probit. Rivers and 7#ong [14] and Smith and 8lundell +,9. suggest two-stage estimators for probit and tobit, respectively. The strategy is to model a con- tinuous endogenous regressor as a linear function of the exogenous regressors and some instruments. Predicted values from this regression are then used in the second stage probit or tobit. These two-step methods are not efficient, &ut are consistent. -
An Empirical Analysis of Factors Affecting Honors Program Completion Rates
An Empirical Analysis of Factors Affecting Honors Program Completion Rates HALLIE SAVAGE CLARION UNIVERSITY OF PENNSYLVANIA AND T H E NATIONAL COLLEGIATE HONOR COUNCIL ROD D. RAE H SLER CLARION UNIVERSITY OF PENNSYLVANIA JOSE ph FIEDOR INDIANA UNIVERSITY OF PENNSYLVANIA INTRODUCTION ne of the most important issues in any educational environment is iden- tifying factors that promote academic success. A plethora of research on suchO factors exists across most academic fields, involving a wide range of student demographics, and the definition of student success varies across the range of studies published. While much of the research is devoted to looking at student performance in particular courses and concentrates on examina- tion scores and grades, many authors have directed their attention to student success in the context of an entire academic program; student success in this context usually centers on program completion or graduation and student retention. The analysis in this paper follows the emphasis of McKay on the importance of conducting repeated research on student completion of honors programs at different universities for different time periods. This paper uses a probit regression analysis as well as the logit regression analysis employed by McKay in order to determine predictors of student success in the honors pro- gram at a small, public university, thus attempting to answer McKay’s call for a greater understanding of honors students and factors influencing their suc- cess. The use of two empirical models on completion data, employing differ- ent base distributions, provides more robust statistical estimates than observed in similar studies 115 SPRING /SUMMER 2014 AN EMPIRI C AL ANALYSIS OF FA C TORS AFFE C TING COMPLETION RATES PREVIOUS LITEratURE The early years of our research was concurrent with the work of McKay, who studied the 2002–2005 entering honors classes at the University of North Florida and published his work in 2009. -
Application of Random-Effects Probit Regression Models
Journal of Consulting and Clinical Psychology Copyright 1994 by the American Psychological Association, Inc. 1994, Vol. 62, No. 2, 285-296 0022-006X/94/S3.00 Application of Random-Effects Probit Regression Models Robert D. Gibbons and Donald Hedeker A random-effects probit model is developed for the case in which the outcome of interest is a series of correlated binary responses. These responses can be obtained as the product of a longitudinal response process where an individual is repeatedly classified on a binary outcome variable (e.g., sick or well on occasion t), or in "multilevel" or "clustered" problems in which individuals within groups (e.g., firms, classes, families, or clinics) are considered to share characteristics that produce similar responses. Both examples produce potentially correlated binary responses and modeling these per- son- or cluster-specific effects is required. The general model permits analysis at both the level of the individual and cluster and at the level at which experimental manipulations are applied (e.g., treat- ment group). The model provides maximum likelihood estimates for time-varying and time-invari- ant covariates in the longitudinal case and covariates which vary at the level of the individual and at the cluster level for multilevel problems. A similar number of individuals within clusters or number of measurement occasions within individuals is not required. Empirical Bayesian estimates of per- son-specific trends or cluster-specific effects are provided. Models are illustrated with data from -
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 9: Logit/Probit Prof
Lecture 9: Logit/Probit Prof. Sharyn O’Halloran Sustainable Development U9611 Econometrics II Review of Linear Estimation So far, we know how to handle linear estimation models of the type: Y = β0 + β1*X1 + β2*X2 + … + ε ≡ Xβ+ ε Sometimes we had to transform or add variables to get the equation to be linear: Taking logs of Y and/or the X’s Adding squared terms Adding interactions Then we can run our estimation, do model checking, visualize results, etc. Nonlinear Estimation In all these models Y, the dependent variable, was continuous. Independent variables could be dichotomous (dummy variables), but not the dependent var. This week we’ll start our exploration of non- linear estimation with dichotomous Y vars. These arise in many social science problems Legislator Votes: Aye/Nay Regime Type: Autocratic/Democratic Involved in an Armed Conflict: Yes/No Link Functions Before plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric model We have one example of this already: logs Start with Y = Xβ+ ε Then change to log(Y) ≡ Y′ = Xβ+ ε Run this like a regular OLS equation Then you have to “back out” the results Link Functions Before plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric model We have one example of this already: logs Start with Y = Xβ+ ε Different β’s here Then change to log(Y) ≡ Y′ = Xβ + ε Run this like a regular OLS equation Then you have to “back out” the results Link Functions If the coefficient on some particular X is β, then a 1 unit ∆X Æ β⋅∆(Y′) = β⋅∆[log(Y))] = eβ ⋅∆(Y) Since for small values of β, eβ ≈ 1+β , this is almost the same as saying a β% increase in Y (This is why you should use natural log transformations rather than base-10 logs) In general, a link function is some F(⋅) s.t. -
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. -
Generalized Linear Models
CHAPTER 6 Generalized linear models 6.1 Introduction Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + + Xkβk.Logistic regression predicts Pr(y =1)forbinarydatafromalinearpredictorwithaninverse-··· logit transformation. A generalized linear model involves: 1. A data vector y =(y1,...,yn) 2. Predictors X and coefficients β,formingalinearpredictorXβ 1 3. A link function g,yieldingavectoroftransformeddataˆy = g− (Xβ)thatare used to model the data 4. A data distribution, p(y yˆ) | 5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution. The options in a generalized linear model are the transformation g and the data distribution p. In linear regression,thetransformationistheidentity(thatis,g(u) u)and • the data distribution is normal, with standard deviation σ estimated from≡ data. 1 1 In logistic regression,thetransformationistheinverse-logit,g− (u)=logit− (u) • (see Figure 5.2a on page 80) and the data distribution is defined by the proba- bility for binary data: Pr(y =1)=y ˆ. This chapter discusses several other classes of generalized linear model, which we list here for convenience: The Poisson model (Section 6.2) is used for count data; that is, where each • data point yi can equal 0, 1, 2, ....Theusualtransformationg used here is the logarithmic, so that g(u)=exp(u)transformsacontinuouslinearpredictorXiβ to a positivey ˆi.ThedatadistributionisPoisson. It is usually a good idea to add a parameter to this model to capture overdis- persion,thatis,variationinthedatabeyondwhatwouldbepredictedfromthe Poisson distribution alone. -
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 -
Week 12: Linear Probability Models, Logistic and Probit
Week 12: Linear Probability Models, Logistic and Probit Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2019 These slides are part of a forthcoming book to be published by Cambridge University Press. For more information, go to perraillon.com/PLH. c This material is copyrighted. Please see the entire copyright notice on the book's website. Updated notes are here: https://clas.ucdenver.edu/marcelo-perraillon/ teaching/health-services-research-methods-i-hsmp-7607 1 Outline Modeling 1/0 outcomes The \wrong" but super useful model: Linear Probability Model Deriving logistic regression Probit regression as an alternative 2 Binary outcomes Binary outcomes are everywhere: whether a person died or not, broke a hip, has hypertension or diabetes, etc We typically want to understand what is the probability of the binary outcome given explanatory variables It's exactly the same type of models we have seen during the semester, the difference is that we have been modeling the conditional expectation given covariates: E[Y jX ] = β0 + β1X1 + ··· + βpXp Now, we want to model the probability given covariates: P(Y = 1jX ) = f (β0 + β1X1 + ··· + βpXp) Note the function f() in there 3 Linear Probability Models We could actually use our vanilla linear model to do so If Y is an indicator or dummy variable, then E[Y jX ] is the proportion of 1s given X , which we interpret as the probability of Y given X The parameters are changes/effects/differences in the probability of Y by a unit change in X or for a small change in X If an indicator variable, then change from 0 to 1 For example, if we model diedi = β0 + β1agei + i , we could interpret β1 as the change in the probability of death for an additional year of age 4 Linear Probability Models The problem is that we know that this model is not entirely correct. -
Probit Model 1 Probit Model
Probit model 1 Probit model In statistics, a probit model is a type of regression where the dependent variable can only take two values, for example married or not married. The name is from probability + unit.[1] A probit model is a popular specification for an ordinal[2] or a binary response model that employs a probit link function. This model is most often estimated using standard maximum likelihood procedure, such an estimation being called a probit regression. Probit models were introduced by Chester Bliss in 1934, and a fast method for computing maximum likelihood estimates for them was proposed by Ronald Fisher in an appendix to Bliss 1935. Introduction Suppose response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes form where Pr denotes probability, and Φ is the Cumulative Distribution Function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood. It is also possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive: The use of the standard normal distribution causes no loss of generality compared with using an arbitrary mean and standard deviation because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount.