Generalized Linear Models Link Function the Logistic Equation Is
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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. -
Analysis of Variance; *********43***1****T******T
DOCUMENT RESUME 11 571 TM 007 817 111, AUTHOR Corder-Bolz, Charles R. TITLE A Monte Carlo Study of Six Models of Change. INSTITUTION Southwest Eduoational Development Lab., Austin, Tex. PUB DiTE (7BA NOTE"". 32p. BDRS-PRICE MF-$0.83 Plug P4istage. HC Not Availablefrom EDRS. DESCRIPTORS *Analysis Of COariance; *Analysis of Variance; Comparative Statistics; Hypothesis Testing;. *Mathematicai'MOdels; Scores; Statistical Analysis; *Tests'of Significance ,IDENTIFIERS *Change Scores; *Monte Carlo Method ABSTRACT A Monte Carlo Study was conducted to evaluate ,six models commonly used to evaluate change. The results.revealed specific problems with each. Analysis of covariance and analysis of variance of residualized gain scores appeared to, substantially and consistently overestimate the change effects. Multiple factor analysis of variance models utilizing pretest and post-test scores yielded invalidly toF ratios. The analysis of variance of difference scores an the multiple 'factor analysis of variance using repeated measures were the only models which adequately controlled for pre-treatment differences; ,however, they appeared to be robust only when the error level is 50% or more. This places serious doubt regarding published findings, and theories based upon change score analyses. When collecting data which. have an error level less than 50% (which is true in most situations)r a change score analysis is entirely inadvisable until an alternative procedure is developed. (Author/CTM) 41t i- **************************************4***********************i,******* 4! Reproductions suppliedby ERRS are theibest that cap be made * .... * '* ,0 . from the original document. *********43***1****t******t*******************************************J . S DEPARTMENT OF HEALTH, EDUCATIONWELFARE NATIONAL INST'ITUTE OF r EOUCATIOp PHISDOCUMENT DUCED EXACTLYHAS /3EENREPRC AS RECEIVED THE PERSONOR ORGANIZATION J RoP AT INC. -
Generalized Linear Models (Glms)
San Jos´eState University Math 261A: Regression Theory & Methods Generalized Linear Models (GLMs) Dr. Guangliang Chen This lecture is based on the following textbook sections: • Chapter 13: 13.1 – 13.3 Outline of this presentation: • What is a GLM? • Logistic regression • Poisson regression Generalized Linear Models (GLMs) What is a GLM? In ordinary linear regression, we assume that the response is a linear function of the regressors plus Gaussian noise: 0 2 y = β0 + β1x1 + ··· + βkxk + ∼ N(x β, σ ) | {z } |{z} linear form x0β N(0,σ2) noise The model can be reformulate in terms of • distribution of the response: y | x ∼ N(µ, σ2), and • dependence of the mean on the predictors: µ = E(y | x) = x0β Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University3/24 Generalized Linear Models (GLMs) beta=(1,2) 5 4 3 β0 + β1x b y 2 y 1 0 −1 0.0 0.2 0.4 0.6 0.8 1.0 x x Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University4/24 Generalized Linear Models (GLMs) Generalized linear models (GLM) extend linear regression by allowing the response variable to have • a general distribution (with mean µ = E(y | x)) and • a mean that depends on the predictors through a link function g: That is, g(µ) = β0x or equivalently, µ = g−1(β0x) Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University5/24 Generalized Linear Models (GLMs) In GLM, the response is typically assumed to have a distribution in the exponential family, which is a large class of probability distributions that have pdfs of the form f(x | θ) = a(x)b(θ) exp(c(θ) · T (x)), including • Normal - ordinary linear regression • Bernoulli - Logistic regression, modeling binary data • Binomial - Multinomial logistic regression, modeling general cate- gorical data • Poisson - Poisson regression, modeling count data • Exponential, Gamma - survival analysis Dr. -
6 Modeling Survival Data with Cox Regression Models
CHAPTER 6 ST 745, Daowen Zhang 6 Modeling Survival Data with Cox Regression Models 6.1 The Proportional Hazards Model A proportional hazards model proposed by D.R. Cox (1972) assumes that T z1β1+···+zpβp z β λ(t|z)=λ0(t)e = λ0(t)e , (6.1) where z is a p × 1 vector of covariates such as treatment indicators, prognositc factors, etc., and β is a p × 1 vector of regression coefficients. Note that there is no intercept β0 in model (6.1). Obviously, λ(t|z =0)=λ0(t). So λ0(t) is often called the baseline hazard function. It can be interpreted as the hazard function for the population of subjects with z =0. The baseline hazard function λ0(t) in model (6.1) can take any shape as a function of t.The T only requirement is that λ0(t) > 0. This is the nonparametric part of the model and z β is the parametric part of the model. So Cox’s proportional hazards model is a semiparametric model. Interpretation of a proportional hazards model 1. It is easy to show that under model (6.1) exp(zT β) S(t|z)=[S0(t)] , where S(t|z) is the survival function of the subpopulation with covariate z and S0(t)isthe survival function of baseline population (z = 0). That is R t − λ0(u)du S0(t)=e 0 . PAGE 120 CHAPTER 6 ST 745, Daowen Zhang 2. For any two sets of covariates z0 and z1, zT β λ(t|z1) λ0(t)e 1 T (z1−z0) β, t ≥ , = zT β =e for all 0 λ(t|z0) λ0(t)e 0 which is a constant over time (so the name of proportional hazards model). -
Logistic Regression, Dependencies, Non-Linear Data and Model Reduction
COMP6237 – Logistic Regression, Dependencies, Non-linear Data and Model Reduction Markus Brede [email protected] Lecture slides available here: http://users.ecs.soton.ac.uk/mb8/stats/datamining.html (Thanks to Jason Noble and Cosma Shalizi whose lecture materials I used to prepare) COMP6237: Logistic Regression ● Outline: – Introduction – Basic ideas of logistic regression – Logistic regression using R – Some underlying maths and MLE – The multinomial case – How to deal with non-linear data ● Model reduction and AIC – How to deal with dependent data – Summary – Problems Introduction ● Previous lecture: Linear regression – tried to predict a continuous variable from variation in another continuous variable (E.g. basketball ability from height) ● Here: Logistic regression – Try to predict results of a binary (or categorical) outcome variable Y from a predictor variable X – This is a classification problem: classify X as belonging to one of two classes – Occurs quite often in science … e.g. medical trials (will a patient live or die dependent on medication?) Dependent variable Y Predictor Variables X The Oscars Example ● A fictional data set that looks at what it takes for a movie to win an Oscar ● Outcome variable: Oscar win, yes or no? ● Predictor variables: – Box office takings in millions of dollars – Budget in millions of dollars – Country of origin: US, UK, Europe, India, other – Critical reception (scores 0 … 100) – Length of film in minutes – This (fictitious) data set is available here: https://www.southampton.ac.uk/~mb1a10/stats/filmData.txt Predicting Oscar Success ● Let's start simple and look at only one of the predictor variables ● Do big box office takings make Oscar success more likely? ● Could use same techniques as below to look at budget size, film length, etc. -
Bayesian Inference Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Aus´ınand Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 1 / 35 Objective AFM Smith Dennis Lindley We analyze the Bayesian approach to fitting normal and generalized linear models and introduce the Bayesian hierarchical modeling approach. Also, we study the modeling and forecasting of time series. Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 2 / 35 Contents 1 Normal linear models 1.1. ANOVA model 1.2. Simple linear regression model 2 Generalized linear models 3 Hierarchical models 4 Dynamic models Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 3 / 35 Normal linear models A normal linear model is of the following form: y = Xθ + ; 0 where y = (y1;:::; yn) is the observed data, X is a known n × k matrix, called 0 the design matrix, θ = (θ1; : : : ; θk ) is the parameter set and follows a multivariate normal distribution. Usually, it is assumed that: 1 ∼ N 0 ; I : k φ k A simple example of normal linear model is the simple linear regression model T 1 1 ::: 1 where X = and θ = (α; β)T . x1 x2 ::: xn Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 4 / 35 Normal linear models Consider a normal linear model, y = Xθ + . A conjugate prior distribution is a normal-gamma distribution: -
Small-Sample Properties of Tests for Heteroscedasticity in the Conditional Logit Model
HEDG Working Paper 06/04 Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole May 2006 ISSN 1751-1976 york.ac.uk/res/herc/hedgwp Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole National Primary Care Research and Development Centre Centre for Health Economics University of York May 16, 2006 Abstract This paper compares the small-sample properties of several asymp- totically equivalent tests for heteroscedasticity in the conditional logit model. While no test outperforms the others in all of the experiments conducted, the likelihood ratio test and a particular variety of the Wald test are found to have good properties in moderate samples as well as being relatively powerful. Keywords: conditional logit, heteroscedasticity JEL classi…cation: C25 National Primary Care Research and Development Centre, Centre for Health Eco- nomics, Alcuin ’A’Block, University of York, York YO10 5DD, UK. Tel.: +44 1904 321404; fax: +44 1904 321402. E-mail address: [email protected]. NPCRDC receives funding from the Department of Health. The views expressed are not necessarily those of the funders. 1 1 Introduction In most applications of the conditional logit model the error term is assumed to be homoscedastic. Recently, however, there has been a growing interest in testing the homoscedasticity assumption in applied work (Hensher et al., 1999; DeShazo and Fermo, 2002). This is partly due to the well-known result that heteroscedasticity causes the coe¢ cient estimates in discrete choice models to be inconsistent (Yatchew and Griliches, 1985), but also re‡ects a behavioural interest in factors in‡uencing the variance of the latent variables in the model (Louviere, 2001; Louviere et al., 2002). -
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. -
Multinomial Logistic Regression
26-Osborne (Best)-45409.qxd 10/9/2007 5:52 PM Page 390 26 MULTINOMIAL LOGISTIC REGRESSION CAROLYN J. ANDERSON LESLIE RUTKOWSKI hapter 24 presented logistic regression Many of the concepts used in binary logistic models for dichotomous response vari- regression, such as the interpretation of parame- C ables; however, many discrete response ters in terms of odds ratios and modeling prob- variables have three or more categories (e.g., abilities, carry over to multicategory logistic political view, candidate voted for in an elec- regression models; however, two major modifica- tion, preferred mode of transportation, or tions are needed to deal with multiple categories response options on survey items). Multicate- of the response variable. One difference is that gory response variables are found in a wide with three or more levels of the response variable, range of experiments and studies in a variety of there are multiple ways to dichotomize the different fields. A detailed example presented in response variable. If J equals the number of cate- this chapter uses data from 600 students from gories of the response variable, then J(J – 1)/2 dif- the High School and Beyond study (Tatsuoka & ferent ways exist to dichotomize the categories. Lohnes, 1988) to look at the differences among In the High School and Beyond study, the three high school students who attended academic, program types can be dichotomized into pairs general, or vocational programs. The students’ of programs (i.e., academic and general, voca- socioeconomic status (ordinal), achievement tional and general, and academic and vocational). test scores (numerical), and type of school How the response variable is dichotomized (nominal) are all examined as possible explana- depends, in part, on the nature of the variable. -
Logit and Ordered Logit Regression (Ver
Getting Started in Logit and Ordered Logit Regression (ver. 3.1 beta) Oscar Torres-Reyna Data Consultant [email protected] http://dss.princeton.edu/training/ PU/DSS/OTR Logit model • Use logit models whenever your dependent variable is binary (also called dummy) which takes values 0 or 1. • Logit regression is a nonlinear regression model that forces the output (predicted values) to be either 0 or 1. • Logit models estimate the probability of your dependent variable to be 1 (Y=1). This is the probability that some event happens. PU/DSS/OTR Logit odelm From Stock & Watson, key concept 9.3. The logit model is: Pr(YXXXFXX 1 | 1= , 2 ,...=k β ) +0 β ( 1 +2 β 1 +βKKX 2 + ... ) 1 Pr(YXXX 1= | 1 , 2k = ,... ) 1−+(eβ0 + βXX 1 1 + β 2 2 + ...βKKX + ) 1 Pr(YXXX 1= | 1 , 2= ,... ) k ⎛ 1 ⎞ 1+ ⎜ ⎟ (⎝ eβ+0 βXX 1 1 + β 2 2 + ...βKK +X ⎠ ) Logit nd probita models are basically the same, the difference is in the distribution: • Logit – Cumulative standard logistic distribution (F) • Probit – Cumulative standard normal distribution (Φ) Both models provide similar results. PU/DSS/OTR It tests whether the combined effect, of all the variables in the model, is different from zero. If, for example, < 0.05 then the model have some relevant explanatory power, which does not mean it is well specified or at all correct. Logit: predicted probabilities After running the model: logit y_bin x1 x2 x3 x4 x5 x6 x7 Type predict y_bin_hat /*These are the predicted probabilities of Y=1 */ Here are the estimations for the first five cases, type: 1 x2 x3 x4 x5 x6 x7 y_bin_hatbrowse y_bin x Predicted probabilities To estimate the probability of Y=1 for the first row, replace the values of X into the logit regression equation. -
Chapter 4: Central Tendency and Dispersion
Central Tendency and Dispersion 4 In this chapter, you can learn • how the values of the cases on a single variable can be summarized using measures of central tendency and measures of dispersion; • how the central tendency can be described using statistics such as the mode, median, and mean; • how the dispersion of scores on a variable can be described using statistics such as a percent distribution, minimum, maximum, range, and standard deviation along with a few others; and • how a variable’s level of measurement determines what measures of central tendency and dispersion to use. Schooling, Politics, and Life After Death Once again, we will use some questions about 1980 GSS young adults as opportunities to explain and demonstrate the statistics introduced in the chapter: • Among the 1980 GSS young adults, are there both believers and nonbelievers in a life after death? Which is the more common view? • On a seven-attribute political party allegiance variable anchored at one end by “strong Democrat” and at the other by “strong Republican,” what was the most Democratic attribute used by any of the 1980 GSS young adults? The most Republican attribute? If we put all 1980 95 96 PART II :: DESCRIPTIVE STATISTICS GSS young adults in order from the strongest Democrat to the strongest Republican, what is the political party affiliation of the person in the middle? • What was the average number of years of schooling completed at the time of the survey by 1980 GSS young adults? Were most of these twentysomethings pretty close to the average on -
Bayesian Methods: Review of Generalized Linear Models
Bayesian Methods: Review of Generalized Linear Models RYAN BAKKER University of Georgia ICPSR Day 2 Bayesian Methods: GLM [1] Likelihood and Maximum Likelihood Principles Likelihood theory is an important part of Bayesian inference: it is how the data enter the model. • The basis is Fisher’s principle: what value of the unknown parameter is “most likely” to have • generated the observed data. Example: flip a coin 10 times, get 5 heads. MLE for p is 0.5. • This is easily the most common and well-understood general estimation process. • Bayesian Methods: GLM [2] Starting details: • – Y is a n k design or observation matrix, θ is a k 1 unknown coefficient vector to be esti- × × mated, we want p(θ Y) (joint sampling distribution or posterior) from p(Y θ) (joint probabil- | | ity function). – Define the likelihood function: n L(θ Y) = p(Y θ) | i| i=1 Y which is no longer on the probability metric. – Our goal is the maximum likelihood value of θ: θˆ : L(θˆ Y) L(θ Y) θ Θ | ≥ | ∀ ∈ where Θ is the class of admissable values for θ. Bayesian Methods: GLM [3] Likelihood and Maximum Likelihood Principles (cont.) Its actually easier to work with the natural log of the likelihood function: • `(θ Y) = log L(θ Y) | | We also find it useful to work with the score function, the first derivative of the log likelihood func- • tion with respect to the parameters of interest: ∂ `˙(θ Y) = `(θ Y) | ∂θ | Setting `˙(θ Y) equal to zero and solving gives the MLE: θˆ, the “most likely” value of θ from the • | parameter space Θ treating the observed data as given.