Estimating the random coefficients logit model of demand using aggregate data
|
|
- Myron Sanders
- 7 years ago
- Views:
Transcription
1 Estimating the random coefficients logit model of demand using aggregate data David Vincent Deloitte Economic Consulting London, UK September 14, 2012
2 Introduction Estimation of consumer demand in differentiated product industries plays a central role in applied economic analysis The conventional approach is to specify a system of demand functions that correspond to a valid preference ordering, and estimate the parameters using aggregate data A popular example is the Almost Ideal Demand System of Deaton(1980), where market shares are linear functions of the logarithm of prices, and real expenditure. A major concern in adopting this approach, is the large number of parameters that need to be estimated, even after the restrictions of adding-up homogeneity a symmetry have been imposed The dimensionality problem can be solved if preferences are assumed to be separable; however, this places severe restrictions on the degree of substitutability between goods in different sub-groups
3 Introduction The logit-demand model (McFadden 1973) is another way to address the dimensionality problem, by assuming instead that consumers have preferences over product characteristics Although easy to estimate, this model again imposes strong a-prior restrictions over the patterns of substitutability The purpose of this presentation is to discuss the random coefficients logit demand model (Berry Levinhson Pakes 1995) This framework accommodates consumer heterogeneity, by allowing taste parameters to vary with individual characteristics and requires market level data for estimation The model produces cross price elasticities that are more realistic and allows for the case where prices are endogenous It is very popular in the Industrial Organization literature and routinely applied by regulatory authorities, yet these is no official BLP Stata command!
4 The Model Following Nevo(2005), assume we observe t = 1,.., T markets consisting of I t consumers and J products. For each market, data is available on total quantities sold, prices and product characteristics of all J products Markets are assumed to be independent and can be cross-sectional (e.g. different cities) or repeated observations let u ijt denote the indirect utility that individual i experiences in market t when consuming product j, and assume this depends on a K 1 vector of product characteristics x jt, price p jt an unobserved component ξ jt, and an idiosyncratic error ɛ ijt. If the utility function is quasi-linear utility, then: u ijt = α i (y i p jt ) + x ijtβ i + ξ jt + ɛ ijt (1) where y i is income, β i is a K 1 vector of coefficients and α i is the marginal utility of income.
5 The Model Consumer i also has the choice to buy the outside product j = 0 with normalized utility u i0t = α i y i + ɛ i0t. Both β i and α i and assumed to be linear functions of characteristics D i and v i of dimensions d 1 and (K + 1) 1: ( ) ( ) βi β0 = + ΠD i + Lvi (2) α i α 0 where v i iid(0, I K+1 ), D i iid(0, Σ D ),Π is a K + 1 d matrix of coefficients, and LL = Σ v Although both D i and v i are unobserved, the distribution of the demographics D i including Σ D is assumed to be known This is not the case for v i where a parametric distribution is assumed (e.g. normal) In practice F D (D) is the empirical non-parametric distribution
6 The Model Define the set: A ijt = {ɛ it : u ijt > u ikt, j k}, then the probability that individual i selects product j in market t is Pr ijt = df (ɛ it D i, v i ) (3) A ijt Integrating over the unobserved variables D i and v i yields: Pr jt = Pr ijt df (D i v i )df (v i ) (4) v i D i where Pr jt is the same for all i and can be estimated by the product share s jt = q jt M t where M t is the market size The error in this approximation is O(I 1/2 t ) and will be negligible for large I t which is often the case
7 The Model: Distributional Assumptions To evaluate the integral in (3) first assume that ɛ ijt are iidd and have a Type I extreme value distribution. Then: exp(x ijt Pr ijt = β i α i p jt + ξ jt ) 1 + k exp(1 + x ijt β i α i p jt + ξ jt ) To evaluate (4), it is necessary to specify the distributions of D i and v i. At one extreme, we could assume Σ D = Σ v = 0 Although appealing, consider the price elasticities: e jkt = { α0 p jt (1 s jt ) if j = k; α 0 p kt s kt if j k. As shares are often small, the own price elasticities will be proportional to price. This is unrealistic Furthermore, the cross price elasticities restrict proportionate increases to be identical for all goods (5)
8 The Model: Distributional Assumptions When preferences are allowed to differ, the elasticities will be: e jkt = { p jt s jt αi Pr ijt (1 Pr ijt )df (D i, v i ) if j = k; p kt s jt αi Pr ijt Pr ikt df (D i, v i ) if j k. The price sensitivity is now a probability weighted average, and can differ over products. As such the model allows for flexible substitution patterns To continue, assume v i iidn(0, I K+1 ), let F (D i ) be the EDF, and denote δ jt = x jt β 0 α 0 + ξ jt as the mean-utility. Then the integral in (4) can be approximated by simulation: s jt = 1 R R exp(δ jt + [p jt, x jt ](ΠD r + Lv i )) r=1 1 + k exp(δ jt + [p kt, x kt ](ΠD r + Lv i )) (6)
9 Estimation As prices p jt may be correlated with error ξ jt, the parameters β 0, α 0, L and Π are estimated by GMM. This is carried out in three steps excluding an initial step 0. Draw R individuals v 1,..., v R, and D 1,..., D R from v i an D i 1. For a given value of Π,L, solve for the vector δ = [δ 11,..., δ JT ] such that predicted shares using (4) equals observed shares 2. Compute the sample-moment conditions T 1 T t=1 Z tξ t where Z t is a J l set of instruments, and form the GMM-objective function. 3. Search for the values β 0, α 0, Π, L that minimize the objective function in step 4 To simplify the notation, let x jt contain all variables, assume L is diagonal and define θ = (θ 1, θ 2 ), where: θ 1 = ( α0 β 0 ) ( ) vec(π ), θ 2 = diag(l) (7)
10 Estimation: Step 1 For each market t = 1,..., T, we need the J 1 vector δ t such that: s(δ t, θ 2 ) = s t (8) where s t = [s 1t,..., s Jt ]. This system of J equations can be solved using the contraction mapping suggested by BLP. For a given vector δ n t, this involves computing: δ n+1 t = δ n t + log s t log (s(δ n t, θ 2 )) (9) Iteration continues using (8) and (9) until δt n δt n 1 is below a specified tolerance level. In the Stata command blp, iteration is over w t = exp(δ t ) and δ t is recovered at convergence. This saves considerable time
11 Estimation: Step 2 Let Z t be a J l matrix of instruments that satisfies E[Z t ξ t ] = 0 and define the GMM-objective function as: Q = h (θ)w T h(θ) (10) where h = T 1 Z ξ are the sample moments based on δ and W T is a positive definite weighting matrix. If the errors are homoskedastic, a consistent estimator of W is: Ŵ = (T 1ˆσ 2 ξ Z Z) 1 (11) If instead the errors are assumed to be correlated over J and heteroskedastic over t, then: Ŵ = (T 1 T t=1 Z ˆξ t t ˆξ tz t ) (12) Estimation using (12) is carried out from an initial estimate of ˆθ based on (11). This is often referred to as the two-step method
12 Estimation: Step 3 The GMM-estimator ˆθ is the vector that minimizes (10), and is the solution to the following first order conditions: Q θ 1 = X ZWZ ξ = 0 (13) Q θ 2 = D θ2 δ ZWZ ξ = 0 (14) To reduce search-time, θ 1 can be written as a function of θ 2 ˆθ 1 = (X ZWZ X ) 1 X ZWZ δ(θ 2 ) (15) The search is now limited to θ 2, but to employ a Newton method, the analytical derivatives D θ2 δ t are required. By the implicit function theorem applied to s(δ t (θ 2 ), θ 2 ) = s t : D θ2 δ t = (D δt s t ) 1 D θ2 s t (16)
13 Estimation: Step 3 The elements inside the matrices of (16) are: ) 1 ( s1t.. σ 1 D θ2 δ t = ( s1t s.. 1t δ 1t δ Jt : : : s Jt δ 1t.. s Jt δ Jt From equation (6), the derivatives are: s jt δ jt = R 1 s jt δ mt = R 1 s jt σ k = R 1 s jt π kd = R 1 R Pr rjt (1 Pr rjt ) r=1 R Pr rjt Pr rmt r=1 R Pr rjt v rk (x jtk r=1 R Pr rjt D rd (x jtk r=1 s 1t, σ K1 s 1t π 11.. : : : : : : s Jt σ 1.. s Jt, σ K1 s Jt π 11.. J x mtk s rmt ) m=1 J x mtk s rmt ) m=1 s 1t π K1 d s Jt π K1 d )
14 Stata Command: blp blp depvar [ indepvars ] [ if ] [ in ] (endogvars=instruments), (stochastic 1 = varlist 1,., stochastic K = varlist K ) markets(string) draws(#),[vce(string),demofile(string),twostep] 1. endogvars - specify endogenous variables and instruments 2. stochastic - specify variable with random coefficient and demographic variables (if used) 3. markets - input the market variable 4. draws - specify the number of simulations 5. vce - robust if one-step else standard demotfile - specify the path and name of the demographics data (optional) 6. twostep - uses optimal weighting matrix
15 Syntax Example blp s cons x1 x2, stochastic(x1=d1,x2,p) endog(p=p x12 x22 expx1 p2 z1 z2) demofile(demodata) markets(mkt) draws(100) Random coefficients logit model estimates Coef. Std. Err. z P> z [95% Conf. Interval] Mean Utility cons x x p x1 x2 p d SD SD SD
16 Monte Carlo Experiments: DGP To exmaine the properties of the estimator, data is generated from the following DGP where J = 25 and T = 30 ( x1jt x 2jt u ijt = 10 + β 1i x 1jt + β 2i x 2jt + α i p jt + ξ jt + ɛ ijt α i N( 1, 0.5) β 1i N(1, 1) β 2i N(1, 1) ɛ ijt EV ) [( ) 10, 10 p jt N(10, 1) x it U(0, 1) ]
17 Monte Carlo Experiments: Parameter Estimates The following table sets out the mean and standard deviation of the parameter estimates σ β1, σ β2, σ α from 50 replications using 500 draws for each. Monte Carlo Results X1 X2 Price True Parameter Mean Standard deviation
18 Monte Carlo Experiments: Logit Elasticities The following table sets out the price elasticities from the logit model assuming homogeneous preferences Logit Price Elasticities Product
19 Monte Carlo Experiments: Random Parameter Logit Elasticities The following table set out the price elasticities from the random-parameters logit model. Logit Price Elasticities Product
Sample Size Calculation for Longitudinal Studies
Sample Size Calculation for Longitudinal Studies Phil Schumm Department of Health Studies University of Chicago August 23, 2004 (Supported by National Institute on Aging grant P01 AG18911-01A1) Introduction
More informationONLINE APPENDIX: Vertical Integration and Exclusivity in Platform and Two-Sided Markets
ONLINE APPENDIX: Vertical Integration and Exclusivity in Platform and Two-Sided Markets Robin S. Lee January 2013 A Additional Details A.1 Consumer Demand A.1.1 Estimation and Computation Recall that ι
More informationFrom the help desk: Demand system estimation
The Stata Journal (2002) 2, Number 4, pp. 403 410 From the help desk: Demand system estimation Brian P. Poi Stata Corporation Abstract. This article provides an example illustrating how to use Stata to
More informationBayesian Statistics in One Hour. Patrick Lam
Bayesian Statistics in One Hour Patrick Lam Outline Introduction Bayesian Models Applications Missing Data Hierarchical Models Outline Introduction Bayesian Models Applications Missing Data Hierarchical
More informationGasoline Taxes and Fuel Economy: A Preference Heterogeneity Approach
Gasoline Taxes and Fuel Economy: A Preference Heterogeneity Approach Mario Samano March, 2012 Abstract This paper estimates an equilibrium model for the U.S. car market to measure the value of two policies
More informationWhat s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationMarkups and Firm-Level Export Status: Appendix
Markups and Firm-Level Export Status: Appendix De Loecker Jan - Warzynski Frederic Princeton University, NBER and CEPR - Aarhus School of Business Forthcoming American Economic Review Abstract This is
More informationStandard errors of marginal effects in the heteroskedastic probit model
Standard errors of marginal effects in the heteroskedastic probit model Thomas Cornelißen Discussion Paper No. 320 August 2005 ISSN: 0949 9962 Abstract In non-linear regression models, such as the heteroskedastic
More informationFrom the help desk: Swamy s random-coefficients model
The Stata Journal (2003) 3, Number 3, pp. 302 308 From the help desk: Swamy s random-coefficients model Brian P. Poi Stata Corporation Abstract. This article discusses the Swamy (1970) random-coefficients
More informationA Practitioner s Guide to Estimation of Random-Coef cients Logit Models of Demand
A Practitioner s Guide to Estimation of Random-Coef cients Logit Models of Demand Aviv Nevo University of California Berkeley, Berkeley, CA 94720-3880 and NBER Estimation of demand is at the heart of many
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationPlease follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its
More informationLab 5 Linear Regression with Within-subject Correlation. Goals: Data: Use the pig data which is in wide format:
Lab 5 Linear Regression with Within-subject Correlation Goals: Data: Fit linear regression models that account for within-subject correlation using Stata. Compare weighted least square, GEE, and random
More informationDepartment of Economics Session 2012/2013. EC352 Econometric Methods. Solutions to Exercises from Week 10 + 0.0077 (0.052)
Department of Economics Session 2012/2013 University of Essex Spring Term Dr Gordon Kemp EC352 Econometric Methods Solutions to Exercises from Week 10 1 Problem 13.7 This exercise refers back to Equation
More informationEconometric Tools for Analyzing Market Outcomes.
Econometric Tools for Analyzing Market Outcomes. Daniel Ackerberg, C. Lanier Benkard, Steven Berry, and Ariel Pakes. June 14, 2006 Abstract This paper outlines recently developed techniques for estimating
More informationEmpirical Models of Consumer Behavior
Empirical Models of Consumer Behavior Aviv Nevo October 10, 2010 Abstract Models of consumer behavior play a key role in modern empirical Industrial Organization. In this paper, I survey some of the models
More informationUsing instrumental variables techniques in economics and finance
Using instrumental variables techniques in economics and finance Christopher F Baum 1 Boston College and DIW Berlin German Stata Users Group Meeting, Berlin, June 2008 1 Thanks to Mark Schaffer for a number
More informationNominal and ordinal logistic regression
Nominal and ordinal logistic regression April 26 Nominal and ordinal logistic regression Our goal for today is to briefly go over ways to extend the logistic regression model to the case where the outcome
More informationAn introduction to GMM estimation using Stata
An introduction to GMM estimation using Stata David M. Drukker StataCorp German Stata Users Group Berlin June 2010 1 / 29 Outline 1 A quick introduction to GMM 2 Using the gmm command 2 / 29 A quick introduction
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationMonte Carlo Simulation
1 Monte Carlo Simulation Stefan Weber Leibniz Universität Hannover email: sweber@stochastik.uni-hannover.de web: www.stochastik.uni-hannover.de/ sweber Monte Carlo Simulation 2 Quantifying and Hedging
More informationMATLAB and Big Data: Illustrative Example
MATLAB and Big Data: Illustrative Example Rick Mansfield Cornell University August 19, 2014 Goals Use a concrete example from my research to: Demonstrate the value of vectorization Introduce key commands/functions
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationFrom the help desk: hurdle models
The Stata Journal (2003) 3, Number 2, pp. 178 184 From the help desk: hurdle models Allen McDowell Stata Corporation Abstract. This article demonstrates that, although there is no command in Stata for
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationIAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results
IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the
More informationPS 271B: Quantitative Methods II. Lecture Notes
PS 271B: Quantitative Methods II Lecture Notes Langche Zeng zeng@ucsd.edu The Empirical Research Process; Fundamental Methodological Issues 2 Theory; Data; Models/model selection; Estimation; Inference.
More informationMonte Carlo-based statistical methods (MASM11/FMS091)
Monte Carlo-based statistical methods (MASM11/FMS091) Jimmy Olsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February 5, 2013 J. Olsson Monte Carlo-based
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics
ESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Quantile Treatment Effects 2. Control Functions
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More informationExample: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.
Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C
More informationVI. Real Business Cycles Models
VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized
More informationLecture notes: single-agent dynamics 1
Lecture notes: single-agent dynamics 1 Single-agent dynamic optimization models In these lecture notes we consider specification and estimation of dynamic optimization models. Focus on single-agent models.
More informationFrom the help desk: Bootstrapped standard errors
The Stata Journal (2003) 3, Number 1, pp. 71 80 From the help desk: Bootstrapped standard errors Weihua Guan Stata Corporation Abstract. Bootstrapping is a nonparametric approach for evaluating the distribution
More informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More informationMultiple Choice Models II
Multiple Choice Models II Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Multiple Choice Models II 1 / 28 Categorical data Categorical
More informationECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
More informationMULTIPLE REGRESSION EXAMPLE
MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationWeb-based Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni
1 Web-based Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed
More informationYou Are What You Bet: Eliciting Risk Attitudes from Horse Races
You Are What You Bet: Eliciting Risk Attitudes from Horse Races Pierre-André Chiappori, Amit Gandhi, Bernard Salanié and Francois Salanié March 14, 2008 What Do We Know About Risk Preferences? Not that
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More information1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
More informationNote on the EM Algorithm in Linear Regression Model
International Mathematical Forum 4 2009 no. 38 1883-1889 Note on the M Algorithm in Linear Regression Model Ji-Xia Wang and Yu Miao College of Mathematics and Information Science Henan Normal University
More informationStandardization and Estimation of the Number of Factors for Panel Data
Journal of Economic Theory and Econometrics, Vol. 23, No. 2, Jun. 2012, 79 88 Standardization and Estimation of the Number of Factors for Panel Data Ryan Greenaway-McGrevy Chirok Han Donggyu Sul Abstract
More informationSimultaneous or Sequential? Search Strategies in the U.S. Auto. Insurance Industry
Simultaneous or Sequential? Search Strategies in the U.S. Auto Insurance Industry Elisabeth Honka 1 University of Texas at Dallas Pradeep Chintagunta 2 University of Chicago Booth School of Business September
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February
More informationSimultaneous or Sequential? Search Strategies in the U.S. Auto. Insurance Industry
Simultaneous or Sequential? Search Strategies in the U.S. Auto Insurance Industry Elisabeth Honka 1 University of Texas at Dallas Pradeep Chintagunta 2 University of Chicago Booth School of Business October
More informationGenerating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte
More informationAuxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus
Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
More informationEffects of Brand Preference, Product Attributes, and Marketing Mix Variables in Technology Product Markets. S. Sriram. Pradeep K.
Effects of Brand Preference, Product Attributes, and Marketing Mix Variables in Technology Product Markets S. Sriram Pradeep K. Chintagunta Ramya Neelamegham First Version: June 2004 Revised: September
More informationAn Internal Model for Operational Risk Computation
An Internal Model for Operational Risk Computation Seminarios de Matemática Financiera Instituto MEFF-RiskLab, Madrid http://www.risklab-madrid.uam.es/ Nicolas Baud, Antoine Frachot & Thierry Roncalli
More informationHow To Find Out What Search Strategy Is Used In The U.S. Auto Insurance Industry
Simultaneous or Sequential? Search Strategies in the U.S. Auto Insurance Industry Current version: April 2014 Elisabeth Honka Pradeep Chintagunta Abstract We show that the search method consumers use when
More informationEditor Executive Editor Associate Editors Copyright Statement:
The Stata Journal Editor H. Joseph Newton Department of Statistics Texas A & M University College Station, Texas 77843 979-845-3142 979-845-3144 FAX jnewton@stata-journal.com Associate Editors Christopher
More informationTesting Cost Inefficiency under Free Entry in the Real Estate Brokerage Industry
Web Appendix to Testing Cost Inefficiency under Free Entry in the Real Estate Brokerage Industry Lu Han University of Toronto lu.han@rotman.utoronto.ca Seung-Hyun Hong University of Illinois hyunhong@ad.uiuc.edu
More informationVertical Channel Analysis of the U.S. Milk Market. The research in this paper extends the standard random coefficient logit demand (S-RCL)
Vertical Channel Analysis of the U.S. Milk Market Abstract The research in this paper extends the standard random coefficient logit demand (S-RCL) framework using a Box-Cox transformation of the attribute
More informationBasics of Statistical Machine Learning
CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar
More informationA Classical Monetary Model - Money in the Utility Function
A Classical Monetary Model - Money in the Utility Function Jarek Hurnik Department of Economics Lecture III Jarek Hurnik (Department of Economics) Monetary Economics 2012 1 / 24 Basic Facts So far, the
More informationA HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA: A FIRST REPORT
New Mathematics and Natural Computation Vol. 1, No. 2 (2005) 295 303 c World Scientific Publishing Company A HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA:
More informationMatching Investment Strategies in General Insurance Is it Worth It? Aim of Presentation. Background 34TH ANNUAL GIRO CONVENTION
Matching Investment Strategies in General Insurance Is it Worth It? 34TH ANNUAL GIRO CONVENTION CELTIC MANOR RESORT, NEWPORT, WALES Aim of Presentation To answer a key question: What are the benefit of
More informationCHARLES J. ROMEO Economist, Antitrust Division, U.S. Department of Justice, Washington, D.C., 20530 E-mail: charles.romeo@usdoj.
Quantitative Marketing and Economics, 3, 71 93, 2005. C 2005 Springer Science + Business Media, Inc. Printed in The United States. Estimating Discrete Joint Probability Distributions for Demographic Characteristics
More informationChapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 )
Chapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 ) and Neural Networks( 類 神 經 網 路 ) 許 湘 伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 10 1 / 35 13 Examples
More informationIDENTIFICATION IN DIFFERENTIATED PRODUCTS MARKETS USING MARKET LEVEL DATA
Econometrica, Vol. 82, No. 5 (September, 2014), 1749 1797 IDENTIFICATION IN DIFFERENTIATED PRODUCTS MARKETS USING MARKET LEVEL DATA BY STEVEN T. BERRY AND PHILIP A. HAILE 1 We present new identification
More information2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
More informationEstimating the Effect of Tax Reform in Differentiated Product Oligopolistic Markets
Estimating the Effect of Tax Reform in Differentiated Product Oligopolistic Markets by Chaim Fershtman, Tel Aviv University & CentER, Tilburg University Neil Gandal*, Tel Aviv University & CEPR, and Sarit
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationPrivate Equity Fund Valuation and Systematic Risk
An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology
More informationNonparametric adaptive age replacement with a one-cycle criterion
Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk
More informationChapter 10: Basic Linear Unobserved Effects Panel Data. Models:
Chapter 10: Basic Linear Unobserved Effects Panel Data Models: Microeconomic Econometrics I Spring 2010 10.1 Motivation: The Omitted Variables Problem We are interested in the partial effects of the observable
More informationTopic 5: Stochastic Growth and Real Business Cycles
Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar
More informationInstitut für Soziologie Eberhard Karls Universität Tübingen www.maartenbuis.nl
from Indirect Extracting from Institut für Soziologie Eberhard Karls Universität Tübingen www.maartenbuis.nl from Indirect What is the effect of x on y? Which effect do I choose: average marginal or marginal
More informationHETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY. Daniel Harenberg daniel.harenberg@gmx.de. University of Mannheim. Econ 714, 28.11.
COMPUTING EQUILIBRIUM WITH HETEROGENEOUS AGENTS AND AGGREGATE UNCERTAINTY (BASED ON KRUEGER AND KUBLER, 2004) Daniel Harenberg daniel.harenberg@gmx.de University of Mannheim Econ 714, 28.11.06 Daniel Harenberg
More informationMISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group
MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could
More informationEfficiency and the Cramér-Rao Inequality
Chapter Efficiency and the Cramér-Rao Inequality Clearly we would like an unbiased estimator ˆφ (X of φ (θ to produce, in the long run, estimates which are fairly concentrated i.e. have high precision.
More informationFinancial Market Microstructure Theory
The Microstructure of Financial Markets, de Jong and Rindi (2009) Financial Market Microstructure Theory Based on de Jong and Rindi, Chapters 2 5 Frank de Jong Tilburg University 1 Determinants of the
More informationThe Dynamic Effects of Bundling as a Product Strategy
The Dynamic Effects of Bundling as a Product Strategy Timothy Derdenger & Vineet Kumar Abstract Several key questions in bundling have not been empirically examined: Is mixed bundling more effective than
More informationThe Use of Event Studies in Finance and Economics. Fall 2001. Gerald P. Dwyer, Jr.
The Use of Event Studies in Finance and Economics University of Rome at Tor Vergata Fall 2001 Gerald P. Dwyer, Jr. Any views are the author s and not necessarily those of the Federal Reserve Bank of Atlanta
More informationThe Elasticity of Taxable Income: A Non-Technical Summary
The Elasticity of Taxable Income: A Non-Technical Summary John Creedy The University of Melbourne Abstract This paper provides a non-technical summary of the concept of the elasticity of taxable income,
More informationFor more information on the Stata Journal, including information for authors, see the web page. http://www.stata-journal.com
The Stata Journal Editor H. Joseph Newton Department of Statistics Texas A & M University College Station, Texas 77843 979-845-3142; FAX 979-845-3144 jnewton@stata-journal.com Associate Editors Christopher
More informationMacroeconomic Effects of Financial Shocks Online Appendix
Macroeconomic Effects of Financial Shocks Online Appendix By Urban Jermann and Vincenzo Quadrini Data sources Financial data is from the Flow of Funds Accounts of the Federal Reserve Board. We report the
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationCOMMITTEE ON COMMODITY PROBLEMS
December 2011 CCP:TE 12/2 E COMMITTEE ON COMMODITY PROBLEMS INTERGOVERNMENTAL GROUP ON TEA TWENTIETH SESSION Colombo, Sri Lanka, 30 January - 1 February 2012 A DEMAND ANALYSIS FOR THE TEA MARKET I. INTRODUCTION
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David
More informationTwo Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering
Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014
More informationThe Cost of Financial Frictions for Life Insurers
The Cost of Financial Frictions for Life Insurers Ralph S. J. Koijen Motohiro Yogo University of Chicago and NBER Federal Reserve Bank of Minneapolis 1 1 The views expressed herein are not necessarily
More informationPortfolio Using Queuing Theory
Modeling the Number of Insured Households in an Insurance Portfolio Using Queuing Theory Jean-Philippe Boucher and Guillaume Couture-Piché December 8, 2015 Quantact / Département de mathématiques, UQAM.
More informationOn Marginal Effects in Semiparametric Censored Regression Models
On Marginal Effects in Semiparametric Censored Regression Models Bo E. Honoré September 3, 2008 Introduction It is often argued that estimation of semiparametric censored regression models such as the
More informationA Basic Introduction to Missing Data
John Fox Sociology 740 Winter 2014 Outline Why Missing Data Arise Why Missing Data Arise Global or unit non-response. In a survey, certain respondents may be unreachable or may refuse to participate. Item
More informationUnderstanding Retail Assortments In Competitive Markets 1
Understanding Retail Assortments In Competitive Markets 1 Kanishka Misra 2 Kellogg School of Management June 26, 2008 1 I wish to thank my advisors Eric Anderson, Karsten Hansen, Igal Handel Aviv Nevo,
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More informationA Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models
Article A Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models Richard A. Ashley 1, and Xiaojin Sun 2,, 1 Department of Economics, Virginia Tech, Blacksburg, VA 24060;
More informationWeather Insurance to Protect Specialty Crops against Costs of Irrigation in Drought Years
Weather Insurance to Protect Specialty Crops against Costs of Irrigation in Drought Years By Edouard K. Mafoua Post-Doctoral Associate Food Policy Institute Rutgers, The State University of New Jersey
More information3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a
More informationCorporate taxes and the location of intellectual property
Corporate taxes and the location of intellectual property Rachel Griffith a,b, Helen Miller a,* and Martin O Connell a,c Abstract April 2011 The income derived from intellectual property is highly mobile;
More information