9. Binary Dependent Variables

Size: px
Start display at page:

Download "9. Binary Dependent Variables"

Transcription

1 9. Bnar Dependent Varables 9. Homogeneous models Log, prob models Inference Tax preparers 9.2 Random effects models 9.3 Fxed effects models 9.4 Margnal models and GEE Appendx 9A - Lkelhood calculatons

2 9. Homogeneous models The response of nterest,, now ma be onl a 0 or a, a bnar dependent varable. Tpcall ndcates whether the h subject possesses an attrbute at tme t. Suppose that the probabl that the response equals s denoted b Prob( p. Then, we ma nterpret the mean response to be the probabl that the response equals, that s, E 0 Prob( 0 + Prob( p. Further, straghtforward calculatons show that the varance s related to the mean through the expresson Var p ( - p.

3 Inadequac of lnear models Homogeneous means that we wll not ncorporate subjectspecfc terms that account for heterogene. Lnear models of the form x β + ε are nadequate because: The expected response s a probabl and thus must var between 0 and although the lnear combnaton, x β, ma var between negatve and posve nfn. Lnear models assume homoscedastc (constant varance et the varance of the response depends on the mean whch vares over observatons. The response must be eher a 0 or although the dstrbuton of the error term s tpcall regarded as contnuous.

4 Usng nonlnear functons of explanator varables In leu of lnear, or addve, functons, we express the probabl of the response beng as a nonlnear functon of explanator varables p π (x β. Two specal cases are: z e π( z + e z e z + the log case π (z as a cumulatve standard normal dstrbuton functon, the prob case. These two functons are smlar. I focus on the log case because perms closed-form expressons unlke the cumulatve normal dstrbuton functon.

5 Threshold nterpretaton Suppose that there exsts an underlng lnear model, * x β + ε *. The response s nterpreted to be the propens to possess a characterstc. We do not observe the propens but we do observe when the propens crosses a threshold, sa 0. * 0 0 We observe * Usng the log dstrbuton functon, Prob (ε * a / ( + exp(-a Note that Prob(-ε * x β Prob(ε * x β. Thus, * * Prob( Prob( > 0 Prob( ε x β π ( x β + exp( x β > 0

6 Random utl nterpretaton In economcs applcatons, we thnk of an ndvdual choosng among c categores. Preferences among categores are ndexed b an unobserved utl functon. We model utl as a functon of an underlng value plus random nose, that s, U j u (V j + ε j, j 0,. If U > U 0, then denote ths choce as. Assumng that u s a strctl ncreasng functon, we have Prob( Prob( U 0 < U ( u ( V + ε < u ( V + ε ( ε < V V Prob ε 0 0 Prob Parameterze the problem b takng V 0 0 and V x β. We ma take the dfference n the errors, ε 0 - ε, to be normal or logstc, correspondng to the prob and log cases. 0 0

7 Logstc regresson Ths s another phrase used to descrbe the log case. Usng p π(z, the nverse of π can be calculated as z π - (p ln ( p/(-p. Defne log (p ln ( p/(-p to be the log functon. Here, p/(-p s known as the odds rato. It has a convenent economc nterpretaton n terms of far games. That s, suppose that p Then, the odds rato s The odds aganst wnnng are to, or to 3. If we bet $, then n a far game we should wn $3. The logstc regresson models the lnear combnaton of explanator varables as the logarhm of the odds rato, x β ln ( p /(-p.

8 Parameter nterpretaton To nterpret β ( β, β 2,, β K, we begn b assumng that jth explanator varable, x j, s eher 0 or. Then, wh the notaton, we ma nterpret β x L L x β x L 0 L x Thus, j ( ( β K K Prob( xj ln Prob( x β e j Prob( Prob( Prob( xj ln Prob( x j x x j j / 0 / 0 0 ( Prob( xj ( Prob( x 0 To llustrate, f β j 0.693, then exp(β j 2. The odds (for are twce as great for x j as for x j 0. j j

9 More parameter nterpretaton Smlarl, assumng that jth explanator varable s contnuous, we have β j d dx j x β d dx j Prob( xj ln Prob( x j d dx j ( Prob( xj /( Prob( xj x /( Prob( x Prob( j j Thus, we ma nterpret β j as the proportonal change n the odds rato, known as an elastc n economcs.

10 E Parameter estmaton The customar estmaton method s maxmum lkelhood. The log lkelhood of a sngle observaton s ln( π( x ln π( x β + ( ln( π( x β The log lkelhood of the data set s ln π( x β { ln π( x β + ( ln( π( x β } Takng partal dervatves wh respect to b elds the score equatons π ( x β x ( π( x β 0 π( x β π( x β ( The soluton of these equatons, sa b MLE, elds the maxmum lkelhood estmate. The score equatons can also be expressed as a generalzed estmatng equaton: ( E ( E ( Var 0 β where π( x β E x π ( x β β Var π( x β π( x β f f 0 ( β

11 For the log functon The normal equatons are: ( π ( x β 0 x The soluton depends on the responses onl through the vector of statstcs Σ x. The soluton of these equatons, sa b MLE, elds the maxmum lkelhood estmate b MLE. Ths method can be extended to provde standard errors for the estmates.

12 9.2 Random effects models We accommodate heterogene b ncorporatng subjectspecfc varables of the form: p π (α + x β. We assume that the ntercepts are realzatons of random varables from a common dstrbuton. We estmate the parameters of the {α } dstrbuton and the K slope parameters β. B usng the random effects specfcaton, we dramatcall reduced the number of parameters to be estmated compared to the Secton 9.3 fxed effects set-up. Ths s smlar to the lnear model case. Ths model s computatonall dffcult to evaluate.

13 Commonl used dstrbutons We assume that subject-specfc effects are ndependent and come from a common dstrbuton. It s customar to assume that the subject-specfc effects are normall dstrbuted. We assume, condonal on subject-specfc effects, that the responses are ndependent. Thus, there s no seral correlaton. There are two commonl used specfcatons of the condonal dstrbutons n the random effects panel data model.. A logstc model for the condonal dstrbuton of a response. That s, Prob( α π( α + x β + exp ( ( α + x β 2. A normal model for the condonal dstrbuton of a response. That s, Prob( α Φ ( α + where Φ s the standard normal dstrbuton functon. x β

14 Lkelhood Let Prob( α π(α + x β denote the condonal probabl for both the logstc and normal models. Condonal on α, the lkelhood for the th observaton s: ( π( α + xβ f 0 π( α + + xβ ( π( α xβ π( α + xβ f Condonal on α, the lkelhood for the h subject s: T t π ( α + x β ( π( α + x β Thus, the (uncondonal lkelhood for the h subject s: l T t π ( a + x β ( π( a + x β Here, φ s the standard normal dens functon. Hence, the total log-lkelhood s Σ ln l. Note: lots of evaluatons of a numercal ntegral. φ( a da

15 Comparng log to prob specfcaton There are no mportant advantages or dsadvantages when choosng the condonal probabl π to be: log functon (log model standard normal (prob model The lkelhood nvolves roughl the same amount of work to evaluate and maxmze, although the log functon s slghtl easer to evaluate than the standard normal dstrbuton functon. The prob model s slghtl easer to nterpret because uncondonal probables can be expressed n terms of the standard normal dstrbuton functon. That s, x β Prob( EΦ( α Φ + x β + 2 σ α

16 9.3 Fxed effects models As wh homogeneous models, we express the probabl of the response beng as a nonlnear functon of lnear combnatons of explanator varables. To accommodate heterogene, we ncorporate subjectspecfc varables of the form: p π (α + x β. Here, the subject-specfc effects account onl for the ntercepts and do not nclude other varables. We assume that {α } are fxed effects n ths secton. In ths chapter, we assume that responses are serall uncorrelated. Important pont: Panel data wh dumm varables provde nconsstent parameter estmates.

17 Maxmum lkelhood estmaton Unlke random effect models, maxmum lkelhood estmators are nconsstent n fxed effects models. The log lkelhood of the data set s ln π ( α + x β + ( ln( π ( α + x β { } Ths log lkelhood can stll be maxmzed to eld maxmum lkelhood estmators. However, as the subject sze n tends to nfn, the number of parameters also tends to nfn. Intuvel, our abl to estmate β s corrupted b our nabl to estmate consstentl the subject-specfc effects {α }. In the lnear case, we had that the maxmum lkelhood estmates are equvalent to the least squares estmates. The least squares estmates of β were consstent. The least squares procedure swept out ntercept estmators when producng estmates of β.

18 Maxmum lkelhood estmaton s nconsstent Example 9.2 (Chamberlan, 978, Hsao 986. Let T 2, K and x 0 and x 2. Take dervatves of the lkelhood functon to get the score functons these are n dspla (9.8. From (9.8, the score functons are and Appendx 9A. Maxmze ths to get b mle Show that the probabl lm of b mle s 2 β, and hence s an nconsstent estmator of β. 0 e e e e β α β α α α α L 0 e e β L β α β α

19 Condonal maxmum lkelhood estmaton Ths estmaton technque provdes consstent estmates of the beta coeffcents. It s due to Chamberlan (980 n the context of fxed effects panel data models. Let s consder the log specfcaton of π, so that p π( α + x β + exp ( ( α + x β Bg dea: Wh ths specfcaton, turns out that Σ t s a suffcent statstc for α. Thus, f we condon on Σ t, then the dstrbuton of the responses wll not depend on α.

20 Example of the suffcenc To llustrate how to separate the ntercept from the slope effects, consder the case T 2. Suppose that the sum, Σ t + 2, equals eher 0 or 2. If sum equals 0, then Prob ( 0, sum. If sum equals 2, then Prob (, sum. Both condonal probables do not depend on α. Both condonal events are certan and wll contrbute nothng to a condonal lkelhood. If sum equals, ( Prob( 0 Prob( + Prob( Prob( 0 Prob exp( α + x β + exp( α + x 2β ( + exp( α + x β ( + exp( α + x β 2

21 Example of the suffcenc Thus, Prob ( 0, Prob ( 0 Prob( 2 ( + Prob 2 exp exp( α + x 2β exp ( x 2β ( α + x β + exp( α + x 2β exp ( x β + exp ( x β Ths does not depend on α. Note that f an explanator varable x j s tme-constant (x j2 x j, then the correspondng parameter β j dsappears from the condonal lkelhood. 2

22 Condonal lkelhood estmaton Let S be the random varable representng Σ t and let sum be the realzaton of Σ t. The condonal lkelhood of the data set s n 2 T p p2 L pt Prob( S sum Note that the rato equals one when sum equal 0 or T. The dstrbuton of S s mess and s dffcult to compute for moderate sze data sets wh T more than 0. Ths provdes a fx for the problem of nfnel man nusance parameters. Computatonall dffcult, hard to extend to more complex models, hard to explan to consumers

23 9.4 Margnal models and GEE Margnal models, also know as populaton-averaged models, onl requre specfcaton of the frst two moments Means, varances and covarances Not a true probabl model Ideal for moment estmaton (GEE, GMM Begn n the context of the random effects bnar dependent varable model The mean s E µ µ ( β, τ π( a + x β d Fα ( a The varance s Var µ (- µ. The covarance s Cov ( r, s ( a + x rβ π( a + x sβ d Fα ( a µ r µ s π

24 GEE generalzed estmatng equatons Ths s a method of moments procedure Essentall the same as generalzed method of moments One matches theoretcal moments to sample moments, wh approprate weghtng. Idea fnd the values of the parameters that satsf 0 K n G ( V ( b, τ µ b EE, τ EE EE EE ( µ ( b EE (, τ We have alread specfed the varance matrx. We also use a K x T matrx of dervatves G µ ( β, τ µ ( β, τ β µ β For bnar varables, we have µ x π a + x β β β µ T L ( d F ( a α EE

25 Margnal Model ( β Choose the mean functon to be µ Φ x Motvated b prob specfcaton Prob( EΦ( + Φ α xβ For the varance functon, consder Var φµ (- µ. Let Corr( r, s denote the correlaton between r and s. Ths s known as a workng correlaton. Use the exchangeable correlaton structure specfed as Corr ( r, s ρ x β Here, the motvaton s that the latent varable α s common to all observatons whn a subject, thus nducng a common correlaton. The parameters τ (φ, ρ constute the varance components. for for r r s s + 2 σ α

26 Robust Standard Errors Model-based standard errors are taken from the square root of the dagonal elements of As an alternatve, robust or emprcal standards errors are from These are robust to msspecfed heterscedastc as well as tme seres correlaton. (, (, (, ( n EE EE EE EE EE EE τ τ τ µ µ b G b V b G ( ( n n n µ µ µ µ µ µ G V G G V µ µ V G G V G

10. Generalized linear models

10. Generalized linear models 10. Generalzed lnear models 10.1 Homogeneous models Exponental famles of dstrbutons, lnk functons, lkelhood estmaton 10.2 Example: Tort flngs 10.3 Margnal models and GEE 10.4 Random effects models 10.5

More information

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

Chapter 13: Multiple Regression

Chapter 13: Multiple Regression Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

Rockefeller College University at Albany

Rockefeller College University at Albany Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models

Maximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Mamum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models for

More information

x i1 =1 for all i (the constant ).

x i1 =1 for all i (the constant ). Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by

More information

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.

More information

e i is a random error

e i is a random error Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have

More information

Chapter 8 Indicator Variables

Chapter 8 Indicator Variables Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n

More information

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise. Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Limited Dependent Variables and Panel Data. Tibor Hanappi

Limited Dependent Variables and Panel Data. Tibor Hanappi Lmted Dependent Varables and Panel Data Tbor Hanapp 30.06.2010 Lmted Dependent Varables Dscrete: Varables that can take onl a countable number of values Censored/Truncated: Data ponts n some specfc range

More information

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2017 Instructor: Victor Aguirregabiria

ECONOMETRICS II (ECO 2401S) University of Toronto. Department of Economics. Winter 2017 Instructor: Victor Aguirregabiria ECOOMETRICS II ECO 40S Unversty of Toronto Department of Economcs Wnter 07 Instructor: Vctor Agurregabra SOLUTIO TO FIAL EXAM Tuesday, Aprl 8, 07 From :00pm-5:00pm 3 hours ISTRUCTIOS: - Ths s a closed-book

More information

2016 Wiley. Study Session 2: Ethical and Professional Standards Application

2016 Wiley. Study Session 2: Ethical and Professional Standards Application 6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Chapter 11: Simple Linear Regression and Correlation

Chapter 11: Simple Linear Regression and Correlation Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests

More information

Composite Hypotheses testing

Composite Hypotheses testing Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter

More information

Tests of Single Linear Coefficient Restrictions: t-tests and F-tests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions

Tests of Single Linear Coefficient Restrictions: t-tests and F-tests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions ECONOMICS 35* -- NOTE ECON 35* -- NOTE Tests of Sngle Lnear Coeffcent Restrctons: t-tests and -tests Basc Rules Tests of a sngle lnear coeffcent restrcton can be performed usng ether a two-taled t-test

More information

STAT 511 FINAL EXAM NAME Spring 2001

STAT 511 FINAL EXAM NAME Spring 2001 STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte

More information

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics

ECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott

More information

1 Binary Response Models

1 Binary Response Models Bnary and Ordered Multnomal Response Models Dscrete qualtatve response models deal wth dscrete dependent varables. bnary: yes/no, partcpaton/non-partcpaton lnear probablty model LPM, probt or logt models

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6 Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.

More information

β0 + β1xi. You are interested in estimating the unknown parameters β

β0 + β1xi. You are interested in estimating the unknown parameters β Revsed: v3 Ordnar Least Squares (OLS): Smple Lnear Regresson (SLR) Analtcs The SLR Setup Sample Statstcs Ordnar Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals)

More information

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson

More information

Basically, if you have a dummy dependent variable you will be estimating a probability.

Basically, if you have a dummy dependent variable you will be estimating a probability. ECON 497: Lecture Notes 13 Page 1 of 1 Metropoltan State Unversty ECON 497: Research and Forecastng Lecture Notes 13 Dummy Dependent Varable Technques Studenmund Chapter 13 Bascally, f you have a dummy

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

Andreas C. Drichoutis Agriculural University of Athens. Abstract

Andreas C. Drichoutis Agriculural University of Athens. Abstract Heteroskedastcty, the sngle crossng property and ordered response models Andreas C. Drchouts Agrculural Unversty of Athens Panagots Lazards Agrculural Unversty of Athens Rodolfo M. Nayga, Jr. Texas AMUnversty

More information

Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis

Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques

More information

Global Sensitivity. Tuesday 20 th February, 2018

Global Sensitivity. Tuesday 20 th February, 2018 Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values

More information

Chapter 9: Statistical Inference and the Relationship between Two Variables

Chapter 9: Statistical Inference and the Relationship between Two Variables Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,

More information

First Year Examination Department of Statistics, University of Florida

First Year Examination Department of Statistics, University of Florida Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve

More information

Basic R Programming: Exercises

Basic R Programming: Exercises Basc R Programmng: Exercses RProgrammng John Fox ICPSR, Summer 2009 1. Logstc Regresson: Iterated weghted least squares (IWLS) s a standard method of fttng generalzed lnear models to data. As descrbed

More information

Exam. Econometrics - Exam 1

Exam. Econometrics - Exam 1 Econometrcs - Exam 1 Exam Problem 1: (15 ponts) Suppose that the classcal regresson model apples but that the true value of the constant s zero. In order to answer the followng questons assume just one

More information

Testing for seasonal unit roots in heterogeneous panels

Testing for seasonal unit roots in heterogeneous panels Testng for seasonal unt roots n heterogeneous panels Jesus Otero * Facultad de Economía Unversdad del Rosaro, Colomba Jeremy Smth Department of Economcs Unversty of arwck Monca Gulett Aston Busness School

More information

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values

However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly

More information

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M

CIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of

Chapter 7 Generalized and Weighted Least Squares Estimation. In this method, the deviation between the observed and expected values of Chapter 7 Generalzed and Weghted Least Squares Estmaton The usual lnear regresson model assumes that all the random error components are dentcally and ndependently dstrbuted wth constant varance. When

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 130. Lecture 4 Simple Linear Regression Continued Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do

More information

Computation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models

Computation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models Computaton of Hgher Order Moments from Two Multnomal Overdsperson Lkelhood Models BY J. T. NEWCOMER, N. K. NEERCHAL Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore,

More information

LECTURE 9 CANONICAL CORRELATION ANALYSIS

LECTURE 9 CANONICAL CORRELATION ANALYSIS LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of

More information

Statistics for Business and Economics

Statistics for Business and Economics Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear

More information

BIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data

BIO Lab 2: TWO-LEVEL NORMAL MODELS with school children popularity data Lab : TWO-LEVEL NORMAL MODELS wth school chldren popularty data Purpose: Introduce basc two-level models for normally dstrbuted responses usng STATA. In partcular, we dscuss Random ntercept models wthout

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed

More information

Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients

Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It

More information

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)

Here is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y) Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,

More information

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston

More information

Primer on High-Order Moment Estimators

Primer on High-Order Moment Estimators Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc

More information

Empirical Methods for Corporate Finance. Identification

Empirical Methods for Corporate Finance. Identification mprcal Methods for Corporate Fnance Identfcaton Causalt Ultmate goal of emprcal research n fnance s to establsh a causal relatonshp between varables.g. What s the mpact of tangblt on leverage?.g. What

More information

Statistics for Economics & Business

Statistics for Economics & Business Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable

More information

Lab 4: Two-level Random Intercept Model

Lab 4: Two-level Random Intercept Model BIO 656 Lab4 009 Lab 4: Two-level Random Intercept Model Data: Peak expratory flow rate (pefr) measured twce, usng two dfferent nstruments, for 17 subjects. (from Chapter 1 of Multlevel and Longtudnal

More information

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA 4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected

More information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9 Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,

More information

Properties of Least Squares

Properties of Least Squares Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures

More information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng

More information

Systems of Equations (SUR, GMM, and 3SLS)

Systems of Equations (SUR, GMM, and 3SLS) Lecture otes on Advanced Econometrcs Takash Yamano Fall Semester 4 Lecture 4: Sstems of Equatons (SUR, MM, and 3SLS) Seemngl Unrelated Regresson (SUR) Model Consder a set of lnear equatons: $ + ɛ $ + ɛ

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrcs of Panel Data Jakub Mućk Meetng # 8 Jakub Mućk Econometrcs of Panel Data Meetng # 8 1 / 17 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random

More information

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010

Parametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010 Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton

More information

The EM Algorithm (Dempster, Laird, Rubin 1977) The missing data or incomplete data setting: ODL(φ;Y ) = [Y;φ] = [Y X,φ][X φ] = X

The EM Algorithm (Dempster, Laird, Rubin 1977) The missing data or incomplete data setting: ODL(φ;Y ) = [Y;φ] = [Y X,φ][X φ] = X The EM Algorthm (Dempster, Lard, Rubn 1977 The mssng data or ncomplete data settng: An Observed Data Lkelhood (ODL that s a mxture or ntegral of Complete Data Lkelhoods (CDL. (1a ODL(;Y = [Y;] = [Y,][

More information

Chapter 20 Duration Analysis

Chapter 20 Duration Analysis Chapter 20 Duraton Analyss Duraton: tme elapsed untl a certan event occurs (weeks unemployed, months spent on welfare). Survval analyss: duraton of nterest s survval tme of a subject, begn n an ntal state

More information

Lecture 10 Support Vector Machines II

Lecture 10 Support Vector Machines II Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

[ ] λ λ λ. Multicollinearity. multicollinearity Ragnar Frisch (1934) perfect exact. collinearity. multicollinearity. exact

[ ] λ λ λ. Multicollinearity. multicollinearity Ragnar Frisch (1934) perfect exact. collinearity. multicollinearity. exact Multcollnearty multcollnearty Ragnar Frsch (934 perfect exact collnearty multcollnearty K exact λ λ λ K K x+ x+ + x 0 0.. λ, λ, λk 0 0.. x perfect ntercorrelated λ λ λ x+ x+ + KxK + v 0 0.. v 3 y β + β

More information

Chapter 5 Multilevel Models

Chapter 5 Multilevel Models Chapter 5 Multlevel Models 5.1 Cross-sectonal multlevel models 5.1.1 Two-level models 5.1.2 Multple level models 5.1.3 Multple level modelng n other felds 5.2 Longtudnal multlevel models 5.2.1 Two-level

More information

Outline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3.

Outline. Zero Conditional mean. I. Motivation. 3. Multiple Regression Analysis: Estimation. Read Wooldridge (2013), Chapter 3. Outlne 3. Multple Regresson Analyss: Estmaton I. Motvaton II. Mechancs and Interpretaton of OLS Read Wooldrdge (013), Chapter 3. III. Expected Values of the OLS IV. Varances of the OLS V. The Gauss Markov

More information

Econ Statistical Properties of the OLS estimator. Sanjaya DeSilva

Econ Statistical Properties of the OLS estimator. Sanjaya DeSilva Econ 39 - Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate

More information

See Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)

See Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition) Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes

More information

Multinomial logit regression

Multinomial logit regression 07/0/6 Multnomal logt regresson Introducton We now turn our attenton to regresson models for the analyss of categorcal dependent varables wth more than two response categores: Y car owned (many possble

More information

January Examinations 2015

January Examinations 2015 24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)

More information

Introduction to Regression

Introduction to Regression Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes

More information

Firm Heterogeneity and its Implications for Efficiency Measurement. Antonio Álvarez University of Oviedo & European Centre for Soft Computing

Firm Heterogeneity and its Implications for Efficiency Measurement. Antonio Álvarez University of Oviedo & European Centre for Soft Computing Frm Heterogeney and s Implcatons for Effcency Measurement Antono Álvarez Unversy of Ovedo & European Centre for Soft Computng Frm heterogeney (I) Defnon Characterstcs of the ndvduals (frms, regons, persons,

More information

The Ordinary Least Squares (OLS) Estimator

The Ordinary Least Squares (OLS) Estimator The Ordnary Least Squares (OLS) Estmator 1 Regresson Analyss Regresson Analyss: a statstcal technque for nvestgatng and modelng the relatonshp between varables. Applcatons: Engneerng, the physcal and chemcal

More information

A Monte Carlo Study for Swamy s Estimate of Random Coefficient Panel Data Model

A Monte Carlo Study for Swamy s Estimate of Random Coefficient Panel Data Model A Monte Carlo Study for Swamy s Estmate of Random Coeffcent Panel Data Model Aman Mousa, Ahmed H. Youssef and Mohamed R. Abonazel Department of Appled Statstcs and Econometrcs, Instute of Statstcal Studes

More information

EEE 241: Linear Systems

EEE 241: Linear Systems EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they

More information

Exercise 1 The General Linear Model : Answers

Exercise 1 The General Linear Model : Answers Eercse The General Lnear Model Answers. Gven the followng nformaton on 67 pars of values on and -.6 - - - 9 a fnd the OLS coeffcent estmate from a regresson of on. Usng b 9 So. 9 b Suppose that now also

More information

Introduction to Generalized Linear Models

Introduction to Generalized Linear Models INTRODUCTION TO STATISTICAL MODELLING TRINITY 00 Introducton to Generalzed Lnear Models I. Motvaton In ths lecture we extend the deas of lnear regresson to the more general dea of a generalzed lnear model

More information

MLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012

MLE and Bayesian Estimation. Jie Tang Department of Computer Science & Technology Tsinghua University 2012 MLE and Bayesan Estmaton Je Tang Department of Computer Scence & Technology Tsnghua Unversty 01 1 Lnear Regresson? As the frst step, we need to decde how we re gong to represent the functon f. One example:

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

8 : Learning in Fully Observed Markov Networks. 1 Why We Need to Learn Undirected Graphical Models. 2 Structural Learning for Completely Observed MRF

8 : Learning in Fully Observed Markov Networks. 1 Why We Need to Learn Undirected Graphical Models. 2 Structural Learning for Completely Observed MRF 10-708: Probablstc Graphcal Models 10-708, Sprng 2014 8 : Learnng n Fully Observed Markov Networks Lecturer: Erc P. Xng Scrbes: Meng Song, L Zhou 1 Why We Need to Learn Undrected Graphcal Models In the

More information

Statistical analysis using matlab. HY 439 Presented by: George Fortetsanakis

Statistical analysis using matlab. HY 439 Presented by: George Fortetsanakis Statstcal analyss usng matlab HY 439 Presented by: George Fortetsanaks Roadmap Probablty dstrbutons Statstcal estmaton Fttng data to probablty dstrbutons Contnuous dstrbutons Contnuous random varable X

More information

Credit Card Pricing and Impact of Adverse Selection

Credit Card Pricing and Impact of Adverse Selection Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n

More information

Estimation: Part 2. Chapter GREG estimation

Estimation: Part 2. Chapter GREG estimation Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the

More information

LOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi

LOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent

More information

A Comparative Study for Estimation Parameters in Panel Data Model

A Comparative Study for Estimation Parameters in Panel Data Model A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and

More information

Chapter 14: Logit and Probit Models for Categorical Response Variables

Chapter 14: Logit and Probit Models for Categorical Response Variables Chapter 4: Logt and Probt Models for Categorcal Response Varables Sect 4. Models for Dchotomous Data We wll dscuss only ths secton of Chap 4, whch s manly about Logstc Regresson, a specal case of the famly

More information

Factor models with many assets: strong factors, weak factors, and the two-pass procedure

Factor models with many assets: strong factors, weak factors, and the two-pass procedure Factor models wth many assets: strong factors, weak factors, and the two-pass procedure Stanslav Anatolyev 1 Anna Mkusheva 2 1 CERGE-EI and NES 2 MIT December 2017 Stanslav Anatolyev and Anna Mkusheva

More information

Problem 3.1: Error autotocorrelation and heteroskedasticity Standard variance components model:

Problem 3.1: Error autotocorrelation and heteroskedasticity Standard variance components model: ECON 510: Panel data econometrcs Semnar 3: October., 007 Problem 3.1: Error autotocorrelaton and heteroskedastcy Standard varance components model: (0.1) y = k+ x β + + u, ε = + u, IID(0, ), u Rewrng the

More information

Durban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications

Durban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha ralyaf@ksu.edu.sa, maomar@ksu.edu.sa, aalshha@ksu.edu.sa Department

More information

x = , so that calculated

x = , so that calculated Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to

More information

0.1 The micro "wage process"

0.1 The micro wage process 0.1 The mcro "wage process" References For estmaton of mcro wage models: John Abowd and Davd Card (1989). "On the Covarance Structure of Earnngs and Hours Changes". Econometrca 57 (2): 411-445. Altonj,

More information

β0 + β1xi. You are interested in estimating the unknown parameters β

β0 + β1xi. You are interested in estimating the unknown parameters β Ordnary Least Squares (OLS): Smple Lnear Regresson (SLR) Analytcs The SLR Setup Sample Statstcs Ordnary Least Squares (OLS): FOCs and SOCs Back to OLS and Sample Statstcs Predctons (and Resduals) wth OLS

More information

β0 + β1xi and want to estimate the unknown

β0 + β1xi and want to estimate the unknown SLR Models Estmaton Those OLS Estmates Estmators (e ante) v. estmates (e post) The Smple Lnear Regresson (SLR) Condtons -4 An Asde: The Populaton Regresson Functon B and B are Lnear Estmators (condtonal

More information

Classification as a Regression Problem

Classification as a Regression Problem Target varable y C C, C,, ; Classfcaton as a Regresson Problem { }, 3 L C K To treat classfcaton as a regresson problem we should transform the target y nto numercal values; The choce of numercal class

More information