Problem of Estimation. Ordinary Least Squares (OLS) Ordinary Least Squares Method. Basic Econometrics in Transportation. Bivariate Regression Analysis
|
|
- Heather Richards
- 5 years ago
- Views:
Transcription
1 1/60 Problem of Estmaton Basc Econometrcs n Transportaton Bvarate Regresson Analyss Amr Samm Cvl Engneerng Department Sharf Unversty of Technology Ordnary Least Squares (OLS) Maxmum Lkelhood (ML) Generally, OLS s used extensvely n regresson analyss: It s ntutvely appealng Mathematcally much smpler than the MLE. Both methods generally gve smlar results n the lnear regresson context. Prmary Source: Basc Econometrcs (Gujarat) 2/60 3/60 Ordnary Least Squares Method To odetermne e PRF: Y 1 2 Populaton Regresson Functon We estmate t from the SRF: Sample Regresson Functon X u We would lke to determne the SRF n a way that t s as close as possble to the actual Y. Y ˆ ˆ 1 2 X.e. sum of the resduals û s as small as possble. uˆ Ordnary Least Squares Method 2 uˆ f ( ˆ, ˆ 1 2) OLS fnds unque estmates of β 1 and β 2 that gve the smallest possble value of the functon above. ˆ 2 ˆ Y ˆ 2 X 1 ( X X )( Y Y ) 2 ( X X ) x y 2 x Why square? More weght to large resduals. Sgn of the resduals. Devaton form: y Y Y x X X
2 4/60 5/60 OLS Propertes: Sample Mean OLS Propertes: Lnearty Regresson lne passes through the sample mean Lnearty It s a lnear functon (a weghted average) of Y. X and thus k are nonstochastc and: Note: 6/60 7/60 OLS Propertes: Unbasedness OLS Propertes: Mean of Estmated Y Unbasedness ess Mean value of the estmated Y s equal to the mean value of the actual Y: Sum both sdes over the sample values and dvde by sample sze
3 8/60 9/60 OLS Propertes: Mean of Resduals OLS Propertes: Uncorrelated Resduals Y The mean value of the resduals s zero: The resduals are uncorrelated wth the predcted Y : 10/60 11/60 OLS Propertes: Uncorrelated Resduals X OLS Assumptons The resduals s are uncorrelated wth X : Our objectve s not only to estmate some coeffcents c e but also to draw nferences about the true coeffcents. Thus certan assumptons are made about the manner n whch Y are generated. Y = β 1 + β 2 X + u Unless we are specfc about how X and u are created, there s no way we can make any statstcal nference about the Y, β 1 and β 2. The Gaussan standard, or classcal lnear regresson model (CLRM), makes 10 assumptons that are extremely crtcal to the vald nterpretaton of the regresson estmates.
4 12/60 13/60 Assumpton 1 Assumpton 2 The regresson model s lnear n the parameters. E(Y X ) = β 1 + β 2 X 2 s a lnear model. E(Y X ) = β 1 + β 22 X s not a lnear model. X values are fxed n repeated samplng. X s assumed to be non-stochastc. Our regresson analyss s condtonal regresson analyss, that s, condtonal on the gven values of the regressor(s) X. 14/60 15/60 Assumpton 3 Assumpton 4 Zero mean value of dsturbance u E(u X ) = 0 Homoscedastcty oscedast c ty or equal varance a of u Factors not explctly ncluded n the model, and therefore subsumed n u, do not systematcally affect the mean value of Y.
5 16/60 17/60 Assumpton 4 Assumpton 5 Heteroscedastcty No autocorrelaton (seral correlaton) between the dsturbances. 18/60 19/60 Assumpton 6 Assumpton 7 Dsturbance u and dexplanatory ato varable ab X are uncorrelated. n must be greater than the number of explanatory ato varables. ab Obvously, we need at least two pars of observatons to estmate the two unknowns! If X and u are correlated, ther ndvdual effects on Y may not be assessed.
6 20/60 21/60 Assumpton 8 Assumpton 9 Varablty n X values. Mathematcally, f all the X values are dentcal, t mpossble to estmate β 2 (the denomnator wll be zero) and therefore β 1. Intutvely, t s obvous as well. The regresson model s correctly specfed. Important questons that arse n the specfcaton of a model: What varables should be ncluded n the model? What s the functonal form of the model? Is t lnear n the parameters, the varables, or both? What are the probablstc assumptons made about the Y, the X, and the u enterng the model? 22/60 23/60 Assumpton 10 Assumptons There e s no perfect ect multcollnearty. ty. No perfect lnear relatonshps among the explanatory varables. Wll be further dscussed n multple regresson models. How realstc are all these assumptons? s? We make certan assumptons because they facltate the study, not because they are realstc. Consequences of volaton of CLRM assumptons wll be examned later. We wll look nto: Precson of OLS estmates, and Statstcal propertes of OLS.
7 24/60 25/60 Precson of OLS Estmates Standard Error of Precson of an estmate s measured by ts standard error. Precson of an estmate s measured by ts standard error. 26/60 27/60 Homoscedastc Varance of u Features of the Varances How to estmate 2 The varance a of s drectly proportonal o to but nversely proportonal to sum of. As n ncreases, the precson wth whch β 2 can be estmated also ncreases. If there s substantal varaton n X, β 2 can be measured more accurately. The varance of s drectly proportonal to σ 2 and X 2, but nversely proportonal to x 2 2 and the sample sze n. Note:. If the slope coeffcent s overestmated, the ntercept wll be underestmated.
8 28/60 29/60 Gauss Markov Theorem Goodness of Ft Gven the assumptons of the classcal lnear regresson model, the OLS estmators, n the class of unbased lnear estmators, have mnmum varance, that s, they are BLUE. The coeffcent of determnaton R 2 s a summary measure that tells how well the sample regresson lne fts the data. The theorem makes no assumptons about the probablty dstrbuton of u, and therefore of Y. Total Sum of Squares (TSS) = Explaned Sum of Squares (ESS) + Resdual Sum of Squares (RSS) 30/60 31/60 Consstency Before Hypothess Testng An asymptotc property. An estmator s consstent f t s unbased and ts varance tends to zero as the sample sze n tends to nfnty. Unbasedness s already proved. Usng the method of OLS we can estmate β 2 1, β 2, and σ. Estmators ( ) are random varables. To draw nferences about PRF, we must fnd out how close s to the true. We need to fnd out PDF of the estmators.
9 32/60 33/60 Probablty Dstrbuton of Dsturbances Why the Normalty Assumpton? s ultmately a lnear functon of the random varable u, whch s random by assumpton. The nature of the probablty dstrbuton of u plays an extremely mportant role n hypothess testng. It s usually assumed that u NID(0, σ 2 ) NID: Normally and Independently Dstrbuted. We hope that the nfluence of these omtted or neglected varables s small and at best random. We show by the CLT, that f there are a large number of IID random varables, the dstrbuton of ther sum tends to a normal dstrbuton as the number of such varables ncrease ndefntely. Central lmt theorem (CLT): Let X 1, X 2,..., X n denote n ndependent random varables, all of whch have the same PDF wth mean = μ and varance = σ 2. Let : 34/60 35/60 Why the Normalty Assumpton? Normalty Test A varant a of the CLT states that, even f the number of varables ab s not very large or f these varables are not strctly ndependent, ther sum may stll be normally dstrbuted. Wth ths assumpton, PDF of OLS estmators can be easly derved, as any lnear functon of normally dstrbuted varables s tself normally dstrbuted. The normal dstrbuton s a comparatvely smple dstrbuton nvolvng only two parameters. It enables us to use the t, F, and χ 2 tests for regresson models. The normalty assumpton plays a crtcal role for small sample sze data. In reasonably large sample sze, we may relax the normalty assumpton. Snce we are mposng the normalty assumpton, t behooves us to fnd out n practcal applcatons nvolvng small sample sze data whether the normalty assumpton s approprate. Later, we wll ntroduces some tests to do just that. We wll come across stuatons where the normalty assumpton may be napproprate. Untl then we wll contnue wth the normalty assumpton.
10 36/60 37/60 Estmators Propertes wth Normalty Assumpton Method of Maxmum Lkelhood They are unbased, wth mnmum varance (effcent), and consstent.,, and s dstrbuted as the χ 2 wth (n 2) df. Wll help us to draw nferences about the true σ 2 from the estmated σ 2.. The mportance of ths wll be explaned later. Estmators have mnmum varance n the entre class of unbased estmators, whether lnear or not. Best Unbased Estmators (BUE) If u are assumed to be normally dstrbuted, the ML and OLS estmators of the regresson coeffcents, are dentcal. The ML estmator of σ 2 s based for small sample sze. Asymptotcally, the ML estmator of σ 2 s unbased. ML method can be appled to regresson models that are nonlnear n the parameters, n whch OLS s generally not used. 38/60 39/60 ML Estmaton ML Estmaton Assume ethe etwova two-varable ab model Y = β 1 + β 2 X + u n whch Havng ndependent Y s, jont PDF of Y 1,...,Y n, can be wrtten as: The method of maxmum lkelhood conssts sts n estmatng the unknown parameters n such a manner that the probablty of observng the gven Y s s as hgh as possble. Where: β 1, β 2, and σ 2 are unknowns n lkelhood functon:
11 40/60 41/60 ML Estmaton Interval Estmaton From the frst-order condton for optmzaton: How relable are the pont estmates? We try to fnd out two postve numbers δ and α, such that: Note how ML underestmates the true σ 2 n small samples. Probablty of constructng an nterval that contans β 2 s 1 α. Such an nterval s known as a confdence nterval. α (0 < α < 1) s known as the level of sgnfcance. How are the confdence ntervals constructed? If the probablty dstrbutons of the estmators are known, the task of constructng confdence ntervals s a smple one. 42/60 43/60 Confdence Intervals for Confdence Intervals for It tcan beshown that atthe ett varable ab follows ows the t dstrbuton wth n 2 df. It can be shown that under the normalty assumpton, followng ow varable follows χ 2 dstrbuton wth n 2 df. Wdth of the confdence nterval s proportonal to the standard error of the estmator. Same for 1 Interpretaton of ths nterval: If we establsh 95% confdence lmts on σ 2 and f we mantan a pror that these lmts wll nclude true σ 2,we shall be rght n the long run 95 percent of the tme.
12 44/60 45/60 Hypothess Testng Confdence Interval Approach Is a gven observaton compatble wth some stated hypothess? In statstcs, the stated hypothess s known as the null hypothess or H 0 (versus an alternatve hypothess or H 1 ). Hypothess testng s developng rules for rejectng or acceptng the null hypothess. Confdence nterval approach, Test of sgnfcance approach. Most of the statstcal hypotheses of our nterest make statements about one or more values of the parameters of some assumed probablty dstrbuton such as the normal, F, t, or χ 2. Decson Rule: Construct a 100(1 α)% confdence nterval for β 2. If the β 2 under H 0 falls wthn ths nterval, do not reject H 0, but f t falls outsde ths nterval, reject H 0. Note: There s a 100α percent chance of commttng a Type I error. If α = 0.05, there s a 5 percent chance that we could reject the null hypothess even though t s true. When we reject the null hypothess, we say that our fndng s statstcally sgnfcant. One-tal or two-tal test: Sometmes we have a strong expectaton that the alternatve hypothess s onesded rather than two-sded. 46/60 47/60 Test of Sgnfcance Approach Practcal Aspects In the econfdence confdence-nterval nterval procedure e we try to establsh s a range that has a probablty of ncludng the true but unknown β 2. In the test-of-sgnfcance approach we hypothesze some value for β 2 and try to see whether the estmated β 2 les wthn confdence lmts around the hypotheszed value. A large t value wll be evdence aganst the null hypothess. Acceptng null hypothess: s: All we can say: based on the sample evdence we have no reason to reject t. Another null hypothess may be equally compatble wth the data. 2-t rule of thumb If df >20 and α = 0.05, then the null hypothess β 2 = 0 can be rejected f t > 2. In these cases we do not even have to refer to the t table to assess the sgnfcance of the estmated slope coeffcent. Formng the null hypotheses Theoretcal expectatons or pror emprcal work can be reled upon to formulate hypotheses. p value The lowest sgnfcance level at whch a null hypothess can be rejected.
13 48/60 49/60 Analyss of Varance Analyss of Varance A study of two components of TSS (= ESS + RSS) s known as ANOVA from the regresson vewpont. If we assume that the dsturbances u are normally dstrbuted, whch we do under the CNLRM, and f the null hypothess (H 0 ) s that β 2 = 0, then t can be shown that the F follows the F dstrbuton wth 1 df n the numerator and (n 2) df n the denomnator. What use can be made of the precedng F rato? 50/60 51/60 F-rato Example It tcan beshown that Compute F rato and obtan p value of the computed F statstc. Note that β 2 2 and σ are the true parameters. If β 2 s zero, both equatons provde us wth dentcal estmates of true σ 2. Thus, X has no lnear nfluence on Y. F rato provdes a test of the null hypothess H 0 : β 2 = 0. p value of ths F statstc wth 1 and 8 df s Therefore, f we reject the null hypothess, the probablty of commttng a Type I error s very small.
14 52/60 53/60 Applcaton Of Regresson Analyss Mean Predcton One use s to predct or forecast the future consumpton expendture Y correspondng to some gven level of ncome X. Now there are two knds of predctons: Predcton of the condtonal mean value of Y correspondng to a chosen X. Predcton of an ndvdual Y value correspondng to a chosen X. Estmator of E(Y X 0 ): It can be shown that: Ths statstc follows the t dstrbuton wth n 2 df and may be used to derve confdence ntervals 54/60 55/60 Indvdual Predcton Confdence Bands Estmator of E(Y ( X 0 0): It can be shown that: Ths statstc follows the t dstrbuton wth n 2 df and may be used to derve confdence ntervals
15 56/60 57/60 Indvdual Versus Mean Predcton Reportng the Results Confdence nterval for ndvdual Y 0 s wder than that for the mean value of Y 0. The wdth of confdence bands s smallest when X 0 = X. 58/60 59/60 Evaluatng the Results Normalty Tests How good s the ftted model? Any standard? d? Are the sgns of estmated coeffcents n accordance wth theoretcal or pror expectatons? How well does the model explan varaton n Y? One can use r 2 Does the model satsfes the assumptons of CNLRM? For now, we would lke to check the normalty of the dsturbance term. Recall that the t and F tests requre that the error term follow the normal dstrbuton. Several tests n the lterature. We look at: Hstogram of resduals: A smple graphc devce to learn about the shape of the PDF. Horzontal axs: the values of OLS resduals are dvded nto sutable ntervals. Vertcal axs: erect rectangles equal n heght to the frequency n that nterval. From a normal populaton we wll get a bell-shape PDF. Normal probablty yp plot (NPP): A smple graphc devce. Horzontal axs: plot values of OLS resduals, Vertcal axs: show expected value of varable f t were normally dstrbuted. From a normal populaton we wll get a straght lne. The Jarque Bera test. An asymptotc test, wth ch-squared dstrbuton and 2 df:
16 60/60 Homework 2 Basc Econometrcs (Gujarat, 2003) 1. Chapter 3, Problem 21 [10 ponts] 2. Chapter 3, Problem 23 [30 ponts] 3. Chapter 5, Problem 9 [30 ponts] 4. Chapter 5, Problem 19 [30 ponts] Assgnment weght factor = 1
Chapter 3. Two-Variable Regression Model: The Problem of Estimation
Chapter 3. Two-Varable Regresson Model: The Problem of Estmaton Ordnary Least Squares Method (OLS) Recall that, PRF: Y = β 1 + β X + u Thus, snce PRF s not drectly observable, t s estmated by SRF; that
More informationStatistics 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 informationECONOMICS 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 informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationEcon107 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 information1. 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 informationEcon 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 informationChapter 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 informationProperties 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 informationEconomics 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 informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More informationDepartment 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 informationx 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 informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationChapter 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 informationChapter 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 informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationThe 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 informationThe 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 informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationChapter 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 informationStatistics 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 informationCorrelation 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 informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationHere 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 informatione 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 information18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
More information28. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted
More informationa. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationEcon107 Applied Econometrics Topic 9: Heteroskedasticity (Studenmund, Chapter 10)
I. Defnton and Problems Econ7 Appled Econometrcs Topc 9: Heteroskedastcty (Studenmund, Chapter ) We now relax another classcal assumpton. Ths s a problem that arses often wth cross sectons of ndvduals,
More information4 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 informationIntroduction 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 informationIII. Econometric Methodology Regression Analysis
Page Econ07 Appled Econometrcs Topc : An Overvew of Regresson Analyss (Studenmund, Chapter ) I. The Nature and Scope of Econometrcs. Lot s of defntons of econometrcs. Nobel Prze Commttee Paul Samuelson,
More informationFirst 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[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
More informationOutline. 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 informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationx = , 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 informationCorrelation and Regression
Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and
More informationChapter 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 informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationLinear regression. Regression Models. Chapter 11 Student Lecture Notes Regression Analysis is the
Chapter 11 Student Lecture Notes 11-1 Lnear regresson Wenl lu Dept. Health statstcs School of publc health Tanjn medcal unversty 1 Regresson Models 1. Answer What Is the Relatonshp Between the Varables?.
More informationProfessor Chris Murray. Midterm Exam
Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam.
More informationY = β 0 + β 1 X 1 + β 2 X β k X k + ε
Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty
More informationAnswers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
More informationJanuary 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 informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationLINEAR 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 informationDiagnostics in Poisson Regression. Models - Residual Analysis
Dagnostcs n Posson Regresson Models - Resdual Analyss 1 Outlne Dagnostcs n Posson Regresson Models - Resdual Analyss Example 3: Recall of Stressful Events contnued 2 Resdual Analyss Resduals represent
More informationCHAPER 11: HETEROSCEDASTICITY: WHAT HAPPENS WHEN ERROR VARIANCE IS NONCONSTANT?
Basc Econometrcs, Gujarat and Porter CHAPER 11: HETEROSCEDASTICITY: WHAT HAPPENS WHEN ERROR VARIANCE IS NONCONSTANT? 11.1 (a) False. The estmators are unbased but are neffcent. (b) True. See Sec. 11.4
More informationChapter 5: Hypothesis Tests, Confidence Intervals & Gauss-Markov Result
Chapter 5: Hypothess Tests, Confdence Intervals & Gauss-Markov Result 1-1 Outlne 1. The standard error of 2. Hypothess tests concernng β 1 3. Confdence ntervals for β 1 4. Regresson when X s bnary 5. Heteroskedastcty
More information2016 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 informationStatistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationSee 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 informationExam. 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 informationContinuous vs. Discrete Goods
CE 651 Transportaton Economcs Charsma Choudhury Lecture 3-4 Analyss of Demand Contnuous vs. Dscrete Goods Contnuous Goods Dscrete Goods x auto 1 Indfference u curves 3 u u 1 x 1 0 1 bus Outlne Data Modelng
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationBiostatistics 360 F&t Tests and Intervals in Regression 1
Bostatstcs 360 F&t Tests and Intervals n Regresson ORIGIN Model: Y = X + Corrected Sums of Squares: X X bar where: s the y ntercept of the regresson lne (translaton) s the slope of the regresson lne (scalng
More informationLOGIT 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 informationChapter 15 Student Lecture Notes 15-1
Chapter 15 Student Lecture Notes 15-1 Basc Busness Statstcs (9 th Edton) Chapter 15 Multple Regresson Model Buldng 004 Prentce-Hall, Inc. Chap 15-1 Chapter Topcs The Quadratc Regresson Model Usng Transformatons
More informationLecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management
Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =
More informationANSWERS. 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 informationPredictive 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 informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationResource 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 informationSTAT 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 informationF8: Heteroscedasticity
F8: Heteroscedastcty Feng L Department of Statstcs, Stockholm Unversty What s so-called heteroscedastcty In a lnear regresson model, we assume the error term has a normal dstrbuton wth mean zero and varance
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More information17 - LINEAR REGRESSION II
Topc 7 Lnear Regresson II 7- Topc 7 - LINEAR REGRESSION II Testng and Estmaton Inferences about β Recall that we estmate Yˆ ˆ β + ˆ βx. 0 μ Y X x β0 + βx usng To estmate σ σ squared error Y X x ε s ε we
More informationDr. 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 information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationANSWERS CHAPTER 9. TIO 9.2: If the values are the same, the difference is 0, therefore the null hypothesis cannot be rejected.
ANSWERS CHAPTER 9 THINK IT OVER thnk t over TIO 9.: χ 2 k = ( f e ) = 0 e Breakng the equaton down: the test statstc for the ch-squared dstrbuton s equal to the sum over all categores of the expected frequency
More informationNow we relax this assumption and allow that the error variance depends on the independent variables, i.e., heteroskedasticity
ECON 48 / WH Hong Heteroskedastcty. Consequences of Heteroskedastcty for OLS Assumpton MLR. 5: Homoskedastcty var ( u x ) = σ Now we relax ths assumpton and allow that the error varance depends on the
More informationChapter 4: Regression With One Regressor
Chapter 4: Regresson Wth One Regressor Copyrght 2011 Pearson Addson-Wesley. All rghts reserved. 1-1 Outlne 1. Fttng a lne to data 2. The ordnary least squares (OLS) lne/regresson 3. Measures of ft 4. Populaton
More informationBasically, 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 informationβ0 + 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 informationUnit 10: Simple Linear Regression and Correlation
Unt 10: Smple Lnear Regresson and Correlaton Statstcs 571: Statstcal Methods Ramón V. León 6/28/2004 Unt 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regresson analyss s a method for studyng the
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan
More informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationSTAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression
STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationCHAPTER 8. Exercise Solutions
CHAPTER 8 Exercse Solutons 77 Chapter 8, Exercse Solutons, Prncples of Econometrcs, 3e 78 EXERCISE 8. When = N N N ( x x) ( x x) ( x x) = = = N = = = N N N ( x ) ( ) ( ) ( x x ) x x x x x = = = = Chapter
More informationMidterm Examination. Regression and Forecasting Models
IOMS Department Regresson and Forecastng Models Professor Wllam Greene Phone: 22.998.0876 Offce: KMC 7-90 Home page: people.stern.nyu.edu/wgreene Emal: wgreene@stern.nyu.edu Course web page: people.stern.nyu.edu/wgreene/regresson/outlne.htm
More informationA 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 informationThe SAS program I used to obtain the analyses for my answers is given below.
Homework 1 Answer sheet Page 1 The SAS program I used to obtan the analyses for my answers s gven below. dm'log;clear;output;clear'; *************************************************************; *** EXST7034
More informationChapter 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 informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationECONOMETRICS - 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 informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationModule Contact: Dr Susan Long, ECO Copyright of the University of East Anglia Version 1
UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 016-17 ECONOMETRIC METHODS ECO-7000A Tme allowed: hours Answer ALL FOUR Questons. Queston 1 carres a weght of 5%; Queston carres 0%;
More information