First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,

Size: px
Start display at page:

Download "First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,"

Transcription

1 0 2. OLS Part II The OLS residuals are orthogoal to the regressors. If the model icludes a itercept, the orthogoality of the residuals ad regressors gives rise to three results, which have limited practical usefuless but are good exercises for uderstadig the algebra of least squares. Partitioed ad partial regressio is ot treated as seriously as it might be i a graduate course, ad is oly give a cursory presetatio. However, the residual maker matrix Mi is preseted, ad is used i to defie R 2, ad i several other parts of the course. 2. Some basic properties of OLS First, ote that the LS residuals are orthogoal to the regressors X Xb X y = 0 ( ormal equatios ; (k ) ) So, X (y Xb) = X e = 0 ; or, X e = 0 Orthogoality implies liear idepedece (but ot vice versa). See Liearly Idepedet, Orthogoal, ad Ucorrelated Variables (Rodgers et al., 984), for the defiitio ad relatioship betwee the three terms. If the model icludes a itercept term, the oe regressor (say, the first colum of X) is a uit vector. I this case we get some further results:. The LS residuals sum to zero From the first elemet: x k e X e = ( ) (e ) = ( ) ( ) x k e x k x k e i e i 0 = (? ) = ( )? 0 i= e i = 0

2 2. Fitted regressio passes through sample mea X y = X Xb, y x k b or, ( ) ( ) = ( ) ( ) ( ). x k x k y x k x k x k b k i? y i i x i2 So, ( ) = (?? ) ( ).??? b k b From the first row of this vector equatio or, i y i = b + b 2 x i2 + + b k x ik i i y = b + b 2 x b k x k 3. Sample mea of the fitted y-values equals sample mea of actual y-values So, y i = x i β + ε i = x i b + e i = y i + e i. or, i=, y i = y i= i + e i= i y = y + 0 = y Note: These last 3 results use the fact that the model icludes a itercept. 2.2 Partitioed ad Partial Regressio We ca solve for blocks of b. Suppose we partitio our model ito 2 blocks: y = X β + X 2 β 2 + ε. ( ) ( k)(k ) ( k2)(k2 ) ( ) We could solve for the OLS estimator of β oly: Whe does b = (X X ) X y? b = (X X ) X y (X X ) X X 2 b 2

3 2 Above we have a solutio for b i terms of b 2. We ca substitute i a similar solutio for b 2 to obtai a solutio for b that is a fuctio of oly the X ad y data. Defie: M 2 = (I X 2 (X 2 X 2 ) X 2 ) M2 is a idempotet matrix M 2 M 2 = M 2 M 2 = M 2 = M 2 M 2. The, we ca write: b = (X M 2 X ) X M 2 y By the symmetry of the problem, we ca iterchage the ad 2 subscripts, ad get: b 2 = (X 2 M X 2 ) X 2 M y So, fially, we ca write: b = (X X ) X y b 2 = (X 2 X 2 ) X 2 y 2 where: X = M 2 X ; X 2 = M X 2 ; y = M 2 y ; y 2 = M y These results are importat for several reasos: Historically, for computatioal reasos. See the Frisch-Waugh-Lovell Theorem. I certai situatios b ad b 2 may have differet properties. This is difficult to show without have separate formulas. Havig established the Mi residual maker matrix, it is used frequetly to derive other results. Why is Mi called a residual maker matrix?

4 3 2.3 Goodess-of-Fit Oe way of measurig the quality of fitted regressio model is by the extet to which the model explais the sample variatio for y. Sample variace of y is ( ) (y i y) 2 i=. Or, we could just use i= (y i y) 2 to measure variability. Our fitted regressio model, usig LS, gives us y = Xb + e = y + e where y = Xb = X(X X) X y Recall that if the model icludes a itercept, the the residuals sum to zero, ad y = y. To simplify thigs, itroduce the followig matrix: M 0 = [I ii ] where: i = ( ) ; ( ) Note that: M 0 is a idempotet matrix. M 0 i = 0. M 0 trasforms elemets of a vector ito deviatios from sample mea. y M 0 y = y M 0 M 0 y = i= (y i y) 2. Let s check the third of these results: 0 / / M 0 y = {[ ] [[ ]]} ( ) 0 / / y y

5 4 y y y 2 y y y = [ ] = ( ). y y y 2 y y y Returig to our fitted model: y = Xb + e = y + e So, we have: M 0 y = M 0 y + M 0 e = M 0 y + e. [M 0 e = e ; because the residuals sum to zero.] The y M 0 y = y M 0 M 0 y = (M 0 y + e) (M 0 y + e) = y M 0 y + e e + 2e M 0 y However, e M 0 y = e M 0 y = (M 0 e) y = e y = e X(X X) X y = 0. So, we have Recall: y = y. y M 0 y = y M 0 y + e e i= (y i y) 2 = i= (y i y) e i i= SST = SSR + SSE This lets us defie the Coefficiet of Determiatio R 2 = ( SSR SST ) = (SSE SST )

6 5 Note: The secod equality i defiitio of R 2 holds oly if model icludes a itercept. R 2 = ( SSR SST ) 0 R 2 = ( SSE SST ) So, 0 R 2 Iterpretatio of 0 ad? R 2 is uitless. What happes if we add ay regressor(s) to the model? y = X β + ε ; [] The: y = X β + X 2 β 2 + u ; [2] (A) Applyig LS to [2]: mi. (u u ) ; u = y X b X 2 b 2 (B) Applyig LS to []: mi. (e e) ; e = y X β Problem (B) is just Problem (A), subject to restrictio: β 2 = 0. Miimized value i (A) must be miimized value i (B). So, u u e e. What does this imply? Addig ay regressor(s) to the model caot icrease (ad typically will decrease) the sum of squared residuals. So, addig ay regressor(s) to the model caot decrease (ad typically will icrease) the value of R 2. Meas that R 2 is ot really a very iterestig measure of the quality of the regressio model, i terms of explaiig sample variability of the depedet variable. For these reasos, we usually use the adjusted Coefficiet of Determiatio. We modify R 2 = [ e e y M 0 y ] to become:

7 6 R 2 = [ e e/( k) y M 0 y/( ) ]. What are we doig here? We re adjustig for degrees of freedom i umerator ad deomiator. Degrees of freedom = umber of idepedet pieces of iformatio. e = y Xb. We estimate k parameters from the data-poits. We have ( k) degrees of freedom associated with the fitted model. I deomiator have costructed y from sample. Lost oe degree of freedom. Possible for R 2 < 0 (eve with itercept i the model). R 2 ca icrease or decrease whe we add regressors. Whe will it icrease (decrease)? I multiple regressio, R 2 will icrease (decrease) if a variable is deleted, if ad oly if the associated t-statistic has absolute value less tha (greater tha) uity. If model does t iclude a itercept, the SST SSR + SSE, ad i this case o loger ay guaratee that 0 R 2. Must be careful comparig R 2 ad R 2 values across models. Example () C i = Y i ; R 2 = 0.90 (2) log (C i ) = Y i ; R 2 = 0.80 Sample variatio is i differet uits.

Algebra of Least Squares

Algebra of Least Squares October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal

More information

Geometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT

Geometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT OCTOBER 7, 2016 LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT Geometry of LS We ca thik of y ad the colums of X as members of the -dimesioal Euclidea space R Oe ca

More information

CLRM estimation Pietro Coretto Econometrics

CLRM estimation Pietro Coretto Econometrics Slide Set 4 CLRM estimatio Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Thursday 24 th Jauary, 2019 (h08:41) P. Coretto

More information

Simple Linear Regression

Simple Linear Regression Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i

More information

(all terms are scalars).the minimization is clearer in sum notation:

(all terms are scalars).the minimization is clearer in sum notation: 7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1

More information

ECON 3150/4150, Spring term Lecture 3

ECON 3150/4150, Spring term Lecture 3 Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

Statistical Properties of OLS estimators

Statistical Properties of OLS estimators 1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of

More information

Regression, Inference, and Model Building

Regression, Inference, and Model Building Regressio, Iferece, ad Model Buildig Scatter Plots ad Correlatio Correlatio coefficiet, r -1 r 1 If r is positive, the the scatter plot has a positive slope ad variables are said to have a positive relatioship

More information

Simple Regression Model

Simple Regression Model Simple Regressio Model 1. The Model y i 0 1 x i u i where y i depedet variable x i idepedet variable u i disturbace/error term i 1,..., Eg: y wage (measured i 1976 dollars per hr) x educatio (measured

More information

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N. 3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear

More information

Correlation Regression

Correlation Regression Correlatio Regressio While correlatio methods measure the stregth of a liear relatioship betwee two variables, we might wish to go a little further: How much does oe variable chage for a give chage i aother

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

Linear Regression Models, OLS, Assumptions and Properties

Linear Regression Models, OLS, Assumptions and Properties Chapter 2 Liear Regressio Models, OLS, Assumptios ad Properties 2.1 The Liear Regressio Model The liear regressio model is the sigle most useful tool i the ecoometricia s kit. The multiple regressio model

More information

Economics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator

Economics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

Linear Regression Models

Linear Regression Models Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect

More information

S Y Y = ΣY 2 n. Using the above expressions, the correlation coefficient is. r = SXX S Y Y

S Y Y = ΣY 2 n. Using the above expressions, the correlation coefficient is. r = SXX S Y Y 1 Sociology 405/805 Revised February 4, 004 Summary of Formulae for Bivariate Regressio ad Correlatio Let X be a idepedet variable ad Y a depedet variable, with observatios for each of the values of these

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

1 General linear Model Continued..

1 General linear Model Continued.. Geeral liear Model Cotiued.. We have We kow y = X + u X o radom u v N(0; I ) b = (X 0 X) X 0 y E( b ) = V ar( b ) = (X 0 X) We saw that b = (X 0 X) X 0 u so b is a liear fuctio of a ormally distributed

More information

Simple Regression. Acknowledgement. These slides are based on presentations created and copyrighted by Prof. Daniel Menasce (GMU) CS 700

Simple Regression. Acknowledgement. These slides are based on presentations created and copyrighted by Prof. Daniel Menasce (GMU) CS 700 Simple Regressio CS 7 Ackowledgemet These slides are based o presetatios created ad copyrighted by Prof. Daiel Measce (GMU) Basics Purpose of regressio aalysis: predict the value of a depedet or respose

More information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j. Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx) Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso

More information

Lecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise)

Lecture 22: Review for Exam 2. 1 Basic Model Assumptions (without Gaussian Noise) Lecture 22: Review for Exam 2 Basic Model Assumptios (without Gaussia Noise) We model oe cotiuous respose variable Y, as a liear fuctio of p umerical predictors, plus oise: Y = β 0 + β X +... β p X p +

More information

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

UNIT 11 MULTIPLE LINEAR REGRESSION

UNIT 11 MULTIPLE LINEAR REGRESSION UNIT MULTIPLE LINEAR REGRESSION Structure. Itroductio release relies Obectives. Multiple Liear Regressio Model.3 Estimatio of Model Parameters Use of Matrix Notatio Properties of Least Squares Estimates.4

More information

Full file at

Full file at Chapter Ecoometrics There are o exercises or applicatios i Chapter. 0 Pearso Educatio, Ic. Publishig as Pretice Hall Chapter The Liear Regressio Model There are o exercises or applicatios i Chapter. 0

More information

STP 226 ELEMENTARY STATISTICS

STP 226 ELEMENTARY STATISTICS TP 6 TP 6 ELEMENTARY TATITIC CHAPTER 4 DECRIPTIVE MEAURE IN REGREION AND CORRELATION Liear Regressio ad correlatio allows us to examie the relatioship betwee two or more quatitative variables. 4.1 Liear

More information

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that

More information

Chimica Inorganica 3

Chimica Inorganica 3 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule

More information

Lecture 11 Simple Linear Regression

Lecture 11 Simple Linear Regression Lecture 11 Simple Liear Regressio Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Midterm 2 mea: 91.2 media: 93.75 std: 6.5 2 Meddicorp

More information

11 Correlation and Regression

11 Correlation and Regression 11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated

More information

Efficient GMM LECTURE 12 GMM II

Efficient GMM LECTURE 12 GMM II DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet

More information

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

More information

CHAPTER I: Vector Spaces

CHAPTER I: Vector Spaces CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig

More information

Chapter Vectors

Chapter Vectors Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

Math 475, Problem Set #12: Answers

Math 475, Problem Set #12: Answers Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Dr. Maddah ENMG 617 EM Statistics 11/26/12. Multiple Regression (2) (Chapter 15, Hines)

Dr. Maddah ENMG 617 EM Statistics 11/26/12. Multiple Regression (2) (Chapter 15, Hines) Dr Maddah NMG 617 M Statistics 11/6/1 Multiple egressio () (Chapter 15, Hies) Test for sigificace of regressio This is a test to determie whether there is a liear relatioship betwee the depedet variable

More information

University of California, Los Angeles Department of Statistics. Practice problems - simple regression 2 - solutions

University of California, Los Angeles Department of Statistics. Practice problems - simple regression 2 - solutions Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 00C Istructor: Nicolas Christou EXERCISE Aswer the followig questios: Practice problems - simple regressio - solutios a Suppose y,

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS

UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY 2009 EXAMINATIONS ECO220Y1Y PART 1 OF 2 SOLUTIONS PART of UNIVERSITY OF TORONTO Faculty of Arts ad Sciece APRIL/MAY 009 EAMINATIONS ECO0YY PART OF () The sample media is greater tha the sample mea whe there is. (B) () A radom variable is ormally distributed

More information

Linear Programming and the Simplex Method

Linear Programming and the Simplex Method Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

Stat 139 Homework 7 Solutions, Fall 2015

Stat 139 Homework 7 Solutions, Fall 2015 Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,

More information

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients. Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

More information

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Matrix Representation of Data in Experiment

Matrix Representation of Data in Experiment Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Regression, Part I. A) Correlation describes the relationship between two variables, where neither is independent or a predictor.

Regression, Part I. A) Correlation describes the relationship between two variables, where neither is independent or a predictor. Regressio, Part I I. Differece from correlatio. II. Basic idea: A) Correlatio describes the relatioship betwee two variables, where either is idepedet or a predictor. - I correlatio, it would be irrelevat

More information

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n. ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic

More information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row: Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,

More information

Random Models. Tusheng Zhang. February 14, 2013

Random Models. Tusheng Zhang. February 14, 2013 Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the

More information

The Method of Least Squares. To understand least squares fitting of data.

The Method of Least Squares. To understand least squares fitting of data. The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve

More information

MAT 271 Project: Partial Fractions for certain rational functions

MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

More information

SIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS

SIMPLE LINEAR REGRESSION AND CORRELATION ANALYSIS SIMPLE LINEAR REGRESSION AND CORRELATION ANALSIS INTRODUCTION There are lot of statistical ivestigatio to kow whether there is a relatioship amog variables Two aalyses: (1) regressio aalysis; () correlatio

More information

4 Multidimensional quantitative data

4 Multidimensional quantitative data Chapter 4 Multidimesioal quatitative data 4 Multidimesioal statistics Basic statistics are ow part of the curriculum of most ecologists However, statistical techiques based o such simple distributios as

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

STP 226 EXAMPLE EXAM #1

STP 226 EXAMPLE EXAM #1 STP 226 EXAMPLE EXAM #1 Istructor: Hoor Statemet: I have either give or received iformatio regardig this exam, ad I will ot do so util all exams have bee graded ad retured. PRINTED NAME: Siged Date: DIRECTIONS:

More information

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1 460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

More information

Kinetics of Complex Reactions

Kinetics of Complex Reactions Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A

More information

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

Asymptotic Results for the Linear Regression Model

Asymptotic Results for the Linear Regression Model Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is

More information

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13 BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the

More information

Chapter 1 Simple Linear Regression (part 6: matrix version)

Chapter 1 Simple Linear Regression (part 6: matrix version) Chapter Simple Liear Regressio (part 6: matrix versio) Overview Simple liear regressio model: respose variable Y, a sigle idepedet variable X Y β 0 + β X + ε Multiple liear regressio model: respose Y,

More information

Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)

Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1) 5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01 ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly

More information

Why learn matrix algebra? Vectors & Matrices with statistical applications. Brief history of linear algebra

Why learn matrix algebra? Vectors & Matrices with statistical applications. Brief history of linear algebra R Vectors & Matrices with statistical applicatios x RXX RXY y RYX RYY Why lear matrix algebra? Simple way to express liear combiatios of variables ad geeral solutios of equatios. Liear statistical models

More information

MATH 304: MIDTERM EXAM SOLUTIONS

MATH 304: MIDTERM EXAM SOLUTIONS MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest

More information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

More information

Lecture 7: Properties of Random Samples

Lecture 7: Properties of Random Samples Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ

More information

Least-Squares Regression

Least-Squares Regression MATH 482 Least-Squares Regressio Dr. Neal, WKU As well as fidig the correlatio of paired sample data {{ x 1, y 1 }, { x 2, y 2 },..., { x, y }}, we also ca plot the data with a scatterplot ad fid the least

More information

Cov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n.

Cov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n. CS 189 Itroductio to Machie Learig Sprig 218 Note 11 1 Caoical Correlatio Aalysis The Pearso Correlatio Coefficiet ρ(x, Y ) is a way to measure how liearly related (i other words, how well a liear model

More information

Week 5-6: The Binomial Coefficients

Week 5-6: The Binomial Coefficients Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers

More information

University of California, Los Angeles Department of Statistics. Simple regression analysis

University of California, Los Angeles Department of Statistics. Simple regression analysis Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100C Istructor: Nicolas Christou Simple regressio aalysis Itroductio: Regressio aalysis is a statistical method aimig at discoverig

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

Unit 4: Polynomial and Rational Functions

Unit 4: Polynomial and Rational Functions 48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad

More information

Chapters 5 and 13: REGRESSION AND CORRELATION. Univariate data: x, Bivariate data (x,y).

Chapters 5 and 13: REGRESSION AND CORRELATION. Univariate data: x, Bivariate data (x,y). Chapters 5 ad 13: REGREION AND CORRELATION (ectios 5.5 ad 13.5 are omitted) Uivariate data: x, Bivariate data (x,y). Example: x: umber of years studets studied paish y: score o a proficiecy test For each

More information

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:

More information