F3: Classical normal linear rgression model distribution, interval estimation and hypothesis testing
|
|
|
- Shon McDonald
- 5 years ago
- Views:
Transcription
1 F3: Classical normal linear rgression model distribution, interval estimation and hypothesis testing Feng Li Department of Statistics, Stockholm University
2 What we have learned last time... 1 Estimating ˆβ 1 and ˆβ and find their variance via OLS. The properties of OLS (have you done the assignments?). 3 The assumptions of linear model. Are they realistic? 4 The measurement of goodness fit. 5 Remember to use the correct notations: 1 The population regression function: Y i = β 1 + β X i + u i We estimate it from sample regression function: Y i = ˆβ 1 + ˆβ X i + û i 3 And the estimated value of Y i : Ŷ i = ˆβ 1 + ˆβ X i Feng Li (Stockholm University) Econometrics / 14
3 Today we are going to learn... 1 Normal assumptions of u i Confidence Intervals for regression coefficients β 1 and β 3 Confidence Intervals for σ 4 Hypothesis testing 5 Predictions 6 Normality tests of residuals Feng Li (Stockholm University) Econometrics 3 / 14
4 Normal assumptions of u i 1 The classical normal linear regression model assumes each u i is distributed normally with E(u i ) =0 Var(u i ) =E(u i E(u i )) = σ cov(u i, u j ) =E[(u i E(u i ))(u j E(u j ))] = E(u i, u j ) = 0, i j. We write u i N(0, σ ) for short. 3 Why normal? 1 Simple. Central limit theorem. Feng Li (Stockholm University) Econometrics 4 / 14
5 The properties under normal assumptions of u i 1 The estimators are unbiased, i.e., E(ˆβ 1 ) = β 1, E(ˆβ ) = β see Appendix 3A. The variance of the estimators (F, p.14) are minimal. 3 The estimators of the parameters also follows normal distribution (suppose σ is known). ř X ˆβ 1 N(β 1, i n ř x σ 1 ), ˆβ N(β, ř i x σ ). i 4 Recall that σ is not known and replaced with its estimator ˆσ = 5 ˆβ 1 is independent of ˆσ, so does ˆβ. (n ) ˆσ σ χ (n ). ř û i n, and Summary: The least estimators ˆβ 1 and ˆβ are best unbiased estimators(bue) in the entire class of unbiased estimators. Feng Li (Stockholm University) Econometrics 5 / 14
6 Confidence Intervals for regression coefficients 1 ˆβ i β i se( ˆβ i ) N(0, 1) for i = 1, when σ is known which is rare. ˆβ i β i se( ˆβ i ) t(n ) when σ is replaced by ˆσ. 1 Given α/ level of significance, [ Pr t α/ ď ˆβ i β i se(ˆβ i ) ď t α/ ] = 1 α which provides 100(1 α) percent confidence interval for β i ] Pr [ˆβ i t α/ se(ˆβ i ) ď β i ď ˆβ i + t α/ se(ˆβ i ) = 1 α or simply ˆβ i t α/ se(ˆβ i ). The interpretation: Right way: Given the confidence coefficient of 1 α, 100(1 α) out of 100 cases the interval will contain the true β i Wrong way: The probability of β i falling into the interval is 100(1 α). Note: The probability of β i falling into the interval is either 0 or 1. Feng Li (Stockholm University) Econometrics 6 / 14
7 Confidence Intervals for σ 1 Given α/ level of significance, ] ˆσ Pr [χ 1 α/ ď (n ) σ ď χ α/ = 1 α which provides 100(1 α) percent confidence interval for σ Pr [ (n ) ˆσ χ α/ ď σ ď (n ) Remember that χ is always positive and skewed. ˆσ χ 1 α/ ] = 1 α Exercise Table 3.: Construct the confidence intervals for β and σ. Feng Li (Stockholm University) Econometrics 7 / 14
8 The significance of coefficients: the t test 1 Significant of a statistic: If the value of the test statistic lies in the critical region. Significance testing: Find the critical region 3 Procedures: 1 Write down the null hypothesis (H 0 ) and alternative hypothesis (H a ) Calculate the test statistic e.g., t = (ˆβ β )/se(ˆβ ) 3 Look up the table and find the critical value 4 Make decision. 4 One-side test vs two-sided test Table Feng Li (Stockholm University) Econometrics 8 / 14
9 The significance of σ : the χ test 1 The testing purpose: if σ = σ 0 or not. The decision rule. Feng Li (Stockholm University) Econometrics 9 / 14
10 The ANOVA table for the two-variable regression model 1 We arrange the sums of squares in the following table (aka ANOVA table.) Then we consider ESS/df ESS ˆβ ř x = ř i F(1, n ) RSS/df RSS û i /(n ) which can be used to test the overall significance of the model. In particular the null hypothesis of β = 0 can also be tested (How?). Feng Li (Stockholm University) Econometrics 10 / 14
11 Predictions 1 Mean predictions: Predict Y 0 given X 0 which is from the observations. 1 [ 1 var(ŷ 0 ) = σ n + (X ] 0 X) ř x i t = Ŷ0 (β 1 + β X 0 ) se(ŷ 0 ) Individual prediction: Predict Y 0 given X 0 which is not from the observations. 1 [ var(y 0 Ŷ 0 ) = σ n + (X ] 0 X) ř x i t = Y 0 Ŷ 0 se(y 0 Ŷ 0 ) Feng Li (Stockholm University) Econometrics 11 / 14
12 Feng Li (Stockholm University) Econometrics 1 / 14
13 Normality tests of residuals 1 Histogram of residuals Anderson Darling test: H 0 : the variable is normal distributed. 3 Jarque-Bera test: H 0 : the variable is normal distributed (skewness(s)=0, kurtosis(k)=3). test statistic: JB = n[s /6 + (K 3) /4] Feng Li (Stockholm University) Econometrics 13 / 14
14 Take home questions 1 Read Appendix A.8, p.831 if you have problems of hypothesis testing. Do the example in p Feng Li (Stockholm University) Econometrics 14 / 14
L2: Two-variable regression model
L2: Two-variable regression model Feng Li feng.li@cufe.edu.cn School of Statistics and Mathematics Central University of Finance and Economics Revision: September 4, 2014 What we have learned last time...
F9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
Simple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
Ch 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
3. Linear Regression With a Single Regressor
3. Linear Regression With a Single Regressor Econometrics: (I) Application of statistical methods in empirical research Testing economic theory with real-world data (data analysis) 56 Econometrics: (II)
Multivariate Regression Analysis
Matrices and vectors The model from the sample is: Y = Xβ +u with n individuals, l response variable, k regressors Y is a n 1 vector or a n l matrix with the notation Y T = (y 1,y 2,...,y n ) 1 x 11 x
L7: Multicollinearity
L7: Multicollinearity Feng Li feng.li@cufe.edu.cn School of Statistics and Mathematics Central University of Finance and Economics Introduction ï Example Whats wrong with it? Assume we have this data Y
Ch 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
Heteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
Multiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
LECTURE 5. Introduction to Econometrics. Hypothesis testing
LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will
Review of Econometrics
Review of Econometrics Zheng Tian June 5th, 2017 1 The Essence of the OLS Estimation Multiple regression model involves the models as follows Y i = β 0 + β 1 X 1i + β 2 X 2i + + β k X ki + u i, i = 1,...,
Simple Linear Regression: The Model
Simple Linear Regression: The Model task: quantifying the effect of change X in X on Y, with some constant β 1 : Y = β 1 X, linear relationship between X and Y, however, relationship subject to a random
The general linear regression with k explanatory variables is just an extension of the simple regression as follows
3. Multiple Regression Analysis The general linear regression with k explanatory variables is just an extension of the simple regression as follows (1) y i = β 0 + β 1 x i1 + + β k x ik + u i. Because
INTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
Two-Variable Regression Model: The Problem of Estimation
Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable
y ˆ i = ˆ " T u i ( i th fitted value or i th fit)
1 2 INFERENCE FOR MULTIPLE LINEAR REGRESSION Recall Terminology: p predictors x 1, x 2,, x p Some might be indicator variables for categorical variables) k-1 non-constant terms u 1, u 2,, u k-1 Each u
Simple linear regression
Simple linear regression Biometry 755 Spring 2008 Simple linear regression p. 1/40 Overview of regression analysis Evaluate relationship between one or more independent variables (X 1,...,X k ) and a single
Statistics II Exercises Chapter 5
Statistics II Exercises Chapter 5 1. Consider the four datasets provided in the transparencies for Chapter 5 (section 5.1) (a) Check that all four datasets generate exactly the same LS linear regression
Linear Regression with 1 Regressor. Introduction to Econometrics Spring 2012 Ken Simons
Linear Regression with 1 Regressor Introduction to Econometrics Spring 2012 Ken Simons Linear Regression with 1 Regressor 1. The regression equation 2. Estimating the equation 3. Assumptions required for
Linear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
Homoskedasticity. Var (u X) = σ 2. (23)
Homoskedasticity How big is the difference between the OLS estimator and the true parameter? To answer this question, we make an additional assumption called homoskedasticity: Var (u X) = σ 2. (23) This
Lecture 6 Multiple Linear Regression, cont.
Lecture 6 Multiple Linear Regression, cont. BIOST 515 January 22, 2004 BIOST 515, Lecture 6 Testing general linear hypotheses Suppose we are interested in testing linear combinations of the regression
Multiple Regression Analysis: Heteroskedasticity
Multiple Regression Analysis: Heteroskedasticity y = β 0 + β 1 x 1 + β x +... β k x k + u Read chapter 8. EE45 -Chaiyuth Punyasavatsut 1 topics 8.1 Heteroskedasticity and OLS 8. Robust estimation 8.3 Testing
The Simple Regression Model. Part II. The Simple Regression Model
Part II The Simple Regression Model As of Sep 22, 2015 Definition 1 The Simple Regression Model Definition Estimation of the model, OLS OLS Statistics Algebraic properties Goodness-of-Fit, the R-square
Homework Set 2, ECO 311, Fall 2014
Homework Set 2, ECO 311, Fall 2014 Due Date: At the beginning of class on October 21, 2014 Instruction: There are twelve questions. Each question is worth 2 points. You need to submit the answers of only
Review of probability and statistics 1 / 31
Review of probability and statistics 1 / 31 2 / 31 Why? This chapter follows Stock and Watson (all graphs are from Stock and Watson). You may as well refer to the appendix in Wooldridge or any other introduction
ECON3150/4150 Spring 2015
ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2
Econometric Methods. Prediction / Violation of A-Assumptions. Burcu Erdogan. Universität Trier WS 2011/2012
Econometric Methods Prediction / Violation of A-Assumptions Burcu Erdogan Universität Trier WS 2011/2012 (Universität Trier) Econometric Methods 30.11.2011 1 / 42 Moving on to... 1 Prediction 2 Violation
Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
Econometrics A. Simple linear model (2) Keio University, Faculty of Economics. Simon Clinet (Keio University) Econometrics A October 16, / 11
Econometrics A Keio University, Faculty of Economics Simple linear model (2) Simon Clinet (Keio University) Econometrics A October 16, 2018 1 / 11 Estimation of the noise variance σ 2 In practice σ 2 too
Lecture 15. Hypothesis testing in the linear model
14. Lecture 15. Hypothesis testing in the linear model Lecture 15. Hypothesis testing in the linear model 1 (1 1) Preliminary lemma 15. Hypothesis testing in the linear model 15.1. Preliminary lemma Lemma
MS&E 226: Small Data
MS&E 226: Small Data Lecture 15: Examples of hypothesis tests (v5) Ramesh Johari ramesh.johari@stanford.edu 1 / 32 The recipe 2 / 32 The hypothesis testing recipe In this lecture we repeatedly apply the
Econometrics Multiple Regression Analysis: Heteroskedasticity
Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties
ECON The Simple Regression Model
ECON 351 - The Simple Regression Model Maggie Jones 1 / 41 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In
The regression model with one fixed regressor cont d
The regression model with one fixed regressor cont d 3150/4150 Lecture 4 Ragnar Nymoen 27 January 2012 The model with transformed variables Regression with transformed variables I References HGL Ch 2.8
Applied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 5 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 44 Outline of Lecture 5 Now that we know the sampling distribution
Peter Hoff Linear and multilinear models April 3, GLS for multivariate regression 5. 3 Covariance estimation for the GLM 8
Contents 1 Linear model 1 2 GLS for multivariate regression 5 3 Covariance estimation for the GLM 8 4 Testing the GLH 11 A reference for some of this material can be found somewhere. 1 Linear model Recall
Multiple Regression Analysis. Part III. Multiple Regression Analysis
Part III Multiple Regression Analysis As of Sep 26, 2017 1 Multiple Regression Analysis Estimation Matrix form Goodness-of-Fit R-square Adjusted R-square Expected values of the OLS estimators Irrelevant
Introduction to Estimation Methods for Time Series models. Lecture 1
Introduction to Estimation Methods for Time Series models Lecture 1 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 1 SNS Pisa 1 / 19 Estimation
Matrices and vectors A matrix is a rectangular array of numbers. Here s an example: A =
Matrices and vectors A matrix is a rectangular array of numbers Here s an example: 23 14 17 A = 225 0 2 This matrix has dimensions 2 3 The number of rows is first, then the number of columns We can write
Econometrics I Lecture 3: The Simple Linear Regression Model
Econometrics I Lecture 3: The Simple Linear Regression Model Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 32 Outline Introduction Estimating
Problems. Suppose both models are fitted to the same data. Show that SS Res, A SS Res, B
Simple Linear Regression 35 Problems 1 Consider a set of data (x i, y i ), i =1, 2,,n, and the following two regression models: y i = β 0 + β 1 x i + ε, (i =1, 2,,n), Model A y i = γ 0 + γ 1 x i + γ 2
Simple Linear Regression for the Climate Data
Prediction Prediction Interval Temperature 0.2 0.0 0.2 0.4 0.6 0.8 320 340 360 380 CO 2 Simple Linear Regression for the Climate Data What do we do with the data? y i = Temperature of i th Year x i =CO
MFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators
MFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators Thilo Klein University of Cambridge Judge Business School Session 4: Linear regression,
Econometrics Review questions for exam
Econometrics Review questions for exam Nathaniel Higgins nhiggins@jhu.edu, 1. Suppose you have a model: y = β 0 x 1 + u You propose the model above and then estimate the model using OLS to obtain: ŷ =
Homework 2: Simple Linear Regression
STAT 4385 Applied Regression Analysis Homework : Simple Linear Regression (Simple Linear Regression) Thirty (n = 30) College graduates who have recently entered the job market. For each student, the CGPA
Introductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna November 23, 2013 Outline Introduction
Simple and Multiple Linear Regression
Sta. 113 Chapter 12 and 13 of Devore March 12, 2010 Table of contents 1 Simple Linear Regression 2 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where
+ Specify 1 tail / 2 tail
Week 2: Null hypothesis Aeroplane seat designer wonders how wide to make the plane seats. He assumes population average hip size μ = 43.2cm Sample size n = 50 Question : Is the assumption μ = 43.2cm reasonable?
Inference in Regression Analysis
Inference in Regression Analysis Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 4, Slide 1 Today: Normal Error Regression Model Y i = β 0 + β 1 X i + ǫ i Y i value
Econometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
BANA 7046 Data Mining I Lecture 2. Linear Regression, Model Assessment, and Cross-validation 1
BANA 7046 Data Mining I Lecture 2. Linear Regression, Model Assessment, and Cross-validation 1 Shaobo Li University of Cincinnati 1 Partially based on Hastie, et al. (2009) ESL, and James, et al. (2013)
Formulary Applied Econometrics
Department of Economics Formulary Applied Econometrics c c Seminar of Statistics University of Fribourg Formulary Applied Econometrics 1 Rescaling With y = cy we have: ˆβ = cˆβ With x = Cx we have: ˆβ
WISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
Applied Econometrics (QEM)
Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42 Outline 1 2 3 t-test P-value Linear
Applied Regression Analysis
Applied Regression Analysis Chapter 3 Multiple Linear Regression Hongcheng Li April, 6, 2013 Recall simple linear regression 1 Recall simple linear regression 2 Parameter Estimation 3 Interpretations of
Multiple Regression Analysis
Chapter 4 Multiple Regression Analysis The simple linear regression covered in Chapter 2 can be generalized to include more than one variable. Multiple regression analysis is an extension of the simple
Linear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
Empirical Economic Research, Part II
Based on the text book by Ramanathan: Introductory Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 7, 2011 Outline Introduction
Statistical Inference. Part IV. Statistical Inference
Part IV Statistical Inference As of Oct 5, 2017 Sampling Distributions of the OLS Estimator 1 Statistical Inference Sampling Distributions of the OLS Estimator Testing Against One-Sided Alternatives Two-Sided
Econometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.
LECTURE 5 HYPOTHESIS TESTING
October 25, 2016 LECTURE 5 HYPOTHESIS TESTING Basic concepts In this lecture we continue to discuss the normal classical linear regression defined by Assumptions A1-A5. Let θ Θ R d be a parameter of interest.
INTRODUCTORY ECONOMETRICS
INTRODUCTORY ECONOMETRICS Lesson 2b Dr Javier Fernández etpfemaj@ehu.es Dpt. of Econometrics & Statistics UPV EHU c J Fernández (EA3-UPV/EHU), February 21, 2009 Introductory Econometrics - p. 1/192 GLRM:
Chapter 3: Multiple Regression. August 14, 2018
Chapter 3: Multiple Regression August 14, 2018 1 The multiple linear regression model The model y = β 0 +β 1 x 1 + +β k x k +ǫ (1) is called a multiple linear regression model with k regressors. The parametersβ
Introduction to Econometrics Midterm Examination Fall 2005 Answer Key
Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is
Econometrics. 4) Statistical inference
30C00200 Econometrics 4) Statistical inference Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Confidence intervals of parameter estimates Student s t-distribution
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
Simple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
Making sense of Econometrics: Basics
Making sense of Econometrics: Basics Lecture 2: Simple Regression Egypt Scholars Economic Society Happy Eid Eid present! enter classroom at http://b.socrative.com/login/student/ room name c28efb78 Outline
LECTURE 10. Introduction to Econometrics. Multicollinearity & Heteroskedasticity
LECTURE 10 Introduction to Econometrics Multicollinearity & Heteroskedasticity November 22, 2016 1 / 23 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists
Association studies and regression
Association studies and regression CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar Association studies and regression 1 / 104 Administration
CHAPTER 2 SIMPLE LINEAR REGRESSION
CHAPTER 2 SIMPLE LINEAR REGRESSION 1 Examples: 1. Amherst, MA, annual mean temperatures, 1836 1997 2. Summer mean temperatures in Mount Airy (NC) and Charleston (SC), 1948 1996 Scatterplots outliers? influential
Lecture 3: Multiple Regression
Lecture 3: Multiple Regression R.G. Pierse 1 The General Linear Model Suppose that we have k explanatory variables Y i = β 1 + β X i + β 3 X 3i + + β k X ki + u i, i = 1,, n (1.1) or Y i = β j X ji + u
Introductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 17, 2012 Outline Heteroskedasticity
Regression Analysis: Basic Concepts
The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance
Lectures 5 & 6: Hypothesis Testing
Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across
Simple Linear Regression
Simple Linear Regression Christopher Ting Christopher Ting : christophert@smu.edu.sg : 688 0364 : LKCSB 5036 January 7, 017 Web Site: http://www.mysmu.edu/faculty/christophert/ Christopher Ting QF 30 Week
Properties of the least squares estimates
Properties of the least squares estimates 2019-01-18 Warmup Let a and b be scalar constants, and X be a scalar random variable. Fill in the blanks E ax + b) = Var ax + b) = Goal Recall that the least squares
From last time... The equations
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
Motivation for multiple regression
Motivation for multiple regression 1. Simple regression puts all factors other than X in u, and treats them as unobserved. Effectively the simple regression does not account for other factors. 2. The slope
Statistical Inference with Regression Analysis
Introductory Applied Econometrics EEP/IAS 118 Spring 2015 Steven Buck Lecture #13 Statistical Inference with Regression Analysis Next we turn to calculating confidence intervals and hypothesis testing
So far our focus has been on estimation of the parameter vector β in the. y = Xβ + u
Interval estimation and hypothesis tests So far our focus has been on estimation of the parameter vector β in the linear model y i = β 1 x 1i + β 2 x 2i +... + β K x Ki + u i = x iβ + u i for i = 1, 2,...,
Homework Set 2, ECO 311, Spring 2014
Homework Set 2, ECO 311, Spring 2014 Due Date: At the beginning of class on March 31, 2014 Instruction: There are twelve questions. Each question is worth 2 points. You need to submit the answers of only
Inference for Regression
Inference for Regression Section 9.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 13b - 3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
The Statistical Property of Ordinary Least Squares
The Statistical Property of Ordinary Least Squares The linear equation, on which we apply the OLS is y t = X t β + u t Then, as we have derived, the OLS estimator is ˆβ = [ X T X] 1 X T y Then, substituting
11 Hypothesis Testing
28 11 Hypothesis Testing 111 Introduction Suppose we want to test the hypothesis: H : A q p β p 1 q 1 In terms of the rows of A this can be written as a 1 a q β, ie a i β for each row of A (here a i denotes
Semester 2, 2015/2016
ECN 3202 APPLIED ECONOMETRICS 2. Simple linear regression B Mr. Sydney Armstrong Lecturer 1 The University of Guyana 1 Semester 2, 2015/2016 PREDICTION The true value of y when x takes some particular
Measuring the fit of the model - SSR
Measuring the fit of the model - SSR Once we ve determined our estimated regression line, we d like to know how well the model fits. How far/close are the observations to the fitted line? One way to do
STAT420 Midterm Exam. University of Illinois Urbana-Champaign October 19 (Friday), :00 4:15p. SOLUTIONS (Yellow)
STAT40 Midterm Exam University of Illinois Urbana-Champaign October 19 (Friday), 018 3:00 4:15p SOLUTIONS (Yellow) Question 1 (15 points) (10 points) 3 (50 points) extra ( points) Total (77 points) Points
13 Simple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 3 Simple Linear Regression 3. An industrial example A study was undertaken to determine the effect of stirring rate on the amount of impurity
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
Violation of OLS assumption - Heteroscedasticity
Violation of OLS assumption - Heteroscedasticity What, why, so what and what to do? Lars Forsberg Uppsala Uppsala University, Department of Statistics October 22, 2014 Lars Forsberg (Uppsala University)
Estadística II Chapter 5. Regression analysis (second part)
Estadística II Chapter 5. Regression analysis (second part) Chapter 5. Regression analysis (second part) Contents Diagnostic: Residual analysis The ANOVA (ANalysis Of VAriance) decomposition Nonlinear
LECTURE 6. Introduction to Econometrics. Hypothesis testing & Goodness of fit
LECTURE 6 Introduction to Econometrics Hypothesis testing & Goodness of fit October 25, 2016 1 / 23 ON TODAY S LECTURE We will explain how multiple hypotheses are tested in a regression model We will define
Estadística II Chapter 4: Simple linear regression
Estadística II Chapter 4: Simple linear regression Chapter 4. Simple linear regression Contents Objectives of the analysis. Model specification. Least Square Estimators (LSE): construction and properties
Ma 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA
Ma 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA March 6, 2017 KC Border Linear Regression II March 6, 2017 1 / 44 1 OLS estimator 2 Restricted regression 3 Errors in variables 4
Quantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017
Summary of Part II Key Concepts & Formulas Christopher Ting November 11, 2017 christopherting@smu.edu.sg http://www.mysmu.edu/faculty/christophert/ Christopher Ting 1 of 16 Why Regression Analysis? Understand
Regression Analysis. y t = β 1 x t1 + β 2 x t2 + β k x tk + ϵ t, t = 1,..., T,
Regression Analysis The multiple linear regression model with k explanatory variables assumes that the tth observation of the dependent or endogenous variable y t is described by the linear relationship