Summer School in Statistics for Astronomers V June 1 - June 6, Regression. Mosuk Chow Statistics Department Penn State University.
|
|
- Madison Bryant
- 5 years ago
- Views:
Transcription
1 Summer School in Statistics for Astronomers V June 1 - June 6, 2009 Regression Mosuk Chow Statistics Department Penn State University. Adapted from notes prepared by RL Karandikar
2 Mean and variance Recall we had defined the Expectation or the mean of a random variable X as i µ = E(X) = x ip(x = x i ) xf (x)dx for discrete X for continuous X with density f (x) and the variance of a random variable X as i σ 2 = Var(X) = E(X 2 ) µ 2 = x2 i P(X = x i) µ 2 x2 f (x)dx µ 2
3 If X and Y are random variables with means µ X and µ Y, then the covariance of X and Y is defined by Cov (X, Y) = E (X µ X ) (Y µ Y ) = E(XY) µ X µ Y The correlation coefficient ρ (X, Y) of X and Y is defined by ρ(x, Y) = Cov (X, Y) Var (X) Var (Y)
4 Properties of mean and variance E (ax + b) = aex + b, Var (ax + b) = a 2 Var (X) E (ax + by + c) = aex + bey + c Var (ax + by + c) = a 2 Var (X) + b 2 Var (Y) + 2abCov (X, Y)
5 It may be shown that for any n n matrix A and n 1 vector b E (AY + b) = AEY + b, Cov (AY + b) = ACov (Y) A T. which is the basic result used in regression.
6 Hubble s data (1929) In 1929 Edwin Hubble investigated the relationship between distance and radial velocity of extra-galactic nebulae (celestial objects). It was hoped that some knowledge of this relationship might give clues as to the way the universe was formed and what may happen later. His findings revolutionized astronomy and are the source of much research today. Given here is the data which Hubble used for 24 nebulae.
7 X = Distance (in Megaparsecs) from earth Y = The recession velocity (in km/sec) X Y X Y X Y X Y lib.stat.cmu.edu/dasl/datafiles/hubble.html
8 From this data-set Hubble obtained the relation Recession Velocity = H 0 Distance where H 0 is Hubble s constant thought to be about 75 km/sec/mpc.
9 Back to Hubble s data Scatterplot of Recession Velocity vs Distance Recession Velocity (km/sec) Distance (megaparsecs)
10 The ML Method for Linear Regression Analysis Scatterplot data: (x 1, y 1 ),..., (x n, y n ) Basic assumption: The x i s are non-random measurements; the y i are observations on Y, a random variable Statistical model: Y i = α + βx i + ɛ i, i = 1,..., n Errors ɛ 1,..., ɛ n : a random sample from N(0, σ 2 ) Parameters: α, β, σ 2 Y i N(α + βx i, σ 2 ): The Y i s are independent The Y i are not identically distributed, because they have differing means
11 The likelihood function is the joint density function of the observed data, Y 1,..., Y n n [ L(α, β, σ 2 1 ) = exp (Y i α βx i ) 2 ] i=1 2πσ 2 2σ 2 n (Y = (2πσ 2 ) n/2 i α βx i ) 2 exp i=1 2σ 2 Use partial derivatives to maximize L over all α, β and σ 2 > 0 (Wise advice: Maximize ln L) The ML estimators are: n i=1 ˆβ = (x i x)(y i Ȳ) n i=1 (x i x) 2, ˆα = Ȳ ˆβ x and ˆσ 2 = 1 n n (Y i ˆα ˆβx i ) 2 i=1
12 Using this on Hubble s data we get ˆβ = , ˆα = where the intercept term is not significant. If we use regression through origin, the result is ˆβ = The result from the historical data set obtained by Hubble is very different from the current estimate of the Hubble s constant due to various issues of measurement errors and censoring, etc. Those issues will be discussed in a subsequent lecture.
13 How do we obtain the least squares estimates? Let Y i be the response for the i th data point and let x i be the p-dimensional (row vector) of the predictors for the ith data point, i = 1,, n. There are p-1 predictors. We assume that Y i = x i β + e i, where β, an unknown parameter, is a p 1 column vector, and ( e i N 1 0, σ 2 ), and the e i are independent. Note that σ 2 is another parameter for this model.
14 We further assume that the predictors are linearly independent. Thus we could have the second predictor be the square of the first predictor, the third one the cube of the first one, etc, so this model includes polynomial regression.
15 We often write this model in matrices. Let Y = Y 1. Y n, X = x 1. x n, e = e 1. e n so that Y and e are n 1 and X is n p. The assumed linear independence of the predictors implies that the columns of X are linearly independent and hence rank(x) = p. x i = (1, x i,1,..., x i,p 1 ).
16 The normal model can be stated more compactly as Note that: β = (β 0, β 1,..., β p 1 ) T. Y = Xβ + e, e N n ( 0, σ 2 I ) The input matrix X is of dimension n (p): 1 x 1,1 x 1,2... x 1,p 1 1 x 2,1 x 2,2... x 2,p x n,1 x n,2... x n,p 1
17 Therefore, using the formula for the multivariate normal density function, we see that the joint density of Y is f β,σ 2 (y) = (2π) n/2 σ 2 I 1/2 exp{ 1 2 (y Xβ)T ( σ 2 I ) 1 (y Xβ)} = (2π) n/2 ( σ 2) n/2 exp{ 1 2σ 2 y Xβ 2 }
18 Therefore the likelihood for this model is L Y ( β,σ 2 ) = (2π) n/2 ( σ 2) n/2 exp{ 1 2σ 2 Y Xβ 2 }
19 Estimation of β First, we note that the assumption on the X matrix implies that X T X is invertible. The ordinary least square (OLS) estimator of β is found by minimizing q (β) = (Y i x i β) 2 = Y Xβ 2 The formula for the OLS estimator of β is β = ( X T X ) 1 X T Y
20 Note that E β = ( X T X ) 1 X T EY = ( X T X ) 1 X T Xβ = β ) Cov ( β = ( X T X ) 1 X T σ 2 IX ( X T X ) 1 = σ 2 ( X T X ) 1 = σ 2 M Therefore β N p ( β, σ 2 M )
21 Properties of the OLS estimator 1. (Gauss-Markov) For the non-normal model the OLS estimator is the best linear unbiased estimator (BLUE), i.e., it has smaller variance than any other linear unbiased estimator. 2. For the normal model, the OLS is the best unbiased estimator i.e., has smaller variance than any other unbiased estimator 3. Typically, the OLS estimator is consistent, i.e. β β
22 The unbiased estimator of σ 2 In regression we typically estimate σ 2 by σ 2 = Y X β 2 / (n p) which is called the unbiased estimator of σ 2. we first state the distribution of σ 2. (n p) σ 2 χ 2 n p independently of β σ 2
23 Properties of σ 2 1. For the general model σ 2 is unbiased. 2. For the normal model σ 2 is the best unbiased estimator. 3. σ 2 is consistent.
24 Interval estimators and tests We first discuss inference about β i the ith component of β. Note that β i the ith component of the OLS estimator is the estimator of β i. Further ) Var ( βi = σ 2 M ii which implies that that the standard error of β i is σ βi = σ M ii Therefore we see that a 1 α confidence interval for β i is β i ( β i t α/2 n p σ βi, β i + t α/2 n p σ βi ).
25 To test the null hypothesis β i = c against one and two-sided alternatives we use the t-statistic t = β i c σ βi t n p.
26 Now consider inference for δ = a T β, let δ = a T ( β N1 δ, σ 2 a Ma ) therefore we see that δ is the estimator of δ, and ) Var ( δ = σ 2 a Ma so that the standard error of δ is = σ a σ δ Ma
27 Therefore the confidence interval for δ is δ ( δ t α/2 n p σ δ, δ + t α/2 n p σ δ) and the test statistic for testing δ = c is given by δ c σ δ t n p under the null hypothesis There are tests and confidence regions for vector generalizations of these procedures.
28 Let x 0 be a row vector of predictors for a new response Y 0. Let µ 0 = x 0 β = EY 0. µ 0 = x 0 β is the obvious estimator of µ0 and Var ( µ 0 ) = σ 2 x 0 Mx 0 σ µ0 = σ x 0 Mx 0 and therefore a confidence interval for µ 0 is µ 0 ( µ 0 t α/2 n p σ µ 0, µ 0 + t α/2 n p σ µ 0 )
29 A 1 α prediction interval for Y 0 is an interval such that P (a (Y) Y 0 b (Y)) = 1 α A 1 α prediction interval for Y 0 is Y 0 ( µ 0 t α/2 n p σ 2 + σ 2 µ 0, µ 0 + t α/2 n p σ 2 + σ 2 µ 0 ) The derivation of this interval is based on the fact that Var (Y 0 µ 0 ) = σ 2 + σ 2 µ 0
30 R 2 Let T 2 = ( Yi Y ) 2, S 2 = Y X β 2 be the numerators of the variance estimators for the regression model and the intercept only model. We think of these as measuring the variation under these two models. Then the coefficient of determination R 2 is defined by Note that R 2 = T2 S 2 T 2 0 R 2 1
31 Note that T 2 S 2 is the amount of variation in the intercept only model which has been explained by including the extra predictors of the regression model and R 2 is the proportion of the variation left in the intercept only model which has been explained by including the additional predictors.
[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More informationRegression Steven F. Arnold Professor of Statistics Penn State University
Regression Steven F. Arnold Professor of Statistics Penn State University Regression is the most commonly used statistical technique. It is primarily concerned with fitting models to data. It is often
More informationLinear 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,
More informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More informationMa 3/103: Lecture 24 Linear Regression I: Estimation
Ma 3/103: Lecture 24 Linear Regression I: Estimation March 3, 2017 KC Border Linear Regression I March 3, 2017 1 / 32 Regression analysis Regression analysis Estimate and test E(Y X) = f (X). f is the
More informationCovariance and Correlation
Covariance and Correlation ST 370 The probability distribution of a random variable gives complete information about its behavior, but its mean and variance are useful summaries. Similarly, the joint probability
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationwhere x and ȳ are the sample means of x 1,, x n
y y Animal Studies of Side Effects Simple Linear Regression Basic Ideas In simple linear regression there is an approximately linear relation between two variables say y = pressure in the pancreas x =
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationProperties 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
More informationPeter 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
More information3. 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)
More informationStatement: With my signature I confirm that the solutions are the product of my own work. Name: Signature:.
MATHEMATICAL STATISTICS Homework assignment Instructions Please turn in the homework with this cover page. You do not need to edit the solutions. Just make sure the handwriting is legible. You may discuss
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationSTA 302f16 Assignment Five 1
STA 30f16 Assignment Five 1 Except for Problem??, these problems are preparation for the quiz in tutorial on Thursday October 0th, and are not to be handed in As usual, at times you may be asked to prove
More informationRegression diagnostics
Regression diagnostics Kerby Shedden Department of Statistics, University of Michigan November 5, 018 1 / 6 Motivation When working with a linear model with design matrix X, the conventional linear model
More information11 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
More informationGeneral Linear Model: Statistical Inference
Chapter 6 General Linear Model: Statistical Inference 6.1 Introduction So far we have discussed formulation of linear models (Chapter 1), estimability of parameters in a linear model (Chapter 4), least
More informationSTAT 540: Data Analysis and Regression
STAT 540: Data Analysis and Regression Wen Zhou http://www.stat.colostate.edu/~riczw/ Email: riczw@stat.colostate.edu Department of Statistics Colorado State University Fall 205 W. Zhou (Colorado State
More informationProblems. 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
More information3 Multiple Linear Regression
3 Multiple Linear Regression 3.1 The Model Essentially, all models are wrong, but some are useful. Quote by George E.P. Box. Models are supposed to be exact descriptions of the population, but that is
More informationMath 423/533: The Main Theoretical Topics
Math 423/533: The Main Theoretical Topics Notation sample size n, data index i number of predictors, p (p = 2 for simple linear regression) y i : response for individual i x i = (x i1,..., x ip ) (1 p)
More information4 Multiple Linear Regression
4 Multiple Linear Regression 4. The Model Definition 4.. random variable Y fits a Multiple Linear Regression Model, iff there exist β, β,..., β k R so that for all (x, x 2,..., x k ) R k where ε N (, σ
More informationMultiple 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
More informationLinear 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
More informationSimple 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
More informationCh 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
More informationMultilevel Models in Matrix Form. Lecture 7 July 27, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2
Multilevel Models in Matrix Form Lecture 7 July 27, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Today s Lecture Linear models from a matrix perspective An example of how to do
More information2.1 Linear regression with matrices
21 Linear regression with matrices The values of the independent variables are united into the matrix X (design matrix), the values of the outcome and the coefficient are represented by the vectors Y and
More information2. A Review of Some Key Linear Models Results. Copyright c 2018 Dan Nettleton (Iowa State University) 2. Statistics / 28
2. A Review of Some Key Linear Models Results Copyright c 2018 Dan Nettleton (Iowa State University) 2. Statistics 510 1 / 28 A General Linear Model (GLM) Suppose y = Xβ + ɛ, where y R n is the response
More informationTopic 16 Interval Estimation
Topic 16 Interval Estimation Additional Topics 1 / 9 Outline Linear Regression Interpretation of the Confidence Interval 2 / 9 Linear Regression For ordinary linear regression, we have given least squares
More informationQuantitative 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
More informationSimple 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)
More informationEstimating Estimable Functions of β. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 17
Estimating Estimable Functions of β Copyright c 202 Dan Nettleton (Iowa State University) Statistics 5 / 7 The Response Depends on β Only through Xβ In the Gauss-Markov or Normal Theory Gauss-Markov Linear
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 47 Outline 1 Regression 2 Simple Linear regression 3 Basic concepts in regression 4 How to estimate unknown parameters 5 Properties of Least Squares Estimators:
More informationLinear Models and Estimation by Least Squares
Linear Models and Estimation by Least Squares Jin-Lung Lin 1 Introduction Causal relation investigation lies in the heart of economics. Effect (Dependent variable) cause (Independent variable) Example:
More informationIntroduction 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
More informationLinear Regression (9/11/13)
STA561: Probabilistic machine learning Linear Regression (9/11/13) Lecturer: Barbara Engelhardt Scribes: Zachary Abzug, Mike Gloudemans, Zhuosheng Gu, Zhao Song 1 Why use linear regression? Figure 1: Scatter
More informationWell-developed and understood properties
1 INTRODUCTION TO LINEAR MODELS 1 THE CLASSICAL LINEAR MODEL Most commonly used statistical models Flexible models Well-developed and understood properties Ease of interpretation Building block for more
More informationLecture 34: Properties of the LSE
Lecture 34: Properties of the LSE The following results explain why the LSE is popular. Gauss-Markov Theorem Assume a general linear model previously described: Y = Xβ + E with assumption A2, i.e., Var(E
More informationThe 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
More informationMatrix Approach to Simple Linear Regression: An Overview
Matrix Approach to Simple Linear Regression: An Overview Aspects of matrices that you should know: Definition of a matrix Addition/subtraction/multiplication of matrices Symmetric/diagonal/identity matrix
More informationST 740: Linear Models and Multivariate Normal Inference
ST 740: Linear Models and Multivariate Normal Inference Alyson Wilson Department of Statistics North Carolina State University November 4, 2013 A. Wilson (NCSU STAT) Linear Models November 4, 2013 1 /
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini June 9, 2018 1 / 56 Table of Contents Multiple linear regression Linear model setup Estimation of β Geometric interpretation Estimation of σ 2 Hat matrix Gram matrix
More informationLinear models. Linear models are computationally convenient and remain widely used in. applied econometric research
Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y
More informationAssociation 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
More informationBIOS 2083 Linear Models c Abdus S. Wahed
Chapter 5 206 Chapter 6 General Linear Model: Statistical Inference 6.1 Introduction So far we have discussed formulation of linear models (Chapter 1), estimability of parameters in a linear model (Chapter
More informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationHypothesis testing Goodness of fit Multicollinearity Prediction. Applied Statistics. Lecturer: Serena Arima
Applied Statistics Lecturer: Serena Arima Hypothesis testing for the linear model Under the Gauss-Markov assumptions and the normality of the error terms, we saw that β N(β, σ 2 (X X ) 1 ) and hence s
More informationLecture 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
More informationBasic Distributional Assumptions of the Linear Model: 1. The errors are unbiased: E[ε] = The errors are uncorrelated with common variance:
8. PROPERTIES OF LEAST SQUARES ESTIMATES 1 Basic Distributional Assumptions of the Linear Model: 1. The errors are unbiased: E[ε] = 0. 2. The errors are uncorrelated with common variance: These assumptions
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More information18.S096 Problem Set 3 Fall 2013 Regression Analysis Due Date: 10/8/2013
18.S096 Problem Set 3 Fall 013 Regression Analysis Due Date: 10/8/013 he Projection( Hat ) Matrix and Case Influence/Leverage Recall the setup for a linear regression model y = Xβ + ɛ where y and ɛ are
More informationSimple 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.
More informationChapter 14. Linear least squares
Serik Sagitov, Chalmers and GU, March 5, 2018 Chapter 14 Linear least squares 1 Simple linear regression model A linear model for the random response Y = Y (x) to an independent variable X = x For a given
More informationMultivariate Regression
Multivariate Regression The so-called supervised learning problem is the following: we want to approximate the random variable Y with an appropriate function of the random variables X 1,..., X p with the
More informationLecture 3: Linear Models. Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012
Lecture 3: Linear Models Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector of observed
More informationLectures on Simple Linear Regression Stat 431, Summer 2012
Lectures on Simple Linear Regression Stat 43, Summer 0 Hyunseung Kang July 6-8, 0 Last Updated: July 8, 0 :59PM Introduction Previously, we have been investigating various properties of the population
More informationData Mining Stat 588
Data Mining Stat 588 Lecture 02: Linear Methods for Regression Department of Statistics & Biostatistics Rutgers University September 13 2011 Regression Problem Quantitative generic output variable Y. Generic
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
More informationLecture 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
More informationApplied Regression. Applied Regression. Chapter 2 Simple Linear Regression. Hongcheng Li. April, 6, 2013
Applied Regression Chapter 2 Simple Linear Regression Hongcheng Li April, 6, 2013 Outline 1 Introduction of simple linear regression 2 Scatter plot 3 Simple linear regression model 4 Test of Hypothesis
More informationTopic 12 Overview of Estimation
Topic 12 Overview of Estimation Classical Statistics 1 / 9 Outline Introduction Parameter Estimation Classical Statistics Densities and Likelihoods 2 / 9 Introduction In the simplest possible terms, the
More informationRegression. Oscar García
Regression Oscar García Regression methods are fundamental in Forest Mensuration For a more concise and general presentation, we shall first review some matrix concepts 1 Matrices An order n m matrix is
More informationApplied 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
More information1 Mixed effect models and longitudinal data analysis
1 Mixed effect models and longitudinal data analysis Mixed effects models provide a flexible approach to any situation where data have a grouping structure which introduces some kind of correlation between
More informationFENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4. Prof. Mei-Yuan Chen Spring 2008
FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4 Prof. Mei-Yuan Chen Spring 008. Partition and rearrange the matrix X as [x i X i ]. That is, X i is the matrix X excluding the column x i. Let u i denote
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationFoundations of Statistical Inference
Foundations of Statistical Inference Julien Berestycki Department of Statistics University of Oxford MT 2015 Julien Berestycki (University of Oxford) SB2a MT 2015 1 / 16 Lecture 16 : Bayesian analysis
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2
MA 575 Linear Models: Cedric E Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2 1 Revision: Probability Theory 11 Random Variables A real-valued random variable is
More informationIntroduction to Simple Linear Regression
Introduction to Simple Linear Regression Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Introduction to Simple Linear Regression 1 / 68 About me Faculty in the Department
More informationBIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation
BIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation Yujin Chung November 29th, 2016 Fall 2016 Yujin Chung Lec13: MLE Fall 2016 1/24 Previous Parametric tests Mean comparisons (normality assumption)
More informationACE 562 Fall Lecture 2: Probability, Random Variables and Distributions. by Professor Scott H. Irwin
ACE 562 Fall 2005 Lecture 2: Probability, Random Variables and Distributions Required Readings: by Professor Scott H. Irwin Griffiths, Hill and Judge. Some Basic Ideas: Statistical Concepts for Economists,
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7
MA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7 1 Random Vectors Let a 0 and y be n 1 vectors, and let A be an n n matrix. Here, a 0 and A are non-random, whereas y is
More informationPart IB Statistics. Theorems with proof. Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua. Lent 2015
Part IB Statistics Theorems with proof Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationIEOR 165 Lecture 7 1 Bias-Variance Tradeoff
IEOR 165 Lecture 7 Bias-Variance Tradeoff 1 Bias-Variance Tradeoff Consider the case of parametric regression with β R, and suppose we would like to analyze the error of the estimate ˆβ in comparison to
More informationIntroduction 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
More informationLinear Models in Machine Learning
CS540 Intro to AI Linear Models in Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu We briefly go over two linear models frequently used in machine learning: linear regression for, well, regression,
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
More informationCAS MA575 Linear Models
CAS MA575 Linear Models Boston University, Fall 2013 Midterm Exam (Correction) Instructor: Cedric Ginestet Date: 22 Oct 2013. Maximal Score: 200pts. Please Note: You will only be graded on work and answers
More informationMATH11400 Statistics Homepage
MATH11400 Statistics 1 2010 11 Homepage http://www.stats.bris.ac.uk/%7emapjg/teach/stats1/ 4. Linear Regression 4.1 Introduction So far our data have consisted of observations on a single variable of interest.
More informationMAT2377. Rafa l Kulik. Version 2015/November/26. Rafa l Kulik
MAT2377 Rafa l Kulik Version 2015/November/26 Rafa l Kulik Bivariate data and scatterplot Data: Hydrocarbon level (x) and Oxygen level (y): x: 0.99, 1.02, 1.15, 1.29, 1.46, 1.36, 0.87, 1.23, 1.55, 1.40,
More informationRestricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model
Restricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model Xiuming Zhang zhangxiuming@u.nus.edu A*STAR-NUS Clinical Imaging Research Center October, 015 Summary This report derives
More informationQuestions and Answers on Unit Roots, Cointegration, VARs and VECMs
Questions and Answers on Unit Roots, Cointegration, VARs and VECMs L. Magee Winter, 2012 1. Let ɛ t, t = 1,..., T be a series of independent draws from a N[0,1] distribution. Let w t, t = 1,..., T, be
More informationChapter 4. Chapter 4 sections
Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation
More informationThis paper is not to be removed from the Examination Halls
~~ST104B ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON ST104B ZB BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
More informationDirection: This test is worth 250 points and each problem worth points. DO ANY SIX
Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes
More informationRandom vectors X 1 X 2. Recall that a random vector X = is made up of, say, k. X k. random variables.
Random vectors Recall that a random vector X = X X 2 is made up of, say, k random variables X k A random vector has a joint distribution, eg a density f(x), that gives probabilities P(X A) = f(x)dx Just
More informationTHE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS We begin with a relatively simple special case. Suppose y ijk = µ + τ i + u ij + e ijk, (i = 1,..., t; j = 1,..., n; k = 1,..., m) β =
More informationLIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
More informationCS281A/Stat241A Lecture 17
CS281A/Stat241A Lecture 17 p. 1/4 CS281A/Stat241A Lecture 17 Factor Analysis and State Space Models Peter Bartlett CS281A/Stat241A Lecture 17 p. 2/4 Key ideas of this lecture Factor Analysis. Recall: Gaussian
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 51 Outline 1 Matrix Expression 2 Linear and quadratic forms 3 Properties of quadratic form 4 Properties of estimates 5 Distributional properties 3 / 51 Matrix
More informationLecture 4 Multiple linear regression
Lecture 4 Multiple linear regression BIOST 515 January 15, 2004 Outline 1 Motivation for the multiple regression model Multiple regression in matrix notation Least squares estimation of model parameters
More informationMultivariate 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
More informationAdvanced Statistics I : Gaussian Linear Model (and beyond)
Advanced Statistics I : Gaussian Linear Model (and beyond) Aurélien Garivier CNRS / Telecom ParisTech Centrale Outline One and Two-Sample Statistics Linear Gaussian Model Model Reduction and model Selection
More informationHT Introduction. P(X i = x i ) = e λ λ x i
MODS STATISTICS Introduction. HT 2012 Simon Myers, Department of Statistics (and The Wellcome Trust Centre for Human Genetics) myers@stats.ox.ac.uk We will be concerned with the mathematical framework
More informationECON 3150/4150, Spring term Lecture 7
ECON 3150/4150, Spring term 2014. Lecture 7 The multivariate regression model (I) Ragnar Nymoen University of Oslo 4 February 2014 1 / 23 References to Lecture 7 and 8 SW Ch. 6 BN Kap 7.1-7.8 2 / 23 Omitted
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More information