Harmonization of Friction Measuring Devices Using Robust Regression Methods
|
|
|
- Anissa Lawson
- 5 years ago
- Views:
Transcription
1 Harmonization of Friction Measuring Devices Using Robust Regression Methods Samer Katicha 09/09/2013 Center for Sustainable Transportation Infrastructure
2 Outline o What is harmonization of devices? Measurement conversion Conversion accuracy o The best case scenario Perfectly identical devices o Failure of Linear regression Ideal (almost utopian) conditions Playing a trick on you Biased and inconsistent estimate
3 Outline o Measurement error models Orthogonal regression The general model o Practical results Repeated measurements from the same device (locked wheel tester) Simulation of two devices Measurements from two different devices (locked wheel testers)
4 o ASTM Harmonization the adjustment of the outputs of different devices used for the measurement of a specific phenomenon so that all devices report the same value o Does not consider agreement Confidence between adjusted outputs Limits of agreement (LOA)
5 Perfectly Identical Devices o This is the best we can hope for o It means: If we have two devices and obtain two sets of measurements (runs) from device 1, this is the same as: Obtaining two sets of measurements from device 2, or Obtaining two set of measurements each from one of the devices
6 Failure of Linear Regression
7 Ideal Conditions o Source of errors restricted to independent identically distributed normal error o The measurements y f i i y i = measurement (with error) f i = true value i = error which is N(0, 2 ) i
8 Playing a Trick on You o You own two locked wheel skid testers o You ask the operator to test the same pavement sections with the two devices (1 test each) o The operator, being lazy, performs the two tests with device 1 o Testing conditions were ideal: nothing affected results other than random error o To compare the devices you do linear regression and find
9 Biased Estimate Measurements Linear Regression Friction (set 2) y = 0.79*x Friction (set 1)
10 Inconsistent Estimate Measurements Linear Regression Linear Regression x = 0.75*y y = 1.33*x Friction (set 2) y = 0.79*x Friction (set 1)
11 Reasons o Ordinary regression assumes: Only one of the variables contains error x variable is measured without error o When both variables contain error: Ordinary regression (least squares) is Biased Bias depends on amount of error compared to range of measurement More error more bias
12 Ordinary Regression Summary o Biased estimate o Inconsistent estimate o This occurs in the most simple case: Error normally distributed and no other factors affecting measurement Real measurements are more complicated o Things can get worse: Speed correction models not perfect: introduce MORE error Texture models introduce MORE error
13 Measurement Error Model
14 Measurement Error Model o Take into account errors in both variables o Simplest case: orthogonal regression o Gives an unbiased estimate
15 Orthogonal Regression
16 Orthogonal Regression Measurements Linear Regression Linear Regression Orthogonal Regression x = 0.75*y y = 1.33*x Friction (set 2) y = 1.03*x y = 0.79*x Friction (set 1)
17 Postulating the Model o The best case only considers measurement error o Real measurements: Measurement error (repeatability) Model/device error: device estimates the true friction Independent of measurement error Experimental evidence of device error: between devices error
18 Postulating the Model Y a X b Y Y Y Y r Y Y = device estimate (with no repeatability error) X = true friction which is not known a y and b y = model parameters Y = model error = device measurement r Y = repeatability error Z a X b Z Z Z Z r Z
19 Postulating the Model Y a X b Y Y Y Y r Y Y = device estimate (with no repeatability error) X = true friction which is not known a y and b y = model parameters Y = model error = device measurement r Y = repeatability error Z a X b Z Z Z Z r Z
20 Harmonization B AZ a a b a a b Z a a Y z Z Y Y z Z Y Y Z Y s s s s s s A 2 4 ˆ AZ Y B ˆ ˆ
21 Harmonization s 2 var s cov, s 2 var 2 Y 2 Z A 2 Y 2 Z = repeatability error standard deviation = model error standard deviation Solution is found iteratively
22 Practical Results
23 Measurements from Same Device Friction Measurement "j" Friction Measurement "i"
24 o True parameters: Slope = 1 Intercept = 0 o Ordinary regression Slope = Intercept = Model Parameters o Measurement error model: Slope = Intercept =
25 Dangers of Speed and Texture 150 Correction Linear Regression x vs. y: y = 1.35x Friction for Run "j" Orthogonal Regression: y = 1.04x-4.56 Linear Regression y vs. x: y = 0.79x Line of Equality: y = x Friction for Run "i"
26 o True parameters: Slope = 1 Intercept = 0 o Ordinary regression Model Parameters Slope = (previous ) Intercept = (previous ) o Measurement error model: Slope = (previous ) Intercept = (previous )
27 Different Devices (Simulation)
28 Different Devices (Simulation) Linear Regression Errors in Variables Regression
29 Different Devices (Simulation) Linear Regression Errors in Variables Regression
30 Different Devices (Simulation) Y X Y Z 0.75X 10 Z o True parameters: A B o Ordinary regression E A ˆ E B ˆ o Measurement error model: A ˆ ˆ 39. E E B 3649
31 Measurements from Different Devices (Locked Wheel) All data 360 measurements Averaged 5 replicates 72 measurements 70 Device 2 Friction Device 1 Friction
32 o Averaged: Model Parameters Ordinary regression: Slope = ; Intercept = Measurement error model: Slope = ;Intercept = o Not Averaged: Ordinary regression: Slope = (bias = = ) Measurement error model: Slope = (bias = = )
33 Limits of Agreement (LOA) o Relationship between devices: Convert between measurements Does NOT give agreement o After conversion: Limits of Agreement (LOA) General: Bland and Altman (1983), Altman and Bland (1986) Friction devices: De Leon et al. (2012)
34 Conclusions o Ordinary regression: Biased estimate of the relationship Inconsistent relationship based on choice of dependent and independent variable o Measurement Error Model: Unbiased estimate of the relationship Relationship independent of choice of dependent and independent variables
35 o ASTM Conclusion the adjustment of the outputs of different devices used for the measurement of a specific phenomenon so that all devices report the same value o Ordinary regression The same device WILL NOT report the same value
36 Reproducible Research o Paper has been submitted to TRB o Paper written under the reproducible research paradigm: All data used in the paper is made available for anybody to use Matlab files used are made available: Calculations organized in functions GUI that can be used to reproduce all Figures showing the results
37 Blacksburg, VA
Psychology 282 Lecture #3 Outline
Psychology 8 Lecture #3 Outline Simple Linear Regression (SLR) Given variables,. Sample of n observations. In study and use of correlation coefficients, and are interchangeable. In regression analysis,
Topic 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
ECON Introductory Econometrics. Lecture 6: OLS with Multiple Regressors
ECON4150 - Introductory Econometrics Lecture 6: OLS with Multiple Regressors Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 6 Lecture outline 2 Violation of first Least Squares assumption
Intro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
Applied Econometrics (QEM)
Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple
Why is the field of statistics still an active one?
Why is the field of statistics still an active one? It s obvious that one needs statistics: to describe experimental data in a compact way, to compare datasets, to ask whether data are consistent with
Statistical View of Least Squares
Basic Ideas Some Examples Least Squares May 22, 2007 Basic Ideas Simple Linear Regression Basic Ideas Some Examples Least Squares Suppose we have two variables x and y Basic Ideas Simple Linear Regression
ECNS 561 Multiple Regression Analysis
ECNS 561 Multiple Regression Analysis Model with Two Independent Variables Consider the following model Crime i = β 0 + β 1 Educ i + β 2 [what else would we like to control for?] + ε i Here, we are taking
Suppose we obtain a MLR equation as follows:
Psychology 8 Lecture #9 Outline Probing Interactions among Continuous Variables Suppose we carry out a MLR analysis using a model that includes an interaction term and we find the interaction effect to
Algebra of Random Variables: Optimal Average and Optimal Scaling Minimising
Review: Optimal Average/Scaling is equivalent to Minimise χ Two 1-parameter models: Estimating < > : Scaling a pattern: Two equivalent methods: Algebra of Random Variables: Optimal Average and Optimal
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
Algebra of Random Variables: Optimal Average and Optimal Scaling Minimising
Review: Optimal Average/Scaling is equivalent to Minimise χ Two 1-parameter models: Estimating < > : Scaling a pattern: Two equivalent methods: Algebra of Random Variables: Optimal Average and Optimal
Chapter 14. Simultaneous Equations Models Introduction
Chapter 14 Simultaneous Equations Models 14.1 Introduction Simultaneous equations models differ from those we have considered in previous chapters because in each model there are two or more dependent
Simple Linear Regression Using Ordinary Least Squares
Simple Linear Regression Using Ordinary Least Squares Purpose: To approximate a linear relationship with a line. Reason: We want to be able to predict Y using X. Definition: The Least Squares Regression
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
Econ 1123: Section 5. Review. Internal Validity. Panel Data. Clustered SE. STATA help for Problem Set 5. Econ 1123: Section 5.
Outline 1 Elena Llaudet 2 3 4 October 6, 2010 5 based on Common Mistakes on P. Set 4 lnftmpop = -.72-2.84 higdppc -.25 lackpf +.65 higdppc * lackpf 2 lnftmpop = β 0 + β 1 higdppc + β 2 lackpf + β 3 lackpf
Regression. Estimation of the linear function (straight line) describing the linear component of the joint relationship between two variables X and Y.
Regression Bivariate i linear regression: Estimation of the linear function (straight line) describing the linear component of the joint relationship between two variables and. Generally describe as a
Approximations - the method of least squares (1)
Approximations - the method of least squares () In many applications, we have to consider the following problem: Suppose that for some y, the equation Ax = y has no solutions It could be that this is an
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,...,
Econ 1123: Section 2. Review. Binary Regressors. Bivariate. Regression. Omitted Variable Bias
Contact Information Elena Llaudet Sections are voluntary. My office hours are Thursdays 5pm-7pm in Littauer Mezzanine 34-36 (Note room change) You can email me administrative questions to ellaudet@gmail.com.
ANCOVA. ANCOVA allows the inclusion of a 3rd source of variation into the F-formula (called the covariate) and changes the F-formula
ANCOVA Workings of ANOVA & ANCOVA ANCOVA, Semi-Partial correlations, statistical control Using model plotting to think about ANCOVA & Statistical control You know how ANOVA works the total variation among
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
2 Regression Analysis
FORK 1002 Preparatory Course in Statistics: 2 Regression Analysis Genaro Sucarrat (BI) http://www.sucarrat.net/ Contents: 1 Bivariate Correlation Analysis 2 Simple Regression 3 Estimation and Fit 4 T -Test:
Consequences of measurement error. Psychology 588: Covariance structure and factor models
Consequences of measurement error Psychology 588: Covariance structure and factor models Scaling indeterminacy of latent variables Scale of a latent variable is arbitrary and determined by a convention
Variance. Standard deviation VAR = = value. Unbiased SD = SD = 10/23/2011. Functional Connectivity Correlation and Regression.
10/3/011 Functional Connectivity Correlation and Regression Variance VAR = Standard deviation Standard deviation SD = Unbiased SD = 1 10/3/011 Standard error Confidence interval SE = CI = = t value for
Chapter 6: Linear Regression With Multiple Regressors
Chapter 6: Linear Regression With Multiple Regressors 1-1 Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution
Intro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
Linear Regression with Multiple Regressors
Linear Regression with Multiple Regressors (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution
IE 316 Exam 1 Fall 2011
IE 316 Exam 1 Fall 2011 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1 1. Suppose the actual diameters x in a batch of steel cylinders are normally
Bias Correction in the Balanced-half-sample Method if the Number of Sampled Units in Some Strata Is Odd
Journal of Of cial Statistics, Vol. 14, No. 2, 1998, pp. 181±188 Bias Correction in the Balanced-half-sample Method if the Number of Sampled Units in Some Strata Is Odd Ger T. Slootbee 1 The balanced-half-sample
22 Approximations - the method of least squares (1)
22 Approximations - the method of least squares () Suppose that for some y, the equation Ax = y has no solutions It may happpen that this is an important problem and we can t just forget about it If we
Applied Quantitative Methods II
Applied Quantitative Methods II Lecture 4: OLS and Statistics revision Klára Kaĺıšková Klára Kaĺıšková AQM II - Lecture 4 VŠE, SS 2016/17 1 / 68 Outline 1 Econometric analysis Properties of an estimator
Steps in Regression Analysis
MGMG 522 : Session #2 Learning to Use Regression Analysis & The Classical Model (Ch. 3 & 4) 2-1 Steps in Regression Analysis 1. Review the literature and develop the theoretical model 2. Specify the model:
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
2. Linear regression with multiple regressors
2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions
ACE 564 Spring Lecture 8. Violations of Basic Assumptions I: Multicollinearity and Non-Sample Information. by Professor Scott H.
ACE 564 Spring 2006 Lecture 8 Violations of Basic Assumptions I: Multicollinearity and Non-Sample Information by Professor Scott H. Irwin Readings: Griffiths, Hill and Judge. "Collinear Economic Variables,
s e, which is large when errors are large and small Linear regression model
Linear regression model we assume that two quantitative variables, x and y, are linearly related; that is, the the entire population of (x, y) pairs are related by an ideal population regression line y
Linear Regression with Multiple Regressors
Linear Regression with Multiple Regressors (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution
LABORATORY NUMBER 9 STATISTICAL ANALYSIS OF DATA
LABORATORY NUMBER 9 STATISTICAL ANALYSIS OF DATA 1.0 INTRODUCTION The purpose of this laboratory is to introduce the student to the use of statistics to analyze data. Using the data acquisition system
2 Prediction and Analysis of Variance
2 Prediction and Analysis of Variance Reading: Chapters and 2 of Kennedy A Guide to Econometrics Achen, Christopher H. Interpreting and Using Regression (London: Sage, 982). Chapter 4 of Andy Field, Discovering
Chapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
Behavioral Data Mining. Lecture 19 Regression and Causal Effects
Behavioral Data Mining Lecture 19 Regression and Causal Effects Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression Counterfactuals
Investigation of the validity of the ASTM standard for computation of International Friction Index
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 2008 Investigation of the validity of the ASTM standard for computation of International Friction Index Kranthi
Multiple Linear Regression
Multiple Linear Regression Asymptotics Asymptotics Multiple Linear Regression: Assumptions Assumption MLR. (Linearity in parameters) Assumption MLR. (Random Sampling from the population) We have a random
Regression of Time Series
Mahlerʼs Guide to Regression of Time Series CAS Exam S prepared by Howard C. Mahler, FCAS Copyright 2016 by Howard C. Mahler. Study Aid 2016F-S-9Supplement Howard Mahler hmahler@mac.com www.howardmahler.com/teaching
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
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
MS&E 226. In-Class Midterm Examination Solutions Small Data October 20, 2015
MS&E 226 In-Class Midterm Examination Solutions Small Data October 20, 2015 PROBLEM 1. Alice uses ordinary least squares to fit a linear regression model on a dataset containing outcome data Y and covariates
A Non-parametric bootstrap for multilevel models
A Non-parametric bootstrap for multilevel models By James Carpenter London School of Hygiene and ropical Medicine Harvey Goldstein and Jon asbash Institute of Education 1. Introduction Bootstrapping is
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
Introduction to Error Analysis
Introduction to Error Analysis This is a brief and incomplete discussion of error analysis. It is incomplete out of necessity; there are many books devoted entirely to the subject, and we cannot hope to
First Year Examination Department of Statistics, University of Florida
First Year Examination Department of Statistics, University of Florida August 20, 2009, 8:00 am - 2:00 noon Instructions:. You have four hours to answer questions in this examination. 2. You must show
Heteroscedasticity 1
Heteroscedasticity 1 Pierre Nguimkeu BUEC 333 Summer 2011 1 Based on P. Lavergne, Lectures notes Outline Pure Versus Impure Heteroscedasticity Consequences and Detection Remedies Pure Heteroscedasticity
1 The Classic Bivariate Least Squares Model
Review of Bivariate Linear Regression Contents 1 The Classic Bivariate Least Squares Model 1 1.1 The Setup............................... 1 1.2 An Example Predicting Kids IQ................. 1 2 Evaluating
5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1)
5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1) Assumption #A1: Our regression model does not lack of any further relevant exogenous variables beyond x 1i, x 2i,..., x Ki and
Monday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties
Amherst College Department of Economics Economics 360 Fall 2012 Monday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties Chapter 18 Outline Review o Regression Model
Measurement Error. Often a data set will contain imperfect measures of the data we would ideally like.
Measurement Error Often a data set will contain imperfect measures of the data we would ideally like. Aggregate Data: (GDP, Consumption, Investment are only best guesses of theoretical counterparts and
APPENDIX 1 BASIC STATISTICS. Summarizing Data
1 APPENDIX 1 Figure A1.1: Normal Distribution BASIC STATISTICS The problem that we face in financial analysis today is not having too little information but too much. Making sense of large and often contradictory
Linear 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
Forecasting the use, costs and benefits of HSR in the years ahead. Samer Madanat UC Berkeley
Forecasting the use, costs and benefits of HSR in the years ahead Samer Madanat UC Berkeley Outline Demand models and ridership forecasts Errors in demand models and consequences Case study: the CA HSR
Basic Probability Reference Sheet
February 27, 2001 Basic Probability Reference Sheet 17.846, 2001 This is intended to be used in addition to, not as a substitute for, a textbook. X is a random variable. This means that X is a variable
Multicollinearity: Don t Throw Out The Baby (Independent Variable) with the Bath Water
Multicollinearity: Don t Throw Out The Baby (Independent Variable) with the Bath Water Jim Byrd Northrop Grumman Corporation Purpose The purpose of this paper is to stimulate thought and suggest alternative
Sigmaplot di Systat Software
Sigmaplot di Systat Software SigmaPlot Has Extensive Statistical Analysis Features SigmaPlot is now bundled with SigmaStat as an easy-to-use package for complete graphing and data analysis. The statistical
Problem 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.
Probability and Statistics Notes
Probability and Statistics Notes Chapter Seven Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Seven Notes Spring 2011 1 / 42 Outline
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
Supporting Information for Estimating restricted mean. treatment effects with stacked survival models
Supporting Information for Estimating restricted mean treatment effects with stacked survival models Andrew Wey, David Vock, John Connett, and Kyle Rudser Section 1 presents several extensions to the simulation
IE 316 Exam 1 Fall 2011
IE 316 Exam 1 Fall 2011 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1 1. Suppose the actual diameters x in a batch of steel cylinders are normally
Multiple Linear Regression CIVL 7012/8012
Multiple Linear Regression CIVL 7012/8012 2 Multiple Regression Analysis (MLR) Allows us to explicitly control for many factors those simultaneously affect the dependent variable This is important for
Write your identification number on each paper and cover sheet (the number stated in the upper right hand corner on your exam cover).
STOCKHOLM UNIVERSITY Department of Economics Course name: Empirical Methods in Economics 2 Course code: EC2402 Examiner: Peter Skogman Thoursie Number of credits: 7,5 credits (hp) Date of exam: Saturday,
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
Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 11 Output Analysis for a Single Model Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose Objective: Estimate system performance via simulation If q is the system performance,
The Simple Linear Regression Model
The Simple Linear Regression Model Lesson 3 Ryan Safner 1 1 Department of Economics Hood College ECON 480 - Econometrics Fall 2017 Ryan Safner (Hood College) ECON 480 - Lesson 3 Fall 2017 1 / 77 Bivariate
Lecture 2 Simple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: Chapter 1
Lecture Simple Linear Regression STAT 51 Spring 011 Background Reading KNNL: Chapter 1-1 Topic Overview This topic we will cover: Regression Terminology Simple Linear Regression with a single predictor
Econometrics Problem Set 11
Econometrics Problem Set WISE, Xiamen University Spring 207 Conceptual Questions. (SW 2.) This question refers to the panel data regressions summarized in the following table: Dependent variable: ln(q
download instant at
Answers to Odd-Numbered Exercises Chapter One: An Overview of Regression Analysis 1-3. (a) Positive, (b) negative, (c) positive, (d) negative, (e) ambiguous, (f) negative. 1-5. (a) The coefficients in
Handout 11: Measurement Error
Handout 11: Measurement Error In which you learn to recognise the consequences for OLS estimation whenever some of the variables you use are not measured as accurately as you might expect. A (potential)
Econometrics Honor s Exam Review Session. Spring 2012 Eunice Han
Econometrics Honor s Exam Review Session Spring 2012 Eunice Han Topics 1. OLS The Assumptions Omitted Variable Bias Conditional Mean Independence Hypothesis Testing and Confidence Intervals Homoskedasticity
Applied 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
ECONOMETFUCS FIELD EXAM Michigan State University May 11, 2007
ECONOMETFUCS FIELD EXAM Michigan State University May 11, 2007 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 100 points possible. Within
Entropy. Probability and Computing. Presentation 22. Probability and Computing Presentation 22 Entropy 1/39
Entropy Probability and Computing Presentation 22 Probability and Computing Presentation 22 Entropy 1/39 Introduction Why randomness and information are related? An event that is almost certain to occur
ECON2228 Notes 8. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 35
ECON2228 Notes 8 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 6 2014 2015 1 / 35 Functional form misspecification Chapter 9: More on specification and data problems
Passing-Bablok Regression for Method Comparison
Chapter 313 Passing-Bablok Regression for Method Comparison Introduction Passing-Bablok regression for method comparison is a robust, nonparametric method for fitting a straight line to two-dimensional
Likely causes: The Problem. E u t 0. E u s u p 0
Autocorrelation This implies that taking the time series regression Y t X t u t but in this case there is some relation between the error terms across observations. E u t 0 E u t E u s u p 0 Thus the error
Applied 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
Graduate Econometrics I: Unbiased Estimation
Graduate Econometrics I: Unbiased Estimation Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: Unbiased Estimation
Next is material on matrix rank. Please see the handout
B90.330 / C.005 NOTES for Wednesday 0.APR.7 Suppose that the model is β + ε, but ε does not have the desired variance matrix. Say that ε is normal, but Var(ε) σ W. The form of W is W w 0 0 0 0 0 0 w 0
The multiple regression model; Indicator variables as regressors
The multiple regression model; Indicator variables as regressors Ragnar Nymoen University of Oslo 28 February 2013 1 / 21 This lecture (#12): Based on the econometric model specification from Lecture 9
Lecture 4: Multivariate Regression, Part 2
Lecture 4: Multivariate Regression, Part 2 Gauss-Markov Assumptions 1) Linear in Parameters: Y X X X i 0 1 1 2 2 k k 2) Random Sampling: we have a random sample from the population that follows the above
More on Venn Diagrams for Regression
More on Venn Diagrams for Regression Peter E. Kennedy Simon Fraser University Journal of Statistics Education Volume 10, Number 1 (2002) Copyright 2002 by Peter E. Kennedy, all rights reserved. This text
Do not copy, post, or distribute
14 CORRELATION ANALYSIS AND LINEAR REGRESSION Assessing the Covariability of Two Quantitative Properties 14.0 LEARNING OBJECTIVES In this chapter, we discuss two related techniques for assessing a possible
The General Linear Model. How we re approaching the GLM. What you ll get out of this 8/11/16
8// The General Linear Model Monday, Lecture Jeanette Mumford University of Wisconsin - Madison How we re approaching the GLM Regression for behavioral data Without using matrices Understand least squares
Simulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
The SuperBall Lab. Objective. Instructions
1 The SuperBall Lab Objective This goal of this tutorial lab is to introduce data analysis techniques by examining energy loss in super ball collisions. Instructions This laboratory does not have to be
Recitation 1: Regression Review. Christina Patterson
Recitation 1: Regression Review Christina Patterson Outline For Recitation 1. Statistics. Bias, sampling variance and hypothesis testing.. Two important statistical theorems: Law of large numbers (LLN)
Simple Linear Regression Analysis
LINEAR REGRESSION ANALYSIS MODULE II Lecture - 6 Simple Linear Regression Analysis Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Prediction of values of study
Research Design: Topic 18 Hierarchical Linear Modeling (Measures within Persons) 2010 R.C. Gardner, Ph.d.
Research Design: Topic 8 Hierarchical Linear Modeling (Measures within Persons) R.C. Gardner, Ph.d. General Rationale, Purpose, and Applications Linear Growth Models HLM can also be used with repeated
Overview. Overview. Overview. Specific Examples. General Examples. Bivariate Regression & Correlation
Bivariate Regression & Correlation Overview The Scatter Diagram Two Examples: Education & Prestige Correlation Coefficient Bivariate Linear Regression Line SPSS Output Interpretation Covariance ou already
The General Linear Model. Monday, Lecture 2 Jeanette Mumford University of Wisconsin - Madison
The General Linear Model Monday, Lecture 2 Jeanette Mumford University of Wisconsin - Madison How we re approaching the GLM Regression for behavioral data Without using matrices Understand least squares
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