Econometrics The Multiple Regression Model: Inference
|
|
- Lucy Merritt
- 7 years ago
- Views:
Transcription
1 Econometrics The Multiple Regression Model: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
2 in the Multiple Linear Regression Model Suppose you want to test whether a variable is important in explaining variation in the dependent variable: E.g., is the effect of tenure on wages statistically significant (ie, different from zero)? Is the effect of height on wages statistically significant? Or suppose you want to test whether a coefficient has a particular value E.g., is the effect of one additional year of schooling on expected monthly wages equal to 200? Need to take into account the sampling distribution of our estimators We will check whether under the maintained hypothesis (or null hypothesis) the observed values of certain test statistics are likely If they are not we reject the null João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
3 in the Multiple Linear Regression Model y = β 0 + β 1 x 1 + β 2 x β k x k + u Assumption MLR.6 (Normality) The distribution of the population error u is independent of x 1, x 2,..., x k and u is normally distributed with mean 0 and variance σ 2 : we write u Normal(0, σ 2 ) Independence assumption is stronger than MLR.4 (Zero Conditional Mean) assumption. Actually, it implies MLR.4 Also, normality and independence imply MLR.5 so that all the results regarding unbiasedness and variance of the estimators remain valid João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
4 Classical Linear Model Assumptions MLR.1 through MLR.6 are the Classical Linear Model (CLM) assumptions Under the CLM assumptions, OLS is not only BLUE, but is the minimum variance unbiased estimator: no other unbiased estimator has a variance smaller than OLS We can summarize the population assumptions of CLM as follows y X Normal(β 0, β 1 x 1, β 2 x 2,..., β k x k, σ 2 ) Normality is unrealistic in many cases (e.g., wages cannot be negative but under the normality assumption of u we can get negative wages) However, most results would hold in large samples without the normality assumption João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
5 Normal Sampling Distribution y f(y x). Normal distributions. E(y x) = b 0 + b 1 x x 1 x 2 Figure: The homoskedastic normal distribution with a single explanatory variable João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
6 Normal Sampling Distribution Since the OLS estimators are a linear function of the error term u, then (conditional on the x s): Theorem Under the CLM assumptions, conditional on the sample values of the independent variables, ˆβ j Normal[β j, Var( ˆβ j )], Therefore, ( ˆβ j β j ) sd( ˆβ Normal(0, 1) j ) where sd stands for standard deviation (squared root of the variance, derived in previous classes) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
7 Normal Sampling Distribution Now, the σ 2 that appears in the expression for the standard deviation of the estimators must be estimated Also, conditional on the x s (n k 1)ˆσ 2 /σ 2 χ 2 n k 1 which implies: ( ˆβ j β j ) se( ˆβ j ) = ( ˆβ j β j ) sd( ˆβ j ) sd( ˆβ j ) se( ˆβ j ) = ( ˆβ j β j ) σ sd( ˆβ j ) ˆσ Normal(0, 1) χ 2 n k 1 n k 1 t n k 1 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
8 Normal Sampling Distribution Theorem Under the CLM assumptions MLR.1 through MLR.6, ( ˆβ j β j ) t n k 1, se( ˆβ j ) where k+1 is the number of unknown parameters in the population model y = β 0 + β 1 x β k x k + u (k slope parameters and the intercept β 0 ) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
9 Performing a test on a coefficient Set the null hypothesis (and the alternative) E.g., H0 : β j = 0 (coefficient on experience in our wage regression) and H 1 : β j > 0 Choose a significance level (Probability of rejecting the null if the null is actually true) E.g., α = 0.05 Look at the sampling distribution of the test statistic t (random variable) involving the parameter: t = ( ˆβ j β j ) se( ˆβ j ) t (n k 1), Under the null hypothesis, the test statistic should be small across samples. Reject the null if the observed value of the test statistic is very unlikely (very large) João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
10 Performing a test on a coefficient One-side Tests For one-sided tests where the alternative is favored if t obs is large and positive (e.g., H 1 : β j > 0), reject the null if the observed test statistic, t obs, is larger than c, where c is implicitly given by: Prob[t > c H 0 is true]=α For one-sided tests where the alternative is favored if t obs is large and negative (e.g., H 1 : β j < 0), reject the null if the observed test statistic, t obs, is smaller than -c, where c is implicitly given by: Prob[t < c H 0 is true]=α For two-sided tests, where the alternative is favored if t obs is large in absolute value (e.g., H 1 : β j 0), reject the null if the absolute value of observed test statistic, t obs, is larger than c, where c is implicitly given by: Prob[ t > c H 0 is true]=α João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
11 One-Sided Alternative H 0 : β j = 0 H 1 : β j > 0 Fail to reject the null (1-α) Reject the null α Figure: Rejection region for a 5% significance level for alternative H 1 : β j > 0 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
12 Two-Sided Alternative H 0 : β j = 0 H 1 : β j 0 Fail to reject the null Reject the null (1-α) Reject the null α/2 α/2 Figure: Rejection region for a 5% significance level for alternative H 1 : β j 0 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
13 Example: Hypothesis Testing Independent Variable Coefficient Estimate Standard Error Intercept Education (in years) Labor Market Experience (in years) Square of Labor Market Experience (in years) n R t ratio Figure: Dependent Variable: Log of Wages The t ratios are the observed values of the test statistic for testing β j = 0 E.g = / João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
14 Example: Hypothesis Testing (Cont.) Choose α = 0.05 Test H 0 : β j = 0 against H 1 : β j 0 (coefficient on education) t obs = = t >1.96 Reject the null: the coefficient for education is significant at 5% significance level We use Normal approximation since n is large Fail to reject the null Reject the null Reject the null -c=-1.96 c=1.96 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
15 Example: Hypothesis Testing (Cont.) Choose α = 0.05 Test H 0 : β j = 0 against H 1 : β j > 0 (clearly more reasonable...) t obs = = t >1.645 Reject the null: the coefficient for education is significant at 5% significance level We use Normal approximation since n is large Fail to reject the null Reject the null c=1.645 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
16 Example: Hypothesis Testing (Cont.) Choose α = 0.05 Test H 0 : β j = 0.07 against H 1 : β j 0.07 (coefficient on education) t obs = = t >1.96 Reject the null: the coefficient for education is significant at 5% significance level We use Normal approximation since n is large Fail to reject the null Reject the null Reject the null -c=-1.96 c=1.96 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
17 p-value p-value: Given the observed value of the t statistic, what would be the smallest significance level at which the null H 0 : β j = 0 would be rejected against the alternative H 1 : β j 0? It is given by: Prob[ t > t obs H 0 true] 1- p-value p-value /2 p-value /2 -t obs t obs João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
18 Confidence Intervals A (1 α)% confidence interval is defined as: ˆβ j ± c se( ˆβ j ) where c is the (1 α 2 ) percentile in a t n k 1 distribution If the hypothesized value of a parameter (b j ) is inside the confidence interval, we would not reject the null β j = b j against β j b j at the significance level α João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
19 Testing multiple exclusion restrictions Unrestricted model: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x β k x k + u H 0 : β k q+1 = β k q+2 =... = β k = 0 H 1 : NotH 0 Restricted model: y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x β k q x k q + u Under the null: Fstatistic = (SSR r SSR ur )/q SSR ur /(n k 1) F (q,n k 1) r stands for restricted and ur for unrestricted, q is number of restrictions Does SSRur decrease enough compared to SSR r? If F obs is too large we reject the null João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
20 Testing multiple exclusion restrictions H 0 : β k q+1 = β k q+2 =... = β k = 0 H 1 : NotH 0 Fstatistic = (SSR r SSR ur )/q SSR ur /(n k 1) F (q,n k 1) Fstatistic = (R 2 ur R 2 r )/q (1 R 2 ur )/(n k 1) F (q,n k 1) Obtained by dividing the numerator and the denominator above by SST This is different from testing significance of each coefficient individually!! It is a test of joint significance João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
21 Testing multiple exclusion restrictions: F test Reject the null if the observed test statistic, F obs, is larger than c, where c is implicitly given by: Prob[F > c H 0 istrue] = α Fail to Reject the null 1-α α Reject the null c João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
22 Example H 0 : β 2 = β 3 = 0 Independent Variable Coefficient Estimate Standard Error Unrestricted model Intercept Education (in years) Labor Market Experience (in years) Square of Labor Market Experience (in years) R Mean Square Error Restricted model Intercept Education (in years) t ratio R Mean Square Error Figure: Dependent Variable: Log of monthly wage, n=11064 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
23 Example (Cont.) α = 0.05 H 0 : β 2 = β 3 = 0 Fstatistic = ( )/2 ( )/( ) = > 3.00 Reject H 0 João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
24 Overall significance of the model Under the null use: H 0 : β 1 = β 2 =... = β k = 0 H 1 : NotH 0 (SST SSR)/k F = SSR/(n k 1) SSE/k = SSR/(n k 1) = R 2 /k (1 R 2 )/(n k 1) F (k,n k 1) Testing general linear restrictions: in the practice sessions! João Valle e Azevedo (FEUNL) Econometrics Lisbon, March / 24
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More information2. 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
More informationSolución del Examen Tipo: 1
Solución del Examen Tipo: 1 Universidad Carlos III de Madrid ECONOMETRICS Academic year 2009/10 FINAL EXAM May 17, 2010 DURATION: 2 HOURS 1. Assume that model (III) verifies the assumptions of the classical
More informationAugust 2012 EXAMINATIONS Solution Part I
August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More informationMultiple Linear Regression in Data Mining
Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple
More informationFactors affecting online sales
Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4
More informationTwo-Sample T-Tests Assuming Equal Variance (Enter Means)
Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of
More informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More information17. SIMPLE LINEAR REGRESSION II
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
More informationBA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420
BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical
More informationECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationChapter 7: Simple linear regression Learning Objectives
Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -
More informationTwo-Sample T-Tests Allowing Unequal Variance (Enter Difference)
Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More information1.5 Oneway Analysis of Variance
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
More informationIAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results
IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the
More information2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or
Simple and Multiple Regression Analysis Example: Explore the relationships among Month, Adv.$ and Sales $: 1. Prepare a scatter plot of these data. The scatter plots for Adv.$ versus Sales, and Month versus
More informationA Primer on Forecasting Business Performance
A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.
More informationMultiple Linear Regression
Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two- Means
Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More information1 Another method of estimation: least squares
1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i
More informationHypothesis testing - Steps
Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationSome Essential Statistics The Lure of Statistics
Some Essential Statistics The Lure of Statistics Data Mining Techniques, by M.J.A. Berry and G.S Linoff, 2004 Statistics vs. Data Mining..lie, damn lie, and statistics mining data to support preconceived
More informationSTAT 350 Practice Final Exam Solution (Spring 2015)
PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects
More informationGood luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
More informationStatistics courses often teach the two-sample t-test, linear regression, and analysis of variance
2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample
More informationChapter 13 Introduction to Linear Regression and Correlation Analysis
Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
More information2013 MBA Jump Start Program. Statistics Module Part 3
2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just
More informationCONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationPenalized regression: Introduction
Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20th-century statistics dealt with maximum likelihood
More informationNonlinear Regression Functions. SW Ch 8 1/54/
Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General
More informationLOGIT AND PROBIT ANALYSIS
LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationComparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples
Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationRegression step-by-step using Microsoft Excel
Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression
More informationIntroduction to Quantitative Methods
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationOne-Way Analysis of Variance (ANOVA) Example Problem
One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means
More informationExample: Boats and Manatees
Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant
More informationCorrelation and Simple Linear Regression
Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a
More informationName: Date: Use the following to answer questions 3-4:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationData Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression
Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction
More informationOutline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares
Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation
More informationStatistics 305: Introduction to Biostatistical Methods for Health Sciences
Statistics 305: Introduction to Biostatistical Methods for Health Sciences Modelling the Log Odds Logistic Regression (Chap 20) Instructor: Liangliang Wang Statistics and Actuarial Science, Simon Fraser
More informationStatistical Functions in Excel
Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.
More informationStat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015
Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation
More informationIntroduction to Regression and Data Analysis
Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance
More informationAn Introduction to Regression Analysis
The Inaugural Coase Lecture An Introduction to Regression Analysis Alan O. Sykes * Regression analysis is a statistical tool for the investigation of relationships between variables. Usually, the investigator
More informationSection Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini
NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building
More informationAnswer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade
Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements
More informationChapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More informationWe extended the additive model in two variables to the interaction model by adding a third term to the equation.
Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic
More informationChapter 5: Bivariate Cointegration Analysis
Chapter 5: Bivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis...
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationOne-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups
One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups In analysis of variance, the main research question is whether the sample means are from different populations. The
More informationConfidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
More informationSection 13, Part 1 ANOVA. Analysis Of Variance
Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability
More informationSIMPLE LINEAR CORRELATION. r can range from -1 to 1, and is independent of units of measurement. Correlation can be done on two dependent variables.
SIMPLE LINEAR CORRELATION Simple linear correlation is a measure of the degree to which two variables vary together, or a measure of the intensity of the association between two variables. Correlation
More informationInternational Statistical Institute, 56th Session, 2007: Phil Everson
Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationBasic Statistics and Data Analysis for Health Researchers from Foreign Countries
Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationStatistics Review PSY379
Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses
More informationMULTIPLE REGRESSION EXAMPLE
MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if
More informationMultinomial and Ordinal Logistic Regression
Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit August 22, 2012 Regression with categorical dependent variables When the dependent variable is categorical,
More informationQuantile Regression under misspecification, with an application to the U.S. wage structure
Quantile Regression under misspecification, with an application to the U.S. wage structure Angrist, Chernozhukov and Fernandez-Val Reading Group Econometrics November 2, 2010 Intro: initial problem The
More informationAn analysis appropriate for a quantitative outcome and a single quantitative explanatory. 9.1 The model behind linear regression
Chapter 9 Simple Linear Regression An analysis appropriate for a quantitative outcome and a single quantitative explanatory variable. 9.1 The model behind linear regression When we are examining the relationship
More informationPremaster Statistics Tutorial 4 Full solutions
Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for
More informationAn Analysis of the Effect of Income on Life Insurance. Justin Bryan Austin Proctor Kathryn Stoklosa
An Analysis of the Effect of Income on Life Insurance Justin Bryan Austin Proctor Kathryn Stoklosa 1 Abstract This paper aims to analyze the relationship between the gross national income per capita and
More informationX X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)
CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.
More informationHow To Run Statistical Tests in Excel
How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting
More informationForecasting the US Dollar / Euro Exchange rate Using ARMA Models
Forecasting the US Dollar / Euro Exchange rate Using ARMA Models LIUWEI (9906360) - 1 - ABSTRACT...3 1. INTRODUCTION...4 2. DATA ANALYSIS...5 2.1 Stationary estimation...5 2.2 Dickey-Fuller Test...6 3.
More informationBiostatistics: Types of Data Analysis
Biostatistics: Types of Data Analysis Theresa A Scott, MS Vanderbilt University Department of Biostatistics theresa.scott@vanderbilt.edu http://biostat.mc.vanderbilt.edu/theresascott Theresa A Scott, MS
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice
More informationPearson's Correlation Tests
Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationWooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions
Wooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions What will happen if we violate the assumption that the errors are not serially
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationHypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes
More informationTesting for Granger causality between stock prices and economic growth
MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.uni-muenchen.de/2962/ MPRA Paper No. 2962, posted
More informationWooldridge, Introductory Econometrics, 4th ed. Chapter 10: Basic regression analysis with time series data
Wooldridge, Introductory Econometrics, 4th ed. Chapter 10: Basic regression analysis with time series data We now turn to the analysis of time series data. One of the key assumptions underlying our analysis
More informationSTA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance
Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis
More informationGeneral Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.
General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationindividualdifferences
1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,
More information