2. Linear regression with multiple regressors


 Silvester Higgins
 3 years ago
 Views:
Transcription
1 2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measuresoffit in multiple regression Assumptions in the multiple regression model Violations of the assumptions (omittedvariable bias, multicollinearity, heteroskedasticity, autocorrelation) 5
2 2.1. The multiple regression model Intuition: A regression model specifies a functional (parametric) relationship between a dependent (endogenous) variable Y and a set of k independent (exogenous) regressors X 1, X 2,..., X k In a first step, we consider the linear multiple regression model 6
3 Definition 2.1: (Multiple linear regression model) The multiple (linear) regression model is given by Y i = β 0 + β 1 X 1i + β 2 X 2i β k X ki + u i, (2.1) i = 1,..., n, where Y i is the i th observation on the dependent variable, X 1i, X 2i,..., X ki are the i th regressors, u i is the stochastic error term. observations on each of the k The population regression line is the relationship that holds between Y and the X s on average: E(Y i X 1i = x 1, X 2i = x 2,..., X ki = x k ) = β 0 +β 1 x β k x k. 7
4 Meaning of the coefficients: The intercept β 0 is the expected value of Y i (for all i = 1,..., n) when all Xregressors equal 0 β 1,..., β k are the slope coefficients on the respective regressors X 1,..., X k β 1, for example, is the expected change in Y i resulting from changing X 1i by one unit, holding constant X 2i,..., X ki (and analogously β 2,..., β k ) Definition 2.2: (Homoskedasticity, Heteroskedasticity) The error term u i is called homoskedastic if the conditional variance of u i given X 1i,..., X ki, Var(u i X 1i,..., X ki ), is constant for i = 1,..., n and does not depend on the values of X 1i,..., X ki. Otherwise, the error term is called heteroskedastic. 8
5 Example 1: (Student performance) Regression of student performance (Y ) in n = 420 USdistricts on distinct school characteristics (factors) Y i : average test score in the i th district (TEST SCORE) X 1i : average class size in the i th district (measured by the studentteacher ratio, STR) X 2i : percentage of English learners in the i th district (PCTEL) Expected signs of the coefficients: β 1 < 0 β 2 < 0 9
6 Example 2: (House prices) Regression of house prices (Y ) recorded for n = 546 houses sold in Windsor (Canada) on distinct housing characteristics Y i : sale price (in Canadian dollars) of the i th house (SALEPRICE) X 1i : lot size (in square feet) of the i th property (LOTSIZE) X 2i : number of bedrooms in the i th house (BEDROOMS) X 3i : number of bathrooms in the i th house (BATHROOMS) X 4i : number of storeys (excluding the basement) in the i th house (STOREYS) Expected signs of the coefficients: β 1, β 2, β 3, β 4 > 0 10
7 2.2. The OLS estimator in multiple regression Now: Estimation of the coefficients β 0, β 1,..., β k in the multiple regression model on the basis of n observations by applying the Ordinary Least Squares (OLS) technique Idea: Let b 0, b 1,..., b k be estimators of β 0, β 1,..., β k We can predict Y i by b 0 + b 1 X 1i b k X ki The prediction error is Y i b 0 b 1 X 1i... b k X ki 11
8 Idea: [continued] The sum of the squared prediction errors over all n observations is n i=1 (Y i b 0 b 1 X 1i... b k X ki ) 2 (2.2) Definition 2.3: (OLS estimators, predicted values, residuals) The OLS estimators ˆβ 0, ˆβ 1,..., ˆβ k are the values of b 0, b 1,..., b k that minimize the sum of squared prediction errors (2.2). The OLS predicted values Ŷ i and residuals û i (for i = 1,..., n) are and Ŷ i = ˆβ 0 + ˆβ 1 X 1i ˆβ k X ki (2.3) û i = Y i Ŷ i. (2.4) 12
9 Remarks: The OLS estimators ˆβ 0, ˆβ 1,..., ˆβ k and the residuals û i are computed from a sample of n observations of (X 1i,..., X ki, Y i ) for i = 1,..., n They are estimators of the unknown true population coefficients β 0, β 1,..., β k and u i There are closedform formulas for calculating the OLS estimates from the data (see the lectures Econometrics I+II) In this lecture, we use the softwarepackage EViews 13
10 Regression estimation results (EViews) for the studentperformance dataset Dependent Variable: TEST_SCORE Method: Least Squares Date: 07/02/12 Time: 16:29 Sample: Included observations: 420 Variable Coefficient Std. Error tstatistic Prob. C STR PCTEL Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood HannanQuinn criter Fstatistic DurbinWatson stat Prob(Fstatistic)
11 Predicted values Ŷ i and residuals û i for the studentperformance dataset Residual Actual Fitted 15
12 Regression estimation results (EViews) for the houseprices dataset Dependent Variable: SALEPRICE Method: Least Squares Date: 07/02/12 Time: 16:50 Sample: Included observations: 546 Variable Coefficient Std. Error tstatistic Prob. C LOTSIZE BEDROOMS BATHROOMS STOREYS Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 1.80E+11 Schwarz criterion Log likelihood HannanQuinn criter Fstatistic DurbinWatson stat Prob(Fstatistic)
13 Predicted values Ŷ i and residuals û i for the houseprices dataset 200, , , ,000 80,000 40,000 80,000 40, ,00080, Residual Actual Fitted 17
14 OLS assumptions in the multiple regression model (2.1): 1. u i has conditional mean zero given X 1i, X 2i,..., X ki : E(u i X 1i, X 2i,..., X ki ) = 0 2. (X 1i, X 2i,..., X ki, Y i ), i = 1,..., n, are independently and identically distributed (i.i.d.) draws from their joint distribution 3. Large outliers are unlikely: X 1i, X 2i,..., X ki and Y i have nonzero finite fourth moments 4. There is no perfect multicollinearity Remarks: Note that we do not assume any specific parametric distribution for the u i The OLS assumptions imply specific distribution results 18
15 Theorem 2.4: (Unbiasedness, consistency, normality) Given the OLS assumptions the following properties of the OLS estimators ˆβ 0, ˆβ 1,..., ˆβ k hold: 1. ˆβ 0, ˆβ 1,..., ˆβ k are unbiased estimators of β 0,..., β k. 2. ˆβ 0, ˆβ 1,..., ˆβ k are consistent estimators of β 0,..., β k. (Convergence in probability) 3. In large samples ˆβ 0, ˆβ 1,..., ˆβ k are jointly normally distributed and each single OLS estimator ˆβ j, j = 0,..., k, is normally distributed with mean β j and variance σ 2ˆβ j, that is ˆβ j N(β j, σ 2ˆβ j ). 19
16 Remarks: In general, the OLS estimators are correlated This correlation among ˆβ 0, ˆβ 1,..., ˆβ k arises from the correlation among the regressors X 1,..., X k The sampling distribution of the OLS estimators will become relevant in Section 3 (hypothesistesting, confidence intervals) 20
17 2.3. Measuresoffit in multiple regression Now: Three wellknown summary statistics that measure how well the OLS estimates fit the data Standard error of regression (SER): The SER estimates the standard deviation of the error term u i (under the assumption of homoskedasticity): SER = 1 n k 1 n û 2 i i=1 21
18 Standard error of regression: [continued] We denote the sum of squared residuals by SSR n i=1 û 2 i so that SER = SSR n k 1 Given the OLS assumptions and homoskedasticity the squared SER, (SER) 2, is an unbiased estimator of the unknown constant variance of the u i SER is a measure of the spread of the distribution of Y i around the population regression line Both measures, SER and SSR, are reported in the EViews regression output 22
19 R 2 : The R 2 is the fraction of the sample variance of the Y i explained by the regressors Equivalently, the R 2 is 1 minus the fraction of the variance of the Y i not explained by the regressors (i.e. explained by the residuals) Denoting the explained sum of squares (ESS) and the total sum of squares (TSS) by ESS = n i=1 (Ŷ i Ȳ ) 2 and TSS = respectively, we define the R 2 as R 2 = ESS TSS = 1 SSR TSS n i=1 (Y i Ȳ ) 2, 23
20 R 2 : [continued] In multiple regression, the R 2 increases whenever an additional regressor X k+1 is added to the regression model, unless the estimated coefficient ˆβ k+1 is exactly equal to zero Since in practice it is extremely unusual to have exactly ˆβ k+1 = 0, the R 2 generally increases (and never decreases) when an new regressor is added to the regression model An increase in the R 2 due to the inclusion of a new regressor does not necessarily indicate an actually improved fit of the model 24
21 Adjusted R 2 : The adjusted R 2 (in symbols: R 2 ), deflates the conventional R 2 : R 2 = 1 n 1 SSR n k 1TSS It is always true that R 2 < R 2 (why?) When adding a new regressor X k+1 to the model, the R 2 can increase or decrease (why?) The R 2 can be negative (why?) 25
22 2.4. Omittedvariable bias Now: Discussion of a phenomenon that implies violation of the first OLS assumption on Slide 18 This issue is known under the phrasing omittedvariable bias and is extremely relevant in practice Although theoretically easy to grasp, avoiding this specification problem turns out to be a nontrivial task in many empirical applications 26
23 Definition 2.5: (Omittedvariable bias) Consider the multiple regression model in Definition 2.1 on Slide 7. Omittedvariable bias is the bias in the OLS estimator ˆβ j of the coefficient β j (for j = 1,..., k) that arises when the associated regressor X j is correlated with an omitted variable. More precisely, for omittedvariable bias to occur, the following two conditions must hold: 1. X j is correlated with the omitted variable. 2. The omitted variable is a determinant of the dependent variable Y. 27
24 Example: Consider the houseprices dataset (Slides 16, 17) Using the entire set of regressors, we obtain the OLS estimate ˆβ 2 = for the BEDROOMScoefficient The correlation coefficients between the regressors are as follows: BEDROOMS BATHROOMS LOTSIZE STOREYS BEDROOMS BATHROOMS LOTSIZE STOREYS
25 Example: [continued] There is positive (significant) correlation between the variable BEDROOMS and all other regressors Excluding the other variables from the regression yields the following OLSestimates: Dependent Variable: SALEPRICE Method: Least Squares Date: 14/02/12 Time: 16:10 Sample: Included observations: 546 Variable Coefficient Std. Error tstatistic Prob. C BEDROOMS Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 3.36E+11 Schwarz criterion Log likelihood HannanQuinn criter Fstatistic DurbinWatson stat Prob(Fstatistic) The alternative OLSestimates of the BEDROOMScoefficient differ substantially 29
26 Intuitive explanation of the omittedvariable bias: Consider the variable LOTSIZE as omitted LOTSIZE is an important variable for explaining SALEPRICE If we omit LOTSIZE in the regression, it will try to enter in the only way it can, namely through its positive correlation with the included variable BEDROOMS The coefficient on BEDROOMS will confound the effect of BED ROOMS and LOTSIZE on SALEPRICE 30
27 More formal explanation: Omittedvariable bias means that the first OLS assumption on Slide 18 is violated Reasoning: In the multiple regression model the error term u i represents all factors other than the included regressors X 1,..., X k that are determinants of Y i If an omitted variable is correlated with at least one of the included regressors X 1,..., X k, then u i (which contains this factor) is correlated with the set of regressors This implies that E(u i X 1i,..., X ki ) 0 31
28 Important result: In the case of omittedvariable bias the OLS estimators on the corresponding included regressors are biased in finite samples this bias does not vanish in large samples the OLS estimators are inconsistent Solutions to omittedvariable bias: To be discussed in Section 5 32
29 2.5. Multicollinearity Definition 2.6: (Perfect multicollinearity) Consider the multiple regression model in Definition 2.1 on Slide 7. The regressors X 1,..., X k are said to be perfectly multicollinear if one of the regressors is a perfect linear function of the other regressors. Remarks: Under perfect multicollinearity the OLS estimates cannot be calculated due to division by zero in the OLS formulas Perfect multicollinearity often reflects a logical mistake in choosing the regressors or some unrecognized feature in the data set 33
30 Example: (Dummy variable trap) Consider the studentperformance dataset Suppose we partition the school districts into the 3 categories (1) rural, (2) suburban, (3) urban We represent the categories by the dummy regressors { 1 if district i is rural RURAL i = 0 otherwise and by SUBURBAN i and URBAN i analogously defined Since each district belongs to one and only one category, we have for each district i: RURAL i + SUBURBAN i + URBAN i = 1 34
31 Example: [continued] Now, let us define the constant regressor X 0 associated with the intercept coefficient β 0 in the multiple regression model on Slide 7 by X 0i 1 for i = 1,... n Then, for i = 1,..., n, the following relationship holds among the regressors: Perfect multicollinearity X 0i = RURAL i + SUBURBAN i + URBAN i To estimate the regression we must exclude either one of the dummy regressors or the constant regressor X 0 (the intercept β 0 ) from the regression 35
32 Theorem 2.7: (Dummy variable trap) Let there be G different categories in the data set represented by G dummy regressors. If 1. each observation i falls into one and only one category, 2. there is an intercept (constant regressor) in the regression, 3. all G dummy regressors are included as regressors, then regression estimation fails because of perfect multicollinearity. Usual remedy: Exclude one of the dummy regressors (G 1 dummy regressors are sufficient) 36
33 Definition 2.8: (Imperfect multicollinearity) Consider the multiple regression model in Definition 2.1 on Slide 7. The regressors X 1,..., X k are said to be imperfectly multicollinear if two or more of the regressors are highly correlated in the sense that there is a linear function of the regressors that is highly correlated with another regressor. Remarks: Imperfect multicollinearity does not pose any (numeric) problems in calculating OLS estimates However, if regressors are imperfectly multicollinear, then the coefficients on at least one individual regressor will be imprecisely estimated 37
34 Remarks: [continued] Techniques for identifying and mitigating imperfect multicollinearity are presented in econometric textbooks (e.g. Hill et al., 2010, pp ) 38
Econometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
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 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 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 DickeyFuller Test...6 3.
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 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 Rsquared? Rsquared Published in Agricultural Economics 0.45 Best article of 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 informationAir passenger departures forecast models A technical note
Ministry of Transport Air passenger departures forecast models A technical note By Haobo Wang Financial, Economic and Statistical Analysis Page 1 of 15 1. Introduction Sine 1999, the Ministry of Business,
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 informationUK GDP is the best predictor of UK GDP, literally.
UK GDP IS THE BEST PREDICTOR OF UK GDP, LITERALLY ERIK BRITTON AND DANNY GABAY 6 NOVEMBER 2009 UK GDP is the best predictor of UK GDP, literally. The ONS s preliminary estimate of UK GDP for the third
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 15 scale to 0100 scores When you look at your report, you will notice that the scores are reported on a 0100 scale, even though respondents
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 informationOn the Degree of Openness of an Open Economy Carlos Alfredo Rodriguez, Universidad del CEMA Buenos Aires, Argentina
On the Degree of Openness of an Open Economy Carlos Alfredo Rodriguez, Universidad del CEMA Buenos Aires, Argentina car@cema.edu.ar www.cema.edu.ar\~car Version1February 14,2000 All data can be consulted
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 informationThe relationship between stock market parameters and interbank lending market: an empirical evidence
Magomet Yandiev Associate Professor, Department of Economics, Lomonosov Moscow State University mag2097@mail.ru Alexander Pakhalov, PG student, Department of Economics, Lomonosov Moscow State University
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 informationHYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION
HYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate
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 informationIMPACT OF WORKING CAPITAL MANAGEMENT ON PROFITABILITY
IMPACT OF WORKING CAPITAL MANAGEMENT ON PROFITABILITY Hina Agha, Mba, Mphil Bahria University Karachi Campus, Pakistan Abstract The main purpose of this study is to empirically test the impact of working
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 informationModule 5: Multiple Regression Analysis
Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College
More informationSection 14 Simple Linear Regression: Introduction to Least Squares Regression
Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship
More informationThe Impact of Privatization in Insurance Industry on Insurance Efficiency in Iran
The Impact of Privatization in Insurance Industry on Insurance Efficiency in Iran Shahram Gilaninia 1, Hosein Ganjinia, Azadeh Asadian 3 * 1. Department of Industrial Management, Islamic Azad University,
More informationRegression Analysis (Spring, 2000)
Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity
More informationThe Effect of Seasonality in the CPI on Indexed Bond Pricing and Inflation Expectations
The Effect of Seasonality in the CPI on Indexed Bond Pricing and Inflation Expectations Roy Stein* *Research Department, Roy Stein roy.stein@boi.org.il, tel: 026552559 This research was partially supported
More information5. Multiple regression
5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful
More informationCompetition as an Effective Tool in Developing Social Marketing Programs: Driving Behavior Change through Online Activities
Competition as an Effective Tool in Developing Social Marketing Programs: Driving Behavior Change through Online Activities Corina ŞERBAN 1 ABSTRACT Nowadays, social marketing practices represent an important
More informationEconometric Principles and Data Analysis
Econometric Principles and Data Analysis product: 4339 course code: c230 c330 Econometric Principles and Data Analysis Centre for Financial and Management Studies SOAS, University of London 1999, revised
More informationLecture 15. Endogeneity & Instrumental Variable Estimation
Lecture 15. Endogeneity & Instrumental Variable Estimation Saw that measurement error (on right hand side) means that OLS will be biased (biased toward zero) Potential solution to endogeneity instrumental
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 informationChapter 3: The Multiple Linear Regression Model
Chapter 3: The Multiple Linear Regression Model Advanced Econometrics  HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics
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 informationPlease follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used
More informationRegression with a Binary Dependent Variable
Regression with a Binary Dependent Variable Chapter 9 Michael Ash CPPA Lecture 22 Course Notes Endgame Takehome final Distributed Friday 19 May Due Tuesday 23 May (Paper or emailed PDF ok; no Word, Excel,
More informationDeterminants of Stock Market Performance in Pakistan
Determinants of Stock Market Performance in Pakistan Mehwish Zafar Sr. Lecturer Bahria University, Karachi campus Abstract Stock market performance, economic and political condition of a country is interrelated
More informationRegression stepbystep using Microsoft Excel
Step 1: Regression stepbystep 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 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 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 informationSource engine marketing: A preliminary empirical analysis of web search data
Source engine marketing: A preliminary empirical analysis of web search data ABSTRACT Bruce Q. Budd Alfaisal University The purpose of this paper is to empirically investigate a website performance and
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 informationEuropean Journal of Business and Management ISSN 22221905 (Paper) ISSN 22222839 (Online) Vol.5, No.30, 2013
The Impact of Stock Market Liquidity on Economic Growth in Jordan Shatha AbdulKhaliq Assistant Professor,AlBlqa Applied University, Jordan * Email of the corresponding author: yshatha@gmail.com Abstract
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 informationIntegrated Resource Plan
Integrated Resource Plan March 19, 2004 PREPARED FOR KAUA I ISLAND UTILITY COOPERATIVE LCG Consulting 4962 El Camino Real, Suite 112 Los Altos, CA 94022 6509629670 1 IRP 1 ELECTRIC LOAD FORECASTING 1.1
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationMISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group
MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could
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 information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
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 informationForecasting Using Eviews 2.0: An Overview
Forecasting Using Eviews 2.0: An Overview Some Preliminaries In what follows it will be useful to distinguish between ex post and ex ante forecasting. In terms of time series modeling, both predict values
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
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 informationThe Relationship between Life Insurance and Economic Growth: Evidence from India
Global Journal of Management and Business Studies. ISSN 22489878 Volume 3, Number 4 (2013), pp. 413422 Research India Publications http://www.ripublication.com/gjmbs.htm The Relationship between Life
More informationUnderstanding Retention among Private Baccalaureate Liberal Arts Colleges
Understanding Retention among Private Baccalaureate Liberal Arts Colleges Thursday April 19, 2012 Author: Katherine S. Hanson 1 Abstract This paper attempts to analyze the explanatory variables that best
More informationRidge Regression. Patrick Breheny. September 1. Ridge regression Selection of λ Ridge regression in R/SAS
Ridge Regression Patrick Breheny September 1 Patrick Breheny BST 764: Applied Statistical Modeling 1/22 Ridge regression: Definition Definition and solution Properties As mentioned in the previous lecture,
More informationCoefficient of Determination
Coefficient of Determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation ŷ = b 0 + b 1 x performs as a predictor of y. R 2 is computed
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 informationQuick Stata Guide by Liz Foster
by Liz Foster Table of Contents Part 1: 1 describe 1 generate 1 regress 3 scatter 4 sort 5 summarize 5 table 6 tabulate 8 test 10 ttest 11 Part 2: Prefixes and Notes 14 by var: 14 capture 14 use of the
More informationPrice volatility in the silver spot market: An empirical study using Garch applications
Price volatility in the silver spot market: An empirical study using Garch applications ABSTRACT Alan Harper, South University Zhenhu Jin Valparaiso University Raufu Sokunle UBS Investment Bank Manish
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More information2. Simple Linear Regression
Research methods  II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
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 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 informationClustering in the Linear Model
Short Guides to Microeconometrics Fall 2014 Kurt Schmidheiny Universität Basel Clustering in the Linear Model 2 1 Introduction Clustering in the Linear Model This handout extends the handout on The Multiple
More informationDepartment of Economics Session 2012/2013. EC352 Econometric Methods. Solutions to Exercises from Week 10 + 0.0077 (0.052)
Department of Economics Session 2012/2013 University of Essex Spring Term Dr Gordon Kemp EC352 Econometric Methods Solutions to Exercises from Week 10 1 Problem 13.7 This exercise refers back to Equation
More informationUniwersytet Ekonomiczny
Uniwersytet Ekonomiczny George Matysiak Introduction to modelling & forecasting December 15 th, 2014 Agenda Modelling and forecasting  Models Approaches towards modelling and forecasting Forecasting commercial
More informationFORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits
Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies
More informationFinancial Risk Management Exam Sample Questions/Answers
Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM99, Question 4 Random walk assumes that returns from one time period
More informationCorrelation of International Stock Markets Before and During the Subprime Crisis
173 Correlation of International Stock Markets Before and During the Subprime Crisis Ioana Moldovan 1 Claudia Medrega 2 The recent financial crisis has spread to markets worldwide. The correlation of evolutions
More informationEDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION
EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 510 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day
More informationReview of Bivariate Regression
Review of Bivariate Regression A.Colin Cameron Department of Economics University of California  Davis accameron@ucdavis.edu October 27, 2006 Abstract This provides a review of material covered in an
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 informationII. DISTRIBUTIONS distribution normal distribution. standard scores
Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,
More informationHow Far is too Far? Statistical Outlier Detection
How Far is too Far? Statistical Outlier Detection Steven Walfish President, Statistical Outsourcing Services steven@statisticaloutsourcingservices.com 30325329 Outline What is an Outlier, and Why are
More informationDirections for using SPSS
Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...
More informationFrom the help desk: Bootstrapped standard errors
The Stata Journal (2003) 3, Number 1, pp. 71 80 From the help desk: Bootstrapped standard errors Weihua Guan Stata Corporation Abstract. Bootstrapping is a nonparametric approach for evaluating the distribution
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 information4. Multiple Regression in Practice
30 Multiple Regression in Practice 4. Multiple Regression in Practice The preceding chapters have helped define the broad principles on which regression analysis is based. What features one should look
More informationChapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
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 informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos (UC3MUPM)
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 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 information3.2 Measures of Spread
3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationModerator and Mediator Analysis
Moderator and Mediator Analysis Seminar General Statistics Marijtje van Duijn October 8, Overview What is moderation and mediation? What is their relation to statistical concepts? Example(s) October 8,
More informationWhat s New in Econometrics? Lecture 8 Cluster and Stratified Sampling
What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and
More informationCausal Forecasting Models
CTL.SC1x Supply Chain & Logistics Fundamentals Causal Forecasting Models MIT Center for Transportation & Logistics Causal Models Used when demand is correlated with some known and measurable environmental
More informationPARTNERSHIP IN SOCIAL MARKETING PROGRAMS. SOCIALLY RESPONSIBLE COMPANIES AND NONPROFIT ORGANIZATIONS ENGAGEMENT IN SOLVING SOCIETY S PROBLEMS
PARTNERSHIP IN SOCIAL MARKETING PROGRAMS. SOCIALLY RESPONSIBLE COMPANIES AND NONPROFIT ORGANIZATIONS ENGAGEMENT IN SOLVING SOCIETY S PROBLEMS Corina Şerban The Bucharest Academy of Economic Studies, Romania
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 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 informationproblem arises when only a nonrandom sample is available differs from censored regression model in that x i is also unobserved
4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a nonrandom
More informationSPSS Guide: Regression Analysis
SPSS Guide: Regression Analysis I put this together to give you a stepbystep guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar
More informationChapter 3 Quantitative Demand Analysis
Managerial Economics & Business Strategy Chapter 3 uantitative Demand Analysis McGrawHill/Irwin Copyright 2010 by the McGrawHill Companies, Inc. All rights reserved. Overview I. The Elasticity Concept
More informationMARKETING COMMUNICATION IN ONLINE SOCIAL PROGRAMS: OHANIAN MODEL OF SOURCE CREDIBILITY
MARKETING COMMUNICATION IN ONLINE SOCIAL PROGRAMS: OHANIAN MODEL OF SOURCE CREDIBILITY Serban Corina The Bucharest Academy of Economic Studies The Faculty of Marketing The development of the Internet as
More informationStepwise Regression. Chapter 311. Introduction. Variable Selection Procedures. Forward (StepUp) Selection
Chapter 311 Introduction Often, theory and experience give only general direction as to which of a pool of candidate variables (including transformed variables) should be included in the regression model.
More informationIntegrating Financial Statement Modeling and Sales Forecasting
Integrating Financial Statement Modeling and Sales Forecasting John T. Cuddington, Colorado School of Mines Irina Khindanova, University of Denver ABSTRACT This paper shows how to integrate financial statement
More informationA General Approach to Variance Estimation under Imputation for Missing Survey Data
A General Approach to Variance Estimation under Imputation for Missing Survey Data J.N.K. Rao Carleton University Ottawa, Canada 1 2 1 Joint work with J.K. Kim at Iowa State University. 2 Workshop on Survey
More information3.1 Stationary Processes and Mean Reversion
3. Univariate Time Series Models 3.1 Stationary Processes and Mean Reversion Definition 3.1: A time series y t, t = 1,..., T is called (covariance) stationary if (1) E[y t ] = µ, for all t Cov[y t, y t
More informationThe Basic TwoLevel Regression Model
2 The Basic TwoLevel Regression Model The multilevel regression model has become known in the research literature under a variety of names, such as random coefficient model (de Leeuw & Kreft, 1986; Longford,
More information