Simple Linear Regression Model & Introduction to. OLS Estimation
|
|
- Rosalyn Robbins
- 6 years ago
- Views:
Transcription
1 Inside ECOOMICS Introduction to Econometrics Simple Linear Regression Model & Introduction to Introduction OLS Estimation We are interested in a model that explains a variable y in terms of other variables x. We are also interested in finding how much y changes as a result of change in x. The simple linear regression model is used to study the relationship between an independent variable and the explanatory variables. For instance we have one explanatory variable x and one dependent variable y as shown below. It is common to include a constant β 0 which indicates the point of intersection on the y axis. The error term denoted by u represents the factors other than x that have an effect on the dependent variable y. Please note in this document we shall be dealing with only cross sectional data y = β 0 + β 1 x 11 + u The β k are unknown coefficients and the x ik are the regressors. For the regressors x ik the i denotes the observation or individual and are indexed from 1 to, where is called the sample size. So for instance the regressor x 13 means that the coefficient relates to the third regressor of the model for individual or observation 1. In the above equation this is the first regressor for individual or observation 1. y 1 = β 0 + β 1 x 11 + β x 1 + β 3 x 13 + β 4 x 14 + u 1 In the equation above we have linear regression for the 1st observation or individual. y = β 0 + β 1 x 1 + β x + β 3 x 3 + β 4 x 4 + u In the equation above we now have an equation for the second observation or individual. Please note that the regressors are the same for both individuals but they may, have different beta coefficients. For example x i1 and x i could be variables such as education and age. Therefore x 11 and x 1 are education and age for the first individual and x 1 and x are the education and age regressors for the second individual. Suppose that for some k = 1,., k, x ik denotes age of individual i and if the individual i were one year older, the value of the dependent variable y i will increase by β k. Matrix otation y i = β 0 + β 1 x i1 + β x i + β 3 x i3 + + β k x ik + u i (1) In Matrix otation we can write the model as Y = Xβ + u () Where Y is a vector of dependent variablesy = (y 1, y,, y n ). X is a matrix of independent variables with dimensions n k + 1. The 1 column of the matrix is there for the intercept term. The error term is also a vector u = (u 1, u,, u n ). 1
2 Inside ECOOMICS y 1 1 x 11 Y =, X = y 1 x n1 x 1 x 1k β 0 u 1, β =, u = x n x K β k u Ordinary Least Squares Estimation There are various methods to estimate the coefficients. Ordinary Least Squares (OLS) is just one these methods. OLS is relatively simple and has some attractive properties that make it a popular estimation method. The OLS estimator minimises the sum of squared residuals. In the diagram above we see that this line is the line that minimises the sum of squared residuals. So the sum of the squared distance between the errors and the line is minimised with this line. If the line was to change then the sum of squared residuals would be larger and would not be the minimum variance estimator. We prefer estimators with OLS Assumptions Assumption 1: Independent and identically distributed (I.I.D) I.I.D observations: (x i, y i ) is independent from, and has the same distribution as, (x j, y j ) for all i j; We do not observe the population but only a sample therefore we assume that an I.I.D sample can be drawn from the population. The I.I.D assumption makes it easier for us to interpret some of the other assumption. It also allows us to use asymptotic results (as the sample size ). Assumption : Linearity The regresssion model is linear in the parameters, (this is evident in the structure of equation (1)) Essentially the response variable is a linear function of the regressors. In the case where the models may not be linear in parameters a linear regression model will be an approximation. However this approximation often results in minimal accuracy.
3 Inside ECOOMICS Assumption 3: Uncorrelatedness E[x i, u i ] = 0 It is assumed that E[u i ] = 0, which means that the errors in the regression should have conditional mean of zero. Therefore assumption 3 is equivalent to errors being uncorrelated with the regressors. If this assumption holds we can call the regressor exogenous variables. If however it does not hold then the regressors that are correlated with the error term are called endogenous variables. If the regression contains endogenous variables the OLS estimates will be invalid and instrumental variables will be required. Assumption 4: Full Rank (OLS) rank E[x i, x i ] = k This assumption eliminates the possibility of collinearity. In practice collinearity is not a large problem, esspecially if the sample size is large. Assumption 5: Homoskedasticity This can be written as E u i X = σ E u i x i x i = E u i E[x i x i ] = σ A, where σ E u i The u i are known as error terms and include all the differences in y i that are not captured by the x variables. Homoskedasticity means that the errors have the same variance σ for each observation. The variance of the error is treated as a constant. This means that the 1st observation and the last observation in the sample will have equal and identical variance for the error. As a result the probability distribution for the dependent variable has the same variance regardless of the values for the explanatory variables. If this assumption is violated we have hetroskedasticity which means that the variance of the error term is not constant and differs across observations. Aside: If hetroskedasticity is present the weighted least squares estimator will be a more efficient estimator and can be used. If the errors have infinite variance robust estimation techniques are preferred. Assumption 6: Exogeneity This implies that E β β x 1,, x = 0 E[u i x i ] = 0 3
4 Inside ECOOMICS This means that β is an unbiased estimator of β conditional on the regressors x 1,, x. As E β = β, irrespective of the value of β. This assumption is similar to assumption 3, however assumption 3 is a stronger assumption of strict exogeneity. Deriving the OLS Estimator (Summation otation) We will now minimise the sum of squared residuals to derive the OLS estimator. The OLS Estimator is BLUE (Best Linear Unbiased Estimator). We will firstly derive the OLS estimator with sigma notation. Assuming we have an intercept and one regressor so K =. Equation y i = β 0 + β 1 x i + u i The data collected on the x s and y s will be used to construct estimates for β 0 and β 1. OLS is on technique of estimation and requires the minimisation of the sum of square residuals. β 0β 1 = min β0,β 1 (y i β 0 + β 1x i ) (1) The First Order Conditions are as follows Please note that this implies that, y i β 0 + β 1x i = 0 x i y i β 0 + β 1x i = 0 u i = 0 and x i u i = 0 where x = 1 x i therefore the above equation holds because x u i = 0 x is the sample average of the independent variable and comes from the sum of all x i divided by the number of observations in the sample. (Remember there are observations i. ). Similarly the sample average of the dependent variable is y = 1 y i Let us turn our attention to the first FOC and solve for β 0 1 y i β 0 + β 1x i = 0 1 y i 1 β β 1x i = 0 y β 0 + β 1x = 0 4
5 Inside ECOOMICS β 0 = y β 1x ow our task is to solve for β 1 x i y i β 0 + β 1x i = 0 First we can eliminate the - as this is just a constant. Then we can start by multiplying out the equation to get the following expression. x i y i β 0x i β 1x i x i = 0 x i y i β 0x i β 1x i = 0 Substitute the expression for β 0 into the above equation x i y i x i (y β 1x ) β 1x i = 0 The summation term applies to everything in the equation so to work out the step it is best to write it out. (Remember that you can always put a constant term out in front of the summation) x i y i y x i β 1x x i β 1 x i = 0 Using the properties x = 1 x i and y = 1 y i x i y i y x β 1x x β 1 x i = 0 x i y i y x β 1 x x i = 0 (x i x )(y i y ) (x i x ) β 1 Rearrange for β 1 β 1 = x iy i x y x i x β 1 = (x i x )(y i y) (x i x ) 5
6 Inside ECOOMICS Finally we have solved for both β 0 and β 1 OLS For an Arbitrary k > β 1 = β 0 = y β 1x (x i x )(y i y) (x i x ) So far we have only had two parameters the intercept β 0 and the explanatory variable β 1. When k > the previous equations are incorrect. If we have an arbitrary k number of variables we need to minimise the sum of squared residuals in terms of k terms. β = min (y i x i β ) x i (y i x i β ) = 0 x i (y i x i β ) = 0, which is the same as x i u i = 0 x i x i β = x i y i β = x i x i 1 x i y i This equation is equivalent to the Matrix otation OLS Estimator equation β = (X X) 1 X Y Deriving the OLS Estimator (Matrix otation) We will now minimise the sum of squared residuals to derive the OLS estimator using Matrix algebra. Y = Xβ + u u = Y Xβ min u u = Y Xβ Y Xβ Use matrix calculus d A A da d Y Xβ Y Xβ dβ = A and d(cb) db = 0 = C 6
7 Inside ECOOMICS ( X) Y Xβ = 0 X Y Xβ = 0 X Y X Xβ = 0 X Xβ = X Y Assuming that (X X) 1 exists (Assumption 4) β = (X X) 1 X Y Where this β is the OLS estimator of the true population beta β Key Equations 1. Linear Model y i = β 0 + β 1 x i1 + β x i + β 3 x i3 + + β k x ik + u i. Linear Model Matrix otation Y = Xβ + u 3. OLS Estimator β 0 = y β 1x 4. OLS Estimator β 1 = (x i x )(y i y) (x i x ) 5. OLS Estimator for Arbitrary k > β = x i x i 1 x i y i 6. OLS Estimator Matrix otation β = (X X) 1 X Y Gauss-Markov Theorem In the classical linear regression model under Assumptions 1,, 4 and 6 the OLS estimator of equation is the minimum variance unbiased estimator of β. The OLS Estimator is BLUE (Best Linear Unbiased Estimator). For the proof of OLS properties please refer to the document labelled Properties of OLS (Proofs). Also for a brief OLS derivation reference refer to the document labelled Derivation of the OLS Estimator. 7
LECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationIntroduction to Econometrics. Heteroskedasticity
Introduction to Econometrics Introduction Heteroskedasticity When the variance of the errors changes across segments of the population, where the segments are determined by different values for the explanatory
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationSimple Linear Regression: The Model
Simple Linear Regression: The Model task: quantifying the effect of change X in X on Y, with some constant β 1 : Y = β 1 X, linear relationship between X and Y, however, relationship subject to a random
More informationReview 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,...,
More informationIntroduction to Econometrics Midterm Examination Fall 2005 Answer Key
Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is
More informationMultiple 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
More informationLinear models. Linear models are computationally convenient and remain widely used in. applied econometric research
Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y
More information1 Motivation for Instrumental Variable (IV) Regression
ECON 370: IV & 2SLS 1 Instrumental Variables Estimation and Two Stage Least Squares Econometric Methods, ECON 370 Let s get back to the thiking in terms of cross sectional (or pooled cross sectional) data
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationLecture 3: Multiple Regression
Lecture 3: Multiple Regression R.G. Pierse 1 The General Linear Model Suppose that we have k explanatory variables Y i = β 1 + β X i + β 3 X 3i + + β k X ki + u i, i = 1,, n (1.1) or Y i = β j X ji + u
More informationMotivation 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
More informationMultiple Regression Analysis
Multiple Regression Analysis y = 0 + 1 x 1 + x +... k x k + u 6. Heteroskedasticity What is Heteroskedasticity?! Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,
More informationQuantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017
Summary of Part II Key Concepts & Formulas Christopher Ting November 11, 2017 christopherting@smu.edu.sg http://www.mysmu.edu/faculty/christophert/ Christopher Ting 1 of 16 Why Regression Analysis? Understand
More informationLeast Squares Estimation-Finite-Sample Properties
Least Squares Estimation-Finite-Sample Properties Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Finite-Sample 1 / 29 Terminology and Assumptions 1 Terminology and Assumptions
More informationIntermediate Econometrics
Intermediate Econometrics Heteroskedasticity Text: Wooldridge, 8 July 17, 2011 Heteroskedasticity Assumption of homoskedasticity, Var(u i x i1,..., x ik ) = E(u 2 i x i1,..., x ik ) = σ 2. That is, the
More informationRepeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data
Panel data Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data - possible to control for some unobserved heterogeneity - possible
More informationIntroduction to Estimation Methods for Time Series models. Lecture 1
Introduction to Estimation Methods for Time Series models Lecture 1 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 1 SNS Pisa 1 / 19 Estimation
More informationOrdinary Least Squares Regression
Ordinary Least Squares Regression Goals for this unit More on notation and terminology OLS scalar versus matrix derivation Some Preliminaries In this class we will be learning to analyze Cross Section
More informationLinear Models in Econometrics
Linear Models in Econometrics Nicky Grant At the most fundamental level econometrics is the development of statistical techniques suited primarily to answering economic questions and testing economic theories.
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationINTRODUCTION TO BASIC LINEAR REGRESSION MODEL
INTRODUCTION TO BASIC LINEAR REGRESSION MODEL 13 September 2011 Yogyakarta, Indonesia Cosimo Beverelli (World Trade Organization) 1 LINEAR REGRESSION MODEL In general, regression models estimate the effect
More informationThe Simple Regression Model. Simple Regression Model 1
The Simple Regression Model Simple Regression Model 1 Simple regression model: Objectives Given the model: - where y is earnings and x years of education - Or y is sales and x is spending in advertising
More informationRegression Analysis for Data Containing Outliers and High Leverage Points
Alabama Journal of Mathematics 39 (2015) ISSN 2373-0404 Regression Analysis for Data Containing Outliers and High Leverage Points Asim Kumer Dey Department of Mathematics Lamar University Md. Amir Hossain
More informationthe error term could vary over the observations, in ways that are related
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance Var(u i x i ) = σ 2 is common to all observations i = 1,..., n In many applications, we may
More informationRecent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data
Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data July 2012 Bangkok, Thailand Cosimo Beverelli (World Trade Organization) 1 Content a) Classical regression model b)
More informationEconomics 113. Simple Regression Assumptions. Simple Regression Derivation. Changing Units of Measurement. Nonlinear effects
Economics 113 Simple Regression Models Simple Regression Assumptions Simple Regression Derivation Changing Units of Measurement Nonlinear effects OLS and unbiased estimates Variance of the OLS estimates
More informationECON 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
More informationHomoskedasticity. Var (u X) = σ 2. (23)
Homoskedasticity How big is the difference between the OLS estimator and the true parameter? To answer this question, we make an additional assumption called homoskedasticity: Var (u X) = σ 2. (23) This
More informationEconometrics I KS. Module 1: Bivariate Linear Regression. Alexander Ahammer. This version: March 12, 2018
Econometrics I KS Module 1: Bivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: March 12, 2018 Alexander Ahammer (JKU) Module 1: Bivariate
More informationEconometrics I Lecture 3: The Simple Linear Regression Model
Econometrics I Lecture 3: The Simple Linear Regression Model Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 32 Outline Introduction Estimating
More informationChapter 2: simple regression model
Chapter 2: simple regression model Goal: understand how to estimate and more importantly interpret the simple regression Reading: chapter 2 of the textbook Advice: this chapter is foundation of econometrics.
More information1. The OLS Estimator. 1.1 Population model and notation
1. The OLS Estimator OLS stands for Ordinary Least Squares. There are 6 assumptions ordinarily made, and the method of fitting a line through data is by least-squares. OLS is a common estimation methodology
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna November 23, 2013 Outline Introduction
More informationRegression #4: Properties of OLS Estimator (Part 2)
Regression #4: Properties of OLS Estimator (Part 2) Econ 671 Purdue University Justin L. Tobias (Purdue) Regression #4 1 / 24 Introduction In this lecture, we continue investigating properties associated
More informationMultiple 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
More informationEssential of Simple regression
Essential of Simple regression We use simple regression when we are interested in the relationship between two variables (e.g., x is class size, and y is student s GPA). For simplicity we assume the relationship
More informationThe Simple Regression Model. Part II. The Simple Regression Model
Part II The Simple Regression Model As of Sep 22, 2015 Definition 1 The Simple Regression Model Definition Estimation of the model, OLS OLS Statistics Algebraic properties Goodness-of-Fit, the R-square
More informationHeteroskedasticity. We now consider the implications of relaxing the assumption that the conditional
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance V (u i x i ) = σ 2 is common to all observations i = 1,..., In many applications, we may suspect
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 11, 2012 Outline Heteroskedasticity
More informationEconometrics Multiple Regression Analysis: Heteroskedasticity
Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties
More informationIntermediate Econometrics
Intermediate Econometrics Markus Haas LMU München Summer term 2011 15. Mai 2011 The Simple Linear Regression Model Considering variables x and y in a specific population (e.g., years of education and wage
More informationECON3150/4150 Spring 2015
ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2
More information1. The Multivariate Classical Linear Regression Model
Business School, Brunel University MSc. EC550/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 08956584) Lecture Notes 5. The
More informationA Course on Advanced Econometrics
A Course on Advanced Econometrics Yongmiao Hong The Ernest S. Liu Professor of Economics & International Studies Cornell University Course Introduction: Modern economies are full of uncertainties and risk.
More information1 The Multiple Regression Model: Freeing Up the Classical Assumptions
1 The Multiple Regression Model: Freeing Up the Classical Assumptions Some or all of classical assumptions were crucial for many of the derivations of the previous chapters. Derivation of the OLS estimator
More informationModel Mis-specification
Model Mis-specification Carlo Favero Favero () Model Mis-specification 1 / 28 Model Mis-specification Each specification can be interpreted of the result of a reduction process, what happens if the reduction
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More informationECNS 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
More informationA Course in Applied Econometrics Lecture 7: Cluster Sampling. Jeff Wooldridge IRP Lectures, UW Madison, August 2008
A Course in Applied Econometrics Lecture 7: Cluster Sampling Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of roups and
More informationThe Multiple Regression Model Estimation
Lesson 5 The Multiple Regression Model Estimation Pilar González and Susan Orbe Dpt Applied Econometrics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 5 Regression model:
More informationstatistical sense, from the distributions of the xs. The model may now be generalized to the case of k regressors:
Wooldridge, Introductory Econometrics, d ed. Chapter 3: Multiple regression analysis: Estimation In multiple regression analysis, we extend the simple (two-variable) regression model to consider the possibility
More informationWooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model
Wooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model Most of this course will be concerned with use of a regression model: a structure in which one or more explanatory
More informationMultivariate Regression Analysis
Matrices and vectors The model from the sample is: Y = Xβ +u with n individuals, l response variable, k regressors Y is a n 1 vector or a n l matrix with the notation Y T = (y 1,y 2,...,y n ) 1 x 11 x
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationY i = η + ɛ i, i = 1,...,n.
Nonparametric tests If data do not come from a normal population (and if the sample is not large), we cannot use a t-test. One useful approach to creating test statistics is through the use of rank statistics.
More informationEMERGING MARKETS - Lecture 2: Methodology refresher
EMERGING MARKETS - Lecture 2: Methodology refresher Maria Perrotta April 4, 2013 SITE http://www.hhs.se/site/pages/default.aspx My contact: maria.perrotta@hhs.se Aim of this class There are many different
More informationAdvanced Quantitative Methods: ordinary least squares
Advanced Quantitative Methods: Ordinary Least Squares University College Dublin 31 January 2012 1 2 3 4 5 Terminology y is the dependent variable referred to also (by Greene) as a regressand X are the
More informationReliability of inference (1 of 2 lectures)
Reliability of inference (1 of 2 lectures) Ragnar Nymoen University of Oslo 5 March 2013 1 / 19 This lecture (#13 and 14): I The optimality of the OLS estimators and tests depend on the assumptions of
More informationEcon 510 B. Brown Spring 2014 Final Exam Answers
Econ 510 B. Brown Spring 2014 Final Exam Answers Answer five of the following questions. You must answer question 7. The question are weighted equally. You have 2.5 hours. You may use a calculator. Brevity
More informationUNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS FINAL EXAM (Type B) 2. This document is self contained. Your are not allowed to use any other material.
DURATION: 125 MINUTES Directions: UNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS FINAL EXAM (Type B) 1. This is an example of a exam that you can use to self-evaluate about the contents of the course Econometrics
More informationLinear Regression with Time Series Data
Econometrics 2 Linear Regression with Time Series Data Heino Bohn Nielsen 1of21 Outline (1) The linear regression model, identification and estimation. (2) Assumptions and results: (a) Consistency. (b)
More informationEnvironmental Econometrics
Environmental Econometrics Syngjoo Choi Fall 2008 Environmental Econometrics (GR03) Fall 2008 1 / 37 Syllabus I This is an introductory econometrics course which assumes no prior knowledge on econometrics;
More informationEconometrics II - EXAM Answer each question in separate sheets in three hours
Econometrics II - EXAM Answer each question in separate sheets in three hours. Let u and u be jointly Gaussian and independent of z in all the equations. a Investigate the identification of the following
More informationBusiness Economics BUSINESS ECONOMICS. PAPER No. : 8, FUNDAMENTALS OF ECONOMETRICS MODULE No. : 3, GAUSS MARKOV THEOREM
Subject Business Economics Paper No and Title Module No and Title Module Tag 8, Fundamentals of Econometrics 3, The gauss Markov theorem BSE_P8_M3 1 TABLE OF CONTENTS 1. INTRODUCTION 2. ASSUMPTIONS OF
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate
More informationEconometrics Master in Business and Quantitative Methods
Econometrics Master in Business and Quantitative Methods Helena Veiga Universidad Carlos III de Madrid Models with discrete dependent variables and applications of panel data methods in all fields of economics
More informationRegression Models - Introduction
Regression Models - Introduction In regression models there are two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent
More informationHeteroscedasticity and Autocorrelation
Heteroscedasticity and Autocorrelation Carlo Favero Favero () Heteroscedasticity and Autocorrelation 1 / 17 Heteroscedasticity, Autocorrelation, and the GLS estimator Let us reconsider the single equation
More informationP1.T2. Stock & Watson Chapters 4 & 5. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
P1.T2. Stock & Watson Chapters 4 & 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal
More informationXβ is a linear combination of the columns of X: Copyright c 2010 Dan Nettleton (Iowa State University) Statistics / 25 X =
The Gauss-Markov Linear Model y Xβ + ɛ y is an n random vector of responses X is an n p matrix of constants with columns corresponding to explanatory variables X is sometimes referred to as the design
More informationBasic Econometrics - rewiev
Basic Econometrics - rewiev Jerzy Mycielski Model Linear equation y i = x 1i β 1 + x 2i β 2 +... + x Ki β K + ε i, dla i = 1,..., N, Elements dependent (endogenous) variable y i independent (exogenous)
More informationEC3062 ECONOMETRICS. THE MULTIPLE REGRESSION MODEL Consider T realisations of the regression equation. (1) y = β 0 + β 1 x β k x k + ε,
THE MULTIPLE REGRESSION MODEL Consider T realisations of the regression equation (1) y = β 0 + β 1 x 1 + + β k x k + ε, which can be written in the following form: (2) y 1 y 2.. y T = 1 x 11... x 1k 1
More informationSimple Linear Regression Estimation and Properties
Simple Linear Regression Estimation and Properties Outline Review of the Reading Estimate parameters using OLS Other features of OLS Numerical Properties of OLS Assumptions of OLS Goodness of Fit Checking
More informationEconometrics - 30C00200
Econometrics - 30C00200 Lecture 11: Heteroskedasticity Antti Saastamoinen VATT Institute for Economic Research Fall 2015 30C00200 Lecture 11: Heteroskedasticity 12.10.2015 Aalto University School of Business
More informationA Course in Applied Econometrics Lecture 14: Control Functions and Related Methods. Jeff Wooldridge IRP Lectures, UW Madison, August 2008
A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. Linear-in-Parameters Models: IV versus Control Functions 2. Correlated
More informationEconomics 308: Econometrics Professor Moody
Economics 308: Econometrics Professor Moody References on reserve: Text Moody, Basic Econometrics with Stata (BES) Pindyck and Rubinfeld, Econometric Models and Economic Forecasts (PR) Wooldridge, Jeffrey
More informationMS&E 226: Small Data
MS&E 226: Small Data Lecture 6: Bias and variance (v5) Ramesh Johari ramesh.johari@stanford.edu 1 / 49 Our plan today We saw in last lecture that model scoring methods seem to be trading off two different
More information1 Introduction to Generalized Least Squares
ECONOMICS 7344, Spring 2017 Bent E. Sørensen April 12, 2017 1 Introduction to Generalized Least Squares Consider the model Y = Xβ + ɛ, where the N K matrix of regressors X is fixed, independent of the
More informationIV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors
IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors Laura Mayoral IAE, Barcelona GSE and University of Gothenburg Gothenburg, May 2015 Roadmap of the course Introduction.
More informationWISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
More informationLab 07 Introduction to Econometrics
Lab 07 Introduction to Econometrics Learning outcomes for this lab: Introduce the different typologies of data and the econometric models that can be used Understand the rationale behind econometrics Understand
More informationLecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)
Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model
More informationEconometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.
More informationPanel Data Models. James L. Powell Department of Economics University of California, Berkeley
Panel Data Models James L. Powell Department of Economics University of California, Berkeley Overview Like Zellner s seemingly unrelated regression models, the dependent and explanatory variables for panel
More informationEconometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
More informationHeteroskedasticity. Part VII. Heteroskedasticity
Part VII Heteroskedasticity As of Oct 15, 2015 1 Heteroskedasticity Consequences Heteroskedasticity-robust inference Testing for Heteroskedasticity Weighted Least Squares (WLS) Feasible generalized Least
More informationMa 3/103: Lecture 24 Linear Regression I: Estimation
Ma 3/103: Lecture 24 Linear Regression I: Estimation March 3, 2017 KC Border Linear Regression I March 3, 2017 1 / 32 Regression analysis Regression analysis Estimate and test E(Y X) = f (X). f is the
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 16, 2013 Outline Introduction Simple
More informationSemester 2, 2015/2016
ECN 3202 APPLIED ECONOMETRICS 5. HETEROSKEDASTICITY Mr. Sydney Armstrong Lecturer 1 The University of Guyana 1 Semester 2, 2015/2016 WHAT IS HETEROSKEDASTICITY? The multiple linear regression model can
More information4.8 Instrumental Variables
4.8. INSTRUMENTAL VARIABLES 35 4.8 Instrumental Variables A major complication that is emphasized in microeconometrics is the possibility of inconsistent parameter estimation due to endogenous regressors.
More informationLecture 8: Instrumental Variables Estimation
Lecture Notes on Advanced Econometrics Lecture 8: Instrumental Variables Estimation Endogenous Variables Consider a population model: y α y + β + β x + β x +... + β x + u i i i i k ik i Takashi Yamano
More informationUNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS Academic year 2009/10 FINAL EXAM (2nd Call) June, 25, 2010
UNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS Academic year 2009/10 FINAL EXAM (2nd Call) June, 25, 2010 Very important: Take into account that: 1. Each question, unless otherwise stated, requires a complete
More informationBootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator
Bootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator by Emmanuel Flachaire Eurequa, University Paris I Panthéon-Sorbonne December 2001 Abstract Recent results of Cribari-Neto and Zarkos
More informationMFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators
MFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators Thilo Klein University of Cambridge Judge Business School Session 4: Linear regression,
More informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationMultiple Regression Analysis
Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 +... β k x k + u 2. Inference 0 Assumptions of the Classical Linear Model (CLM)! So far, we know: 1. The mean and variance of the OLS estimators
More informationIntroductory Econometrics
Introductory Econometrics Violation of basic assumptions Heteroskedasticity Barbara Pertold-Gebicka CERGE-EI 16 November 010 OLS assumptions 1. Disturbances are random variables drawn from a normal distribution.
More informationECONOMETRICS FIELD EXAM Michigan State University May 9, 2008
ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 00 points possible. Within
More informationEconomics 241B Estimation with Instruments
Economics 241B Estimation with Instruments Measurement Error Measurement error is de ned as the error resulting from the measurement of a variable. At some level, every variable is measured with error.
More information