ECON The Simple Regression Model
|
|
- Margery Harmon
- 5 years ago
- Views:
Transcription
1 ECON The Simple Regression Model Maggie Jones 1 / 41
2 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In general, more complicated econometric models are used for empirical analysis, but this provides a good starting point Suppose we have two variables, x and y, and we are interested in the relationship between the two Specifically, we care about the question, how does x affect y? Typically, we don t observe the full population of y or the full population of x so we can think of y and x as random samples 2 / 41
3 The Simple Regression Model In determining the relationship between x and y, we should keep three questions in mind: 1 How do we allow for factors other than x that might affect y? 2 What is the functional relationship between x and y? 3 How can we be certain we are capturing the ceteris paribus relationship between x and y? We resolve these questions by writing down an equation relating y to x 3 / 41
4 The Simple Regression Model y = β 0 + β 1 x + u (1) We call equation 1 the simple linear regression model y is called the dependent variable x is called the independent variable u is called the error term, it represents everything else that helps to explain y, but is not contained in x 4 / 41
5 The Simple Regression Model Equation 1 assumes a linear functional form, i.e. it assumes that the relationship between x and y is linear β 0 is the intercept term/parameter β 1 is the slope parameter - it measures the effect of x on y, holding all other factors constant: y = β 0 + β 1 x + u Note: in what instances would a linear functional form be a poor choice? 5 / 41
6 The Simple Regression Model Equation 1 assumes a linear functional form, i.e. it assumes that the relationship between x and y is linear β 0 is the intercept term/parameter β 1 is the slope parameter 6 / 41
7 More on the Error Term As long as β 0 is included in the equation, we can assume that the average value of u in the population is zero E(u) = 0 (2) A crucial assumption is that the average value of u does not depend on x, this is known as mean independence E(u x) = E(u) (3) Combining equation 2 and 3 yields one of the most important assumptions in regression analysis, the zero conditional mean assumption E(u x) = 0 (4) 7 / 41
8 The Simple Regression Model Equation 1 assumes a linear functional form, i.e. it assumes that the relationship between x and y is linear β 0 is the intercept term/parameter β 1 is the slope parameter 8 / 41
9 The Simple Regression Model The zero conditional mean assumption gives β 1 another interpretation Taking conditional expectations of equation 1 yields: E(y x) = β 0 + β 1 x (5) which is known as the population regression function We interpret β 1 as, a 1 unit increase in x increases the expected value of y by β 1 units 9 / 41
10 The Simple Regression Model We can now re-consider equation 1 y = β 0 + β 1 x }{{} explained part y can be decomposed into + u }{{} unexplained part the explained part - part of y explained by x the unexplained portion - part of y that can t be explained by x 10 / 41
11 Ordinary Least Squares Now we can begin to discuss the way to estimate β 0 and β 1 given a random sample of y and x Let {(x i, y i ) : i = 1,..., n} be a random sample of size n drawn from the population (x, y) y i = β 0 + β 1 x i + u i (6) How do we use the data to obtain parameter estimates of the population intercept and slope? 11 / 41
12 Ordinary Least Squares We begin with the zero conditional mean assumption of equation 4, which implies: Cov(x, u) = E(ux) = 0 (7) And the zero mean assumption of equation 2 E(u) = 0 (8) These two equations are known as moment conditions 12 / 41
13 Ordinary Least Squares We then define u in terms of the simple regression equation and our moment conditions become E(ux) = E [(y β 0 β 1 x)x] = 0 (9) And the zero mean assumption of equation 2 E(u) = E(y β 0 β 1 x) = 0 (10) 13 / 41
14 Ordinary Least Squares Given our sample of x and y, using the method of moments, we choose our parameter estimates, ˆβ 0 and ˆβ 1 to solve the system of equations E(ux) = 1 n E(u) = 1 n n (y i ˆβ 0 ˆβ 1 x i )x i = 0 (11) i=1 n (y i ˆβ 0 ˆβ 1 x i ) = 0 (12) i=1 14 / 41
15 Ordinary Least Squares Solving yields the parameter estimate for β 0 ˆβ 0 = ȳ + ˆβ 1 x (13) And the estimate for β 1 ˆβ 1 = n i=1 (x i x)(y i ȳ) n i=1 (x i x) 2 (14) Equation 14 is actually just the sample covariance between x and y divided by the sample variance of x 15 / 41
16 Ordinary Least Squares The method of moments is not the only way to arrive at these equations for parameter estimates of β 0 and β 1 The focus of Econ 351 will be on the method of Ordinary Least Squares Our estimates ˆβ 0 and ˆβ 1 are also called the ordinary least squares estimates 16 / 41
17 Ordinary Least Squares To see why, define a fitted value as the value of y i that we obtain from combining the sample x i with our parameter estimates, ˆβ 0 and ˆβ 1 ŷ i = ˆβ 0 + ˆβ 1 x i Define the residual as the difference between the actual value of y i and the fitted value ŷ i û i = y i ŷ i = y i ˆβ 0 ˆβ 1 x i 17 / 41
18 Chapter 2 Ordinary Least Squares Figure 2.4 Fitted values and residuals. The Simple Regression Model y y i û i residual y ˆ ˆ 0 ˆ 1 x y 1 yˆ i Fitted value x 1 x i x as small as possible. The appendix to this chapter shows that the conditions necessary 18 / 41
19 Ordinary Least Squares It seems reasonable to want parameter values that minimize the difference between the true y i and the fitted value ŷ i Sometimes û i will be positive and sometimes it will be negative, thus in theory summing over all residuals could equal zero However, if we square the residuals, we have a more accurate summary of the total error in the regression residuals 19 / 41
20 Ordinary Least Squares Choosing parameter values for β 0 and β 1 that minimize the sum of squared residuals is the basic principle behind ordinary least squares n û 2 i = i=1 n i=1 ( y i ˆβ 0 ˆβ 1 x i ) 2 (15) To minimize equation 15 we set the first order conditions with respect to each of the ˆβs equal to zero 20 / 41
21 Ordinary Least Squares The fitted values and parameter values form the OLS regression line ŷ = ˆβ 0 + ˆβ 1 x (16) The slope estimate tells us the amount by which ŷ changes when x changes by one unit ˆβ 1 = ŷ x 21 / 41
22 Useful Properties of OLS Estimates 1 The sum of the OLS residuals is zero n û i = 0 i=1 2 The sample covariance between x and û is zero n x i û i = 0 3 The point ( x, ū) is always on the OLS regression line i=1 22 / 41
23 Useful Properties of OLS Estimates Re-writing y i in terms of its fitted value and its residual is useful y i = ŷ i + û i From here we see that If 1 n n i=1 ûi = 0 then ȳ i = ŷ i The covariance of ŷi and û i is zero OLS decomposes y i into two parts: a fitted value and a residual, both of which are uncorrelated 23 / 41
24 Sum of Squares 1 Total Sum of Squares SST = n (y i ȳ) 2 i=1 2 Explained Sum of Squares SSE = n (ŷ i ȳ) 2 i=1 3 Residual Sum of Squares SSR = n (y i ŷ i ) 2 i=1 24 / 41
25 Sum of Squares 1 Total Sum of Squares: measures the total sample variation in the y i (measures how spread out the y i are in the sample) 2 Explained Sum of Squares: measures the sample variation in the fitted values, ŷ i 3 Residual Sum of Squares: measures the sample variation in the residuals, û i Note that the total variation can be expressed as the sum of the explained and unexplained variation: SST = SSE + SSR 25 / 41
26 Goodness of Fit One of the most common ways to measure how well a regression fits the data is to use the R-squared R 2 = SSE/SST = 1 SSR/SST (17) It tells us the ratio of the explained variation compared to the total variation So if the majority of y is explained by unobserved factors, the R 2 tends to be very low R 2 is always between 0 and 1 26 / 41
27 Notes on the R 2 A low R 2 does not necessarily mean that the regression is bad and shouldn t be used It simply means that the variable x does not explain much of the variation in the variable y i.e. there are other variables that might help to explain y The regression may still provide an accurate summary of the relationship between x and y 27 / 41
28 Functional Form Level-Level: dependent and independent variables are in levels and related linearly y = β 0 + β 1 x + u Log-Level: dependent variable is in log form, independent variable in levels log(y) = β 0 + β 1 x + u Log-Log: dependent and independent variables are in log form - can be interpreted as an elasticity log(y) = β 0 + β 1 log(x) + u Level-Log: dependent variable is in levels and independent variable in log form y = β 0 + β 1 log(x) + u 28 / 41
29 Functional Form Model Equation Y X β 1 Lev-Lev y = β 0 + β 1 x + u y x y = β 1 x Log-Lev log(y) = β 0 + β 1 x + u log(y) x % y = (100β 1 ) x Log-Log log(y) = β 0 + β 1 log(x) + u log(y) log(x) % y = β 1 % x Lev-Log y = β 0 + β 1 log(x) + u y log(x) y = (β 1 /100)% x 29 / 41
30 Unbiasedness of OLS Unbiasedness is a statistical property that we will examine in the context of our simple linear regression model We require four assumptions to establish the unbiasedness of OLS parameters SLR. 1 - Linear in Parameters: needs to be in the form y = β 0 + β 1 x + u SLR. 2 - Random Sampling: {(xi, y i ) : i = 1,..., n} must be drawn from a random sample SLR. 3 - Variation in x: the sample outcomes on x are not all the same value SLR. 4 - Zero Conditional Mean: our previous assumption E(u x) = 0 holds 30 / 41
31 Unbiasedness of OLS Now consider rewriting ˆβ 1 as ˆβ 1 = n i=1 (x i x)y i n i=1 (x i x) 2 Recall from the review that a parameter is unbiased if its expectation equals its true value Substituting in the regression equation for y i yields ˆβ 1 = n i=1 (x i x)(β 0 + β 1 x i + u i ) n i=1 (x i x) 2 31 / 41
32 Unbiasedness of OLS Which, cancelling terms that equal 0, is ˆβ 1 = β 1 + n i=1 (x i x)u i n i=1 (x i x) 2 Checking unbiasedness: [ n E( ˆβ i=1 1 ) = E(β 1 ) + E (x ] i x)u i }{{} n i=1 (x i x) 2 =β 1 }{{} And since E(u i ) = 0, we have: 1 = n ni=1 (x i x) 2 i=1 (x i x)e(u i ) E( ˆβ 1 ) = β 1 32 / 41
33 Unbiasedness of OLS Now to verify the unbiasedness of ˆβ 0 ˆβ 0 = ȳ ˆβ 1 x = β 0 + β 1 x + ū ˆβ 1 x E( ˆβ 0 ) = E(β 0 ) }{{} + E(β }{{ 1 x) } =β 0 =β 1 x E( ˆβ 0 ) = β 0 + E(ū) }{{} =0 So ˆβ 0 is also unbiased under SLR. 1 - SLR. 4 E( ˆβ 1 x) }{{} =β 1 x 33 / 41
34 Variance of the OLS Estimate We also wish to know how far we can expect ˆβ 1 to be from β 1 on average We can compute the variance of the OLS estimators under assumptions SLR. 1 - SLR. 4, plus one additional assumption SLR. 5 - Homoskedasticity: the error term has the same variance given any value of the explanatory variable Var(u x) = σ 2 x 34 / 41
35 Variance of the OLS Estimate Under SLR. 1 - SLR. 5, the variance of the OLS estimators are: Var( ˆβ 1 ) = σ 2 n i=1 (x i x) 2 And Var( ˆβ 0 ) = σ 2 n n i=1 x2 i n i=1 (x i x) 2 35 / 41
36 Estimating the Error Variance Typically, we don t know the true value of σ 2, so we need to obtain an estimate of it The errors are never observed, but the regression residuals are Note that E(u 2 ) = σ 2 Thus, an unbiased estimator of σ 2 is 1 n n i=1 u2 i However, we do not observe u i, we observe û i 36 / 41
37 Estimating the Error Variance Replacing u i with û i yields the estimator n ˆσ 2 = 1 n i=1 û 2 i However, this estimator is biased Recall the two restrictions from the first order conditions: n i=1 ûi = 0 and n i=1 x iû i = 0 If we observed n 2 residuals, we could always use the above conditions to back out the remaining two residuals 37 / 41
38 Estimating the Error Variance Our estimate of the error variance makes an adjustment for the degrees of freedom Is ˆσ 2 unbiased? Yes! ˆσ 2 = 1 n 2 n û 2 i (18) i=1 38 / 41
39 Estimators of the OLS Parameter Variances We can use equation 18 in Var( ˆβ 0 ) and Var( ˆβ 1 ) to obtain an estimate of the variances of ˆβ 0 and ˆβ 1 Var( ˆβ 1 ) = 1 n 2 n i=1 û2 i n i=1 (x i x) 2 Var( ˆβ 0 ) = 1 n n 2 i=1 û2 i n n i=1 x2 i n i=1 (x i x) 2 39 / 41
40 Additional Notes on Variance Estimates We call the square root of the estimate of the variance of the errors the standard error of the regression ˆσ = ˆσ 2 ˆσ is used to compute the standard error of ˆβ 1 se( ˆβ 1 ) = ˆσ n i=1 (x i x) 2 40 / 41
41 Regression Through the Origin In some instances it makes sense to exclude the constant term from the model This regression equation is called a regression through the origin since we are imposing the intercept to be equal to 0 y = β 1 x + u (19) Minimizing the sum of squared residuals for this regression yields the following estimate for β 1 β 1 = n i=1 x iy i n i=1 x2 i 41 / 41
The 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 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 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 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 informationStatistics II. Management Degree Management Statistics IIDegree. Statistics II. 2 nd Sem. 2013/2014. Management Degree. Simple Linear Regression
Model 1 2 Ordinary Least Squares 3 4 Non-linearities 5 of the coefficients and their to the model We saw that econometrics studies E (Y x). More generally, we shall study regression analysis. : The regression
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 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 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 informationECON2228 Notes 2. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 47
ECON2228 Notes 2 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 2 2014 2015 1 / 47 Chapter 2: The simple regression model Most of this course will be concerned with
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 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 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 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 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 informationApplied Econometrics (QEM)
Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple
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 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 informationMultiple Regression Analysis. Part III. Multiple Regression Analysis
Part III Multiple Regression Analysis As of Sep 26, 2017 1 Multiple Regression Analysis Estimation Matrix form Goodness-of-Fit R-square Adjusted R-square Expected values of the OLS estimators Irrelevant
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 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 informationECON3150/4150 Spring 2016
ECON3150/4150 Spring 2016 Lecture 4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo Last updated: January 26, 2016 1 / 49 Overview These lecture slides covers: The linear regression
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 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 informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More informationBasic econometrics. Tutorial 3. Dipl.Kfm. Johannes Metzler
Basic econometrics Tutorial 3 Dipl.Kfm. Introduction Some of you were asking about material to revise/prepare econometrics fundamentals. First of all, be aware that I will not be too technical, only as
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More 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 informationChapter 2 The Simple Linear Regression Model: Specification and Estimation
Chapter The Simple Linear Regression Model: Specification and Estimation Page 1 Chapter Contents.1 An Economic Model. An Econometric Model.3 Estimating the Regression Parameters.4 Assessing the Least Squares
More informationLECTURE 6. Introduction to Econometrics. Hypothesis testing & Goodness of fit
LECTURE 6 Introduction to Econometrics Hypothesis testing & Goodness of fit October 25, 2016 1 / 23 ON TODAY S LECTURE We will explain how multiple hypotheses are tested in a regression model We will define
More informationThe Simple Linear Regression Model
The Simple Linear Regression Model Lesson 3 Ryan Safner 1 1 Department of Economics Hood College ECON 480 - Econometrics Fall 2017 Ryan Safner (Hood College) ECON 480 - Lesson 3 Fall 2017 1 / 77 Bivariate
More informationL2: Two-variable regression model
L2: Two-variable regression model Feng Li feng.li@cufe.edu.cn School of Statistics and Mathematics Central University of Finance and Economics Revision: September 4, 2014 What we have learned last time...
More informationMaking sense of Econometrics: Basics
Making sense of Econometrics: Basics Lecture 2: Simple Regression Egypt Scholars Economic Society Happy Eid Eid present! enter classroom at http://b.socrative.com/login/student/ room name c28efb78 Outline
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 informationEstimating σ 2. We can do simple prediction of Y and estimation of the mean of Y at any value of X.
Estimating σ 2 We can do simple prediction of Y and estimation of the mean of Y at any value of X. To perform inferences about our regression line, we must estimate σ 2, the variance of the error term.
More informationMultiple Regression Analysis: Heteroskedasticity
Multiple Regression Analysis: Heteroskedasticity y = β 0 + β 1 x 1 + β x +... β k x k + u Read chapter 8. EE45 -Chaiyuth Punyasavatsut 1 topics 8.1 Heteroskedasticity and OLS 8. Robust estimation 8.3 Testing
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More 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 informationThe general linear regression with k explanatory variables is just an extension of the simple regression as follows
3. Multiple Regression Analysis The general linear regression with k explanatory variables is just an extension of the simple regression as follows (1) y i = β 0 + β 1 x i1 + + β k x ik + u i. Because
More informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationRegression Analysis: Basic Concepts
The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42 Outline 1 2 3 t-test P-value Linear
More informationLinear Regression with 1 Regressor. Introduction to Econometrics Spring 2012 Ken Simons
Linear Regression with 1 Regressor Introduction to Econometrics Spring 2012 Ken Simons Linear Regression with 1 Regressor 1. The regression equation 2. Estimating the equation 3. Assumptions required for
More informationEstadística II Chapter 4: Simple linear regression
Estadística II Chapter 4: Simple linear regression Chapter 4. Simple linear regression Contents Objectives of the analysis. Model specification. Least Square Estimators (LSE): construction and properties
More informationDiagnostics of Linear Regression
Diagnostics of Linear Regression Junhui Qian October 7, 14 The Objectives After estimating a model, we should always perform diagnostics on the model. In particular, we should check whether the assumptions
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 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 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 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 informationIntro to Applied Econometrics: Basic theory and Stata examples
IAPRI-MSU Technical Training Intro to Applied Econometrics: Basic theory and Stata examples Training materials developed and session facilitated by icole M. Mason Assistant Professor, Dept. of Agricultural,
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 informationECON Introductory Econometrics. Lecture 4: Linear Regression with One Regressor
ECON4150 - Introductory Econometrics Lecture 4: Linear Regression with One Regressor Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 4 Lecture outline 2 The OLS estimators The effect of
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 informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More information3. Linear Regression With a Single Regressor
3. Linear Regression With a Single Regressor Econometrics: (I) Application of statistical methods in empirical research Testing economic theory with real-world data (data analysis) 56 Econometrics: (II)
More 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 informationECON Program Evaluation, Binary Dependent Variable, Misc.
ECON 351 - Program Evaluation, Binary Dependent Variable, Misc. Maggie Jones () 1 / 17 Readings Chapter 13: Section 13.2 on difference in differences Chapter 7: Section on binary dependent variables Chapter
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within
More informationFinal Exam. Economics 835: Econometrics. Fall 2010
Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each
More informationEmpirical Application of Simple Regression (Chapter 2)
Empirical Application of Simple Regression (Chapter 2) 1. The data file is House Data, which can be downloaded from my webpage. 2. Use stata menu File Import Excel Spreadsheet to read the data. Don t forget
More informationTwo-Variable Regression Model: The Problem of Estimation
Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable
More informationIntroduction to Econometrics
Introduction to Econometrics Lecture 3 : Regression: CEF and Simple OLS Zhaopeng Qu Business School,Nanjing University Oct 9th, 2017 Zhaopeng Qu (Nanjing University) Introduction to Econometrics Oct 9th,
More informationSimple and Multiple Linear Regression
Sta. 113 Chapter 12 and 13 of Devore March 12, 2010 Table of contents 1 Simple Linear Regression 2 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where
More informationApplied Regression. Applied Regression. Chapter 2 Simple Linear Regression. Hongcheng Li. April, 6, 2013
Applied Regression Chapter 2 Simple Linear Regression Hongcheng Li April, 6, 2013 Outline 1 Introduction of simple linear regression 2 Scatter plot 3 Simple linear regression model 4 Test of Hypothesis
More informationMeasuring the fit of the model - SSR
Measuring the fit of the model - SSR Once we ve determined our estimated regression line, we d like to know how well the model fits. How far/close are the observations to the fitted line? One way to do
More informationCorrelation and Regression
Correlation and Regression October 25, 2017 STAT 151 Class 9 Slide 1 Outline of Topics 1 Associations 2 Scatter plot 3 Correlation 4 Regression 5 Testing and estimation 6 Goodness-of-fit STAT 151 Class
More informationECON3150/4150 Spring 2016
ECON3150/4150 Spring 2016 Lecture 6 Multiple regression model Siv-Elisabeth Skjelbred University of Oslo February 5th Last updated: February 3, 2016 1 / 49 Outline Multiple linear regression model and
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More 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 informationSection 3: Simple Linear Regression
Section 3: Simple Linear Regression Carlos M. Carvalho The University of Texas at Austin McCombs School of Business http://faculty.mccombs.utexas.edu/carlos.carvalho/teaching/ 1 Regression: General Introduction
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 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 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 informationOrdinary Least Squares (OLS): Multiple Linear Regression (MLR) Analytics What s New? Not Much!
Ordinary Least Squares (OLS): Multiple Linear Regression (MLR) Analytics What s New? Not Much! OLS: Comparison of SLR and MLR Analysis Interpreting Coefficients I (SRF): Marginal effects ceteris paribus
More informationReview of Statistics
Review of Statistics Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and
More informationReview of probability and statistics 1 / 31
Review of probability and statistics 1 / 31 2 / 31 Why? This chapter follows Stock and Watson (all graphs are from Stock and Watson). You may as well refer to the appendix in Wooldridge or any other introduction
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More information4.1 Least Squares Prediction 4.2 Measuring Goodness-of-Fit. 4.3 Modeling Issues. 4.4 Log-Linear Models
4.1 Least Squares Prediction 4. Measuring Goodness-of-Fit 4.3 Modeling Issues 4.4 Log-Linear Models y = β + β x + e 0 1 0 0 ( ) E y where e 0 is a random error. We assume that and E( e 0 ) = 0 var ( e
More informationLecture 2 Linear Regression: A Model for the Mean. Sharyn O Halloran
Lecture 2 Linear Regression: A Model for the Mean Sharyn O Halloran Closer Look at: Linear Regression Model Least squares procedure Inferential tools Confidence and Prediction Intervals Assumptions Robustness
More informationRegression Analysis with Cross-Sectional Data
89782_02_c02_p023-072.qxd 5/25/05 11:46 AM Page 23 PART 1 Regression Analysis with Cross-Sectional Data P art 1 of the text covers regression analysis with cross-sectional data. It builds upon a solid
More informationAdvanced Econometrics I
Lecture Notes Autumn 2010 Dr. Getinet Haile, University of Mannheim 1. Introduction Introduction & CLRM, Autumn Term 2010 1 What is econometrics? Econometrics = economic statistics economic theory mathematics
More informationEcon 3790: Statistics Business and Economics. Instructor: Yogesh Uppal
Econ 3790: Statistics Business and Economics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 14 Covariance and Simple Correlation Coefficient Simple Linear Regression Covariance Covariance between
More informationTHE MULTIVARIATE LINEAR REGRESSION MODEL
THE MULTIVARIATE LINEAR REGRESSION MODEL Why multiple regression analysis? Model with more than 1 independent variable: y 0 1x1 2x2 u It allows : -Controlling for other factors, and get a ceteris paribus
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More 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 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 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 informationThe Classical Linear Regression Model
The Classical Linear Regression Model ME104: Linear Regression Analysis Kenneth Benoit August 14, 2012 CLRM: Basic Assumptions 1. Specification: Relationship between X and Y in the population is linear:
More informationSimple Linear Regression. Material from Devore s book (Ed 8), and Cengagebrain.com
12 Simple Linear Regression Material from Devore s book (Ed 8), and Cengagebrain.com The Simple Linear Regression Model The simplest deterministic mathematical relationship between two variables x and
More informationECON 450 Development Economics
ECON 450 Development Economics Statistics Background University of Illinois at Urbana-Champaign Summer 2017 Outline 1 Introduction 2 3 4 5 Introduction Regression analysis is one of the most important
More information5.1 Model Specification and Data 5.2 Estimating the Parameters of the Multiple Regression Model 5.3 Sampling Properties of the Least Squares
5.1 Model Specification and Data 5. Estimating the Parameters of the Multiple Regression Model 5.3 Sampling Properties of the Least Squares Estimator 5.4 Interval Estimation 5.5 Hypothesis Testing for
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 informationECONOMETRICS Introduction & First Principles
ECONOMETRICS Introduction & First Principles DA V I D C. BR O A D S T O C K Research Institute of Economics & Management, China. OSEC Pre-Master Course, 2015. COURSE OUTLINE Part 1. Introduction to econometrics:
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 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 informationWooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares
Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares Many economic models involve endogeneity: that is, a theoretical relationship does not fit
More information1 A Non-technical Introduction to Regression
1 A Non-technical Introduction to Regression Chapters 1 and Chapter 2 of the textbook are reviews of material you should know from your previous study (e.g. in your second year course). They cover, in
More informationRegression #3: Properties of OLS Estimator
Regression #3: Properties of OLS Estimator Econ 671 Purdue University Justin L. Tobias (Purdue) Regression #3 1 / 20 Introduction In this lecture, we establish some desirable properties associated with
More informationLectures 5 & 6: Hypothesis Testing
Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across
More information