Economics 113. Simple Regression Assumptions. Simple Regression Derivation. Changing Units of Measurement. Nonlinear effects
|
|
- Delilah Anthony
- 5 years ago
- Views:
Transcription
1 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 Book chapters Chapters 1, 2 and 3 are relevant for the next few lectures Books have been placed on reserve at the Science and Engineering Library.
2 Econometrics Intro Statistics applied to non-experimental data Estimate relationships and describe behaviors Usually comes from an economic model Utility Maximization Profit Maximization Government Objectives Political Objectivs
3 Econometrics Example Model of criminal activity y = Hours spent breaking into cars y = f(x 1, x 2,..., x m 1, x m ) What should be included in the function? What do we do with our model and estimates? Form individual or joint hypotheses about the variables and their effects. Generate predictions
4 Econometrics Causality Causality: The most important thing you will learn in this class! Very hard to determine in a non-experimental setting. Examples: Maternal smoking and infant birth weight Smoking vs non-smoking mothers Are they the same? Wages and Labor supply You only receive a wage if you have a job People who are in the labor force receive a job for a reason Temperatures and CO2 More CO2 leads to higher temperatures Higher temperatures lead to more CO2
5 Econometrics Data Cross-sectional Data Sample of agents taken at one point in time. Ideally, the data is a random sample and observations are independent. Are cross-sectional observation independent? Time-series Data Repeat observations on specific agents. Are time-series observations are independent? Panel Data Have repeat observations for the same agents in different time periods. Ideal data, but difficult to get. Panel data can be used to analyze individual-specific differences Are panel observations are independent?
6 Simple Regression Model Intro How does y change with x. y can be called: Dependent variable Explained variable LHS variable x can be called: Independent variable Explanatory variable RHS variable y = β 0 + β 1 x + u u is the error term, or "disturbance" term u contains everything that we don t control for, both observed and unobserved β 1 is the slope parameter β 0 is the intercept parameter
7 Simple Regression Model An Example Example: Class attendance and grades grade i = β 0 + β 1 Attend i + u i How do we interpret β 0 and β 1. Suppose we estimate: grade i = Attend i Each additional class attended is associated with a higher grade of Is this causal? When does β 1 summarize a causal relationship between Attend and grade?
8 Simple Regression Model The Assumptions General framework: y i = β 0 + β 1 x i + u i Assumption 1: E(u) = 0 This is innocuous as long as we have an intercept in the model. Assumption 2: E(u x) = E(u) Combined with assumption 1 this gives us E(u x) = 0 This means that given any x, the value of u we expect will be 0. This is not necessarily realistic. This is the hard assumption to satisfy.
9 Simple Regression Model The Example Example: Class attendance and grades grade i = β 0 + β 1 Attend i + u i The key: u contains all the variables, other than Attend, that help determine your grade!!!! Can you list some of these variables? For example, for A2 to hold, we would need E(u Attend = 32) = E(u Attend = 10) What does this mean? Is this likely?
10 Least Squares Regression The Derivation How do we estimate β 0 and β 1? Predicted Value: y i = β 0 + β 1 x i Residual: u i = y i y i Suppose that we choose to minimize the sum of squared errors min β 0 β1 n i=1 u 2 i Thus: Take derivatives! min β 0 β1 n yi β 0 β 2 1 x i (1) i=1
11 Least Squares Regression The Derivation Differentiate n i=1 yi β 0 β 1 x i 2 with respect to β0 : n 2 y i β 0 β 1 x i = 0 i=1 Divide by 2, divide by n 1 n n i=1 yi β 0 β 1 x i = 0 To which assumption does this equation correspond? E(u) = 0
12 Least Squares Regression The Derivation Differentiate n i=1 yi β 0 β 1 x i 2 with respect to β1 : n 2x i yi β 0 β 1 x i = 0 i=1 Divide by 2, divide by n 1 n n i=1 x i yi β 0 β 1 x i = 0 To which assumption does this equation correspond? E(u x) = 0
13 Least Squares Regression The Derivation Combining the two equations, and after lots of algebra, we get: n i=1 xi µ x (yi µ y ) β 1 = n 2 i=1 xi µ x Another way to write this β 1 = σ xy σ 2 x To solve for β 0, take means of y i = β 0 + β 1 x i + u i and rearrange: We can also solve for the residuals: β 0 = µ y β 1 µ x u i = y i y i u i = y i ( β 0 + β 1 x i )
14 Simple Regression Model Diagnostic Measures SST: Total sum of squares measures the total amount of variability in the dependent variable. n 2 SST = yi µ y SSR: Sum of squared residuals measures the total amount of variability that the model does not explain n 2 SSR = ui R-Squared: R 2 i=1 i=1 R 2 = 1 SSR SST Measures the variation "explained" by the model Often misinterpreted as "goodness of fit"
15 OLS Changing Units of Measurement Data Scaling Predictions in different units Different interpretations Example: Estimates: wage = β 0 + β 1 educ + β 2 tenure + u - educ is in years - tenure is years on the job - wage is in dollars wage = β 0 + β 1 educ + β 2 tenure Again, the u vanishes since E[u educ, tenure] = 0.
16 OLS Changing Units of Measurement Wage in cents rather than dollars? = wage dollars = wage cents Original Equation: Substitute: 1 wage dollars = β 0 + β 1 educ + β 2 tenure 100 wage cents = β 0 + β 1 educ + β 2 tenure wage cents = 100 β β 1 educ β 2 tenure What if we want to measure tenure in months? = tenure years = 1 12 tenure months Substitute 1 wage = β 0 + β 1 educ + β 2 12 tenure months 1 wage = β 0 + β 1 educ + β 12 2 tenure months
17 OLS Handling non-linearity Not everything linear in real life. Relationship between education and wage is linear? No. Which has the higher benefit? 3 more years after 6th grade? 3 more years after undergrad? Common ways to easily handle non-linearity 1 Take logs of the dependent variable 2 Take logs of the independent variable 3 Take logs of both
18 OLS Wage in Levels wage educ
19 OLS Wage in logs log(wage) educ
20 OLS Handling non-linearity If data are in levels: wage = β 0 + β 1 educ How do we interpret β 1? Totally differentiate. Simplify Interpret β 1 wage = β 1 educ wage educ = β 1 wage = 15, , 324educ For each additional year of education, you earn $1,324 more.
21 OLS Handling non-linearity If wage is in logs log( wage) = β 0 + β 1 educ How do we interpret β 1? Totally differentiate. Simplify wage wage 100 }{{} % change wage wage = β 1 educ Interpret β 1 in the following results = β1 100 educ }{{} unit change log( wage) = educ A one-year increase in education yields an 8% increase in wage
22 OLS Handling non-linearity If wage and educ in logs How do we interpret β 1? Totally differentiate. Simplify log( wage) = β 0 + β 1 log(educ) wage wage = β educ 1 educ wage wage 100 }{{} % change Interpret β 1 in the following results = β 1 educ educ 100 }{{} % change log( wage) = log(educ) A 1% increase in education yields an 0.5% increase in wage
23 Simple Regression Model Biased or unbiased When is β 1 a good estimate, where "good" is defined as unbiased? By unbiased, E β1 x = β 1 β 1 s are centered around β 1 β 1 Unbiased if the following assumptions hold! 1 Linear in parameters: y i = β 0 + β 1 x i 2 Random sample of size n. {(x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 )... (x n, y n )} 3 Zero conditional mean: E(u x) = 0 4 σ 2 x > 0.
24 Simple Regression Model Biased or unbiased Simple example Suppose that the population is characterized by: y = 3 2x 1 + u - β 0 = 3 - β 1 = 2 - u distributed normal, mean 0 and sd 3 - x s are between 0.01 and 10, spaced evenly people Estimate using: y = β 0 + β 1 x 1 + u Plot y on x
25 y x
26 Simple Regression Model Biased or unbiased Suppose that we sample 30 people from the population, and estimate β 1 via OLS First sample: β 1 = Second sample: β 1 = Third sample: β 1 = They re all wrong. Is this a problem? Keep sampling!! Sample 1000 times Plot a histogram of the estimates of β 1 How does the distribution of estimates compare to 2?
27 Histogram of Beta1 Density Beta1
28 OLS - Variance Basics If assumptions 1-4 hold, β 1 is centered around β 1. Central tendency says nothing about dispersion. We are also interested in estimating Var( β 1 ) From the previous histogram, there is variance in the estimate β 1 Is the estimate of β 1 precise/reliable? Assumption 5 - Homoskedastic Errors: Var [u x] = σ 2 Variance of errors is common across x. Assumptions 1-5 are called the "Gauss-Markov Assumptions" If Var [u x] Var [u], errors are heteroskedastic.
29 y x
30 y x
31 OLS - Variance Estimate Variance Variance of the slope parameter: var β1 = σ 2 n i=1 xi µ x 2 What do I need for these variance estimates? An estimate of σ 2 : σ 2 = 1 n 2 n i=1 u 2 i Why n 2? σ 2 requires estimating β 0 and β 1.
32 OLS Estimate Variance Standard error of β 1 : se β1 = σ 2 n i=1 xi µ x 2 Dispersion of β 1 around β 1, same scale as β 1 How does σ effect the precision of our estimates? Why? Higher σ yields higher standard errors (lower precision). With higher σ, there is more noise, and thus it is harder to get a precise estimate of β 1 Using the original example, compare the following two situations: u distributed normal, mean 0 and sd 10 u distributed normal, mean 0 and sd 3
33 Histogram of Beta1 - SD(u)=10 Density Beta1
34 Histogram of Beta1 - Adding SE(u)=3 Density Beta1
ECON3150/4150 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 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 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 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 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 information2) For a normal distribution, the skewness and kurtosis measures are as follows: A) 1.96 and 4 B) 1 and 2 C) 0 and 3 D) 0 and 0
Introduction to Econometrics Midterm April 26, 2011 Name Student ID MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. (5,000 credit for each correct
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 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 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 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 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 informationInference in Regression Model
Inference in Regression Model Christopher Taber Department of Economics University of Wisconsin-Madison March 25, 2009 Outline 1 Final Step of Classical Linear Regression Model 2 Confidence Intervals 3
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 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. 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 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 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 informationPanel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43
Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression
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 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 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 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 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 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 informationx i = 1 yi 2 = 55 with N = 30. Use the above sample information to answer all the following questions. Show explicitly all formulas and calculations.
Exercises for the course of Econometrics Introduction 1. () A researcher is using data for a sample of 30 observations to investigate the relationship between some dependent variable y i and independent
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 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 informationMidterm 1 ECO Undergraduate Econometrics
Midterm ECO 23 - Undergraduate Econometrics Prof. Carolina Caetano INSTRUCTIONS Reading and understanding the instructions is your responsibility. Failure to comply may result in loss of points, and there
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 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 informationLECTURE 10. Introduction to Econometrics. Multicollinearity & Heteroskedasticity
LECTURE 10 Introduction to Econometrics Multicollinearity & Heteroskedasticity November 22, 2016 1 / 23 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists
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 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 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 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 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 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 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 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 informationMultiple Regression Analysis: Inference MULTIPLE REGRESSION ANALYSIS: INFERENCE. Sampling Distributions of OLS Estimators
1 2 Multiple Regression Analysis: Inference MULTIPLE REGRESSION ANALYSIS: INFERENCE Hüseyin Taştan 1 1 Yıldız Technical University Department of Economics These presentation notes are based on Introductory
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 informationRegression with a Single Regressor: Hypothesis Tests and Confidence Intervals
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (SW Chapter 5) Outline. The standard error of ˆ. Hypothesis tests concerning β 3. Confidence intervals for β 4. Regression
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 information5. Let W follow a normal distribution with mean of μ and the variance of 1. Then, the pdf of W is
Practice Final Exam Last Name:, First Name:. Please write LEGIBLY. Answer all questions on this exam in the space provided (you may use the back of any page if you need more space). Show all work but do
More informationVariance Decomposition and Goodness of Fit
Variance Decomposition and Goodness of Fit 1. Example: Monthly Earnings and Years of Education In this tutorial, we will focus on an example that explores the relationship between total monthly earnings
More informationVariance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017
Variance Decomposition in Regression James M. Murray, Ph.D. University of Wisconsin - La Crosse Updated: October 04, 2017 PDF file location: http://www.murraylax.org/rtutorials/regression_anovatable.pdf
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 informationCHAPTER 6: SPECIFICATION VARIABLES
Recall, we had the following six assumptions required for the Gauss-Markov Theorem: 1. The regression model is linear, correctly specified, and has an additive error term. 2. The error term has a zero
More informationStatistical Inference with Regression Analysis
Introductory Applied Econometrics EEP/IAS 118 Spring 2015 Steven Buck Lecture #13 Statistical Inference with Regression Analysis Next we turn to calculating confidence intervals and hypothesis testing
More informationSimple Linear Regression Model & Introduction to. OLS Estimation
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
More informationMaking sense of Econometrics: Basics
Making sense of Econometrics: Basics Lecture 4: Qualitative influences and Heteroskedasticity Egypt Scholars Economic Society November 1, 2014 Assignment & feedback enter classroom at http://b.socrative.com/login/student/
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 informationInstrumental Variables
Instrumental Variables Department of Economics University of Wisconsin-Madison September 27, 2016 Treatment Effects Throughout the course we will focus on the Treatment Effect Model For now take that to
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 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 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 informationApplied Quantitative Methods II
Applied Quantitative Methods II Lecture 4: OLS and Statistics revision Klára Kaĺıšková Klára Kaĺıšková AQM II - Lecture 4 VŠE, SS 2016/17 1 / 68 Outline 1 Econometric analysis Properties of an estimator
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 informationNotes 6: Multivariate regression ECO 231W - Undergraduate Econometrics
Notes 6: Multivariate regression ECO 231W - Undergraduate Econometrics Prof. Carolina Caetano 1 Notation and language Recall the notation that we discussed in the previous classes. We call the outcome
More informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
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 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 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 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 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 informationWooldridge, Introductory Econometrics, 4th ed. Chapter 6: Multiple regression analysis: Further issues
Wooldridge, Introductory Econometrics, 4th ed. Chapter 6: Multiple regression analysis: Further issues What effects will the scale of the X and y variables have upon multiple regression? The coefficients
More informationProblem 13.5 (10 points)
BOSTON COLLEGE Department of Economics EC 327 Financial Econometrics Spring 2013, Prof. Baum, Mr. Park Problem Set 2 Due Monday 25 February 2013 Total Points Possible: 210 points Problem 13.5 (10 points)
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 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 informationMathematics for Economics MA course
Mathematics for Economics MA course Simple Linear Regression Dr. Seetha Bandara Simple Regression Simple linear regression is a statistical method that allows us to summarize and study relationships between
More informationWrite your identification number on each paper and cover sheet (the number stated in the upper right hand corner on your exam cover).
Formatmall skapad: 2011-12-01 Uppdaterad: 2015-03-06 / LP Department of Economics Course name: Empirical Methods in Economics 2 Course code: EC2404 Semester: Spring 2015 Type of exam: MAIN Examiner: Peter
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 informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 5 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 44 Outline of Lecture 5 Now that we know the sampling distribution
More informationECONOMETRICS FIELD EXAM Michigan State University August 21, 2009
ECONOMETRICS FIELD EXAM Michigan State University August 21, 2009 Instructions: Answer all four (4) questions. Point totals for each question are given in parentheses; there are 100 points possible. Within
More informationUnless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold.
Economics 345: Applied Econometrics Section A01 University of Victoria Midterm Examination #2 Version 1 SOLUTIONS Spring 2015 Instructor: Martin Farnham Unless provided with information to the contrary,
More informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
More informationInference in Regression Analysis
ECNS 561 Inference Inference in Regression Analysis Up to this point 1.) OLS is unbiased 2.) OLS is BLUE (best linear unbiased estimator i.e., the variance is smallest among linear unbiased estimators)
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 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 informationSimple Linear Regression for the MPG Data
Simple Linear Regression for the MPG Data 2000 2500 3000 3500 15 20 25 30 35 40 45 Wgt MPG What do we do with the data? y i = MPG of i th car x i = Weight of i th car i =1,...,n n = Sample Size Exploratory
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 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 informationSample Problems. Note: If you find the following statements true, you should briefly prove them. If you find them false, you should correct them.
Sample Problems 1. True or False Note: If you find the following statements true, you should briefly prove them. If you find them false, you should correct them. (a) The sample average of estimated residuals
More informationMultiple linear regression
Multiple linear regression Course MF 930: Introduction to statistics June 0 Tron Anders Moger Department of biostatistics, IMB University of Oslo Aims for this lecture: Continue where we left off. Repeat
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 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 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 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 informationHeteroskedasticity (Section )
Heteroskedasticity (Section 8.1-8.4) Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Heteroskedasticity 1 / 44 Consequences of Heteroskedasticity for OLS Consequences
More informationAn overview of applied econometrics
An overview of applied econometrics Jo Thori Lind September 4, 2011 1 Introduction This note is intended as a brief overview of what is necessary to read and understand journal articles with empirical
More informationMultiple Regression. Midterm results: AVG = 26.5 (88%) A = 27+ B = C =
Economics 130 Lecture 6 Midterm Review Next Steps for the Class Multiple Regression Review & Issues Model Specification Issues Launching the Projects!!!!! Midterm results: AVG = 26.5 (88%) A = 27+ B =
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 7 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 68 Outline of Lecture 7 1 Empirical example: Italian labor force
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 informationHeteroscedasticity 1
Heteroscedasticity 1 Pierre Nguimkeu BUEC 333 Summer 2011 1 Based on P. Lavergne, Lectures notes Outline Pure Versus Impure Heteroscedasticity Consequences and Detection Remedies Pure Heteroscedasticity
More informationMultiple Regression Analysis: Further Issues
Multiple Regression Analysis: Further Issues Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) MLR: Further Issues 1 / 36 Effects of Data Scaling on OLS Statistics Effects
More informationContest Quiz 3. Question Sheet. In this quiz we will review concepts of linear regression covered in lecture 2.
Updated: November 17, 2011 Lecturer: Thilo Klein Contact: tk375@cam.ac.uk Contest Quiz 3 Question Sheet In this quiz we will review concepts of linear regression covered in lecture 2. NOTE: Please round
More informationLECTURE 2: SIMPLE REGRESSION I
LECTURE 2: SIMPLE REGRESSION I 2 Introducing Simple Regression Introducing Simple Regression 3 simple regression = regression with 2 variables y dependent variable explained variable response variable
More informationRewrap ECON November 18, () Rewrap ECON 4135 November 18, / 35
Rewrap ECON 4135 November 18, 2011 () Rewrap ECON 4135 November 18, 2011 1 / 35 What should you now know? 1 What is econometrics? 2 Fundamental regression analysis 1 Bivariate regression 2 Multivariate
More information