Wednesday, September 19 Handout: Ordinary Least Squares Estimation Procedure The Mechanics
|
|
- Kelley Blake
- 5 years ago
- Views:
Transcription
1 Amherst College Department of Economics Economics Fall 2012 Wednesday, September 19 Handout: Ordinary Least Squares Estimation Procedure he Mechanics Preview Best Fitting Line: Income and Savings Clint s Assignment Simple Regression Model o Parameters of the Model o Error erm o Best Fitting Line Ordinary Least Squares (OLS) Estimation Procedure o Sum of Squared Residuals Criterion o Finding the Best Fitting Line Importance of the Error erm o o Absence of Random Influences Presence of Random Influences: Constant and Coefficient of Best Fitting Line Are Random Variables Error erms and Random Influences: A Closer Look Clint s Assignment: he wo Parts Income and Savings he following table reports on the (after tax) income of Americans and their savings between 19 and 1975 in billions of dollars: Year Income Savings Year Income Savings Year Income Savings Economic theory suggests that as Americans earn more income, we will save more. heory: Additional income increases savings.
2 2 Question: Do the data support the theory? Question: How can we estimate the relationship between savings and income more precisely? hat is, what equation describes the best fitting line? We estimate that an additional $1 of income increases savings by $ ; or equivalently, an an additional $1,000 of income increase savings by $. Aside: Random Influences Clint s Assignment: Effect of Studying on Quiz Scores Background: hree students are enrolled in Professor Jeff Lord s 8:0 am class. Every week, he gives a quiz. Professor Lord asks his students to report the number of minutes they studied; the students always respond honestly. Std heory: Additional studying increases quiz scores. Professor Lord s First Quiz: Student Minutes Score Question: Do the data support the theory?
3 he Regression Model y t β Const + β x x t + e t where y t Quiz score received by student t: x t Number of minutes studied by student t: e t Error term for student t: Interpretation of the parameters, β Const and β x : β Const represents the number of points Professor Lord gives students just for showing up; β x represents the number of additional points earned for each additional minute of study. Interpretation of the error term, e t : he error term, e t, is a random variable; it represents random influences, the factors that cannot be anticipated and/or determined before the quiz is given. wo implicit assumptions: Professor Lord gives each student the same number of points for showing up. he number of additional points earned for an additional minute of study is the same for each student. Clint s Assignment: Find β Const and β x. But, β Const and β x are unobservable. What can Clint do? Econometrician s Philosophy: If you lack the information to determine the value directly, estimate the value to the best of your ability using the information you do have. Strategy: Use the intercept and slope of the best fitting line to estimate β Const and β x. b Const Intercept of the best fitting line b Const estimates the value of β Const b x Slope of the best fitting line b x estimates the value of β x Problem: How can we decide on the best fitting line? Std
4 4 he Ordinary Least Squares (OLS) Estimation Procedure Ordinary Least Squares (OLS) Criterion: Minimize the sum of squared residuals. he following two equations achieve this objective: Σ b Const y b t1 (yt y )(x t x x b x Σ t1 (xt 2 Step 1: Define the sum of squared residuals (SSR) he Model: y t β Const + β x x t + e t y t Actual quiz score received by student t: Dependent variable x t Actual number of minutes studied by student t: Explanatory variable e t Actual error for student t β Const Actual constant: Points awarded for showing up β x Actual coefficient: Additional points received for each additional minute studied he Estimate: Esty t b Const x t Esty t Estimated quiz score for student t b Const Estimated constant; that is, b Const estimates the value of β Const b x Estimated coefficient; that is, b x estimates the value of β x he Residual: Res t y t Esty t Res t Residual for student t Res t Actual quiz score for student t Estimated quiz score for student t Strategy: Determine the best fitting line by minimizing the sum of squared residuals. Esty 1 b Const Esty 2 b Const Esty b Const Res 1 y 1 Esty 1 Res 2 y 2 Esty 2 Res y Esty Res 1 y 1 Res 2 y 2 Res y SSR Res Res2 2 + Res2 ) 2 + (y 2 ) 2 + (y ) 2
5 5 Step 2: Differentiate the sum of squared residuals (SSR) with respect to b Const dssr db 2 ) 2(y 2 ) 2(y ) 0 Const ) + (y 2 ) + (y ) 0 + y 2 + y ) + ( ) + ( b x ) 0 + y 2 + y ) b Const ( + + ) 0 y 1 + y 2 + y y x 0 y b Const x Note that b Const y x (x, y ) Std OLS Estimate: y b Const x Step : Differentiate the sum of squared residuals (SSR) with respect to b x SSR ) 2 + (y 2 ) 2 + (y ) 2 [y 1 (y x ) ] 2 + [y 2 (y x ) ] 2 + [y (y x ) ] 2 [y 1 y x ] 2 + [y 2 y x ] 2 + [y y x ] 2 [y 1 y x ] 2 + [y 2 y x ] 2 + [y y x ] 2 [ y ) ( ] 2 + [(y 2 y ) ( ] 2 + [(y y ) ( ] 2 dssr db x 2[ y ) ( ]( 2[(y 2 y ) ( ]( 2[(y y ) ( ]( 0 [ y ) ( ]( + [(y 2 y ) ( ]( + [(y y ) ( ]( 0 y )( ( 2 + (y 2 y )( ( 2 + (y y )( ( 2 0 y )( + (y 2 y )( + (y y )( b x ( 2 ( 2 ( 2 y )( + (y 2 y )( + (y y )( b x [( 2 + ( 2 + ( 2 ] b x y )( + (y 2 y )( + (y y )( ( 2 + ( 2 + ( 2 Σ t1 (yt y )(x t Σ t1 (xt 2
6 6 Ordinary Least Squares Estimates Calculations he Data: Student x y x Minutes Studied y Quiz score 25 he equations: b Const y b x x b x y )( + (y 2 y )( + (y y )( ( 2 + ( 2 + ( 2 Σ t1 (yt y )(x t Σ t1 (xt 2 he means: y y 1 + y 2 + y + + x Deviations from the means: Student y t y y t y x t x x t x Product of the x and y deviations and squared x deviations. Student (y t y)(x t x) (x t x) 2 1 ( )( ) ( ) 2 2 ( )( ) ( ) 2 ( )( ) ( ) 2 Sum Sum Σ t1 (yt y )(x t Applying the formulas: b x Σ t1 (xt x ) 2 b Const y x Ordinary Least Squares (OLS) Best Fitting Line: y + x Std OLS Estimate: y + x (x, y )
7 7 he sum of squared residuals for the best fitting line Student x t y t Esty t x t x t Res t y t Esty 2 t Res t SSR Simulation to Check Our Calculations for the OLS Best Fitting Line EViews Dependent Variable: Y Included observations: Variable Coefficient Std. Error t-statistic Prob. X C Sum squared resid Schwarz criterion Best Fitting Line: y + x Summary he Regression Model Consider the following equation: y t β Const + β x x t + e t where y t Quiz score received by student t x t Minutes studied by student t e t Error term for student t β Const and β x are called the parameters of the model. Before interpreting the parameters recall that it is generally believed that Professor Lord gives students some credit just for showing up for the quiz; Studying more will improve a student s score. Interpreting β Const and β x : β Const represents ; β x represents. Interpreting the Ordinary Least Squares Estimates: Esty x We estimate that Professor Lord gives students points for showing up for the quiz. Studying one additional minute results in additional points.
8 8 Importance of the Error erm Regression Model: y t β Const + β x x t + e t where y t Quiz score of student t x t Minutes studied by student t e t Error term for student t For the moment, suppose that β Const equals and β x equals 2. In words, this means: Professor Lord gives each student points for showing up. Each additional minute of study provides 2 additional points. he regression model is: y t + 2x t + e t he actual constant would be and the actual coefficient would be 2. Error erm Represents Random Influences: e t he error term reflects all the factors that cannot be anticipated or determined before the quiz is given; that is the error term represents all random influences. WHA IF Question: What if there were no random influences? hat is, what if there were no error term? In the absence of an error term, y t + 2x t ; that is, in the absence of an error term there would be no random influences: Actual: y + 2x Absence of Random Influences Student Minutes (x t ) Score (y t + 2x t ) Claim: In the absence of random influences, it would be trivial to compute the actual value of the constant and coefficient.
9 9 Coefficient Estimate Simulation: Absence of Random Influences Absence of Error erm o address this question, we shall begin by using Act Const our simulation to Actual 40 illustrate the importance Constant: No error term of the error term. β Const NB: We can view each week s quiz as one repetition of an experiment. Actual Coefficient: β x Act Coef Our simulation allows us to do something we Repetition cannot do in the real world. It allows us to Coef Est specify the constant and coefficient of our model; that is, we can select β Const and β x. hat is, we can specify the points Professor Lord gives students just for showing up, ; additional points earned for an additional minute of study, 2. Err erm Note that initially the Err erm checkbox is checked indicating that the error term and hence random influences are present. o eliminate the error term and random influences, the Err erm checkbox is cleared Estimated coefficient value calculated from this repetition: Σ t1 (yt y )(x t b x Σ t1 (xt 2 Coefficient Estimate: Estimate of Coefficient Value Repetition No Error erm Std In the absence of random influences, the best fitting line fits the data perfectly. he best fitting line coincides with the actual line. We can determine the actual value of the coefficient by calculating the slope of the line using any two points. Actual: y + 2x But remember that the absence of random influences is unrealistic. In the real world, random influences are inevitably present. We shall now use a simulation to illustrate how the error term in the model captures the random influences
10 10 Random Influences Are Present in the Real World But the real world is not this simple; random influences play an important role in the real world. Presence of Random Influences Student he red points represent the actual scores from the first quiz; that is, the red points include the random influences. As a consequence of the random influences, Students 1 and 2 over perform while Student under performs. hat is, Student 1: e 1 is Student 2: e 2 is Student : e is Coefficient Estimate Simulation: Presence of Random Influences Presence of Error erm Coefficient Estimate: Estimate of Coefficient Value Repetition No Error erm Act Err Var Actual: y + 2x Std As a consequence of the random influences, the line which best fits the data does not have an intercept of, the actual intercept; also, the best fitting line does not have a coefficient of 2, the actual coefficient. he simulation is reporting on the coefficient estimates. Actual Constant: β Const Actual Coefficient: β x Repetition Coef Est Act Const 40 Act Coef Act Err Var Err erm Variance Error of Error erm Probability Distribution: Var[e] Estimated coefficient value calculated from this repetition: Σ t1 (yt y )(x t b x Σ t1 (xt 2
11 11 Key Point: he constant and coefficient estimates are a random variable. Real world Random influences are We expect the intercept and slope of the best fitting line to equal the actual constant and coefficient In fact, even if we know the actual values of the constant and coefficient, β Const and β x, we predict the constant and coefficient of the best fitting line, b Const and b x, with certainty before the quiz was given. he intercept and slope of the best fitting line, b Const and b x, are. he Error erm and Random Influences: A Closer Look Actual: y + 2x Std OLS Estimate: y x he Model: y t β Const + β x x t + e t he error term, e t, is a random variable. Intuition: What happens after many, many quizzes? Since the error term represents the random influences, a student s error term should be: positive about half the time indicating that the student performs than usual; negative about half the time indicating that the student performs than usual. In the long run, however, the error terms should average out to. Random Influence Error erm Simulation Initially, the Pause Err Var checkbox is checked and variance of the error Repetition Pause 200 term s probability distribution is 0. Click 0 Start and record error term for each of the three students in the first repetition Repetition Student 1 Student 2 Student 1 2 Actual Variance of Error erm s Probability Distribution: Var[e] Can you predict the numerical value of a student s error term beforehand?.
12 12 Next, clear the Pause checkbox and click Continue. After 1,000,000 or so repetitions, click Stop. Mean[e 1 ] Mean[e 2 ] Mean[e ] e 1 is positive about e 2 is positive about e is positive about the time and negative the time and negative the time and negative about the time about the time about the time e 1 has systematic e 2 has systematic e has systematic effect on Student 1 s score effect on Student 2 s score effect on Student s score e 1 represents e 2 represents e represents a influence a influence a influence Summary he mean of the probability distribution for each student s error term equals 0. he chances that a student s error term will be positive in any one quiz are about equal to the chances that it will be negative. A student s error term has no systematic effect on a his/her quiz score. A student s error term represents a random influence. Clint s Assignment: Where Do We Stand? Summary he OLS estimate for the value of the coefficient is 1.2; Clint estimates that an additional minute of studying results in 1.2 additional points suggesting that the theory is correct. But, since random influences are present in the real world, we know that the coefficient estimate is a random variable. We are all but certain that the numerical value of the coefficient estimate, 1.2, does NO equal the actual value of the coefficient. What should Clint do? We shall proceed by dividing Clint s assignment into two related parts: Coefficient Reliability: How reliable is the coefficient estimate calculated from the results of the first quiz? hat is, how confident should Clint be that the coefficient estimate, 1.2, will be close to the actual value? heory Confidence: How much confidence should Clint have in the theory that additional studying increases quiz scores?
Chapter 5: Ordinary Least Squares Estimation Procedure The Mechanics Chapter 5 Outline Best Fitting Line Clint s Assignment Simple Regression Model o
Chapter 5: Ordinary Least Squares Estimation Procedure The Mechanics Chapter 5 Outline Best Fitting Line Clint s Assignment Simple Regression Model o Parameters of the Model o Error Term and Random Influences
More informationWednesday, September 26 Handout: Estimating the Variance of an Estimate s Probability Distribution
Amherst College Department of Economics Economics 60 Fall 2012 Wednesday, September 26 Handout: Estimating the Variance of an Estimate s Probability Distribution Preview: Review: Ordinary Least Squares
More informationChapter 8 Handout: Interval Estimates and Hypothesis Testing
Chapter 8 Handout: Interval Estimates and Hypothesis esting Preview Clint s Assignment: aking Stock General Properties of the Ordinary Least Squares (OLS) Estimation Procedure Estimate Reliability: Interval
More informationMonday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties
Amherst College Department of Economics Economics 360 Fall 2012 Monday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties Chapter 18 Outline Review o Regression Model
More information[Mean[e j ] Mean[e i ]]
Amherst College Department of Economics Economics 360 Fall 202 Solutions: Wednesday, September 26. Assume that the standard ordinary least square (OLS) premises are met. Let (x i, y i ) and (, y j ) be
More informationWednesday, October 17 Handout: Hypothesis Testing and the Wald Test
Amherst College Department of Economics Economics 360 Fall 2012 Wednesday, October 17 Handout: Hypothesis Testing and the Wald Test Preview No Money Illusion Theory: Calculating True] o Clever Algebraic
More informationMonday, September 10 Handout: Random Processes, Probability, Random Variables, and Probability Distributions
Amherst College Department of Economics Economics 360 Fall 202 Monday, September 0 Handout: Random Processes, Probability, Random Variables, and Probability Distributions Preview Random Processes and Probability
More informationChapter 11 Handout: Hypothesis Testing and the Wald Test
Chapter 11 Handout: Hypothesis Testing and the Wald Test Preview No Money Illusion Theory: Calculating True] o Clever Algebraic Manipulation o Wald Test Restricted Regression Reflects Unrestricted Regression
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 informationECO220Y Simple Regression: Testing the Slope
ECO220Y Simple Regression: Testing the Slope Readings: Chapter 18 (Sections 18.3-18.5) Winter 2012 Lecture 19 (Winter 2012) Simple Regression Lecture 19 1 / 32 Simple Regression Model y i = β 0 + β 1 x
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 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 informationAt this point, if you ve done everything correctly, you should have data that looks something like:
This homework is due on July 19 th. Economics 375: Introduction to Econometrics Homework #4 1. One tool to aid in understanding econometrics is the Monte Carlo experiment. A Monte Carlo experiment allows
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 informationMeasurement Error. Often a data set will contain imperfect measures of the data we would ideally like.
Measurement Error Often a data set will contain imperfect measures of the data we would ideally like. Aggregate Data: (GDP, Consumption, Investment are only best guesses of theoretical counterparts and
More informationWednesday, November 7 Handout: Heteroskedasticity
Amhers College Deparmen of Economics Economics 360 Fall 202 Wednesday, November 7 Handou: Heeroskedasiciy Preview Review o Regression Model o Sandard Ordinary Leas Squares (OLS) Premises o Esimaion Procedures
More informationChapter 13: Dummy and Interaction Variables
Chapter 13: Dummy and eraction Variables Chapter 13 Outline Preliminary Mathematics: Averages and Regressions Including Only a Constant An Example: Discrimination in Academia o Average Salaries o Dummy
More informationChapter 15: Other Regression Statistics and Pitfalls
Chapter 15: Other Regression Statistics and Pitfalls Chapter 15 Outline Two-Tailed Confidence Intervals o Confidence Interval Approach: Which Theories Are Consistent with the Data? o A Confidence Interval
More informationMonday, October 15 Handout: Multiple Regression Analysis Introduction
Amherst College Department of Economics Economics 360 Fall 2012 Monday, October 15 Handout: Multiple Regression Analysis Introduction Review Simple and Multiple Regression Analysis o Distinction between
More informationHandout 11: Measurement Error
Handout 11: Measurement Error In which you learn to recognise the consequences for OLS estimation whenever some of the variables you use are not measured as accurately as you might expect. A (potential)
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 informationAn Introduction to Econometrics. A Self-contained Approach. Frank Westhoff. The MIT Press Cambridge, Massachusetts London, England
An Introduction to Econometrics A Self-contained Approach Frank Westhoff The MIT Press Cambridge, Massachusetts London, England How to Use This Book xvii 1 Descriptive Statistics 1 Chapter 1 Prep Questions
More informationWednesday, October 10 Handout: One-Tailed Tests, Two-Tailed Tests, and Logarithms
Amherst College Department of Economics Economics 360 Fall 2012 Wednesday, October 10 Handout: One-Tailed Tests, Two-Tailed Tests, and Logarithms Preview A One-Tailed Hypothesis Test: The Downward Sloping
More informationHint: The following equation converts Celsius to Fahrenheit: F = C where C = degrees Celsius F = degrees Fahrenheit
Amherst College Department of Economics Economics 360 Fall 2014 Exam 1: Solutions 1. (10 points) The following table in reports the summary statistics for high and low temperatures in Key West, FL from
More informationChapter 14: Omitted Explanatory Variables, Multicollinearity, and Irrelevant Explanatory Variables
Chapter 14: Omitted Explanatory Variables, Multicollinearity, and Irrelevant Explanatory Variables Chapter 14 Outline Review o Unbiased Estimation Procedures Estimates and Random Variables Mean of the
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 informationAnswers to Problem Set #4
Answers to Problem Set #4 Problems. Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated variance covariance matrix are given by: bβ bβ 2 bβ 3 = 2
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More informationAmherst College Department of Economics Economics 360 Fall 2012
Amherst College Department of Economics Economics 360 Fall 2012 Monday, December 3: Omitted Variables and the Instrumental Variable Estimation Procedure Chapter 20 Outline Revisit Omitted Explanatory Variable
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 information4. Nonlinear regression functions
4. Nonlinear regression functions Up to now: Population regression function was assumed to be linear The slope(s) of the population regression function is (are) constant The effect on Y of a unit-change
More informationLECTURE 15: SIMPLE LINEAR REGRESSION I
David Youngberg BSAD 20 Montgomery College LECTURE 5: SIMPLE LINEAR REGRESSION I I. From Correlation to Regression a. Recall last class when we discussed two basic types of correlation (positive and negative).
More informationSolutions: Monday, October 15
Amherst College Department of Economics Economics 360 Fall 2012 1. Consider Nebraska petroleum consumption. Solutions: Monday, October 15 Petroleum Consumption Data for Nebraska: Annual time series data
More informationAP Statistics L I N E A R R E G R E S S I O N C H A P 7
AP Statistics 1 L I N E A R R E G R E S S I O N C H A P 7 The object [of statistics] is to discover methods of condensing information concerning large groups of allied facts into brief and compendious
More informationLecture 5. In the last lecture, we covered. This lecture introduces you to
Lecture 5 In the last lecture, we covered. homework 2. The linear regression model (4.) 3. Estimating the coefficients (4.2) This lecture introduces you to. Measures of Fit (4.3) 2. The Least Square Assumptions
More informationApplied Econometrics. Applied Econometrics Second edition. Dimitrios Asteriou and Stephen G. Hall
Applied Econometrics Second edition Dimitrios Asteriou and Stephen G. Hall MULTICOLLINEARITY 1. Perfect Multicollinearity 2. Consequences of Perfect Multicollinearity 3. Imperfect Multicollinearity 4.
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 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 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 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 informationSolutions: Monday, October 22
Amherst College Department of Economics Economics 360 Fall 2012 1. Focus on the following agricultural data: Solutions: Monday, October 22 Agricultural Production Data: Cross section agricultural data
More informationEconometrics Midterm Examination Answers
Econometrics Midterm Examination Answers March 4, 204. Question (35 points) Answer the following short questions. (i) De ne what is an unbiased estimator. Show that X is an unbiased estimator for E(X i
More informationChapter 10: Multiple Regression Analysis Introduction
Chapter 10: Multiple Regression Analysis Introduction Chapter 10 Outline Simple versus Multiple Regression Analysis Goal of Multiple Regression Analysis A One-Tailed Test: Downward Sloping Demand Theory
More informationChapter 3. Introduction to Linear Correlation and Regression Part 3
Tuesday, December 12, 2000 Ch3 Intro Correlation Pt 3 Page: 1 Richard Lowry, 1999-2000 All rights reserved. Chapter 3. Introduction to Linear Correlation and Regression Part 3 Regression The appearance
More informationEconometrics Review questions for exam
Econometrics Review questions for exam Nathaniel Higgins nhiggins@jhu.edu, 1. Suppose you have a model: y = β 0 x 1 + u You propose the model above and then estimate the model using OLS to obtain: ŷ =
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 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 informationIntroduction to Simple Linear Regression
Introduction to Simple Linear Regression 1. Regression Equation A simple linear regression (also known as a bivariate regression) is a linear equation describing the relationship between an explanatory
More informationSolutions: Wednesday, December 12
Amherst College Department of Economics Economics 360 Fall 2012 Solutions: Wednesday, December 12 Beef Market Data: Monthly time series data relating to the market for beef from 1977 to 1986 Q t P t FeedP
More informationChapter 14. Statistical versus Deterministic Relationships. Distance versus Speed. Describing Relationships: Scatterplots and Correlation
Chapter 14 Describing Relationships: Scatterplots and Correlation Chapter 14 1 Statistical versus Deterministic Relationships Distance versus Speed (when travel time is constant). Income (in millions of
More informationMBF1923 Econometrics Prepared by Dr Khairul Anuar
MBF1923 Econometrics Prepared by Dr Khairul Anuar L4 Ordinary Least Squares www.notes638.wordpress.com Ordinary Least Squares The bread and butter of regression analysis is the estimation of the coefficient
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 information1 Quantitative Techniques in Practice
1 Quantitative Techniques in Practice 1.1 Lecture 2: Stationarity, spurious regression, etc. 1.1.1 Overview In the rst part we shall look at some issues in time series economics. In the second part we
More informationAnswer Key: Problem Set 6
: Problem Set 6 1. Consider a linear model to explain monthly beer consumption: beer = + inc + price + educ + female + u 0 1 3 4 E ( u inc, price, educ, female ) = 0 ( u inc price educ female) σ inc var,,,
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 informationProblem Set 10: Panel Data
Problem Set 10: Panel Data 1. Read in the data set, e11panel1.dta from the course website. This contains data on a sample or 1252 men and women who were asked about their hourly wage in two years, 2005
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 informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
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 informationLecture (chapter 13): Association between variables measured at the interval-ratio level
Lecture (chapter 13): Association between variables measured at the interval-ratio level Ernesto F. L. Amaral April 9 11, 2018 Advanced Methods of Social Research (SOCI 420) Source: Healey, Joseph F. 2015.
More informationIntro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
More informationNov 13 AP STAT. 1. Check/rev HW 2. Review/recap of notes 3. HW: pg #5,7,8,9,11 and read/notes pg smartboad notes ch 3.
Nov 13 AP STAT 1. Check/rev HW 2. Review/recap of notes 3. HW: pg 179 184 #5,7,8,9,11 and read/notes pg 185 188 1 Chapter 3 Notes Review Exploring relationships between two variables. BIVARIATE DATA Is
More informationIntro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
More informationLecture 14. More on using dummy variables (deal with seasonality)
Lecture 14. More on using dummy variables (deal with seasonality) More things to worry about: measurement error in variables (can lead to bias in OLS (endogeneity) ) Have seen that dummy variables are
More informationInteractions. Interactions. Lectures 1 & 2. Linear Relationships. y = a + bx. Slope. Intercept
Interactions Lectures 1 & Regression Sometimes two variables appear related: > smoking and lung cancers > height and weight > years of education and income > engine size and gas mileage > GMAT scores and
More informationSemester 2, 2015/2016
ECN 3202 APPLIED ECONOMETRICS 2. Simple linear regression B Mr. Sydney Armstrong Lecturer 1 The University of Guyana 1 Semester 2, 2015/2016 PREDICTION The true value of y when x takes some particular
More informationSIMPLE REGRESSION ANALYSIS. Business Statistics
SIMPLE REGRESSION ANALYSIS Business Statistics CONTENTS Ordinary least squares (recap for some) Statistical formulation of the regression model Assessing the regression model Testing the regression coefficients
More informationStatistics and Quantitative Analysis U4320. Segment 10 Prof. Sharyn O Halloran
Statistics and Quantitative Analysis U4320 Segment 10 Prof. Sharyn O Halloran Key Points 1. Review Univariate Regression Model 2. Introduce Multivariate Regression Model Assumptions Estimation Hypothesis
More informationEcon 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu Sampling Distribution of b 1 Expected value of b 1 : Variance of b 1 : E(b 1 ) = 1 Var(b 1 ) = σ 2 /SS x Estimate of
More informationBusiness Statistics. Lecture 10: Course Review
Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,
More informationMultiple Regression Analysis
Chapter 4 Multiple Regression Analysis The simple linear regression covered in Chapter 2 can be generalized to include more than one variable. Multiple regression analysis is an extension of the 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 informationChapter 1 Handout: Descriptive Statistics
Preview Chapter 1 Handout: Descriptive Statistics Describing a Single Data Variable o Introduction to Distributions o Measure of the Distribution Center: Mean (Average) o Measures of the Distribution Spread:
More informationLinear Regression with Multiple Regressors
Linear Regression with Multiple Regressors (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution
More informationGov 2000: 9. Regression with Two Independent Variables
Gov 2000: 9. Regression with Two Independent Variables Matthew Blackwell Fall 2016 1 / 62 1. Why Add Variables to a Regression? 2. Adding a Binary Covariate 3. Adding a Continuous Covariate 4. OLS Mechanics
More informationLab 11 - Heteroskedasticity
Lab 11 - Heteroskedasticity Spring 2017 Contents 1 Introduction 2 2 Heteroskedasticity 2 3 Addressing heteroskedasticity in Stata 3 4 Testing for heteroskedasticity 4 5 A simple example 5 1 1 Introduction
More informationGeneral Linear Model (Chapter 4)
General Linear Model (Chapter 4) Outcome variable is considered continuous Simple linear regression Scatterplots OLS is BLUE under basic assumptions MSE estimates residual variance testing regression coefficients
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 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 informationExercise sheet 3 The Multiple Regression Model
Exercise sheet 3 The Multiple Regression Model Note: In those problems that include estimations and have a reference to a data set the students should check the outputs obtained with Gretl. 1. Let 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 informationWednesday, December 12 Handout: Simultaneous Equations Identification
Amherst College epartment of Economics Economics 360 Fall 2012 Wednesday, ecember 12 Handout: imultaneous Equations Identification Preview Review o emand and upply Models o Ordinary Least quares (OL) Estimation
More informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationStatistical Inference. Part IV. Statistical Inference
Part IV Statistical Inference As of Oct 5, 2017 Sampling Distributions of the OLS Estimator 1 Statistical Inference Sampling Distributions of the OLS Estimator Testing Against One-Sided Alternatives Two-Sided
More informationAnswer Key. 9.1 Scatter Plots and Linear Correlation. Chapter 9 Regression and Correlation. CK-12 Advanced Probability and Statistics Concepts 1
9.1 Scatter Plots and Linear Correlation Answers 1. A high school psychologist wants to conduct a survey to answer the question: Is there a relationship between a student s athletic ability and his/her
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationDo not copy, post, or distribute
14 CORRELATION ANALYSIS AND LINEAR REGRESSION Assessing the Covariability of Two Quantitative Properties 14.0 LEARNING OBJECTIVES In this chapter, we discuss two related techniques for assessing a possible
More informationLECTURE 5. Introduction to Econometrics. Hypothesis testing
LECTURE 5 Introduction to Econometrics Hypothesis testing October 18, 2016 1 / 26 ON TODAY S LECTURE We are going to discuss how hypotheses about coefficients can be tested in regression models We will
More informationConditions for Regression Inference:
AP Statistics Chapter Notes. Inference for Linear Regression We can fit a least-squares line to any data relating two quantitative variables, but the results are useful only if the scatterplot shows a
More informationHandout 12. Endogeneity & Simultaneous Equation Models
Handout 12. Endogeneity & Simultaneous Equation Models In which you learn about another potential source of endogeneity caused by the simultaneous determination of economic variables, and learn how to
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 informationregression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist
regression analysis is a type of inferential statistics which tells us whether relationships between two or more variables exist sales $ (y - dependent variable) advertising $ (x - independent variable)
More informationLecture 24: Partial correlation, multiple regression, and correlation
Lecture 24: Partial correlation, multiple regression, and correlation Ernesto F. L. Amaral November 21, 2017 Advanced Methods of Social Research (SOCI 420) Source: Healey, Joseph F. 2015. Statistics: A
More informationStat 101 L: Laboratory 5
Stat 101 L: Laboratory 5 The first activity revisits the labeling of Fun Size bags of M&Ms by looking distributions of Total Weight of Fun Size bags and regular size bags (which have a label weight) of
More informationRegression Analysis. BUS 735: Business Decision Making and Research
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals and Agenda Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn
More information2 Prediction and Analysis of Variance
2 Prediction and Analysis of Variance Reading: Chapters and 2 of Kennedy A Guide to Econometrics Achen, Christopher H. Interpreting and Using Regression (London: Sage, 982). Chapter 4 of Andy Field, Discovering
More informationAmherst College Department of Economics Economics 360 Fall 2015 Monday, December 7 Problem Set Solutions
Amherst College epartment of Economics Economics 3 Fall 2015 Monday, ecember 7 roblem et olutions 1. Consider the following linear model: tate residential Election ata: Cross section data for the fifty
More informationReview of Multiple Regression
Ronald H. Heck 1 Let s begin with a little review of multiple regression this week. Linear models [e.g., correlation, t-tests, analysis of variance (ANOVA), multiple regression, path analysis, multivariate
More informationAP CALCULUS BC 2010 SCORING GUIDELINES
AP CALCULUS BC 2010 SCORING GUIDELINES Question 3 2 A particle is moving along a curve so that its position at time t is ( x() t, y() t ), where xt () = t 4t+ 8 and yt () is not explicitly given. Both
More informationCorrelation and Linear Regression
Correlation and Linear Regression Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means
More information