Wednesday, September 26 Handout: Estimating the Variance of an Estimate s Probability Distribution
|
|
- Stella Ryan
- 5 years ago
- Views:
Transcription
1 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 (OLS) Estimation Procedure o General Properties of the Ordinary Least Squares (OLS) Estimation Procedure o Importance of the Coefficient Estimate s Probability Distribution Mean (Center) of the Coefficient Estimate s Probability Distribution Variance (Spread) of the Coefficient Estimate s Probability Distribution Estimating the Variance of the Coefficient Estimate s Probability Distribution o Step 1: Estimate the Variance of the Error erm s Probability Distribution First Attempt: Variance of the Error erm s Numerical Values Second Attempt: Variance of the Residual s Numerical Values hird Attempt: Adjusted Variance of the Residual s Numerical Values o Step 2: Use the Estimated Variance of the Error erm s Probability Distribution to Estimate the Variance of the Coefficient Estimate s Probability Distribution Degrees of Freedom Summary: he Ordinary Least Squares (OLS) Estimation Procedure o hree Important Parts Value of the Coefficient Variance of the Error erm s Probability Distribution Variance of the Coefficient Estimate s Probability Distribution o Regression Printouts Review: General Properties of the Ordinary Least Squares (OLS) Estimation Procedure When the standard ordinary least squares premises are met, the following equations describe the coefficient estimate s general properties, the estimate s probability distribution: Var[e] Mean[b x ] β x Var[b x ] Σ t1 (xt x ) 2 Importance of the Probability Distribution s Mean (Center) and Variance (Spread) Mean: When the mean of the estimate s probability distribution, Mean[b x ], equals the actual value of the coefficient, β x, the estimation procedure is unbiased. he estimation procedure does not systematically underestimate or overestimate the actual value. Variance: When the estimation procedure is unbiased, the variance of the estimate s probability distribution, Var[b x ], determines the reliability of the estimate. As the variance decreases, the probability distribution becomes more tightly cropped around the actual value making it more likely for the coefficient estimate to be close to the actual value. Var[e] Mean[b x ] β x Var[b x ] Σ t1 (xt x ) 2 Estimation Procedure Determines the Reliability As Var[b x ] Decreases Is Unbiased of the Estimate Reliability of b x Increases he Problem: But there is a problem here, isn t there?
2 2 We need to know the variance of the error term s probability distribution to calculate the variance of the coefficient estimate s probability distribution. Unfortunately, the variance of the error term s probability distribution is unobservable. In reality, we can never know the actual variance of the error term s probability distribution. How can Clint proceed? 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. Estimating the Variance of the Coefficient Estimate s Probability Distribution Strategy Step 1: Estimate the variance of the error term s Step 2: Apply the relationship between the probability distribution from the available variances of the coefficient estimate s and information information from the first quiz: the error term s probability distributions: Var[e] EstVar[e] Var[b x ] Σ t1 (xt x ) 2 EstVar[e] EstVar[b x ] Σ t1 (xt x ) 2 wo Steps Step 1: Estimate of the variance of the error term s probability distribution. Step 2: Use the estimate of the variance of the error term s probability distribution to estimate the variance of the coefficient estimate s probability distribution. Step 1: Estimating the Variance of the Error erm s Probability Distribution We will now describe three attempts to estimate the variance using the results of Professor Lord s first quiz by calculating the: 1. Variance of the error term s numerical values from the first quiz. 2. Variance of the residual s numerical values from the first quiz. Adjusted variance of the residual s numerical values from the first quiz. We shall use simulations to assess these attempts by exploiting the relative frequency interpretation of probability: Relative Frequency Interpretation of Probability: After many, many repetitions of the experiment, the distribution of the numerical values from the experiments mirrors the random variable s probability distribution; the two distributions are identical: Applying this to the variance Distribution of the Numerical Values Variance of the Numerical Values After many, many repetitions Probability Distribution Variance of the Probability Distribution Preview: While the first two attempts fail for different reasons, they provide the motivation for the third attempt which succeeds. herefore, it is useful to explore the first two attempts even though they will fail.
3 Estimating the Variance of the Error erm s Probability Distribution, Var[e] First Attempt Strategy: Use the variance of the three error terms numerical values from the first quiz, Var[e 1, e 2, and e 1 st Quiz], to estimate the variance of the error term s probability distribution, Var[e]. First Quiz: Variance of Error erms Numerical Values. Var[e 1, e 2, and e 1 st Quiz] o Estimate Variance of Error erm s Probability Distribution, Var[e] Using the regression model, solve for the error term: y t β Const + β x x t + e t e t y t (β Const + β x x t ) Recall that the variance is the average of the squared deviations from the mean: Compute the deviations from the mean; Square the deviations; Calculate the average of the squared deviations. We now can calculate the variance of the error terms numerical values from the first quiz: Var[e 1, e 2, and e 1 st Quiz] (e 1 - Mean[e])2 + (e 2 - Mean[e]) 2 + (e - Mean[e]) 2 Since the error term reflects random influences: Mean[e]. First Quiz β Const 50 β x 2 e t y t (β Const + β x x t ) Student x t y t β Const + β x x t x t e t y t (50 + 2x t ) 2 e t Var[e 1, e 2, and e 1 st Quiz] e e e 2 1 st Quiz SSE 1st Quiz SSE Question: As a consequence of random influences, can we expect variance of the numerical values from one repetition, the first quiz to equal the actual variance of the coefficient estimate s probability distribution? Answer: Question: What can we hope for then? Answer:
4 4 Econometrics Lab Simulation Unbiased Estimation Procedure: After many, many repetitions of the experiment the average (mean) of the estimates equals the actual value. Mean (average) of the error term s numerical values from all repetitions. Sum of Squared Residuals Sum of Squared Errors Divide by sample size or degrees of freedom? Use errors or residuals? Estimate of the variance for the error s term probability distribution calculated from this repetition, EstVar[e] Repetition Error erms Mean Var SSE Divide by 2 Use Err Res Error Var Est Mean Act Err Var Does the error term represent a random influence? Actual Variance of Error erm s Probability Distribution, Var[e] Does the simulation represent the variance of the error term s probability distribution accurately? Variance of the error term s numerical values from all repetitions. Is the estimation procedure for the variance of the error term s probability distribution unbiased? Average of the variance estimates from all repetitions. Estimate for the Variance Actual Value of the Error erm s Var[e] Repetition e 1 e 2 e SSE Probability Distribution Observations Can we expect the estimate to equal the actual value?. In fact, we all but certain that the estimate will not equal the actual value. Sometimes the estimate is than the actual value and sometimes it is.
5 5 Question: What is the best we can hope for? An unbiased estimation procedure does not systematically underestimate or overestimate the actual value. When the experiment is repeated many, many times, the average of the numerical values of the estimates will equal the actual value. Question: How can we determine whether or not the estimation procedure for variance of the error term s probability distribution unbiased? Answer: Compare the actual variance of the error term s probability distribution and the mean (average) of the variance estimates after many, many repetitions. Mean (Average) of the Estimates for the Variance of the Error erm s Actual Value Simulation Probability Distribution Var[e] Repetitions SSE Divided by Good news: his procedure is. Bad news: Does this procedure help Clint?. Why? NB: Nevertheless, keep in mind that the sum of squared errors based on the actual constant and coefficient, β Const or β x, provides an unbiased estimate of variance the error term s probability distribution. Sum of Squared Errors (SSE) Versus Sum of Squared Residuals () Sum of Squared Errors (SSE) Sum of Squared Residuals () Based on the error terms Based on the residuals y t β Const + β x x t + e t Res t y t Esty t where Esty t b Const + b x x t e t y t (β Const + β x x t ) Res t y t (b Const + b x x t ) Need the actual constant and coefficient, β Const and β x, to calculate the sum or squared errors But β Const and β x are unobservable; that is the whole problem Use the OLS procedure to calculate the estimates of the constant and coefficient, b Const and b x Use the estimates to calculate the sum of squared residuals We can think of the sum of squared residuals () as an estimate of the sum of squared errors (SSE).
6 6 Estimating the Variance of the Error erm s Probability Distribution, Var[e] Second Attempt Calculate the variance of the three residuals, Res 1, and Res ; that is, the variance of the actual value of y less the estimated value of y. Use this variance to estimate Var[e]: First Quiz: Variance of Residuals Numerical Value, Var[Res 1, and Res 1 st Quiz] Estimates Variance of Error erm s Probability Distribution: Var[e] First, we calculate the residuals from the first quiz Res t y t Est t y t (b Const + b x x t ) NB: Here we use the estimated constant and coefficient. and then their variance: Var[Res 1, and Res 1 st Quiz] (Res 1 - Mean[Res])2 + (Res 2 - Mean[Res]) 2 + (Res - Mean[Res]) 2 First Quiz b Const 6 b x Res t y t Esty t Student x t y t Est t b Const + b x x t x t Res t y t ( x t ) Res 2 t Sum Mean[Res] Mean[Res 1, and Res 1 st Quiz] Res 1 + Res 2 + Res In fact, we can provide that the mean of the residuals will always equal 0. Var[Res 1, and Res 1 st Quiz] Res Res Res 2 1 st Quiz 1st Quiz Econometrics Lab Simulation Estimate for the Variance Actual Value of the Error erm s Var[e] Repetition Res 1 Res 2 Res Probability Distribution Observations Can we expect the estimate to equal the actual value?. In fact, we all but certain that the estimate will not equal the actual value. Sometimes the estimate is than the actual value and sometimes it is.
7 7 Question: Is this estimation procedure unbiased? Mean (Average) of the Estimates for the Variance of the Error erm s Actual Value Simulation Probability Distribution Var[e] Repetitions Divided by Good news: Clint the information to perform this calculation. Bad news: he procedure is. It systematically the variance of the error term s probability distribution. Why Is Our Second Attempt Biased? Recall the difference between the error terms and the residuals: Error term Residual e t y t (β Const + β x x t ) Res t y t Res t y t (b Const + b x x t ) Var[e 1, e 2, and e 1 st Quiz] SSE Var[Res 1, and Res 1 st Quiz] Estimation procedure based on Estimation procedure based on the SSE s is. the s is. SSE 1 st Quiz e 1 + e 2 + e [y 1 (β Const + β x x 1 )] 2 + [y 2 (β Const + β x x 2 )] 2 + [y (β Const + β x x )] 2 1 st Quiz Res Res Res 2 [y 1 (b Const + b x x 1 )] 2 + [y 2 (b Const + b x x 2 )] 2 + [y (b Const + b x x )] 2 Difference between the equations: SSE uses β Const and β x. uses b Const and b x. Question: How were b Const and b x chosen? Answer: o minimize the sum of squared residuals,. he sum of squared residuals will equal the sum of squared errors only if the estimates equal the actual values of β Const and β x. But we can never expect the estimates to equal the actual values: Sum using b s Sum using β s In all likelihood, β Const b Const and β x b x. SSE SSE When the actual Var[Res 1, and Res 1 st Quiz] Var[e 1, e 2, and e 1 st Quiz] constant and coefficient are used Procedure systematically Unbiased estimation the procedure is the variance of the procedure. error term s probability distribution
8 8 Estimating the Variance of the Error erm s Probability Distribution, Var[e] hird Attempt It can be shown that an unbiased estimate of the variance error term s probability distribution results when we divide the sum of squared residuals by the degrees of freedom rather than the sample size: Res 2 AdjVar[Res 1, and Res 1 st 1 + Res Res Quiz] Degrees of Freedom Degrees of Freedom Number of Degrees of Freedom Sample Size Estimated 2 1 Parameters AdjVar[Res 1, and Res 1 st Quiz] 1st Quiz 1 1 Econometrics Lab Simulation Estimate for the Variance Actual Value of the Error erm s Var[e] Repetition Res 1 Res 2 Res Probability Distribution Observations Can we expect the estimate to equal the actual value?. In fact, we all but certain that the estimate will not equal the actual value. Sometimes the estimate is than the actual value and sometimes it is. Question: Is this estimation procedure unbiased? Mean (Average) of the Estimates for the Variance of the Error erm s Actual Value Simulation Probability Distribution Var[e] Repetitions Divided by Good news: Clint the information to perform this calculation. he procedure is.
9 9 Step 2: Use the estimate for the variance of the error term s probability distribution to estimate the variance for the coefficient estimate s probability distribution. Step 1: Estimate the variance of the error term s Step 2: Apply the relationship between the probability distribution from the available variances of the coefficient estimate s and information information from the first quiz: the error term s probability distributions: EstVar[e] AdjVar[Res] Degrees of Freedom Var[b x ] Var[e] Σ t1 (xt x ) 2 EstVar[e] EstVar[b x ] Σ t1 (xt x ) 2 x 1 5 x 2 15 x 25 Σ t1 (xt x ) SE[b x ] EstVar[b x ] NB: he square root of the estimated variance is called the standard error. What do we know about the estimation procedure for the error term s probability distribution?. What can we hope to be able to say about the estimation procedure for the coefficient estimate s probability distribution?.
10 10 Econometrics Lab Simulation Unbiased estimation procedure: After many, many repetitions of the experiment the average of the estimates equals the actual value. Estimated coefficient value from this repetition: Σ t1 (yt y )(x t x ) b x Σ t1 (xt x ) 2 EstVar[e] Act Coef Degrees of Freedom EstVar[e] EstVar[b x ] Σ t1 (xt x ) 2 Estimate of the variance for the coefficient estimate s probability distribution calculated from this repetition Repetition Coef Value Est Mean Var Sum Sqr XDev Coef Var Est Mean Act Err Var Variance of the estimated coefficient values from all repetitions. Actual Variance of Error erm s Probability Distribution: Var[e] Actual Variance of Coefficient Estimate s Probability Distribution: Var[b x ] Var[e] Σ t1 (xt x ) 2 Is the estimation procedure for the variance of the coefficient estimate s probability distribution unbiased? Average of the variance estimates from all repetitions. Estimate for the Variance Actual Value of the Coefficient Estimate s Var[e] Repetition Probability Distribution Is the estimation procedure for the variance of the coefficient estimate s probability distribution unbiased? Variance of the Coefficent Mean (Average) of the Estimates Actual Estimate s Probability for the Variance of the Coefficient Var[e] Distribution: Var[b x ] Estimate s Probability Distribution
11 11 Degrees of Freedom Recall Attempts 2 and to estimate the variance of the error terms probability distribution Clint is trying to estimate the variance of the error term from the residual information: Error terms Residuals e t y t (β Const + β x x t ) Res t y t (b Const + b x x t ) We can interpret the residuals as the estimated error terms. Strategy: Use the residuals ( estimated errors ) to estimate the variance of the error term s probability distribution. Question: Why does dividing by the sample size fail, but dividing by the degrees of freedom succeed? Attempt 2: We divided by the sample size: Var[Res 1, and Res ] (Res 1 - Mean[Res])2 + (Res 2 - Mean[Res]) 2 + (Res - Mean[Res]) 2 Sample Size Since Mean[Res] 0: Var[Res 1, and Res ] Res Res Res Sample Size Sample Size Attempt : We divided by the degrees of freedom rather than the sample size: Res Res Res AdjVar[Res 1, and Res ] Degrees of Freedom Degrees of Freedom 1 Degrees of Freedom Sample Size Number of Estimated Parameters 2 1 Dividing by the degrees of freedom rather than the sample size worked. Question: Why does dividing by 1 rather than work? How Do We Calculate an Average? Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Mean (Average) for June Each of the 100 Junes in the twentieth century provide one piece of information in calculating the average. Consequently, to calculate an average we divide the sum by the number of pieces of information. Key Principle: o calculate an average we divide the sum by the number of pieces of information: Sum Mean (Average) Number of Pieces of Infomration
12 12 Claim: he degrees of freedom equal the number of pieces of information that are available to estimate the error term s variance. y Question: Why does subtracting 2 from the sample size make sense? o understand why, suppose that the sample size were 2. Plot the scatter diagram. With only two observations, we only have points. he best fitting line passes through each of the points on the scatter diagram. Consequently, the two residuals, estimated errors, for each observation must always equal when the sample size is 2 regardless of what the variance of the error term s probability distribution actually equals: Sample Size of 2: Res 1 and Res 2 Question: Do the first two residuals, the first two estimated errors, provide regardless of what Var[e] equals information about the actual variance of error term s probability distribution? x Question: Which observation provides the first piece of information about of the actual variance of the error term s probability distribution? Summary he first two observations provide no information about the variance. he third observation provides the piece of information about the variance. y Res Res 2 y Consequently, in Clint s case, Res 1 when there are three observations, we should divide by to calculate the average of the squared Suggests large error term variance x deviations because we really only have piece of information. x Suggests small error term variance In general, we should divide by the degrees of freedom, the sample size less the number of estimated parameters: Degrees of Freedom Sample Size Number of Estimated Parameters
13 1 Summary of Ordinary Least Squares (OLS) Calculations and the Regression Printout he ordinary least squares (OLS) estimation procedure actually includes three procedures: A Procedure to Estimate the Value of the Parameters Σ t1 (yt y )(x t x ) o b x Σ t1 (xt x 240 ) o b Const y b x x A Procedure to Estimate the Variance of the Error erm s Probability Distribution o Σ 2 t1 Rest Σ t1 (yt Esty t ) 2 Σ t1 (yt b Const b x x t ) 2 54 o EstVar[e] AdjVar[Res 1, and Res ] Degrees of Freedom o S.E. of regression EstVar[e] he square root of the estimated value of the error terms probability distribution is call the standard error of the regression A Procedure to Estimate the Variance of the Coefficient Estimate s Probability Distribution EstVar[e] o EstVar[b x ] Σ t1 (xt x 54 ) o SE[b x ] EstVar[b x ] Good News: When the standard ordinary least squares (OLS) premises are satisfied: Each of these procedures is unbiased. he procedure to estimate the value of the parameters is the best linear unbiased estimation procedure. EViews performs these calculations for us thereby saving us the laborious task of performing all the arithmetic: Dependent Variable: Y Included observations: Variable Coefficient Std. Error t-statistic Prob. X C S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion
Chapter 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 informationWednesday, September 19 Handout: Ordinary Least Squares Estimation Procedure The Mechanics
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
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 informationChapter 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, 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 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 informationGAMINGRE 8/1/ of 7
FYE 09/30/92 JULY 92 0.00 254,550.00 0.00 0 0 0 0 0 0 0 0 0 254,550.00 0.00 0.00 0.00 0.00 254,550.00 AUG 10,616,710.31 5,299.95 845,656.83 84,565.68 61,084.86 23,480.82 339,734.73 135,893.89 67,946.95
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 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 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 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 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 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 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 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 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 informationBrief Suggested Solutions
DEPARTMENT OF ECONOMICS UNIVERSITY OF VICTORIA ECONOMICS 366: ECONOMETRICS II SPRING TERM 5: ASSIGNMENT TWO Brief Suggested Solutions Question One: Consider the classical T-observation, K-regressor linear
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 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 informationTechnical note on seasonal adjustment for M0
Technical note on seasonal adjustment for M0 July 1, 2013 Contents 1 M0 2 2 Steps in the seasonal adjustment procedure 3 2.1 Pre-adjustment analysis............................... 3 2.2 Seasonal adjustment.................................
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 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 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 informationLab 6 - Simple Regression
Lab 6 - Simple Regression Spring 2017 Contents 1 Thinking About Regression 2 2 Regression Output 3 3 Fitted Values 5 4 Residuals 6 5 Functional Forms 8 Updated from Stata tutorials provided by Prof. Cichello
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 informationSuan Sunandha Rajabhat University
Forecasting Exchange Rate between Thai Baht and the US Dollar Using Time Series Analysis Kunya Bowornchockchai Suan Sunandha Rajabhat University INTRODUCTION The objective of this research is to forecast
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 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 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 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 informationDummy Variables. Susan Thomas IGIDR, Bombay. 24 November, 2008
IGIDR, Bombay 24 November, 2008 The problem of structural change Model: Y i = β 0 + β 1 X 1i + ɛ i Structural change, type 1: change in parameters in time. Y i = α 1 + β 1 X i + e 1i for period 1 Y i =
More informationMultivariate Regression Model Results
Updated: August, 0 Page of Multivariate Regression Model Results 4 5 6 7 8 This exhibit provides the results of the load model forecast discussed in Schedule. Included is the forecast of short term system
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 informationDAILY QUESTIONS 28 TH JUNE 18 REASONING - CALENDAR
DAILY QUESTIONS 28 TH JUNE 18 REASONING - CALENDAR LEAP AND NON-LEAP YEAR *A non-leap year has 365 days whereas a leap year has 366 days. (as February has 29 days). *Every year which is divisible by 4
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 informationTime Series Analysis
Time Series Analysis A time series is a sequence of observations made: 1) over a continuous time interval, 2) of successive measurements across that interval, 3) using equal spacing between consecutive
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 informationECON Introductory Econometrics. Lecture 17: Experiments
ECON4150 - Introductory Econometrics Lecture 17: Experiments Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 13 Lecture outline 2 Why study experiments? The potential outcome framework.
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 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 informationAutocorrelation. Think of autocorrelation as signifying a systematic relationship between the residuals measured at different points in time
Autocorrelation Given the model Y t = b 0 + b 1 X t + u t Think of autocorrelation as signifying a systematic relationship between the residuals measured at different points in time This could be caused
More informationJayalath Ekanayake Jonas Tappolet Harald Gall Abraham Bernstein. Time variance and defect prediction in software projects: additional figures
Jayalath Ekanayake Jonas Tappolet Harald Gall Abraham Bernstein TECHNICAL REPORT No. IFI-2.4 Time variance and defect prediction in software projects: additional figures 2 University of Zurich Department
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 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 information6. Assessing studies based on multiple regression
6. Assessing studies based on multiple regression Questions of this section: What makes a study using multiple regression (un)reliable? When does multiple regression provide a useful estimate of the causal
More informationEconometrics. 8) Instrumental variables
30C00200 Econometrics 8) Instrumental variables Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Thery of IV regression Overidentification Two-stage least squates
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 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 informationTOPIC: Descriptive Statistics Single Variable
TOPIC: Descriptive Statistics Single Variable I. Numerical data summary measurements A. Measures of Location. Measures of central tendency Mean; Median; Mode. Quantiles - measures of noncentral tendency
More informationExercises (in progress) Applied Econometrics Part 1
Exercises (in progress) Applied Econometrics 2016-2017 Part 1 1. De ne the concept of unbiased estimator. 2. Explain what it is a classic linear regression model and which are its distinctive features.
More informationModel Specification and Data Problems. Part VIII
Part VIII Model Specification and Data Problems As of Oct 24, 2017 1 Model Specification and Data Problems RESET test Non-nested alternatives Outliers A functional form misspecification generally means
More informationProblem set 1: answers. April 6, 2018
Problem set 1: answers April 6, 2018 1 1 Introduction to answers This document provides the answers to problem set 1. If any further clarification is required I may produce some videos where I go through
More informationNATCOR Regression Modelling for Time Series
Universität Hamburg Institut für Wirtschaftsinformatik Prof. Dr. D.B. Preßmar Professor Robert Fildes NATCOR Regression Modelling for Time Series The material presented has been developed with the substantial
More informationTime series and Forecasting
Chapter 2 Time series and Forecasting 2.1 Introduction Data are frequently recorded at regular time intervals, for instance, daily stock market indices, the monthly rate of inflation or annual profit figures.
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 information7. Prediction. Outline: Read Section 6.4. Mean Prediction
Outline: Read Section 6.4 II. Individual Prediction IV. Choose between y Model and log(y) Model 7. Prediction Read Wooldridge (2013), Chapter 6.4 2 Mean Prediction Predictions are useful But they are subject
More informationACE 564 Spring Lecture 8. Violations of Basic Assumptions I: Multicollinearity and Non-Sample Information. by Professor Scott H.
ACE 564 Spring 2006 Lecture 8 Violations of Basic Assumptions I: Multicollinearity and Non-Sample Information by Professor Scott H. Irwin Readings: Griffiths, Hill and Judge. "Collinear Economic Variables,
More informationEconometrics Lab Hour Session 6
Econometrics Lab Hour Session 6 Agustín Bénétrix benetria@tcd.ie Office hour: Wednesday 4-5 Room 3021 Martin Schmitz schmitzm@tcd.ie Office hour: Monday 5-6 Room 3021 Outline Importing the dataset Time
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 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 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 informationProblem Set 2: Box-Jenkins methodology
Problem Set : Box-Jenkins methodology 1) For an AR1) process we have: γ0) = σ ε 1 φ σ ε γ0) = 1 φ Hence, For a MA1) process, p lim R = φ γ0) = 1 + θ )σ ε σ ε 1 = γ0) 1 + θ Therefore, p lim R = 1 1 1 +
More informationLecture Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University
Lecture 15 20 Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University Modeling for Time Series Forecasting Forecasting is a necessary input to planning, whether in business,
More informationIntroduction to Forecasting
Introduction to Forecasting Introduction to Forecasting Predicting the future Not an exact science but instead consists of a set of statistical tools and techniques that are supported by human judgment
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 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 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 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 informationECON Introductory Econometrics. Lecture 5: OLS with One Regressor: Hypothesis Tests
ECON4150 - Introductory Econometrics Lecture 5: OLS with One Regressor: Hypothesis Tests Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 5 Lecture outline 2 Testing Hypotheses about one
More informationWHEN IS IT EVER GOING TO RAIN? Table of Average Annual Rainfall and Rainfall For Selected Arizona Cities
WHEN IS IT EVER GOING TO RAIN? Table of Average Annual Rainfall and 2001-2002 Rainfall For Selected Arizona Cities Phoenix Tucson Flagstaff Avg. 2001-2002 Avg. 2001-2002 Avg. 2001-2002 October 0.7 0.0
More informationECON 366: ECONOMETRICS II. SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued. Brief Suggested Solutions
DEPARTMENT OF ECONOMICS UNIVERSITY OF VICTORIA ECON 366: ECONOMETRICS II SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued Brief Suggested Solutions 1. In Lab 8 we considered the following
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 information28. SIMPLE LINEAR REGRESSION III
28. SIMPLE LINEAR REGRESSION III Fitted Values and Residuals To each observed x i, there corresponds a y-value on the fitted line, y = βˆ + βˆ x. The are called fitted values. ŷ i They are the values of
More informationLesson 8: Variability in a Data Distribution
Classwork Example 1: Comparing Two Distributions Robert s family is planning to move to either New York City or San Francisco. Robert has a cousin in San Francisco and asked her how she likes living in
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 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 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 informationAnnual Average NYMEX Strip Comparison 7/03/2017
Annual Average NYMEX Strip Comparison 7/03/2017 To Year to Year Oil Price Deck ($/bbl) change Year change 7/3/2017 6/1/2017 5/1/2017 4/3/2017 3/1/2017 2/1/2017-2.7% 2017 Average -10.4% 47.52 48.84 49.58
More informationFinal Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008,
Professor Dr. Roman Liesenfeld Final Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008, 10.00 11.30am 1 Part 1 (38 Points) Consider the following
More informationLong-term Water Quality Monitoring in Estero Bay
Long-term Water Quality Monitoring in Estero Bay Keith Kibbey Laboratory Director Lee County Environmental Laboratory Division of Natural Resource Management Estero Bay Monitoring Programs Three significant
More informationMr. XYZ. Stock Market Trading and Investment Astrology Report. Report Duration: 12 months. Type: Both Stocks and Option. Date: Apr 12, 2011
Mr. XYZ Stock Market Trading and Investment Astrology Report Report Duration: 12 months Type: Both Stocks and Option Date: Apr 12, 2011 KT Astrologer Website: http://www.softwareandfinance.com/magazine/astrology/kt_astrologer.php
More informationAlterations to the Flat Weight For Age Scale BHA Data Published 22 September 2016
Alterations to the Flat Weight For Age Scale BHA Data Published 22 September 2016 Introduction What is weight for age? It is an allowance given to younger horses, usually three-year-olds, to enable them
More information5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1)
5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1) Assumption #A1: Our regression model does not lack of any further relevant exogenous variables beyond x 1i, x 2i,..., x Ki and
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 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 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 informationChapter 3. Regression-Based Models for Developing Commercial Demand Characteristics Investigation
Chapter Regression-Based Models for Developing Commercial Demand Characteristics Investigation. Introduction Commercial area is another important area in terms of consume high electric energy in Japan.
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 informationStatistical Models for Rainfall with Applications to Index Insura
Statistical Models for Rainfall with Applications to April 21, 2008 Overview The idea: Insure farmers against the risk of crop failure, like drought, instead of crop failure itself. It reduces moral hazard
More informationMultiple Regression Analysis: Estimation. Simple linear regression model: an intercept and one explanatory variable (regressor)
1 Multiple Regression Analysis: Estimation Simple linear regression model: an intercept and one explanatory variable (regressor) Y i = β 0 + β 1 X i + u i, i = 1,2,, n Multiple linear regression model:
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 informationFORECASTING COARSE RICE PRICES IN BANGLADESH
Progress. Agric. 22(1 & 2): 193 201, 2011 ISSN 1017-8139 FORECASTING COARSE RICE PRICES IN BANGLADESH M. F. Hassan*, M. A. Islam 1, M. F. Imam 2 and S. M. Sayem 3 Department of Agricultural Statistics,
More informationLecture Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University
Lecture 15 20 Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University Modeling for Time Series Forecasting Forecasting is a necessary input to planning, whether in business,
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 informationPractical Econometrics. for. Finance and Economics. (Econometrics 2)
Practical Econometrics for Finance and Economics (Econometrics 2) Seppo Pynnönen and Bernd Pape Department of Mathematics and Statistics, University of Vaasa 1. Introduction 1.1 Econometrics Econometrics
More informationLecture 8. Using the CLR Model
Lecture 8. Using the CLR Model Example of regression analysis. Relation between patent applications and R&D spending Variables PATENTS = No. of patents (in 1000) filed RDEXP = Expenditure on research&development
More informationF9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
More informationIntroduction to Statistical modeling: handout for Math 489/583
Introduction to Statistical modeling: handout for Math 489/583 Statistical modeling occurs when we are trying to model some data using statistical tools. From the start, we recognize that no model is perfect
More informationEuro-indicators Working Group
Euro-indicators Working Group Luxembourg, 9 th & 10 th June 2011 Item 9.4 of the Agenda New developments in EuroMIND estimates Rosa Ruggeri Cannata Doc 309/11 What is EuroMIND? EuroMIND is a Monthly INDicator
More information