The covariance is the two variable analog to the variance. The formula for the covariance between two variables is


 Deirdre Bennett
 1 years ago
 Views:
Transcription
1 Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables. To do ths there are several concepts that t would be good to have a gasp of Covarance and Correlaton Recall that Var( ) = σ = E( µ ) whch we estmate wth the sample varance ( ) S = n The covarance s the two varable analog to the varance. The formula for the covarance between two varables s Cov( X, Y ) = σ = E( µ )( y µ ) y y whch we estmate wth the sample covarance ( )( y y) Sy = n When two varables are negatvely related the covarance wll be negatve. When two varables are postvely related the covarance wll be postve. Go through some scatter plots. One potental problem wth the covarance s that we do not have any dea how bg t was. Wth the varance we could fnd the standard devaton. If we knew the was appromately normal ths standard devaton conveyed useful nformaton. We cannot do ths for the covarance. Another problem wth the covarance s that t depends on unts of measurement. If we are nterested n the relatonshp between ncome and years educaton we would get a dfferent covarance f we defned ncome n thousands of dollars than we would defnng ncome n dollars.
2 To correct these problems we can use correlaton between and y. The correlaton s gven by ρ y σ y = σ σ y whch we estmate wth the sample correlaton ρ y Sy ( )( y y) = = SS y ( ) ( y y) Ths s often called the Person Product Moment Correlaton Coeffcent. Both the correlaton and the sample correlaton must be between and. If two varables are perfectly negatvely correlated then the correlaton wll be. If they are perfectly postvely correlated t wll be. The covarance and correlaton descrbe the strength of the lnear relatonshp between two varables. Nether measure, however, descrbes ths relatonshp. To descrbe the lnear relatonshp between two varables we need smple lnear regresson analyss. Smple Lnear Regresson Analyss What s gong n the populaton? There are two varables that are related. y s the dependent varable. s the ndependent varable. In theory y should be caused by (or dependent on). Suppose that there s a lnear relatonshp between and y such that y = + + e. The way to thnk about ths s that n the populaton y s beng generated as the sum of (an ntercept term), (a functon of the ndependent varable), and e (an error term).
3 We can thnk of and as populaton parameters the same way we thought about µ as a populaton parameter. Note that and E y ( ) = + E( ) = + µ E( y ) = + We have already dealt wth a specal case of ths model. y = + e In ths case E y µ ( ) = = y and E y = = µ. ( ) y Suppose that we had a sample that looked somethng lke the followng y y.. y.... n n Then we would be able to estmate the populaton parameters and. Lets consder the data that you had last week for a homework assgnment. Ths was the data on unemployment duraton and age that you used n the case problem. The sample covarance between these two varables s 76.6 and the correlaton coeffcent s.66. So we know (from lookng at the scatter plot) and from these measures that there s a postve relatonshp between weeks unemployed and age. What sorts of thngs would lead to ths type of postve assocaton? Whch of these varables would be the ndependent varable and whch would be the dependent varable f we were settng ths up as a smple lnear regresson?
4 To ft a lne to ths data would bascally have to pck values for (or estmate) and. The estmate of the parameter wll gve us our yntercept and the estmate of the parameter wll gve us the slope? How can I go about fndng estmates for and? Scatterplot of Weeks Unemployed and Age Weeks Unemployed S r S y y = 76.6 = 65.9 = 4.69 = Age Scatterplot of Weeks Unemployed and Age Weeks Unemployed Age
5 If we have sample data we can wrte our orgnal equaton as e y e y.. =.... e y n n We want these errors to be as small as possble n absolute value. The way that we make them as small as possble n absolute value s to fnd and that mnmze n = ( ) y wth respect to and. When we do ths we would fnd that So Sy 76.6 = = =.537 S 4.69 = y = = 4. WU = Age
6 A Few Specal Propertes of Least Squares Estmates ) The least squares estmate always goes through the means. Ths means that (, y ) s always on the least squares lne. Algebracally, t means that y = + Graphcally t means that Scatterplot of Weeks Unemployed and Age 46.6 Weeks Unemployed Age ) e = ( y ) =. Ths s just a just a product of how we chose the and. It s a mathematcal fact. I cannot really gve you any ntuton for t wthout confusng you.
7 The Coeffcent of Determnaton We mght be nterested n how precse our estmate s. In partcular we mght be nterested n determnng how much better our lne ft than the sample mean of y. We could estmate y wth ts sample mean (one parameter) or we could ft a lne (by estmatng parameters, ). How much better do we do n terms of ft by estmatng two parameters? Note that ( ) ( ) y y = + + e + = + e Usng the above we can wrte ( y y) = ( ) + e + ( ) e = ( ) + e where SST = SSR + SSE SST = ( y y ) SSR = ( ) SSE = e The coeffcent of determnaton ( Rsquared ), defned as R SSR SSE = = SST SST s a measure of goodness of ft. It bascally tells us how much better than lne fts than the sample mean of y. Another way to thnk about Rsquared s that t tells us the percentage of the varaton n y away from ts mean s eplaned by versus unknown factors (the resduals). It s useful to thnk of a couple specfc cases.
8 Scatterplot of Weeks Unemployed and Age 46.6 Weeks Unemployed Age Ok Lets thnk about the case when and y are not correlated. The mean of y does just as good as any lne. Indeed the slope estmate should be close to zero so the mean of y s close to the least squares lne. One thng you should know for smple lnear regresson analyss (wth one ndependent varable) the square root of Rsquared s the sample correlaton.
9 Inference and Smple Lnear Regresson Thus far we have talked only about the mechancs of least squares. We have not talked about how to use estmates to make nferences about populaton parameters. Ths s the net step. Before we do ths we need to make some assumptons Assumptons ) We got the functonal form correct: y = + + e ) Zero mean dsturbance: Ee ( ) = 3) Constant varance dsturbance: Var( e ) 4) Errors not autocorrelated: Cov( e, e ) = j 5) Regressors are nonstochastc: Var( ) = j = σ  Constant var 6) e N(, σ )  Ths s not strctly necessary but does allow us to make nferences n the case when sample szes are not very large. The Samplng Dstrbuton of If we want to fnd the samplng dstrbuton of we need to specfy the mean, the varance, and the dstrbuton of. ) Lets workng on fndng the mean. ( )( y y) ( )( ( ) + e) = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e e = + = + Takng the epected value we fnd ( ) e E( ) E = + = ( ) So has mean. It s an unbased estmator. Equvalent to E ( ) = µ
10 ) Lets fnd the varance ( ) ( ) e Var( ) = E = E ( ) = E ( ) e = ( ) ( ) = = ( ) Equvalent to σ σ ( ) Var( ) = σ n 3) What s the dstrbuton of? If the dsturbances are normal then s lnear n the dsturbances so s normal If the dsturbances are not normal and the sample sze s large enough we have a central lmt theorem that says the dstrbuton of s appromately normal. Ok so all that we need to do s fnd an estmate of σ and we are n busness. Remember that σ =Var( e ). Thus the approprate estmator s S e SSE = = n n Now we can make nferences about.
11 Confdence Intervals and Hypothess Testng From our estmate wth the BLS data Note that.537 = 4.6 S = 56.9 Now we can estmate Var( ) S S σ σ 56.9 = = = ( ) 56.9 = = ( n ) S 669 Eample) Lets use ths to test the hypothess H H : = a : In words  we are testng the null that age as no effect on the duraton of unemployment spells aganst the alternatve that t does. Choose level of sgnfcance (. ) Get the value of the test statstc.537 z = =5.97 we would reject the null at almost any sgnfcance level..9 Eample) We could also construct a 95% confdence nterval.
12 Multple Regresson Here the model s smlar to the smple lnear model we already talked about. The only dfference beng that y s a lnear functon of a bunch more than one varable not just. That s y = kk + e For ths model E( y,..., k ) = kk E( y) = + µ + µ µ k 3 Var( y ) = Var( e ) = σ Instead of just estmatng 3 populaton parameters Var( e ) = σ we wll be estmatng k+ parameters through k  k+ parameters Var( e ) = σ We wll use least squares to ft our lne (actually t s a plane of dmenson k n k+ dmensonal space). Ths means that through k are the partcular s that mnmze ( y... ) k k Unlke the case of the smple lnear regresson model we are gong to be unable to wrte down formulas for through k. As you shall see we wll stll be able to wrte out a formula for.
13 Goodness of ft and Coeffcent of Determnaton Almost everythng that I sad about the Rsquared n the bvarate regresson model holds for the multvarate model. That s where R SSR SSE = = SST SST SST = ( y y) SSR = ( y y) SSE = e Because you can always mprove Rsquared by addng more varables to the model there s somethng called the adjusted Rsquared n Ra = ( R ) n k I do not epect you to remember ths, but you should understand the purpose for the adjusted Rsquared. Propertes of Multvarate Least Squares ) The least squares lne goes through the means y = k k Here s were the formula for comes n = y... k k ) The least squares resduals wpe out the s e = = j j,..., k
14 Assumptons ) We got the functonal form correct: y = e ) Zero mean dsturbance: Ee ( ) = 3) Constant varance dsturbance: Var( e ) 4) Errors not autocorrelated: Cov( e, e ) = j j k k = σ  Constant varance 5) Regressors are nonstochastc: Var( ) = =,..., n and j =,..., k 6) e N(, σ )  Ths s not strctly necessary but does allow us to make nferences n the case when sample szes are not very large. j Statstcal Inference and the Lnear Regresson Model We are gong to want to characterze the dstrbuton of,..., k. To do ths we need three thngs () The mean of our estmates. As n the case of the bvarate model the assumptons above we can say that E( ) = for all j =,..., k j j () We want to characterze the varance of our estmates. Whle t s possble to wrte down formulas here they are ecessvely complcated. Suffce to know that j σ Var( ) = f ( the ' s, ) for all j =,..., k (3) The dstrbuton of our estmates. It s gong to be normal for the same reason that the dstrbuton was normal n the bvarate case (assumpton 6). If the sample sze s small we need to make a normalty assumpton to conduct statstcal nference If the sample sze s large then we have a central lmt theorem that tells that the s are appromately normal. Note that we wll need an estmate of σ to conduct statstcal nference. We can estmateσ wth e SSE σ = = n ( k + ) n ( k + )
15 The FTest of a Set of Lnear Restrctons Eample : Human Captal Model. Are returns to potental eperence dfferent for men and women? There are two relevant models here Unrestrcted Model: ln( wage ) = + ed + e + esq + fem e + fem esq black + hsp + female + e Restrcted Model: ln( wage ) = + ed + e + esq black + 7 hsp + 8 female + e The restrctons mpled by the restrcted model are 4 = 5 =, therefore my hypothess test can be constructed as H : = = 4 5 H :, a 4 5
16 To evaluate the hypothess I must estmate both the restrcted and unrestrcted models Restrcted Model Estmates. reg lnwage ed e esq black hsp fem f year==987 Source SS df MS Number of obs = F( 6, 4993) = 447. Model Prob > F =. Resdual Rsquared = Adj Rsquared =.3488 Total Root MSE = lnwage Coef. Std. Err. t P> t [95% Conf. Interval] ed e esq black hsp fem _cons Unrestrcted Model Estmates. reg lnwage ed e esq fe fesq black hsp fem f year==987 Source SS df MS Number of obs = F( 8, 499) = Model Prob > F =. Resdual Rsquared = Adj Rsquared =.36 Total Root MSE = lnwage Coef. Std. Err. t P> t [95% Conf. Interval] ed e esq fe fesq black hsp fem _cons Comparson of Rsquared terms from the regressons. F R R * r = = 54 R.363 n ( k ) 4,99 +
17 Alternatvely, a seres of commands n STATA wll gve you the test statstc the pvalue assocated wth the test statstc.. reg lnwage ed e esq fe fesq black hsp fem f year==987. test fe== ( ) fe =. F(, 499) =.7 Prob > F =.. test fesq==, accum ( ) fe =. ( ) fesq =. F(, 499) = 53.7 Prob > F =.
18 Eample: Layoffs (From Chapter 6 of the tet). Do manageral and sales workers have dfferent epected unemployment duratons than productons workers? Age  age of the worker n years Educaton  hghest school grade completed Marred  a dummy varable equal to f worker s marred, otherwse Head  a dummy varable equal to f a worker s the head of household, otherwse Tenure The number of years on the last job Manager a dummy varable equal to f a worker was employed as a manager on hs last job, otherwse. Sales a dummy varable equal to f a worker was employed as a sales worker on hs last job, otherwse. Model: Weeks Unem = + Age + Educaton + Marred 3 + Head + Tenure + Manager + Sales + e Below s a verson of what STATA prnts out wth ts regresson tool.. reg weeks age educ marred head tenure manager sales Source SS df MS Number of obs = F( 7, 4) = 8.68 Model Prob > F =. Resdual Rsquared = Adj Rsquared =.533 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] age educ marred head tenure manager sales _cons
19 Weeks Unem = γ + γ Marred + γ Head + γ3 Tenure Restrcted Model: + γ Manager + γ Sales + e 4 5. reg weeks marred head tenure manager sales Source SS df MS Number of obs = F( 5, 44) = 4.59 Model Prob > F =.9 Resdual Rsquared = Adj Rsquared =.683 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] marred head tenure manager sales _cons * In actualty you would not need to estmate the restrcted model to test the hypothess that = =. In practce, the test statstc and pvalue assocated wth ths test obtaned wth the followng seres of commands n STATA. reg weeks age educ marred head tenure manager sales. test marred==. test educ==, accum
20 Eample: Unemployed Workers (From Chapter 6 of the tet) Age  age of the worker n years Educaton  hghest school grade completed Marred  a dummy varable equal to f worker s marred, otherwse Head  a dummy varable equal to f a worker s the head of household, otherwse Tenure The number of years on the last job Manager a dummy varable equal to f a worker was employed as a manager on hs last job, otherwse. Sales a dummy varable equal to f a worker was employed as a sales worker on hs last job, otherwse. Model: Weeks Unem = + Age + Educaton + Marred 3 + Head + Tenure + Manager + Sales + e
21 . reg weeks age educ marred head tenure manager sales Source SS df MS Number of obs = F( 7, 4) = 8.68 Model Prob > F =. Resdual Rsquared = Adj Rsquared =.533 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] age educ marred head tenure manager sales _cons How can we nterpret the coeffcent on the educaton varable? Lets test the hypothess that educaton has no effect on unemployment duraton of manufacturng workers. How would I state ths hypothess n terms of the populaton parameters? What should the frst step n testng ths hypothess be? What do I do net? What conclusons can I draw?
22 Testng a Set of Lnear Restrctons (The FTest) What I am gong to gve you s a more general verson of the FTest that s n the book. What I gve you s way more powerful that what s n the book so please pay attenton. Ths s open materal for the test. Consder the model y = kk + e Suppose that we want to test the hypothess that some subset of the coeffcents s zero. We can form an Fstatstc for ths test.
23 F * SSE SSE = r SSE F(, r n k) n k where r = the number of restrctons SSE * = the error sum of squares from a restrcted model SSE = the error sum of squares from the unrestrcted model Just for the sake of beng concrete lets suppose that y = e and want to test the null hypothess that = = aganst the alternatve that and. To test ths hypothess t turns out we wll need an F statstc. Ths Fstatstc can be computed as F * SSE SSE = r SSE n k Where SSE = e = ( y ) * SSE = e = ( y γ γ γ ) * 3 4
24 Where the γ s are least squares coeffcent estmates from the model y = γ + 3γ3 + γ4 + v The mportant thng about ths model s that t s a restrcted verson of the frst model. It s essentally the frst model wth the coeffcents on the frst two varables set to zero. If the null hypothess that = = s correct, I would epect that the sum of squared resduals from the restrcted model would not be much larger than the sum of squares from the unrestrcted model. It wll be larger because degrees of freedom due to fewer coeffcents, but t should not be much larger. There s an easy way to compute the test statstc F * SSE SSE = r SSE n k If I dvde the numerator and denomnator by SST I get F r SST SST ( R ) r r = = = SSE R R n k SST n k n k * SSE SSE R R * R * Bascally all I need to do to conduct the test s estmate both the restrcted and the unrestrcted models, get the Rsquareds from these regressons, form the Fstatstc and go to the tables. Eample: Unemployment Duratons Lets test the hypothess that the age and educaton jontly have not effect on weeks unemployed for lad off manufacturng workers. H H : = = a :, I need to compute estmates from the restrcted model
25 Weeks Unem = γ + γ Marred + γ Head + γ3 Tenure Model: + γ Manager + γ Sales + e 4 5. reg weeks marred head tenure manager sales Source SS df MS Number of obs = F( 5, 44) = 4.59 Model Prob > F =.9 Resdual Rsquared = Adj Rsquared =.683 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] marred head tenure manager sales _cons Alternatvely, a seres of commands n STATA wll gve you the test statstc the pvalue assocated wth the test statstc. reg weeks marred head tenure manager sales test marred== test educ==, accum
Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationExamples of Multiple Linear Regression Models
ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A crosssectonal sample of 7 cars sold n
More informationIntroduction to Regression
Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. 
More informationChapter 14 Simple Linear Regression
Sldes Prepared JOHN S. LOUCKS St. Edward s Unverst Slde Chapter 4 Smple Lnear Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationEconomic Interpretation of Regression. Theory and Applications
Economc Interpretaton of Regresson Theor and Applcatons Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationCHAPTER 7 THE TWOVARIABLE REGRESSION MODEL: HYPOTHESIS TESTING
CHAPTER 7 THE TWOVARIABLE REGRESSION MODEL: HYPOTHESIS TESTING QUESTIONS 7.1. (a) In the regresson contet, the method of least squares estmates the regresson parameters n such a way that the sum of the
More informationH 1 : at least one is not zero
Chapter 6 More Multple Regresson Model The Ftest Jont Hypothess Tests Consder the lnear regresson equaton: () y = β + βx + βx + β4x4 + e for =,,..., N The tstatstc gve a test of sgnfcance of an ndvdual
More informationThe Probit Model. Alexander Spermann. SoSe 2009
The Probt Model Aleander Spermann Unversty of Freburg SoSe 009 Course outlne. Notaton and statstcal foundatons. Introducton to the Probt model 3. Applcaton 4. Coeffcents and margnal effects 5. Goodnessofft
More informationLecture 10: Linear Regression Approach, Assumptions and Diagnostics
Approach to Modelng I Lecture 1: Lnear Regresson Approach, Assumptons and Dagnostcs Sandy Eckel seckel@jhsph.edu 8 May 8 General approach for most statstcal modelng: Defne the populaton of nterest State
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationQuestions that we may have about the variables
Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent
More informationSTATISTICAL DATA ANALYSIS IN EXCEL
Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 1401013 petr.nazarov@crpsante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationAnalysis of Covariance
Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationPRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.
PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and mfle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationDescriptive Statistics (60 points)
Economcs 30330: Statstcs for Economcs Problem Set 2 Unversty of otre Dame Instructor: Julo Garín Sprng 2012 Descrptve Statstcs (60 ponts) 1. Followng a recent government shutdown, Mnnesota Governor Mark
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationMULTIPLE REGRESSION EXAMPLE
MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationChapter 7. RandomVariate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 RandomVariate Generation
Chapter 7 RandomVarate Generaton 7. Contents Inversetransform Technque AcceptanceRejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton
More informationTexas Instruments 30Xa Calculator
Teas Instruments 30Xa Calculator Keystrokes for the TI30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check
More informationThe Classical Model. GaussMarkov Theorem, Specification, Endogeneity
The Classcal Model GaussMarkov Theorem, Specfcaton, Endogenety Propertes of Least Squares Estmators Here s the model: Y = β + β + β + β + L+ β + ε 0 For the case wth regressor and constant, I showed some
More informationRegression Models for a Binary Response Using EXCEL and JMP
SEMATECH 997 Statstcal Methods Symposum Austn Regresson Models for a Bnary Response Usng EXCEL and JMP Davd C. Trndade, Ph.D. STATTECH Consultng and Tranng n Appled Statstcs San Jose, CA Topcs Practcal
More informationWeek 4 Lecture: PairedSample Hypothesis Tests (Chapter 9)
Week 4 Lecture: PareSample Hypothess Tests (Chapter 9) The twosample proceures escrbe last week only apply when the two samples are nepenent. However, you may want to perform a hypothess tests to ata
More information5 Multiple regression analysis with qualitative information
5 Multple regresson analyss wth qualtatve nformaton Ezequel Urel Unversty of Valenca Verson: 913 5.1 Introducton of qualtatve nformaton n econometrc models. 1 5. A sngle dummy ndependent varable 5.3 Multple
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationECONOMICS 351*  Stata 10 Tutorial 7. Stata 10 Tutorial 7. TOPIC: Estimation and Hypothesis Testing in Multiple Linear Regression Models
ECONOMICS 5*  Stata 0 Tutoral 7 Stata 0 Tutoral 7 TOPIC: Estmaton and Hypothess Testng n Multple Lnear Regresson Models DATA: autodta (a Stataformat data fle you created n Stata Tutoral ) TASKS: Stata
More informationI. SCOPE, APPLICABILITY AND PARAMETERS Scope
D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationMULTIPLE LINEAR REGRESSION IN MINITAB
MULTIPLE LINEAR REGRESSION IN MINITAB Ths document shows a complcated Mntab multple regresson. It ncludes descrptons of the Mntab commands, and the Mntab output s heavly annotated. Comments n { } are used
More informationProblem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.
Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationLinear Regression Analysis for STARDEX
Lnear Regresson Analss for STARDEX Malcolm Halock, Clmatc Research Unt The followng document s an overvew of lnear regresson methods for reference b members of STARDEX. Whle t ams to cover the most common
More informationLatent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006
Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationREGRESSION LINES IN STATA
REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationInternational University of Japan Public Management & Policy Analysis Program
Internatonal Unversty of Japan Publc Management & Polcy Analyss Program Practcal Gudes To Panel Data Modelng: A Step by Step Analyss Usng Stata * Hun Myoung Park, Ph.D. kucc65@uj.ac.jp 1. Introducton.
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More information14.74 Lecture 5: Health (2)
14.74 Lecture 5: Health (2) Esther Duflo February 17, 2004 1 Possble Interventons Last tme we dscussed possble nterventons. Let s take one: provdng ron supplements to people, for example. From the data,
More informationCalibration and Linear Regression Analysis: A SelfGuided Tutorial
Calbraton and Lnear Regresson Analyss: A SelfGuded Tutoral Part The Calbraton Curve, Correlaton Coeffcent and Confdence Lmts CHM314 Instrumental Analyss Department of Chemstry, Unversty of Toronto Dr.
More informationQuality Adjustment of Secondhand Motor Vehicle Application of Hedonic Approach in Hong Kong s Consumer Price Index
Qualty Adustment of Secondhand Motor Vehcle Applcaton of Hedonc Approach n Hong Kong s Consumer Prce Index Prepared for the 14 th Meetng of the Ottawa Group on Prce Indces 20 22 May 2015, Tokyo, Japan
More informationRegression Analysis. Data Calculations Output
Regression Analysis In an attempt to find answers to questions such as those posed above, empirical labour economists use a useful tool called regression analysis. Regression analysis is essentially a
More informationFORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationPortfolio Risk Decomposition (and Risk Budgeting)
ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen RSquared Rsk Management Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationLecture 14: Implementing CAPM
Lecture 14: Implementng CAPM Queston: So, how do I apply the CAPM? Current readng: Brealey and Myers, Chapter 9 Reader, Chapter 15 M. Spegel and R. Stanton, 2000 1 Key Results So Far All nvestors should
More informationLinear Regression, Regularization BiasVariance Tradeoff
HTF: Ch3, 7 B: Ch3 Lnear Regresson, Regularzaton BasVarance Tradeoff Thanks to C Guestrn, T Detterch, R Parr, N Ray 1 Outlne Lnear Regresson MLE = Least Squares! Bass functons Evaluatng Predctors Tranng
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationThe Choice of Direct Dealing or Electronic Brokerage in Foreign Exchange Trading
The Choce of Drect Dealng or Electronc Brokerage n Foregn Exchange Tradng Mchael Melvn & Ln Wen Arzona State Unversty Introducton Electronc Brokerage n Foregn Exchange Start from a base of zero n 1992
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM BARRIOT JeanPerre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jeanperre.barrot@cnes.fr 1/Introducton The
More information1 De nitions and Censoring
De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationChapter 15 Multiple Regression
Chapter 5 Multple Regresson In chapter 9, we consdered one dependent varable (Y) and one predctor (regressor or ndependent varable) (X) and predcted Y based on X only, whch also known as the smple lnear
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the ageold queston: When the hell
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationPart 1: quick summary 5. Part 2: understanding the basics of ANOVA 8
Statstcs Rudolf N. Cardnal Graduatelevel statstcs for psychology and neuroscence NOV n practce, and complex NOV desgns Verson of May 4 Part : quck summary 5. Overvew of ths document 5. Background knowledge
More informationLecture 15. Endogeneity & Instrumental Variable Estimation
Lecture 15. Endogeneity & Instrumental Variable Estimation Saw that measurement error (on right hand side) means that OLS will be biased (biased toward zero) Potential solution to endogeneity instrumental
More informationDescribing Communities. Species Diversity Concepts. Species Richness. Species Richness. SpeciesArea Curve. SpeciesArea Curve
peces versty Concepts peces Rchness pecesarea Curves versty Indces  mpson's Index  hannonwener Index  rlloun Index peces Abundance Models escrbng Communtes There are two mportant descrptors of a communty:
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationEstimation and Robustness of Linear Mixed Models in Credibility Context
Estmaton and Robustness of Lnear Mxed Models n Credblty Context by Wng Kam Fung and Xao Chen Xu ABSTRACT In ths paper, lnear mxed models are employed for estmaton of structural parameters n credblty context.
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationChapter 2. Determination of appropriate Sample Size
Chapter Determnaton of approprate Sample Sze Dscusson of ths chapter s on the bass of two of our publshed papers Importance of the sze of sample and ts determnaton n the context of data related to the
More information4 Hypothesis testing in the multiple regression model
4 Hypothess testng n the multple regresson model Ezequel Urel Unversdad de Valenca Verson: 913 4.1 Hypothess testng: an overvew 1 4.1.1 Formulaton of the null hypothess and the alternatve hypothess 4.1.
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationEcon 371 Problem Set #3 Answer Sheet
Econ 371 Problem Set #3 Answer Sheet 4.1 In this question, you are told that a OLS regression analysis of third grade test scores as a function of class size yields the following estimated model. T estscore
More informationAn Analysis of Factors Influencing the SelfRated Health of Elderly Chinese People
Open Journal of Socal Scences, 205, 3, 520 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/ss http://dx.do.org/0.4236/ss.205.35003 An Analyss of Factors Influencng the SelfRated Health of
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationLecture 9: Logit/Probit. Prof. Sharyn O Halloran Sustainable Development U9611 Econometrics II
Lecture 9: Logt/Probt Prof. Sharyn O Halloran Sustanable Development U96 Econometrcs II Revew of Lnear Estmaton So far, we know how to handle lnear estmaton models of the type: Y = β 0 + β *X + β 2 *X
More informationSimon Acomb NAG Financial Mathematics Day
1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Comovement between the ponts Postve
More informationECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
More information