DEPARTMENT OF ECONOMICS

Size: px
Start display at page:

Download "DEPARTMENT OF ECONOMICS"

Transcription

1 ISSN ISBN THE UNIVERSITY OF MELBOURNE DEPARTMENT OF ECONOMICS RESEARCH PAPER NUMBER 95 NOVEMBER 005 INTERACTIONS IN REGRESSIONS by Joe Hirschberg & Jenny Lye Deparmen of Economics The Universiy of Melbourne Melbourne Vicoria 3010 Ausralia.

2 Ineracions in Regressions J. Hirschberg and J. Lye 1 Deparmen of Economics Universiy of Melbourne November 005 Absrac Regression specificaions frequenly employ regressors ha are defined as he produc of wo oher regressors o form an ineracion. Unforunaely, hese models have a number of poenial difficulies when i comes o inerpreaion. In his paper, we discuss wo common aspecs of hese specificaions ha can lead o difficulies. The firs is he change in parameer esimaes and - saisics when differen definiions of dummy variables are used. The second involves an applicaion of Fieller s heorem o draw appropriae inferences from a regression wih ineracion variables. Key Words: Ineracion effecs; dummy variables; linear ransformaion, Fieller mehod JEL classificaion: C1; C51 1 Corresponding auhor, jnlye@unimelb.edu.au, Deparmen of Economics, Universiy of Melbourne, Melbourne, Vicoria 3010, Ausralia. The research carried ou in his paper was parially funded by a gran from he Faculy of Economics and Commerce a The Universiy of Melbourne. We wish o hank David Moreon for helpful commens. 1

3 1 Inroducion Regression models wih ineracion effecs are ofen proposed o allow for he marginal effec of one explanaory variable o depend on anoher. In his paper, we discuss wo aspecs of ineracions in regression specificaions. In Secion, we show how dummy variables are prone o linear affine ransformaions and discuss he implicaions for inerpreing an ineracion variable in he regression equaion when he ineracion variable involves a dummy variable. An example is included o highligh he impac on parameer esimaes and -saisics. In Secion 3, he inerpreaion of regressions wih ineracions is proposed by using an applicaion of Fieller s heorem. Two empirical examples are presened. Conclusions are presened in Secion. Variable Transformaions, Ineracions and Dummy Variables Griepenrog e al. (198) (GRS) show ha for models ha include an ineracive erm an affine linear ransformaion of one variable in he ineracive erm affecs he -saisic for he lower order of he oher variable. In his paper, we relae his resul o he case in which he ineracive erm involves a dummy variable. Following GRS, consider a linear model of he form: y = X β + ε (1) where y is an n 1 marix of observaions on he dependen variable, ε is an n 1 marix of disurbances, X is an n k marix of observaions on all of he explanaory variables, ha is, [ ] and = [ β β β β β β ] X 1 x w xw v v = 1 k coefficiens. akes he form: β is a k 1 vecor of k Suppose ha z, he h observaion, is formed by an affine linear ransformaion of x ha z = a + bx. () where a and b are consans. Consider an alernaive model o (1) defined in erms of Z as: y = Zγ + ε (3)

4 where Z is an n k marix of observaions on all of he explanaory variables, ha is, [ ] and = [ γ γ γ γ γ γ ] Z 1 z w zw v v = 1 k γ is a k 1 vecor of k coefficiens. The relaionship beween (1) and (3) is given by: y = XAγ = Xβ + ε () where A is he non-singular marix, A11 A1 Α1 A, in which A 11= 1 a b 0 0, a b A = 0, A = I. From () i follows ha 1, k k-,k- γˆ = -1ˆ A β, where ˆβ is he ordinary leas squares esimaor of β. We can define he - saisics for esing he null hypoheses H 0 : β i = 0, i = 1,, k in (1) as i βˆ i =, where c ij is he ijh elemen of he marix ( ) 1 σ c ii XX. The corresponding saisics for γˆ esing he null hypohesis H 0 : γ i = 0in (3) are * i i =, where d ii is he diagonal elemen of he σ d marix 1 1 ( ) ( 1 ) A XX A. I can be shown ha * i = sign( b) i when i = and and * i = i when i = 5 k. However, he -saisics for i = 1 and 3 are defined as ii * i β ˆ a ( ) ˆ i b β i + 1 a a ( b) ( b) = σ c c + c ii i( i+ 1) ( i+ 1)( i+ 1). In paricular, he -saisic for esing he hypohesis H : γ = 0, which is a es of he significance of he coefficien for variable w in (3), is no 0 3 equivalen o he -saisic for esing he hypohesis H 0: β 3 = 0, ha is, he es saisic for he same variable in (1). Thus, a ransformaion of one of he variables in he ineracive erm has implicaions for he -saisic relaing o he linear effec of he oher variable in he ineracive erm. This resul is of paricular ineres when he ineraced variable is a dummy variable since hese are ofen subjeced o linear affine ransformaions. In paricular, if x = 1 when observaions 3

5 represen he presence of a paricular characerisic, we could jus as easily define anoher variable z = 1 when he characerisic is no presen, ha is, z = 1 x or in his case, a = 1 and b = 1..1 An Applicaion of Models wih Ineracive Dummy Variables This example is based on daa from Bernd (1991, p. 193) in he form of observaions on 550 individuals from he May 1985 Curren Populaion Survey. The naural logarihm of average hourly earnings in dollars is he dependen variable, ED is years of schooling, EX is poenial years of experience, MR is a dummy variable ha akes he value 1 if he individual is married and F is a dummy variable ha akes he value 1 if he individual is female. The following wo models are esimaed in which being female and being male are used as indicaors of gender, respecively: ( ) Log( Wage ) =β +β F +β MR +β F MR +λ ED +λ EX +λ EX +ε (5) ( ) (( ) ) Log( Wage ) =γ +γ 1 F +γ MR +γ 1 F MR +λ ED +λ EX +λ EX +ε (6) The resuls are repored in Table 1. Table 1. A comparison of he resuls of he specificaions (5) and (6) Dependen Variable: LOG(WAGE) Sample: 1 53 Gender =F (5) Gender = (1 F) (6) Variable Coeff -sa Coeff -sa C Gender MR Gender MR ED EX EX^ R-squared S.E. of regression 0. Adjused R-squared 0.98 Sum squared resid A comparison of he coefficien esimaes and -saisics for he Gender variable and he ineracion beween Gender and MR in Table 1 indicaes ha hey are of he same magniude bu of opposie sign. However, in model (5), he coefficien esimae and he -saisic associaed wih MR indicae ha marriage has a posiive and significan impac whereas in (6), he esimaed parameer for he marriage dummy is negaive and insignifican. Thus, while he coefficien on he redefined dummy variable is expeced o change sign, he change in he effec of he unchanged variable (MR)

6 is no inuiive. From Secion.1, alhough he parameer on MR is expeced o change according o γ =β a 3 3 b β (in his case,.05 = ), he change in he -saisic is inversely relaed o he covariance beween ˆβ 3 and ˆβ. 3 The Parial Influence of Ineraced Regressors Consider he model: y = β + β x + β w + β xw + β v + β v + ε (7) k k- where y is he dependen variable, x, w, xw, v j are explanaory variables andε is he k j= 5 disurbance erm. When x and w are boh coninuous variables, he marginal effec of a change in x is: ( ) E yxw, x = β + β w (8) If x is a dummy variable and w is a coninuous variable, we define he difference in he regression equaion when x =1 and when x = 0 as: ΔE, Δx = β + β w (9) In he conex of (8) and (9), he difficuly of choosing an appropriae value of he oher regressor a which o evaluae hese compuaions arises. One approach is o se he value of his oher regressor o a paricular value, such as he mean, and hen make he compuaion. Anoher approach is o deermine he value of he oher regressor a which hese definiions become zero or change sign. The value of w ha resuls in x = 0 or E, * β * βˆ Δ x = 0 is w = β and is esimaed by wˆ = ˆ, ΔE, β where β ˆ i are he OLS esimaes of β, i =,. A confidence bound for i deermine if he impac of x on y is significan over he possible values of w. * ŵ can be consruced o Noe ha a similar analysis can be performed on E, w 5

7 3.1 Confidence Inervals for he Parial Influence Funcion For he regression model in (7), or E, x Δ is defined as β + β w. An esimae of ΔE, x his can be ploed wih a 100(1 α)% confidence inerval given by: where ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) CI = β + β ± σ + σ + w σ (10) σ ˆ i is he esimaed variance of ˆ β i, i =,, ˆσ is he esimaed covariance beween ˆβ and * ˆβ. An esimae of he value of w ( ŵ ), where = 0 or E, x Δ = 0 is found by solving ΔE, x βˆ + β ˆ w = 0. Similarly, he bounds defining he 100(1 α )% confidence inerval on * ŵ are found by solving: ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) β +β ± σ + σ + w σ = 0 (11) This is equivalen o solving he roos of he equaion: ( ˆ ˆ w) ( ˆ ˆ ˆ α w ) β +β - σ + σ + w σ = 0 (1) Equaion (1) is equivalen o he quadraic equaion, ( ax + bx + c = 0 ), where a, b and c are ( ˆ a= β ˆ α σ ), b ( ˆ ˆ ˆ α ) = β β σ and c= βˆ σ ˆ, α wih an α ( ) level of significance and T k degrees of freedom. α is he value from he -disribuion The confidence limis obained in his way are idenical o hose found applying a version of he Fieller mehod (Fieller 193, 195) o a raio of linear combinaions of regression parameers (Zerbe 1978). In order o have real roos, ha is, o have a finie inerval, we require a > 0, which implies rejecion of he null hypohesis ha β = 0 based on he -saisic (Buonaccorsi 1979). Besides he finie inerval, he resuling confidence inerval may be he complemen of a finie inerval, (b ac > 0, a < 0), or he whole real line, (b ac < 0, a < 0). These condiions are discussed in Scheffé (1970) and Zerbe (198). If x or E, Δ x is nonlinear and defined as g(w), hen is esimae, say gw, ˆ( ) can be ΔE, ploed along wih a 100(1 α)% confidence inerval given by: 6

8 where var ˆ { ˆ( )} α { ˆ } CI = gw ˆ( ) ± z var ˆ gw ( ) (13) gw is he esimaed variance of gw ˆ( ) and z α is he value from he sandardized normal disribuion wih an α ( ) level of significance. An esimae (or esimaes) of he value of w, where x or E, Δ x is found by solving gw= ˆ( ) 0; similarly, he bounds defining he ΔE, 100(1 α)% confidence inerval a his poin(s) are found by solving (13) equaed o zero. Because gw ˆ( ) is nonlinear, here may be more han one soluion and, hence, more han one se of confidence bounds. In many insances, i is possible o rule ou a number of he soluions by examining he range of he daa. 3. Example 1: The Demand for Economic Journals Sock and Wason (003, p. 7) analyse he relaionship beween he number of subscripions o a journal a US libraries (Y) and is library subscripions price by using daa from 000 on 180 economics journals. Price is measured in prices per ciaion, which represens an approximaion of dollars per idea. Oher explanaory variables include he age of he journal (Age) and he number of characers per year in he journal (Char). The regression equaion is: ( ) Log( Y ) = β + β Log( Price) + β Log( Age) + β Log( Price) Log( Age) +β Log( Char) + u (1) The regression resuls are repored in Table, which shows ha he coefficien on he ineracive erm is highly significan. The price elasiciy is: which is esimaed as: ( Age ) η =β 1+ β 3 Log( ) (15) η ˆ = Log( Age) (16) The raio ( 0.11 ) defines he value of Log(Age) a which η ˆ = 0. 7

9 Table : Regression Resuls, Demand For Economic Journals Dependen Variable: Log(Y) Sample: Variable Coefficien -Saisic 3 C Log(Price) Log(Age) Log(Price)*Log(Age) Log(Char) R-squared 0.633S.E. of regression Adjused R-squared 0.66 Sum squared resid Figure 1 plos he esimaed price elasiciy and is 95% corresponding confidence inerval. Since he acual values of Age in he daa have a minimum value of (Log(Age)=0.60) and a maximum value of 156 (Log(Age)=.193), we conclude from Figure 1 ha he price elasiciy is significanly differen from 0 for all values of Age in his daa se. Figure 1: Parial Influence Funcion, Demand For Economics Journals Example Elasiciy < Fieller Inerval >.5% Lower bound 97.5% Upper bound Prediced Log of Age However, i is common pracice o repor a summary price elasiciy such as he value of he mean of all he price elasiciies calculaed by using (16), which in his example equals Figure 1 plos a reference line a his value from which we can deermine he values of Age for which he elasiciy is significanly differen from he mean of all he price elasiciies. In his 3 The -saisics are based on Whie Heeroskedasic-Consisen Sandard errors 8

10 example, his is he case for young and old journals, ha is, hose for which Age is less han (Log(Age) =.) or greaer han 9. (Log(Age) = 3.9), respecively. 3.3 Example : California Tes Daa Scores In his example, daa from Sock and Wason (003, p. 30) is used on es scores from 0 California school disrics in The es score measure (TESTSCR) is he disric-wide average of reading and mah scores for fifh graders. Addiional variables are he disric-wide suden eacher raio (STR); he percenage of sudens qualifying for a subsidized lunch (Meal_pc); and he average annual per capia income in he school disric measured in housands of 1998 dollars (Avginc). Furher, HIEL is a dummy variable ha equals 1 if he percenage of sudens sill learning English in he disric is greaer han 10%. The regression equaion examines wheher he effec of he suden-eacher raio depends no only on he value of he suden-eacher raio bu also on he percenage of English learners. Ineracions beween HIEL and STR, STR and STR 3 are included o allow he regression funcions relaing es scores and STR o be differen for low and high percenages of English learners. The regression equaion is: ( ) ( ) ( ) TESTSCR = β + β STR+ β STR + β STR + β HIEL +β STR HIEL +β STR HIEL +β STR HIEL +β Meal _ Pc +β log( Avginc) +ε (17) Table 3 repors he regression resuls. Individually, β, β5, β6, β 7 are significan a he 5% level and a join F-es of H 0 : β =β5= β6= β 7=0 resuls in a p-value of less han is defined as: The difference in he es scores when HIEL akes he value 1 and when i akes he value 0 which is esimaed as: ( Tesscr x) Δ HIEL =β +β 5 STR +β 6 STR +β 7 STR (18) ( ) ( ) ( ) ΔE 3 Δ E ( Tesscr x) 3 Δ HIEL = STR STR 0.10 STR (19) Equaion (19) only has a real roo when STR =

11 Table 3: Regression Resuls, California Tes Daa Scores Dependen Variable: TESTSCR Sample: 1 0 Variable Coefficien -Saisic C STR STR^ STR^ HIEL STR*HIEL STR^*HIEL STR^3*HIEL Meal_pc LOG(Avginc) R-squared S.E. of regression 8.57 Adjused R-squared Sum squared resid Figure () plos (19) and he corresponding 95% confidence inerval. We find ha Δ E Tesscr x ( ) Δ is HIEL significanly differen from zero for values of STR ha are greaer han 16.7 and less han However, only he former range (values from 16.7) overlaps wih he acual range of STR values (1.0 o 5.8). Thus, we can conclude ha he effec of suden-eacher raios (STRs) in schools wih a high proporion of sudens learning English is only negaive for hose schools ha have STRs of over Figure : Parial Influence Funcion, California Tes Daa Scores Example Differences < Fieller > Inerval 97.5% Upper bound.5% Lower bound Prediced Difference Suden Teacher Raio The -saisics are based on Whie Heeroskedasic-Consisen Sandard errors. 10

12 Conclusions Ineracive erms are ofen included in regression equaions o deermine how he effec on he dependen variable of one independen variable depends on anoher independen variable. However, an affine linear ransformaion of one of he variables in he ineracive erm has he counerinuiive effec of changing he -saisic on he linear effec of he oher variable. We have shown he implicaions of his when ineracion erms are based on dummy variables, which are prone o affine linear ransformaions. In addiion, we have also shown ha inferences can be drawn from a regression wih ineracion variables by examining he confidence bounds of he parial influence funcion. In he conex of linear regression, his is equivalen o he applicaion of Fieller s Mehod for he consrucion of confidence bounds for raios of random variables. 11

13 References Bernd, E. R., 1991, The Pracice of Economerics: Classic and Conemporary, Addison Wesley, Massachuses, USA. Buonaccorsi, J. P., (1979), On Fieller s Theorem and he General Linear Model, The American Saisician, 33, 16. Fieller, E. C., (193), The Disribuion of he Index in a Normal Bivariae Populaion, Biomerika,, 8 0. Fieller, E. C., (195), Some Problems in Inerval Esimaion, Journal of he Royal Saisical Sociey. Series B, 16, Griepenrog, G. L., J. M. Ryan and D. Smih (198), Linear Transformaions of Polynomial Regression Models, The American Saisician, 36, Scheffé, H., (1970), Muliple Tesing versus Muliple Esimaion. Improper Confidence Ses. Esimaion of Direcions and Raios, The Annals of Mahemaical Saisics, 1, 1 9. Sock, J. H. and M. W. Wason, (003), Inroducion To Economerics, Addison Wesley, Boson, USA. Zerbe, G., (1978), On Fieller s Theorem and he General Linear Model, The American Saisician, 3, Zerbe, G., (198), On Mulivariae Confidence Regions and Simulaneous Confidence Limis for Raios, Communicaions in Saisics Theory and Mehods, 11,

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,

More information

Solutions to Odd Number Exercises in Chapter 6

Solutions to Odd Number Exercises in Chapter 6 1 Soluions o Odd Number Exercises in 6.1 R y eˆ 1.7151 y 6.3 From eˆ ( T K) ˆ R 1 1 SST SST SST (1 R ) 55.36(1.7911) we have, ˆ 6.414 T K ( ) 6.5 y ye ye y e 1 1 Consider he erms e and xe b b x e y e b

More information

ACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin

ACE 562 Fall Lecture 4: Simple Linear Regression Model: Specification and Estimation. by Professor Scott H. Irwin ACE 56 Fall 005 Lecure 4: Simple Linear Regression Model: Specificaion and Esimaion by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Simple Regression: Economic and Saisical Model

More information

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8) I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression

More information

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H. ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models

More information

Outline. lse-logo. Outline. Outline. 1 Wald Test. 2 The Likelihood Ratio Test. 3 Lagrange Multiplier Tests

Outline. lse-logo. Outline. Outline. 1 Wald Test. 2 The Likelihood Ratio Test. 3 Lagrange Multiplier Tests Ouline Ouline Hypohesis Tes wihin he Maximum Likelihood Framework There are hree main frequenis approaches o inference wihin he Maximum Likelihood framework: he Wald es, he Likelihood Raio es and he Lagrange

More information

Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A

Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In

More information

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models

More information

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,

More information

Robust estimation based on the first- and third-moment restrictions of the power transformation model

Robust estimation based on the first- and third-moment restrictions of the power transformation model h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,

More information

How to Deal with Structural Breaks in Practical Cointegration Analysis

How to Deal with Structural Breaks in Practical Cointegration Analysis How o Deal wih Srucural Breaks in Pracical Coinegraion Analysis Roselyne Joyeux * School of Economic and Financial Sudies Macquarie Universiy December 00 ABSTRACT In his noe we consider he reamen of srucural

More information

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H. ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple

More information

ACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H.

ACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H. ACE 56 Fall 5 Lecure 8: The Simple Linear Regression Model: R, Reporing he Resuls and Predicion by Professor Sco H. Irwin Required Readings: Griffihs, Hill and Judge. "Explaining Variaion in he Dependen

More information

DEPARTMENT OF STATISTICS

DEPARTMENT OF STATISTICS A Tes for Mulivariae ARCH Effecs R. Sco Hacker and Abdulnasser Haemi-J 004: DEPARTMENT OF STATISTICS S-0 07 LUND SWEDEN A Tes for Mulivariae ARCH Effecs R. Sco Hacker Jönköping Inernaional Business School

More information

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1 Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies

More information

Wednesday, November 7 Handout: Heteroskedasticity

Wednesday, 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 information

Unit Root Time Series. Univariate random walk

Unit Root Time Series. Univariate random walk Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he

More information

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070

More information

The Simple Linear Regression Model: Reporting the Results and Choosing the Functional Form

The Simple Linear Regression Model: Reporting the Results and Choosing the Functional Form Chaper 6 The Simple Linear Regression Model: Reporing he Resuls and Choosing he Funcional Form To complee he analysis of he simple linear regression model, in his chaper we will consider how o measure

More information

Comparing Means: t-tests for One Sample & Two Related Samples

Comparing Means: t-tests for One Sample & Two Related Samples Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion

More information

Testing the Random Walk Model. i.i.d. ( ) r

Testing the Random Walk Model. i.i.d. ( ) r he random walk heory saes: esing he Random Walk Model µ ε () np = + np + Momen Condiions where where ε ~ i.i.d he idea here is o es direcly he resricions imposed by momen condiions. lnp lnp µ ( lnp lnp

More information

Solutions: Wednesday, November 14

Solutions: Wednesday, November 14 Amhers College Deparmen of Economics Economics 360 Fall 2012 Soluions: Wednesday, November 14 Judicial Daa: Cross secion daa of judicial and economic saisics for he fify saes in 2000. JudExp CrimesAll

More information

The General Linear Test in the Ridge Regression

The General Linear Test in the Ridge Regression ommunicaions for Saisical Applicaions Mehods 2014, Vol. 21, No. 4, 297 307 DOI: hp://dx.doi.org/10.5351/sam.2014.21.4.297 Prin ISSN 2287-7843 / Online ISSN 2383-4757 The General Linear Tes in he Ridge

More information

Econ Autocorrelation. Sanjaya DeSilva

Econ Autocorrelation. Sanjaya DeSilva Econ 39 - Auocorrelaion Sanjaya DeSilva Ocober 3, 008 1 Definiion Auocorrelaion (or serial correlaion) occurs when he error erm of one observaion is correlaed wih he error erm of any oher observaion. This

More information

ECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates)

ECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates) ECON 48 / WH Hong Time Series Daa Analysis. The Naure of Time Series Daa Example of ime series daa (inflaion and unemploymen raes) ECON 48 / WH Hong Time Series Daa Analysis The naure of ime series daa

More information

4.1 Other Interpretations of Ridge Regression

4.1 Other Interpretations of Ridge Regression CHAPTER 4 FURTHER RIDGE THEORY 4. Oher Inerpreaions of Ridge Regression In his secion we will presen hree inerpreaions for he use of ridge regression. The firs one is analogous o Hoerl and Kennard reasoning

More information

Regression with Time Series Data

Regression with Time Series Data Regression wih Time Series Daa y = β 0 + β 1 x 1 +...+ β k x k + u Serial Correlaion and Heeroskedasiciy Time Series - Serial Correlaion and Heeroskedasiciy 1 Serially Correlaed Errors: Consequences Wih

More information

Testing for a Single Factor Model in the Multivariate State Space Framework

Testing for a Single Factor Model in the Multivariate State Space Framework esing for a Single Facor Model in he Mulivariae Sae Space Framework Chen C.-Y. M. Chiba and M. Kobayashi Inernaional Graduae School of Social Sciences Yokohama Naional Universiy Japan Faculy of Economics

More information

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number

More information

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance

More information

Distribution of Estimates

Distribution of Estimates Disribuion of Esimaes From Economerics (40) Linear Regression Model Assume (y,x ) is iid and E(x e )0 Esimaion Consisency y α + βx + he esimaes approach he rue values as he sample size increases Esimaion

More information

The Effect of Nonzero Autocorrelation Coefficients on the Distributions of Durbin-Watson Test Estimator: Three Autoregressive Models

The Effect of Nonzero Autocorrelation Coefficients on the Distributions of Durbin-Watson Test Estimator: Three Autoregressive Models EJ Exper Journal of Economi c s ( 4 ), 85-9 9 4 Th e Au h or. Publi sh ed by Sp rin In v esify. ISS N 3 5 9-7 7 4 Econ omics.e xp erjou rn a ls.com The Effec of Nonzero Auocorrelaion Coefficiens on he

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

GMM - Generalized Method of Moments

GMM - Generalized Method of Moments GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................

More information

Hypothesis Testing in the Classical Normal Linear Regression Model. 1. Components of Hypothesis Tests

Hypothesis Testing in the Classical Normal Linear Regression Model. 1. Components of Hypothesis Tests ECONOMICS 35* -- NOTE 8 M.G. Abbo ECON 35* -- NOTE 8 Hypohesis Tesing in he Classical Normal Linear Regression Model. Componens of Hypohesis Tess. A esable hypohesis, which consiss of wo pars: Par : a

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

20. Applications of the Genetic-Drift Model

20. Applications of the Genetic-Drift Model 0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0

More information

Chapter 11. Heteroskedasticity The Nature of Heteroskedasticity. In Chapter 3 we introduced the linear model (11.1.1)

Chapter 11. Heteroskedasticity The Nature of Heteroskedasticity. In Chapter 3 we introduced the linear model (11.1.1) Chaper 11 Heeroskedasiciy 11.1 The Naure of Heeroskedasiciy In Chaper 3 we inroduced he linear model y = β+β x (11.1.1) 1 o explain household expendiure on food (y) as a funcion of household income (x).

More information

(a) Set up the least squares estimation procedure for this problem, which will consist in minimizing the sum of squared residuals. 2 t.

(a) Set up the least squares estimation procedure for this problem, which will consist in minimizing the sum of squared residuals. 2 t. Insrucions: The goal of he problem se is o undersand wha you are doing raher han jus geing he correc resul. Please show your work clearly and nealy. No credi will be given o lae homework, regardless of

More information

10. State Space Methods

10. State Space Methods . Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he

More information

The expectation value of the field operator.

The expectation value of the field operator. The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining

More information

Solutions to Exercises in Chapter 12

Solutions to Exercises in Chapter 12 Chaper in Chaper. (a) The leas-squares esimaed equaion is given by (b)!i = 6. + 0.770 Y 0.8 R R = 0.86 (.5) (0.07) (0.6) Boh b and b 3 have he expeced signs; income is expeced o have a posiive effec on

More information

On Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction

On Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction On Mulicomponen Sysem Reliabiliy wih Microshocks - Microdamages Type of Componens Ineracion Jerzy K. Filus, and Lidia Z. Filus Absrac Consider a wo componen parallel sysem. The defined new sochasic dependences

More information

DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Asymmery and Leverage in Condiional Volailiy Models Michael McAleer WORKING PAPER

More information

Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013

Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 www.iise.org The ffec of Inverse Transformaion on he Uni Mean and Consan Variance Assumpions of a Muliplicaive rror Model

More information

Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality

Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]

More information

Computer Simulates the Effect of Internal Restriction on Residuals in Linear Regression Model with First-order Autoregressive Procedures

Computer Simulates the Effect of Internal Restriction on Residuals in Linear Regression Model with First-order Autoregressive Procedures MPRA Munich Personal RePEc Archive Compuer Simulaes he Effec of Inernal Resricion on Residuals in Linear Regression Model wih Firs-order Auoregressive Procedures Mei-Yu Lee Deparmen of Applied Finance,

More information

Department of Economics East Carolina University Greenville, NC Phone: Fax:

Department of Economics East Carolina University Greenville, NC Phone: Fax: March 3, 999 Time Series Evidence on Wheher Adjusmen o Long-Run Equilibrium is Asymmeric Philip Rohman Eas Carolina Universiy Absrac The Enders and Granger (998) uni-roo es agains saionary alernaives wih

More information

Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1

Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1 Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien

More information

AN EXACT TEST FOR THE CHOICE OF THE COMBINATION OF FIRST DIFFERENCES AND PERCENTAGE CHANGES IN LINEAR MODELS

AN EXACT TEST FOR THE CHOICE OF THE COMBINATION OF FIRST DIFFERENCES AND PERCENTAGE CHANGES IN LINEAR MODELS AN EXACT TEST FOR THE CHOICE OF THE COMBINATION OF FIRST DIFFERENCES AND PERCENTAGE CHANGES IN LINEAR MODELS Wai Cheung Ip Deparmen of Applied Mahemaics The Hong Kong Polyechnic Universiy Hung Hom Kowloon

More information

Generalized Least Squares

Generalized Least Squares Generalized Leas Squares Augus 006 1 Modified Model Original assumpions: 1 Specificaion: y = Xβ + ε (1) Eε =0 3 EX 0 ε =0 4 Eεε 0 = σ I In his secion, we consider relaxing assumpion (4) Insead, assume

More information

Forecasting optimally

Forecasting optimally I) ile: Forecas Evaluaion II) Conens: Evaluaing forecass, properies of opimal forecass, esing properies of opimal forecass, saisical comparison of forecas accuracy III) Documenaion: - Diebold, Francis

More information

OBJECTIVES OF TIME SERIES ANALYSIS

OBJECTIVES OF TIME SERIES ANALYSIS OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging

More information

Chapter 16. Regression with Time Series Data

Chapter 16. Regression with Time Series Data Chaper 16 Regression wih Time Series Daa The analysis of ime series daa is of vial ineres o many groups, such as macroeconomiss sudying he behavior of naional and inernaional economies, finance economiss

More information

Has the Business Cycle Changed? Evidence and Explanations. Appendix

Has the Business Cycle Changed? Evidence and Explanations. Appendix Has he Business Ccle Changed? Evidence and Explanaions Appendix Augus 2003 James H. Sock Deparmen of Economics, Harvard Universi and he Naional Bureau of Economic Research and Mark W. Wason* Woodrow Wilson

More information

1. Diagnostic (Misspeci cation) Tests: Testing the Assumptions

1. Diagnostic (Misspeci cation) Tests: Testing the Assumptions Business School, Brunel Universiy MSc. EC5501/5509 Modelling Financial Decisions and Markes/Inroducion o Quaniaive Mehods Prof. Menelaos Karanasos (Room SS269, el. 01895265284) Lecure Noes 6 1. Diagnosic

More information

y = β 1 + β 2 x (11.1.1)

y = β 1 + β 2 x (11.1.1) Chaper 11 Heeroskedasiciy 11.1 The Naure of Heeroskedasiciy In Chaper 3 we inroduced he linear model y = β 1 + β x (11.1.1) o explain household expendiure on food (y) as a funcion of household income (x).

More information

ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING

ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING Chung-ki Min Professor

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

Lecture 15. Dummy variables, continued

Lecture 15. Dummy variables, continued Lecure 15. Dummy variables, coninued Seasonal effecs in ime series Consider relaion beween elecriciy consumpion Y and elecriciy price X. The daa are quarerly ime series. Firs model ln α 1 + α2 Y = ln X

More information

Wednesday, December 5 Handout: Panel Data and Unobservable Variables

Wednesday, December 5 Handout: Panel Data and Unobservable Variables Amhers College Deparmen of Economics Economics 360 Fall 0 Wednesday, December 5 Handou: Panel Daa and Unobservable Variables Preview Taking Sock: Ordinary Leas Squares (OLS) Esimaion Procedure o Sandard

More information

Measurement Error 1: Consequences Page 1. Definitions. For two variables, X and Y, the following hold: Expectation, or Mean, of X.

Measurement Error 1: Consequences Page 1. Definitions. For two variables, X and Y, the following hold: Expectation, or Mean, of X. Measuremen Error 1: Consequences of Measuremen Error Richard Williams, Universiy of Nore Dame, hps://www3.nd.edu/~rwilliam/ Las revised January 1, 015 Definiions. For wo variables, X and Y, he following

More information

Dynamic Econometric Models: Y t = + 0 X t + 1 X t X t k X t-k + e t. A. Autoregressive Model:

Dynamic Econometric Models: Y t = + 0 X t + 1 X t X t k X t-k + e t. A. Autoregressive Model: Dynamic Economeric Models: A. Auoregressive Model: Y = + 0 X 1 Y -1 + 2 Y -2 + k Y -k + e (Wih lagged dependen variable(s) on he RHS) B. Disribued-lag Model: Y = + 0 X + 1 X -1 + 2 X -2 + + k X -k + e

More information

Volatility. Many economic series, and most financial series, display conditional volatility

Volatility. Many economic series, and most financial series, display conditional volatility Volailiy Many economic series, and mos financial series, display condiional volailiy The condiional variance changes over ime There are periods of high volailiy When large changes frequenly occur And periods

More information

State-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter

State-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when

More information

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec

More information

(10) (a) Derive and plot the spectrum of y. Discuss how the seasonality in the process is evident in spectrum.

(10) (a) Derive and plot the spectrum of y. Discuss how the seasonality in the process is evident in spectrum. January 01 Final Exam Quesions: Mark W. Wason (Poins/Minues are given in Parenheses) (15) 1. Suppose ha y follows he saionary AR(1) process y = y 1 +, where = 0.5 and ~ iid(0,1). Le x = (y + y 1 )/. (11)

More information

STATE-SPACE MODELLING. A mass balance across the tank gives:

STATE-SPACE MODELLING. A mass balance across the tank gives: B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing

More information

MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM. Dimitar Atanasov

MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM. Dimitar Atanasov Pliska Sud. Mah. Bulgar. 20 (2011), 5 12 STUDIA MATHEMATICA BULGARICA MANY FACET, COMMON LATENT TRAIT POLYTOMOUS IRT MODEL AND EM ALGORITHM Dimiar Aanasov There are many areas of assessmen where he level

More information

Speaker Adaptation Techniques For Continuous Speech Using Medium and Small Adaptation Data Sets. Constantinos Boulis

Speaker Adaptation Techniques For Continuous Speech Using Medium and Small Adaptation Data Sets. Constantinos Boulis Speaker Adapaion Techniques For Coninuous Speech Using Medium and Small Adapaion Daa Ses Consaninos Boulis Ouline of he Presenaion Inroducion o he speaker adapaion problem Maximum Likelihood Sochasic Transformaions

More information

Estimation Uncertainty

Estimation Uncertainty Esimaion Uncerainy The sample mean is an esimae of β = E(y +h ) The esimaion error is = + = T h y T b ( ) = = + = + = = = T T h T h e T y T y T b β β β Esimaion Variance Under classical condiions, where

More information

Granger Causality Among Pre-Crisis East Asian Exchange Rates. (Running Title: Granger Causality Among Pre-Crisis East Asian Exchange Rates)

Granger Causality Among Pre-Crisis East Asian Exchange Rates. (Running Title: Granger Causality Among Pre-Crisis East Asian Exchange Rates) Granger Causaliy Among PreCrisis Eas Asian Exchange Raes (Running Tile: Granger Causaliy Among PreCrisis Eas Asian Exchange Raes) Joseph D. ALBA and Donghyun PARK *, School of Humaniies and Social Sciences

More information

Lecture 4. Classical Linear Regression Model: Overview

Lecture 4. Classical Linear Regression Model: Overview Lecure 4 Classical Linear Regression Model: Overview Regression Regression is probably he single mos imporan ool a he economerician s disposal. Bu wha is regression analysis? I is concerned wih describing

More information

Modeling and Forecasting Volatility Autoregressive Conditional Heteroskedasticity Models. Economic Forecasting Anthony Tay Slide 1

Modeling and Forecasting Volatility Autoregressive Conditional Heteroskedasticity Models. Economic Forecasting Anthony Tay Slide 1 Modeling and Forecasing Volailiy Auoregressive Condiional Heeroskedasiciy Models Anhony Tay Slide 1 smpl @all line(m) sii dl_sii S TII D L _ S TII 4,000. 3,000.1.0,000 -.1 1,000 -. 0 86 88 90 9 94 96 98

More information

Innova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*)

Innova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*) Soluion 3 x 4x3 x 3 x 0 4x3 x 4x3 x 4x3 x 4x3 x x 3x 3 4x3 x Innova Junior College H Mahemaics JC Preliminary Examinaions Paper Soluions 3x 3 4x 3x 0 4x 3 4x 3 0 (*) 0 0 + + + - 3 3 4 3 3 3 3 Hence x or

More information

5. NONLINEAR MODELS [1] Nonlinear (NL) Regression Models

5. NONLINEAR MODELS [1] Nonlinear (NL) Regression Models 5. NONLINEAR MODELS [1] Nonlinear (NL) Regression Models General form of nonlinear or linear regression models: y = h(x,β) + ε, ε iid N(0,σ ). Assume ha he x and ε sochasically independen. his assumpion

More information

Exponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits

Exponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits DOI: 0.545/mjis.07.5009 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy,

More information

Time Series Test of Nonlinear Convergence and Transitional Dynamics. Terence Tai-Leung Chong

Time Series Test of Nonlinear Convergence and Transitional Dynamics. Terence Tai-Leung Chong Time Series Tes of Nonlinear Convergence and Transiional Dynamics Terence Tai-Leung Chong Deparmen of Economics, The Chinese Universiy of Hong Kong Melvin J. Hinich Signal and Informaion Sciences Laboraory

More information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

1 Differential Equation Investigations using Customizable

1 Differential Equation Investigations using Customizable Differenial Equaion Invesigaions using Cusomizable Mahles Rober Decker The Universiy of Harford Absrac. The auhor has developed some plaform independen, freely available, ineracive programs (mahles) for

More information

Learning a Class from Examples. Training set X. Class C 1. Class C of a family car. Output: Input representation: x 1 : price, x 2 : engine power

Learning a Class from Examples. Training set X. Class C 1. Class C of a family car. Output: Input representation: x 1 : price, x 2 : engine power Alpaydin Chaper, Michell Chaper 7 Alpaydin slides are in urquoise. Ehem Alpaydin, copyrigh: The MIT Press, 010. alpaydin@boun.edu.r hp://www.cmpe.boun.edu.r/ ehem/imle All oher slides are based on Michell.

More information

CHAPTER 17: DYNAMIC ECONOMETRIC MODELS: AUTOREGRESSIVE AND DISTRIBUTED-LAG MODELS

CHAPTER 17: DYNAMIC ECONOMETRIC MODELS: AUTOREGRESSIVE AND DISTRIBUTED-LAG MODELS Basic Economerics, Gujarai and Porer CHAPTER 7: DYNAMIC ECONOMETRIC MODELS: AUTOREGRESSIVE AND DISTRIBUTED-LAG MODELS 7. (a) False. Economeric models are dynamic if hey porray he ime pah of he dependen

More information

Stochastic Model for Cancer Cell Growth through Single Forward Mutation

Stochastic Model for Cancer Cell Growth through Single Forward Mutation Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com

More information

The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information

The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information Chaper 8 The Muliple Regression Model: Hypohesis Tess and he Use of Nonsample Informaion An imporan new developmen ha we encouner in his chaper is using he F- disribuion o simulaneously es a null hypohesis

More information

Linear Combinations of Volatility Forecasts for the WIG20 and Polish Exchange Rates

Linear Combinations of Volatility Forecasts for the WIG20 and Polish Exchange Rates Eliza Buszkowska Universiy of Poznań, Poland Linear Combinaions of Volailiy Forecass for he WIG0 and Polish Exchange Raes Absrak. As is known forecas combinaions may be beer forecass hen forecass obained

More information

5.1 - Logarithms and Their Properties

5.1 - Logarithms and Their Properties Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We

More information

Stationary Time Series

Stationary Time Series 3-Jul-3 Time Series Analysis Assoc. Prof. Dr. Sevap Kesel July 03 Saionary Time Series Sricly saionary process: If he oin dis. of is he same as he oin dis. of ( X,... X n) ( X h,... X nh) Weakly Saionary

More information

You must fully interpret your results. There is a relationship doesn t cut it. Use the text and, especially, the SPSS Manual for guidance.

You must fully interpret your results. There is a relationship doesn t cut it. Use the text and, especially, the SPSS Manual for guidance. POLI 30D SPRING 2015 LAST ASSIGNMENT TRUMPETS PLEASE!!!!! Due Thursday, December 10 (or sooner), by 7PM hrough TurnIIn I had his all se up in my mind. You would use regression analysis o follow up on your

More information

References are appeared in the last slide. Last update: (1393/08/19)

References are appeared in the last slide. Last update: (1393/08/19) SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be

More information

A note on spurious regressions between stationary series

A note on spurious regressions between stationary series A noe on spurious regressions beween saionary series Auhor Su, Jen-Je Published 008 Journal Tile Applied Economics Leers DOI hps://doi.org/10.1080/13504850601018106 Copyrigh Saemen 008 Rouledge. This is

More information

Math 10B: Mock Mid II. April 13, 2016

Math 10B: Mock Mid II. April 13, 2016 Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.

More information

A Dynamic Model of Economic Fluctuations

A Dynamic Model of Economic Fluctuations CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model

More information

Learning a Class from Examples. Training set X. Class C 1. Class C of a family car. Output: Input representation: x 1 : price, x 2 : engine power

Learning a Class from Examples. Training set X. Class C 1. Class C of a family car. Output: Input representation: x 1 : price, x 2 : engine power Alpaydin Chaper, Michell Chaper 7 Alpaydin slides are in urquoise. Ehem Alpaydin, copyrigh: The MIT Press, 010. alpaydin@boun.edu.r hp://www.cmpe.boun.edu.r/ ehem/imle All oher slides are based on Michell.

More information

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time. Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike

More information

5 The fitting methods used in the normalization of DSD

5 The fitting methods used in the normalization of DSD The fiing mehods used in he normalizaion of DSD.1 Inroducion Sempere-Torres e al. 1994 presened a general formulaion for he DSD ha was able o reproduce and inerpre all previous sudies of DSD. The mehodology

More information

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

More information

Robust critical values for unit root tests for series with conditional heteroscedasticity errors: An application of the simple NoVaS transformation

Robust critical values for unit root tests for series with conditional heteroscedasticity errors: An application of the simple NoVaS transformation WORKING PAPER 01: Robus criical values for uni roo ess for series wih condiional heeroscedasiciy errors: An applicaion of he simple NoVaS ransformaion Panagiois Manalos ECONOMETRICS AND STATISTICS ISSN

More information

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3 and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or

More information

Ready for euro? Empirical study of the actual monetary policy independence in Poland VECM modelling

Ready for euro? Empirical study of the actual monetary policy independence in Poland VECM modelling Macroeconomerics Handou 2 Ready for euro? Empirical sudy of he acual moneary policy independence in Poland VECM modelling 1. Inroducion This classes are based on: Łukasz Goczek & Dagmara Mycielska, 2013.

More information

New Challenges for Longitudinal Data Analysis Joint modelling of Longitudinal and Competing risks data

New Challenges for Longitudinal Data Analysis Joint modelling of Longitudinal and Competing risks data New Challenges for Longiudinal Daa Analysis Join modelling of Longiudinal and Compeing risks daa Ruwanhi Kolamunnage-Dona Universiy of Liverpool Acknowledgmen Paula Williamson, Pee Philipson, Tony Marson,

More information