APPLIED ECONOMETRIC TIME SERIES (2nd edition)

Transcription

1 INSTRUCTOR S RESOURCE GUIDE TO ACCOMPANY APPLIED ECONOMETRIC TIME SERIES (2nd ediion) Waler Enders Universiy of Alabama Preared by Pin Chung American Reinsurance Comany and Iowa Sae Universiy Waler Enders Universiy of Alabama Ling Shao Universiy of Alabama Jingan Yuan Universiy of Alabama

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3 PREFACE This Insrucor s Manual is designed o accomany he second ediion of Waler Enders Alied Economeric Time Series (AETS). As in he firs ediion, he ex insrucs by inducion. The mehod is o ake a simle examle and build owards more general models and economeric rocedures. A large number of examles are included in he body of each chaer. Many of he mahemaical roofs are erformed in he ex and deailed examles of each esimaion rocedure are rovided. The aroach is one of learning-by-doing. As such, he mahemaical quesions and he suggesed esimaions a he end of each chaer are imoran. In addiion, i is useful o have sudens erform he ye of semeser rojec described a he end of his manual. One aim of his manual is o rovide he answers o each of he mahemaical roblems. Many of hese quesions are answered in grea deail. Our goal was no o rovide he mos mahemaically elegan soluion echniques. Someimes a long and drawn-ou answer rovides more insigh han a concise roof. This second aim is o rovide samle rograms ha can be used o obain he resuls reored in he Quesions and Exercises secions of AETS. Sudens should be encouraged o work hrough as many of hese exercises as ossible. In order o work hrough he exercises, i is necessary o have access o a saisical ackage such as EViews, Microfi, PC-GIVE, or RATS, SAS, SHAZAM or STATA. Marix ackages such as MATLAB, and GAUSS are no as convenien for univariae models. Some of hese ackages, such as EViews, allow you o erform mos of he exercises using ull-down menus. Ohers, such as GAUSS, need o be rogrammed o erform relaively simle asks. I is no ossible o include rograms for each of hese ackages wihin his small manual. There were several facors leading me o rovide rograms wrien for RATS and STATA. Firs, he RATS Programming Manual can be downloaded (a no charge) from The Programming Manual rovides a comlee discussion of many of he rogramming asks used in ime-series economerics. STATA was included since i is a oular ackage ha mos would no consider o be a ime-series ackage. Neverheless, as shown below, STATA can roduce almos all of he resuls obained in he ex. Adobe Acroba allows you o coy a rogram from he.df version of his manual and ase i direcly ino STATA or RATS. The languages used in RATS and STATA are relaively ransaren. As such, users of oher ackages should be able o ranslae he rograms rovided here. As saed in he Preface of AETS, he ex is cerain o conain a number of errors. If he firs ediion is any guide, he number is embarrassingly large. I will kee a lis of yos and correcions on my Web age: Moreover, ime-series mehods and echniques kee evolving very raidly. I will ry o kee you udaed by osing research noes and clarificaions on my Web age. I would be hay o os any useful rograms or communicaions you migh have; my address is wenders@cba.ua.edu.

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5 CONTENTS 1. Difference Equaions Lecure Suggesions 1 Answers o Quesions 2 2. Saionary Time-Series Models Lecure Suggesions 17 Answers o Quesions Modeling Volailiy Lecure Suggesions 41 Answers o Quesions Models Wih Trend Lecure Suggesions 59 Answers o Quesions Muliequaion Time-Series Models Lecure Suggesions 81 Answers o Quesions Coinegraion and Error-Correcion Models Lecure Suggesions 107 Answers o Quesions Nonlinear Tine-Series Models Lecure Suggesions 127 Answers o Quesions 128 Semeser Projec 137

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7 CHAPTER 1 DIFFERENCE EQUATIONS 1. Time-Series Models 1 2. Difference Equaions and Their Soluions 6 3. Soluion by Ieraion 9 4. An Alernaive Soluion Mehodology The Cobweb Model Solving Homogeneous Difference Equaions Paricular Soluions for Deerminisic Processes The Mehod of Undeermined Coefficiens Lag Oeraors Summary 41 Quesions and Exercises 42 APPENDIX 1.1 Imaginary Roos and de Moivre s Theorem 44 APPENDIX 1.2 Characerisic Roos in Higher-Order Equaions 46 Lecure Suggesions Nearly all sudens will have some familiariy he conces develoed in he chaer. A firs course in inegral calculus makes reference o convergen versus divergen soluions. I draw he analogy beween he aricular soluion o a difference equaion and indefinie inegrals. I is imoran o sress he disincion beween convergen and divergen soluions. Be sure o emhasize he relaionshi beween characerisic roos and he convergence or divergence of a sequence. Much of he curren ime-series lieraure focuses on he issue of uni roos. I is wise o inroduce sudens o he roeries of difference equaions wih uniary characerisic roos a his early sage in he course. Quesion 5 a he end of his chaer is designed o review his imoran issue. The ools o emhasize are he mehod of undeermined coefficiens and lag oeraors. Few sudens will have been exosed o hese mehods in oher classes. I use overheads o show he sudens several daa series and ask hem o discuss he ye of difference equaion model ha migh caure he roeries of each. The figure below shows hree of he real exchange rae series used in Chaer 4. Some sudens see a endency for he series o rever o a long-run mean value. The classroom discussion migh cener on he aroriae way o model he endency for he levels o meander. A his sage, he recise models are no imoran. The objecive is for sudens o conceualize economic daa in erms of difference equaions. Page 1: Difference Equaions

8 Real Exchange Raes (Panel.xls) (1996 = 100) U.S. Canada Germany Answers o Quesions 1. Consider he difference equaion: y = a 0 + a 1 y -1 wih he iniial condiion y 0. Jill solved he difference equaion by ieraing backwards: y = a 0 + a 1 y -1 = a 0 + a 1 [a 0 + a 1 y -2 ] = a 0 + a 0 a 1 + a 0 (a 1 ) a 0 (a 1 ) -1 + (a 1 ) y 0 Bill added he homogeneous and aricular soluions o obain: y = a 0 /(1 - a 1 ) + (a 1 ) [y 0 - a 0 /(1 - a 1 )]. A. Show ha he wo soluions are idenical for a 1 < 1. Answer: The key is o demonsrae: a 0 + a 0 a 1 + a 0 (a 1 ) a 0 (a 1 ) -1 + (a 1 ) y 0 = a 0 /(1 - a 1 ) + (a 1 ) [y 0 - a 0 /(1 - a 1 )] Firs, cancel (a 1 ) y 0 from each side and hen divide by a 0. The wo sides of he equaion are idenical if: 1 + a 1 + (a 1 ) (a 1 ) -1 = 1/(1 - a 1 ) - (a 1 ) /(1 - a 1 ) Page 2: Difference Equaions

9 Now, mulily each side by (1 - a 1 ) o obain: (1 - a 1 )[1 + a 1 + (a 1 ) (a 1 ) -1 ] = 1 - (a 1 ) Mulily he wo exressions in arenheses o obain: 1 - (a 1 ) = 1 - (a 1 ) The wo sides of he equaion are idenical. Hence, Jill and Bob obained he idenical answer. B. Show ha for a 1 = 1, Jill's soluion is equivalen o: y = a 0 + y 0. How would you use Bill's mehod o arrive a his same conclusion in he case a 1 = 1. Answer: When a 1 = 1, Jill's soluion can be wrien as: y = a 0 ( ) + y 0 = a 0 + y 0 To use Bill's mehod, find he homogeneous soluion from he equaion y = y -1. Clearly, he homogeneous soluion is any arbirary consan A. The key in finding he aricular soluion is o realize ha he characerisic roo is uniy. In his insance, he aricular soluion has he form a 0. Adding he homogeneous and aricular soluions, he general soluion is y = a 0 + A To eliminae he arbirary consan, imose he iniial condiion. The general soluion mus hold for all including = 0. Hence, a = 0, y 0 = a 0 + A so ha A = y 0. Hence, Bill's mehod yields: y = a 0 + y 0 2. The Cobweb model in secion 5 assumed saic rice execaions. Consider an alernaive formulaion called adaive execaions. Le he execed rice in (denoed by ) be a weighed average of he rice in -1 and he rice execaion of he revious eriod. Formally: = α -1 + (1 - α) 1 0 < α 1. Clearly, when α = 1, he saic and adaive execaions schemes are equivalen. An ineresing feaure of his model is ha i can be viewed as a difference equaion exressing he execed rice as a funcion of is own lagged value and he forcing variable -1. Page 3: Difference Equaions

10 A. Find he homogeneous soluion for Answer: Form he homogeneous equaion The homogeneous soluion is: - (1 - α) = A(1-α) 1 = 0. where A is an arbirary consan and (1-α) is he characerisic roo. B. Use lag oeraors o find he aricular soluion. Check your answer by subsiuing your answer ino he original difference equaion. Answer: The aricular soluion can be wrien as [ 1 - (1-α)L ] = α -1 or = α -1 /[ 1 - (1-α)L ] so ha = α[ -1 + (1-α) -2 + (1-α) ] To check he answer, subsiue he aricular soluion ino he original difference equaion α[ -1 + (1-α) -2 + (1-α) ] = α -1 + (1-α)α[ -2 + (1-α) -3 + (1-α) ] I should be clear ha he equaion holds as an ideniy. 3. Suose ha he money suly rocess has he form m = m + ρm -1 + ε where m is a consan and 0 < ρ < 1. A. Show ha i is ossible o exress m +n in erms of he known value m and he sequence {ε +1, ε +2,..., ε +n ). Answer: One mehod is o use forward ieraion. Udaing he money suly rocess one eriod yields m +1 = m + ρm + ε +1. Udae again o obain m +2 = m + ρm +1 + ε +2 = m + ρ[m + ρm + ε +1 ] + ε +2 = m + ρm + ε +2 + ρε +1 + ρ 2 m Reeaing he rocess for m +3 m +3 = m + ρm +2 + ε +3 Page 4: Difference Equaions