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Applied Econometric Time SeriesJ

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WALTER ENDERSloWaState University

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PREFACE.' ,q' r'''

'This book was borne out of fnlstration. After returning from aI1 cnjoyable and productivesabbatical at the University of California at San Diego, I bcgan expanding classcs in macroecohomics and intelma7 the empirical content of my graduate-level . tional t'inance. tudents' interest surged as lhey began to understand th concunznt S .;; developmcntof macroeconomic theot'y and time-series econometrcs. The differ' ence between Keynesins. monetarists. the rational expectations school. and thc real business cycle approach could best be understood by the ir ability to explain me . empiricalregularities in the economy. Old-style .macroeconomic mtdels were discarded becausc of their empirical inadequacies, not because of any logicltl inconsis' M' --.

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EE Iowa State University has a world-class Statistics Department, and most of our .'. r r, y . economics students take thrce of four statistics classes. Nevertheless. students' q). 'rj t ! the empirical portion of my courses. 1 needed to . ttk backgroundswere inadequate for q .( ;)' . ? present number of lectures on the topics covered in this book. M7 A feasmable .7 t E5. frustrationwas that the journal articles were written for those already technically . kt' ( ) proficient in time-selies' economctrics. The existing time-sdries texts were inade. econometric quate to the task. Some focsed on forecasting, others on theoretical j. .. issues, and still others on tecbniques that are intrequently used in the economics liti C erature. The idea for this text began as my class notes and use of handouts grew inl . .' ,

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: FprtuMy orignal intent was to write a text on time-serits macroeconometrics. L ;' colleagues at Iowa State convinced me to broaden the focus; applied minately,my : r. ;,q croeconomistswere also cmbracing time-series methods. I decided to include ex- . g. y i $.76 amples drawn from agricultural economics, international finance. and some of my j.gt t, with Todd Sandlcr on thc study of transnational terrorism. You should find' . .)yk.. j.) : . work r jjjt: examples in the text to provide a reasonable balance between macfoeconomic . y' the 1. iri . . 4 t p. j and microeconomic applications. )! k '. r :-j Thc text is intended for those with some background in fultiple regresslon jy j. .!:.jj()y.) .r.rj y of L:).!. ,) analysis. I presulne the reder understands the assumptions underlying the tlse . ? of my students are familiar with the concepts of correla- .' l j . )t.. ordina:yleast squarcs. X11 ):y ( :. .y.; Sj. tiin and covariation', they also know how to tlse t-tests and Fltests within a regresh ;. . 7t j i sion framework. I lgnycance fev'f. . terms juch as titean square drr/r, unanq k use # tL'7 .# js !) . . 's. . j biased estimate without cxplaining thcir meaning. The last two chplers o the text 'it' 3. 'examine multiple time-jeries techniques. To work through these chapters, it is nectf .) ),.ry j :.. f )jy . . .( . . know bow to solve a system of equations . using mtrx algebra. Cbapter 1, . k.' tft tt essae to ! . entitled tDifference. Equations. is the cornerstone of the text. In my cxperience.) ). . t 4 '. . 1) . i .) ;jT . this material and a knowledge of regressipn . are sufcient to bring. .students to th . . yj. . . . point where they are able to read (he profesional jouimalsand to embark on a seri- , . rt lr. ous applird s tu dy ),

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build towards more general and more complicattld models and econometric procedures. Detailed examples of each procedure are provided. Each concludcs with a step-by-stepsummary of tlle stages typically employed in using that procedure. approach is one of learning by doing. A large number of solved problems are ina cluded in the body of each chapter. The Questionsnd Exercises at the end of each chapter are especially important. ney have been designed to complement the matedal in the text. In order to work through the excrcises, it is necessary to have aceae

I bclicvc in tcaching by induction. Thc mcthotl is to take a simple example and

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cess to a soware package such as RATS. SAS, SHAZAM, or TSP. Matrix packages stlch as MARAB and GAUSS are not as convenient for univariate models. Packages such as MINITAB. SPSSX. and MICROFIT can perform many of the procedurescovered in te exercises. You are encouraged to work through as many of the examples and exercises as possible. The answers to all qnestions are contained in the lnstructor's Manual. Most of thc questions are answered in great detail. In addition, the Insructor's Manual contaias the data disk and the computer programs that can be used to answer the end of chapter exerciscs. Programs are provided for the most popular software packages. In spite of a11 my efforts. some errors have undoubtedly crept into thc tcxt. Portions of the manuscript that are crystal clear to me, will surely be opaque to others. Towards this end, I plan to keep a list of corrections and clalifications. You can receive a copy (of what I hope is a short list) from my lnternet address ENDERS@valuable suggestions for improving the manuscript. I am . grateful to my students who kept me challenge and were quick to point out errors. Pin Chung was especially helpful in carefully reading the many dras of the manuscript and ferreting out numerous mistakes. Selahattin Dibooglu at the University of Illinois at Carbondalc and Harvey Cutlcr at Colorado State University used portions of thc text in their own courses; thcir comments concerning the organization, style. and clarity of presentation are mucl) appreciated. My collengue Ban'y Falk was more than willing to answcr my questions und make helpful suggestions. HaeShin Hwang, Texas A and M University; Paul D. McNelis, Georgetown University; H a d i E s t a l! a n U n i v e r s i t y o f I ll i n o i s ; M D a n i e l W e s t b r o o k G e o r g e t o w n University'. Beth Ingram. University of Iowa'. and Subhash C. Ray, University of Connecticut al1 providcd insightful reviews of various stages of the manuscript. Julio Herrera and Nifacio Velasco, the gurus'' at the. Univers'ity of Valladolid. hclped me survive the final stages of proofreading. Most of all, I would like to thank my lovillg wife Linda for putting up with me while 1 was working on the text.,.

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CHAPTER 1: Difference Equations 1. Time-series Modeis 2. Difference Equations and Their Solutions 3. Solution by Iteration 4. An Alternative Solution Methodology 5. ne Cobweb Model 6. Solving Homogeneous Diference Equations 7. Finding Pmicular Solutions for Deterministic Processes 8. The Method of Undetermined'cocfficients 9 Lag Operators l0. Forward-versus Backward-Looking Solutions(' . ? . . .

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Summary and Conclusions i 1 li l o Questns an0 6 xercises Endnotes Appendix l : Imaginary Roots and de Moivre's Theorem Appendix 2: Characteristic Roots in Higher-order Equations

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CHAPTER 2: Stationary Time-series Models 1. Stochastic Difference Equation Models 2. ARMA Models 3. Stationarity Model 4. Stationazity Restrictions for an Function 5. ne Autocorrelation 6. ne Partial Autocorrelation Function 7. Sample Autocorrelations of Stationa:y Series 8. Box-lenkins Model Selection 9. The Forccast Function 10. A Model of the WPl 11. SeasonalityARMAWV

Summary and Conclusions and Questions Exercises Endnotes Appendix: Expected Values and Variance

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2. ARCH Proc ssts

3: Modeling Economic Time Sries: Trnds nd Volatility Economic Tim Series: The Stylized F@cts @.

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4 Es t i m :1l ing :) GAR C H N1odel 5 A GA 1) l 1 N1ol)tt 1 ( l ' I-tis k . 6. Thc ARCH-M Model

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Endnotes Appendix: Signl Extraction and Minimum Mcan Square Errors

anu ARcu-k. Estimation osoAucu uikelihoou uaximum : Detenministic and Stochastic Trends Removing the Trend Are There Business Cycles? Stochastic Trends and Univariate Decompositions Surllary and Conclusions a Questionsnd Exercises' .

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355 356 363365 373 377

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38l 385 293 3964O0 4O4405 4l0 412

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211 2t2 22l 225 233 239 243 251 260 26l 265 265

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REFERENCESAUTHORINDEX

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SUBJECTINDEX

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Chapter 1DIFFERENCE EQUATIONS' . . . ...... . . . ..

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he theory of difference equations underlies all the time-sefies methods employed in later chapters of this text. lt is fair to say that time-series econometrics is concerned with thc estimation of differcnce equatjons containing stochastic components. The traditional use of tim-series analysis was to forecast thu time path of a vmiable.Uncovering the dynamic path of a series improves forecasts since the predictable components of the series can be extrapolated into the futurc. The growing interest in economic dynamics has given a new emphasis to time-scries economstlics. Stochastic difference equations arise quire naturally from 'dynamic economic models. Appropriately estimated equations can be used for the intepretation of . economicdata and for hypotesis testing.'I'he aims of this introductory chapter are to:, . . .

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yjku) . k . .y yj . ' i'z' 2. Explain what it means to slve a difference equation. The solution will deter).y j jy . mine whether a variable has a stable or an explosive tifne path. A knwledge of r. :. ) ' . . F stability conditions is essential to undetstanding the recent innovaiions in ) the q..J. . . . .. .. time-selies econometricsk' The contemporary time-series literature pays. special t( (:; ' . . . . .. ) attention to the issue ot- stationary versus nonstationary variablesr The stability '. . : . jt.j ' ' conditionsunderlie the conditions for statioarity. : ; ) :$ ''-..

l Explain how stochastic difference equations can be used for forecasting and to illustrate how such equaiions can arise from familiar economic models. Tlie 'chapter is not meant to be a treatise on the theory:of differenc quations. Only . j essential to the appropriate estimation f Iiner timthose techniques that are models are presented. This chapter focus L 'on single-eqation ' models: series s . : multivariatemodels are considered in Chapters 5 and 6.. . . . . ,. '' ' ''m . . .

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equation. There how to fld the solution to ' . are Several differcnt techniques that can be used: each haj its own relative mer- ' its. A nuniber of examplej are presented to hlp yo uriderstand th: different methods. Try to' .work throuch each example carfully. For extra practice. you.

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Tme-series Models:' .q' :''''

1. TIME-SERIES MODELSThe task facing thc modern time-serics economtrtrician is to develop reasonably simple models capable o forecasting, interpreting. nnd testing hypotheses conccrning economic data. The challenge has grown ovcr time'. the original use of timeseries analysis was primarily as an aid to forccastillg. As such. a methodology was devcloped to decompose a scfies into a trend. seasonal, cyclical, and an irregular component. Uncovcring thc dynamic path of a st'ries improves forecast accuracy since each of the predictable components can 1,e extrapolated into the future. Suppose you observe the 50 data points show 11 il1 Figure 1 l and are interested in the forecastingthc subsequent values. ausing time-series methods discussed in the next several chapters. it is possible to decomplpse this series into the trend. sea.

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Notice that the irregular component, whilc not having a well-dened pattem. is somewhat predictablc. lf you examine thc figure closcll'. you will see that thc positivc and negative values occur in runs', the occurrence of a large value in any period tends to be followed by another lftrge value. Short-run forcasts will make use of this positive correlation in the irregular component. Ovcr the entire span, however. the irregular component exhibits a te dency to revert to zero. As shown in the lower part of the figure, the projection ot- the irregular component past period 50 overall forecast, shown in the top part of the f'igrapidly klecays toward zero. is the sum of each forecasted component. ure, The general methodology used to make such forecasts entails i'inding the tion of motion'' driving a stochastic process and nsing that equation to predict subsenuent outcomes. Let y, j enote the value of a data point at neriod t. jj- wg use tjjjs 1 notation. the cxnmple in Figure 1 asFumed wa observed y! throtlgh yso. For l 1 ; . : to 50. the enuations of motion used to construct components of the y, series are'rhe

sonal, and irregular components shown in the lower part of tbe figure. As you can see, the trend changes the mean of the seres and the seasonal component imparts a regularcylical pattern with peaks occurring every 12 units ot-time. ln practice, the trend and seasonal components will not be the simplistic detenninistic functions shown in .the Ggure. With economic data, it is typical to t'ind that a series contains stochasticelements in the trend, seasonal, and irregular components. For the time being. it is wise to sidestep these complications so that the projection of the trend and seasonal compon...