Identification of Trends and Cycles in Economic Time Series. by Bora Isik

Transcription

1 Idenificaion of Trends and Cycles in Economic Time Series by Bora Isik An Honours essay submied o Carleon Universiy in fulfillmen of he requiremens for he course ECON 498, as credi oward he degree of Bachelor of Ars wih Honours in Economics. Deparmen of Economics Carleon Universiy Oawa, Onario June, 4

2 TABLE OF CONTENTS. Inroducion. Conceps, Definiions and Environmen. Definiions of ime series. Auocorrelaion Funcions (ACF).3 Definiions of Saionariy.4 Auoregressive Moving Average (ARMA) Models.4. General linear process.4. Moving average processes.4.. The firs order moving average process (MA ()).4.. The q h Order Moving Average Process (MA (q).4.3 Auoregressive Processes.4.3. The firs order auoregressive process (AR ()).4.3. The second order auoregressive process (AR ()) The p h order auoregressive process (AR (p)).4.4 Auoregressive Moving Average Processes.4.4. ARMA (, ).4.4. ARMA (p, q).5 Trend Saionary versus Sochasic Trends.5. Trend Saionary Models.5. Sochasic Trends.6 Uni Roo Tess.6. A es for AR () wihou any consan (Dickey-Fuller).6. A es for AR () wih ime componens.6.3 Augmened Dickey-Fuller Uni Roo Tess.6.4 Phillips-Perron Uni Roo Tess

3 3. An Applicaion o Macroeconomic Time Series 3. Lieraure Review 3. Dynamics of U.S GDP series 3.3 Uni Roo ess 3.4 Possible Model Specificaions 3.4. Inspecion of ACF and PACF 3.4. Residual Diagnosics 3.5 Forecass 3.5. Forecas for ARIMA (,, ) and ARIMA (,, ) 3.5. Forecas for rend-saionary and sochasic rend models 4. Conclusions 5. References 6. Figures Figure Quarerly and seasonally adused U.S GDP series in log scale Figure Auocorrelaion funcion of quarerly and seasonally adused GDP series Figure 3 The firs difference of quarerly and seasonally adused U.S. GDP in log scale Figure 4 ACF and PACF of he firs difference of U.S. GDP series Figure 5 Residual Diagnosic for ARIMA(,,) Figure 6 Residual Diagnosic for ARIMA(,,) Figure 7 Forecas wih ARIMA (,, ) for he ou-of sample series Figure 8 Forecas wih ARIMA (,, ) for he ou-of sample series Figure 9 Forecas wih AR() rend-saionary model for he ou-of sample series Figure Forecas wih a random walk for he ou-of sample series 7. Tables Table The resuls of Augmened Dickey-Fuller Tes for he maximum lag lengh of 5 Table The resuls of Augmened Dickey-Fuller Tes for he maximum lag lengh of 4 Table 3 The resuls of Augmened Dickey-Fuller Tes for he maximum lag lengh of 3 3

4 . Inroducion In macroeconomics, i is acceped ha economies undergo shor-run variaion apar from heir long-run general behavior bu he relaion beween he wo has no ye been fully idenified. The relaionship beween he shor-run and long-run could be explained if he characerisics of shor-run variaion are fully known. Several hypoheses have been proposed o idenify he characerisics of shor-run variaion. One of hese hypoheses of shor-run aciviy, as sudied in convenional macroeconomic heory, is a shock originaing from he imperfecions in he marke ha causes he economy o deviae from is he long-run behavior (Keynes, 936). Imperfecions prevail because of saggered price adusmens, wage rigidiies, price sickiness, and asymmery of informaion (for deails see, Romer, ). On he oher hand, real business cycle heory claims ha a shor-run variaion is also a shor-run aciviy ha causes he economy o deviae from is long-run behavior bu is based on echnological innovaions insead of he presence of imperfecions in he marke. Wheher i is he convenional macroeconomic heory or he real business cycle approach, he economy is assumed o undergo shocks ha have ransiory effecs on he long-run rend of he economy. An alernaive perspecive for idenificaion of he characerisics of he shorrun variaion is o consider hem as innovaions ha permanenly effec he oupu (Nelson and Plosser, 98). Under his perspecive, he economy is assumed o undergo shocks ha have permanen affecs on he long-run rend of he economy. All business cycle heories are based on ime series conceps in idenifying wheher he shocks are permanen or ransiory. If he shocks are permanen, he See Kydland and Presco (98); Long and Plosser (983); Presco (986); and Black (98). 4

5 macroeconomic series inheris a sochasic rend or else if he shocks are ransiory, he macroeconomic series inheris a deerminisic rend. My analysis indicaes ha he shor-run variaions have permanen effecs on he long-run rend for he U.S GDP series and his is suppored by an ARIMA (,, ) process, which indicaes ha he series has a uni roo. The analysis is pursued in wo secions. In secion, brief definiions, explanaions and environmen for he ime series conceps of auocorrelaions, saionariy, auoregressive moving average models, nonsaionariy and uni roo es are provided. In Secion 3, a lieraure review on U.S GDP series is provided. Then, he dynamics of he U.S GDP series is analyzed by sudying he auocorrelaion (ACF) and parial auocorrelaion funcion (PACF) of he series. This examinaion indicaes ha he GDP series is non-saionary. Aferwards, Augmened Dickey-Fuller (984) and Phillips-Perron (988) uni roo ess are implemened and i is found ha he series has a uni roo. Nex, ACF, PACF and residual diagnosic for he firs differenced saionary series are analyzed and he diagnosics indicae owards an ARIMA (,, ) and ARIMA (,, ) model. Laer, a forecas environmen is designed o compare he performance of he ARIMA (,, ) and ARIMA (,, ) models. I is found ha ARIMA (,, ) has more forecas accuracy. Nex, he same forecas echnique is implemened o compare ARIMA (,, ) wih he random walk model of Nelson and Plosser (98) and he rend-saionary model of Perron (989). The comparison suggesed ha he ARIMA (,, ) model has he bes forecas accuracy. Hence, he U.S GDP has a sochasic rend componen as indicaed by he ARIMA (,, ) model. I conclude wih Secion 4. 5

6 . Conceps, Definiions and Environmen. Definiions of ime series A serial daa over an ordered ime sequence is called ime series daa. A ime series of an even may convey he possibiliy ha a ime paern amongs observaions is apparen. The ime paern is formed from dependence amongs observaions. The dependence should be such ha a leas one of he lags is correlaed o any oher lag. This dependence characerizes he underlying ime series process and allows an analyzer o specify a model o undersand he implicaions and also o underake fuure predicions. Amongs all oher ime series, his research examines macroeconomic ime series. Such ime series may include quarerly consumpion figures, seasonally adused GNP or GDP figures, money demand, annual invesmen and consumpion. Time series analysis of such macroeconomic indicaors would reveal he paricular behavior of he economy by explaining wheher he shocks o a paricular economy are permanen or ransiory and would allow he analyzer o underake fuure predicions. 6

7 . Auocorrelaion Funcion (ACF) The foundaion of a ime series analysis is ha here are dependencies in he daa and hence in pursing a ime series analysis, he examiner should firs examine wheher here are auocorrelaions visible in he daa. This could simply be examined by calculaing he correlaion coefficien of ρ which measures he degree of correlaion beween any lag. If he coefficiens are saisically significan, hen he series is no independenly and idenically disribued bu correlaions are apparen amongs observaions. There are wo ways by which an examiner may verify he exisence of auocorrelaions in he daa. Firs, one may consruc a confidence inerval for he sample ρ and es if ρ is off he boundaries or no, and second, one may design a hypohesis o es for he degree of he saisical significance. In order o illusrae he former, consider he equaion given below where { } {.. y =... y, y, y+. } is a ime series as such and ρ is he correlaion coefficien of any wo lags cov( y, y ) ρ = = var( y ) var( y ) cov( y, y var( y ) ) γ =. (..) γ Equaion (..) could be esimaed from he sample as by, T ( y y)( y y) = + ˆ ρ = T (..) ( y y) = Saionariy condiion has been used where var( y ) = var( ). y 7

8 where T is he sample size and T. If y is independenly and idenically disribued, hen he cenral limi heorem would imply ha ρ has zero mean and variance of /T. A 95% confidence inerval can be used o verify if here are auocorrelaions in he daa. Based on he cenral limi heorem, acual confidence inervals are ±.96 / T (Zivo, ). If for insance, he sample ρ is ouside such confidence inervals, he es would imply ha y is no independenly and idenically disribued and hence, saisically significan correlaion beween hose lags exiss. Alernaively, auocorrelaions could be esed by he Lung-Box (978) es 3. Under he null hypohesis, designed such ha H : ρ = ρ =... = ρ m H : ρ a b y is independenly and idenically disribued. The es is where m is he degrees of freedom and b is any ime wihin he boundaries of m. The es saisic is given below Q( m) = T ( T + ) ˆ ρ m l= T (..3) where Q(m) has an asympoic chi-squared disribuion wih m degrees of freedom. When pursuing his paricular es he examiner should pay aenion o wo imporan issues. A firs, he examiner should keep in mind ha under he reecion of 3 Lung Box (978) es is a modified Box and Pierce (97), wih a greaer power han he oher. See Tsay (), Secion.. 8

9 he null hypohesis, he es would no reveal which lags are correlaed. I only indicaes ha here is auocorrelaion. Secondly, he examiner should be careful in choosing he degrees of freedom. The lieraure indicaes ha he choice of he degrees of freedom is approximaely equal o ln(t ) for a reliable power (for deails see Tsay, ). Parial auocorrelaion funcions (PACF) demonsrae he saisical significance of he lags in he ime series process. They invesigae he dependence of each individual lag wih he original series. Analogous o auocorrelaions, parial auocorrelaions can eiher be esed by confidence inervals or by hypohesis esing..3 Definiions of Saionariy Apar from auocorrelaions in he series, he examiner should sudy wheher he daa is saionary hrough he necessary condiions for saionariy. These necessary condiions are cerain resricions on he momens of he underlying disribuion. There are wo ypes of saionariy. The firs ype is sric saionariy, and he second ype is weak saionariy. In order o prove under which condiions such properies are revealed, consider a ime series{ } { y, y, y...} y... and = + { } { y, y, y... y... } + k = + k + k + k + where k is any ime shif from he origin. If he y } { } probabiliy disribuion of { is idenical o ha of y +, hen y } is sricly saionary. Tha is a very srong condiion, requiring ha all momens of he wo disribuions be idenical. A weaker condiion is us o verify wheher he firs wo momens of he disribuions are he same. In oher words, a weak saionariy crierion requires ha he firs wo momens of he series be ime invarian, so ha hey are idenical o each oher. Tha would imply he following: k { 9

10 E ( y y + k var( ) = E( ) (.3.) ) = var( ). (.3.) y y + k Because of he difficuly of esing sric saionariy, weak saionariy is used commonly as a es for saionariy in economic ime series..4 Auoregressive Moving Average (ARMA) Models.4. General linear process A ime series represened by a weighed sum of pas and presen random shocks is called a general linear sochasic process. In such a process, all previous shocks have a specified long lasing effec on he process. The only shock ha is no fully known is he presen shock which is required o be esimaed. The process is illusraed below, where a y is a whie noise 4, μ + a + ψ a + ψ a... (.4.) = + y = 5 = μ + a + ψ a. (.4.).4. Moving average processes A moving average process of an order q is a linear process having a finie ime domain. The process remembers he shocks ha occurred in pas periods down o lag q and assigns a value o each one. The only shock ha he process fails o assign a cerain 4 E ( ) = and a var( a ) =. σ a 5 I is worhwhile o noe ha in order o saisfy he saionary condiions, = ψ <.

11 value o is he shock ha is coming from he presen ime. The presen shock is a random shock ha could aain any unprediced value due o limied or insignifican informaion available. Therefore, an expecaion of a curren shock should be formed based on he informaion available along wih he knowledge of previous shocks as deermined by lag q in order o specify he model properly..4.. The firs order moving average process (MA ()) The firs order moving average process uilizes he informaion gahered from he presen and previous shocks. The process has informaion down o he firs lag only a which some value is assigned o i as deermined byθ. In order o undersand he dynamics of he model properly, an expecaion of he curren shock should be formed along wih he knowledge of he previous shocks. The process is shown in equaion form below y μ θ. (.4.3) = + a + a Taking he firs and he second order expecaions and calculaing he firs auocovariance would yield he following hree ideniies 6 E ( y ) = μ, (.4.4) var( y ) = ( +θ ) σ, (.4.5) cov( y, y ) = θσ. (.4.6) Considering he essenials of saionary condiions and using he above ideniies, we obain he following from equaion (..) for he firs auo-correlaion for an MA () process, 6 For deails see Hamilon (994), Secion 3.3.

12 θ σ θ ρ =. (.4.7) = ( + θ ) σ + θ Auocorrelaion funcion of MA () would depend on he sign ofθ. Posiive values of θ indicae posiive auocorrelaion, whereas negaive values of θ indicae negaive auocorrelaion. In oher words, a posiive sign of θ induces large values of y o be followed by larger values of and a negaive value of θ induces large values of y+ y o be followed by small values of y (Hamilon, 994)..4.. The q h Order Moving Average Process (MA (q)) As menioned previously, he q h order moving average process remembers he previous shocks down o q h lag and forms expecaions for he curren shock. I is shown in equaion form below y = μ + a + θ... θ a + θ a + + qa q. (.4.8) Taking he firs and he second order expecaions and calculaing he firs auocovariance would yield he following four ideniies 7 E ( y ) = μ, (.4.9) var( y + θ... ) = ( + θ + θ ), (.4.) q cov( y { θ + θ θ + θ θ θ θ } σ, y ) q =, for =,,... q (.4.) cov( y, ) = for > q. (.4.) y q Considering he essenials of saionary condiions and using he above ideniies, we obain he following from equaion (..), 7 For deails see Hamilon (994).

13 ρ q = ( θ + θ θ + θ θ θ θ ) + ( + θ + + θ θ ) q q q. (.4.3) The dynamics of he auocorrelaion funcion of MA (q) would rely on he order q. As explained previously, if he order is one, hen he sign of θ would be indicaive of he auocorrelaion funcion. If he order is wo, hen he auocorrelaion funcion lies wihin he segmens of hree curves as deermined by he same crierion for he correlaion coefficiens Auoregressive Processes An auoregressive process of order p uilizes he informaion gahered from pas evens. All of he informaion abou pas evens is known down o lag p and a proporion of each has been refleced in he process. Analogous o moving average processes, he only segmen which remains o be forecased is he curren shock. Thus, in order o specify he model properly, an expecaion of a curren shock should be formed based on he informaion available along wih he knowledge of previous evens as deermined by lag q The firs order auoregressive process (AR ()) The firs order auoregressive process uilizes he informaion gahered from he previous even. The process assigns a value o he previous even only based on he informaion available. As menioned earlier, in order o undersand he dynamics of he 8 For more deails on his subec see Box and Jenkins (994), Chaper 3. 3

14 model properly, an expecaion of he curren shock should be formed along wih he knowledge of he previous even. The process is shown in equaion form below y φ + a (.4.4) = + φ y wih he characerisic equaion of φ( B) = φ B (.4.5) where B is he lag operaor. Taking he expecaion of equaion (.4.4) and solving for E( y ) φ φ E ( y ) =. (.4.6) Taking he square of equaion (.4.4) and aking he expecaion would yield he following σ var( y ) = (.4.7) ( φ ) where φ <. Also, muliplying equaion (.4.4) by following y k and aking he expecaion would yield he = φγ γ (.4.8) and hence ρ. (.4.9) = φρ I is also known ha ρ =, and hence ρ = φ. Therefore, he auocorrelaion funcion of an AR () process decays a an exponenial rae of φ (Tsay, ). 4

15 .4.3. The second order auoregressive process (AR ()) The second order auoregressive process uilizes he informaion gahered from he las wo evens as measured by he coefficiens, φ and φ. As menioned earlier, in order o undersand he dynamics of he model properly, an expecaion of he curren shock should be formed along wih he knowledge of he pas wo evens. The process is shown in equaion form below y φ + a (.4.) = + φ y + φ y wih he characerisic equaion of φ( B) φ φ B = B (.4.) where B is he lag operaor. Taking he expecaion of he above and solving for E( y ) ( φ ) = φ φ E y (.4.) where φ + φ for saionary condiions o saisfy. Also, muliplying equaion (.4.) by y k and aking he expecaion would yield he following where γ denoes he lag covariance, γ (.4.3) = φγ + φγ and hence ρ (.4.4) = φρ + φρ wih saring values of ρ = and ρ φ =. φ 5

16 k k Equaion (.4.4) could also be wrien as ρ = C R + CR where R and R are he roos of he characerisic equaion (.4.) (Tsay, ), 4 φ + φ + φ x =. (.4.5) The auocorrelaion funcion eiher exhibis damped exponenials or pseudo periodic behavior depending on he sign of φ + 4φ. If φ + 4φ, hen he ACF consiss of a composiion of damped exponenials. Tha is, i eiher decays smoohly or decays as alernaing signs depending on he signs of φ and φ (Box and Jenkins, 97). On he oher hand, in case φ + φ 4 i exhibis damping sine and cosine waves (Tsay, )., ACF exhibis pseudo periodic behavior. Tha is, The p h order auoregressive process (AR (p)) As menioned earlier, he p h order auoregressive process remembers he previous evens down o he p h lag and forms expecaions for he curren shock. I is shown in equaion form below y + = φ + φ y + φ y φ p y p a (.4.6) wih he characerisic equaion of φ ( B) = φ B φ B φ pb p (.4.7) where B is he lag operaor. Taking he expecaion of (.4.6) and solving for E( y ) E( y φ ) = φ... φ p (.4.8) 6

17 where φ... + φ for saionary condiions o saisfy. + p Also, muliplying equaion (.4.6) by y k and aking he expecaion would yield he following γ = φ... γ γ + + φ p p (.4.9) and hence ρ = φ... ρ ρ + + φ p p. (.4.3) Equaion (.4.3) canno be wrien as roo form unless p is known. Therefore, you may no come up wih a firm conclusion regarding he shape of he ACF wihou knowing he order of AR process. Forunaely, he order of AR may be checked by inspecing parial auocorrelaion coefficiens of he equaions below y φ + a = + φ y φ φ φ y = + y + y + a.. y y y y + a = φ + φ p + φ p φ pp p p where φ represens he coefficiens of parial auocorrelaions. For insance, if he process is AR (), hen φ is nonzero and he res of he coefficiens are zero. Or if he process is AR (), hen φ is nonzero and he res of he coefficiens are zero. 7

18 .4.4 Auoregressive Moving Average Processes Auoregressive moving average processes are he mixure of auoregressive processes and he moving average process. Tha is, hey remember he informaion gahered from pas evens as deermined by he memory index of lag p, and pas shocks as deermined by a memory index of lag q. All of he informaion abou pas evens and shocks is known and some proporion of each has been carried in he process. Analogous o moving average processes and auoregressive processes, he only par which remains o be forecased is he curren shock. Thus, in order o specify he model properly, an expecaion of a curren shock should be formed based on he informaion available along wih he memory of index p of previous evens and he memory of index q of he pas shocks ARMA (, ) ARMA (, ) uilizes he informaion gahered from he previous even and previous shock. The process has informaion down o he firs lag on pas evens and pas shocks carrying ou a proporion of each. As menioned earlier, in order o specify he model correcly, an expecaion of he curren shock should be formed along wih he knowledge of he previous even and previous shock. The process is shown in equaion form below, y φ + a. (.4.3) = + φ y + θa Normalizing y, muliplying above by and aking expecaions, he following hree ideniies are obained y k γ φγ + ( θ( φ θ ) σ a = for =, (.4.3) 8

19 γ = φγ θσ for =, (.4.33) a γ for. (.4.34) = φγ Solving equaions (.4.3) and (.4.33) for γ (var( y )) yields γ ( θ φ + θ ) = σ a. (.4.35) φ Each recursive subsiuion would yield 9 ( θϕ )( φ θ) k φ θ φ + θ γ = for k. (.4.36) k Equaion (.4.36) illusraes ha he auocorrelaion funcion of ARMA (, ) decays a an exponenial rae of ϕ saring a an iniial value of ρ depending on he valueθ (Cryer, 985) ARMA (p, q) As menioned earlier, he ARMA (p, q) remembers he pas evens down o he p h lag and remembers he pas shocks down o he q h lag and forms expecaions for he curren shock. I is shown in equaion form below y = φ + φ y φ p y p + θa θ pa p... + a. (.4.37) Normalizing y, muliplying above by and aking expecaions y k γ = φ... γ γ + φγ + + φ p p for =q+, q+, (.4.38) and hence γ = φ... φ ρ + φ p + + p p p. (.4.39) 9 For deailed explanaion see Cryer (985). 9

20 .6 Trend Saionary versus Sochasic Trends ARMA models are saionary, however many financial and economic daa exhibi non-saionary dynamics. Examples include ineres raes, exchange raes, gross domesic produc (GDP), consumpion and invesmen series. There are wo ypes of non-saionary models. The firs ype is rend-saionary; he second ype is sochasic rend. They boh have a rend componen bu rendsaionary models inheri rend revering dynamics, whereas sochasic rends do no. The behavior of rend-saionary series could be anicipaed, whereas sochasic rends could no..5. Trend Saionary Models A simple rend saionary model is illusraed below where α is a consan; ν is he rae of ime and e is an error erm ha is independenly and idenically disribued wih a mean of zero and a variance of. σ a y = TR +, (.5.) e TR = α + v. (.5.) Taking he expecaion and variaion of equaion (.5.) would yield he following wo equaions E( y ) = α + v (.5.3) a var( y ) = σ. (.5.4) e The above equaions prevail because E ( ) =, and he variaion of is ha of. Equaion (.5.3) is due o he assumpion ha e is independenly and idenically e y

21 disribued wih a zero mean. Equaion (.5.4) is due o he only random variable in he e process, having a variaion of. σ a The findings so far illusrae ha a rend-saionary model passes on a smooh rend on average. Any movemen around a rend is emporary and he series comes back o he rend. In a macroeconomic daa such as he gross domesic produc, a emporary movemen is called shor-run economic aciviy and can be reaed as noise. The emporary shor-run aciviy arises because here are rigidiies in he economy. The rigidiies prevail because here are marke imperfecions such as, missing markes, asymmeric informaion, and saggered price adusmens. On he oher hand, in he longrun he series always comes back o is rend because all of he imperfecions in he marke canno las forever. Also, he series should be ransformed o a saionary series if furher analysis is desired, such as an examinaion of auocorrelaion funcion. In order o ransform models ino saionary ime series, de-rending is required. Tha simply means aking he rend ou from he original series as illusraed below so ha he series revers o consanα. y v = α +. (.5.4) e.5. Sochasic Trends In order o undersand he dynamics of a sochasic rend, a simple rend sochasic model is illusraed below where he noaions are idenical o he previous secion. y = TR + (.5.5) b For more informaion, see Chaper 4 and Chaper 5 in Romer ().

22 where b = b + e (.5.6) and TR = α + v. (.5.7) By using he properies of AR (), equaion (.5.5) could be re-wrien as, y = α + (.5.8) + v + b e y = α α v + b + e + e (.5.9) + v + b + e = + y = TR + b + e + e + e... + e. (.5.) Taking he firs and he second order expecaions of equaion (.5.) would yield he following, E( y ) = α + v + (.5.) b var( y a ) = σ. (.5.) Equaion (.5.) is due o he fac ha E ( ) = and he expecaion of is iself ( E ( b ) = b ). Therefore he expecaion of y is TR plus b ( E( y ) = TR + b ). e b Equaion (.5.) is because he only random variable in he process is e having a variaion of σ. Since here are numbers of e in he process, he variaion of y is imes ( var( y ) = σ ). σ a a a The resul of equaion.5. illusraes ha he second momen is no well defined due o he presence of he ime componen. This poins ou ha he process is unpredicable. In macroeconomic lieraure, his siuaion arises when he shocks are permanen so ha he series would no rever o he iniial rend disabling he analyzer o

23 observe he general dynamics of he model. In macroeconomic lieraure i is acceped ha such shocks are no simple demand and supply shocks bu complicaed shocks like echnology Furhermore, he series should be convered o saionary if furher analysis is desired. In order o conver such models o a saionary ime series, he examiner should ake he differences of lags unil he series is saionary..6 Uni Roo Tess Uni roo ess exploi he mean-revering dynamics of he ime series daa. Under he null hypohesis, he series has no mean-revering dynamics and hence he process has a uni roo. Conversely, in he alernaive hypoheses, he series has a mean-revering dynamic and hence he process has no uni roo..6. A es for AR () wihou any consan (Dickey-Fuller) A design of a uni roo -es for a possible AR () model wihou any consan is called he Dickey-Fuller es and is shown below y φ + a (.6.) = y H H φ = () (.6.) : I φ < () (.6.3) : I where ˆ φ is he leas square esimae and SE ˆ φ ) is he sandard error. Dickey and Fuller (979) showed ha under he null hypohesis, he process is non-saionary wih a ( See Romer (), Chaper 5. 3

24 variance of σ and firs differencing is appropriae o conver he series o a saionary series. The es is shown below φ ˆ φ = =. (.6.4) SE( ˆ φ ) However, he above equaion is no susainable since under he null hypohesis sample momens converge o fixed consans. Insead, Phillips (987) showed ha he correc es saisic is W ( r) dw ( r) φ = (.6.5) W ( r) dr a which sample momens converge o Brownian Moion (Zivo and Wang, )..6. A es for AR () wih ime componens A design of a uni roo -es for a possible AR () wih rend is shown below y φ + a (.6.6) = + φ y where φ exhibis a ime componen. H H φ = () wih rend (.6.7) : I φ < () wih deerminisic rend. (.6.8) : I Under he null hypohesis, he process is non-saionary wih a ime rend componen and firs differencing is appropriae o conver he series o a saionary series. The -es However, he rend componen is no deerminisic. There is no rend revering dynamics. In Economerics lieraure such a rend componen is named as drif. 4

25 design differs from equaion (.6.5) mainly in how i accommodaes for he consan erm Augmened Dickey-Fuller Uni Roo Tess So far, he uni roo ess explained were for simple AR () models bu were no suiable for ARMA (p, q) models. The model s ess are modified by Said and Dickey (984) o capure he dynamics of ARMA (p, q) for an unknown order of p and q. Two ypes of es regression are considered, y = ω + φy p + λ Δy + = e (.6.9) Δy p + λδy + = = ω + δy e (.6.) where ω is a vecor ha capures he consan or rend componen and erm ha is independenly and idenically disribued. e is an error Under regression (.6.9), he null hypohesis is φ which implies ha he series has a uni roo. The es saisic is ha of (.6.). The normalized bias saisic is shown below = n T ( ˆ φ ) = λ... λ p. (.6.) Under regression (.6.), he null hypohesis is δ = which ess he significance of each lag. The es saisic is he usual es saisic whenδ =. The normalized bias saisic is shown 3 For deails see Hamilon (994), Secion 7.3 Case 4. 5

26 n = Tδ λ... λ p. The suble par in his analysis is he selecion of lag p. Dickey and Fuller (984) suggesed ha a possible lag p is chosen and a model is specified by using regressions (.6.9) and (.6.). Afer, anoher model is specified wih lag p-. If he wo models specified exhibi similar dynamics, hen lag p chosen may be correc. (Hamilon, 994) However, he selecion of p described above is arbirary and may bias he es if p is chosen o be very large or very small. For ha reason, Zivo and Wang () considered anoher ype of procedure. Firs choose 4 4 p = / max ( T ). (.6.) Aferwards, an ADF es regression is performed as in regression (.6.) esing for he significance of each lag up o p max. If he las lag is saisically significan as graned by he es saisics greaer han he absolue value of.6, hen choose p = p max and perform a regression as in (.6.9). Oherwise, reduce he lag lengh by one and sar he procedure again..6.4 Phillips-Perron Uni Roo Tess The design of Phillips-Perron (988) s uni roo ess are similar o ha of Augmened Dickey-Fuller (984) s uni roo ess bu differen in accommodaing moving average erms and heeroskedasiciy in he errors. They boh have he same ype of es saisic asympoic disribuions under he null and hey boh deal wih he deerminisic erms in he same way. Augmened Dickey-Fuller (ADF) ess 4 For deails see Schwer (989). 6

27 accommodae moving average erms bu disregard heeroskedasiciy in he error erms. Conversely, Phillips-Perron (PP) ess accommodae heeroskedasiciy in he error erms bu disregard moving average erms. Furhermore, lag specificaion is necessary for ADF bu no necessary for he PP es. The es regression for he PP is shown below Δy = ω + δy + u (.6.) where u is an error componen ha may be heeroskedasic. Under he null hypohesis, δ = and he firs difference of is a saionary process. Zivo and Wang () shows y ha he es saisic and normalized bias saisics are modificaions of and Tδˆ. They are shown below ˆ γ ( ˆ τ ˆ γ ) = T * SE( ˆ) δ Z * δ = * ˆ τ ˆ τ (.6.3) ˆ γ ˆ T * SE( ˆ) δ ( ˆ Z ˆ δ = Tδ λ γ ) (.6.4) ˆ γ δ = where ˆγ and τˆ denoe he leas square esimae sample variance of on a differen basis 5. u 5 For deails, see Hamilon (994), Secion

28 3. An Applicaion o Macroeconomic Time Series 3. Lieraure Review The applicaions of ime series conceps o macroeconomic series in lieraure focus on idenificaion of rends and cycles in he series. Nelson and Plosser (98) performs Dickey Fuller s (979) uni roo es on he logarihm of real quarerly U.S GNP from 99 o 97. They formally accep he null ha he GNP series has a uni roo and hence he process is similar o a random walk. Even hough hey accep he fac he economy experiences moneary and fiscal shocks ha have ransiory effecs on he rend, hey claim ha hey are dominaed by real shocks ha have permanen effecs. Perron (989) designs a uni roo es for U.S real quarerly GNP 6 wih he null hypohesis ha he series has a uni roo wih possibly a non-zero drif for U.S GNP. He allows for a srucural change 7 in his regression by adding dummy variables ha would become zero in imes of ha change. Unlike Nelson and Plosser (98), he formally reecs he null and concludes ha he rend is saionary bu here is a break in slope when he srucural changes occur, posulaing ha hose shocks were no a realizaion of he series. Thus, he concludes ha he shocks ha he economy experiences are ransiory. Zivo and Andrews (99) modify Perron s (989) uni roo es accommodaing a srucural change ha is unknown for U.S GNP. This accommodaion makes his resul 6 He uses he daa se of he Nelson and Poller (98) and poswar quarerly GNP series. 7 He refers o he Grea crash of 99 and he oil price shock of

29 robus o any srucural change. Zivo and Andrews (99) conclude ha he rend is deerminisic, finding even more evidence han Perron (989). Leybourne and McCabe (994) develop a uni roo es in which he null is ha of he saionary ARIMA (p,, ) 8 process agains he alernaive hypohesis of a nonsaionary ARIMA (p,, ) for U.S GNP series. Unlike he Dickey Fuller es, hey claim ha he selecion of p would affec he es. Nelson and Murray (997) choose p=, and conduc he Leybourne and McCabe es for he poswar U.S real GDP. They reec he null and conclude ha he model is ha of ARIMA (,, ). Thus, he series conains a uni roo. 3. Dynamics of U.S GDP series. The dynamics of he U.S.GDP series are analyzed by conducing a Lung Box (978) es, inspecing he plo of he series and analyzing he plo of ACF and PACF. For he U.S GDP series, he Lung Box (978) es is implemened for 5 degrees of freedom 9. The p-value for his es is zero which srongly suggess ha he null hypohesis canno be reeced. Thus, here are auocorrelaions presen in he daa. As shown in Figure, he U.S GDP series clearly exhibis non-saionariy. The non-saionariy dynamics may resul from he exisence of a ime componen in is mean or he presence of a ime componen in is variance or boh. Since i is observed in he series ha he daa is increasing over ime, he mean embodies a ime componen. Thus, 8 I is worhwhile o noe ha under he null, MA componen has a uni roo. 9 As explained in secion. he degrees of freedom should be seleced so ha i is approximaely equal o ln(m). ln a sample size of 8 ha corresponds o five. 9

30 he series has a rend componen. However, no furher deails could be revealed regarding he dynamics of he series. The ACF on he op of Figure no only suggess ha here are auocorrelaions apparen in he daa, bu ha he series is non-saionary. The auocorrelaions are apparen because auocorrelaion coefficiens are ouside of confidence bands. Also, he daa is non-saionary because here is persisence in auocorrelaions. However, he figures do no reveal any furher informaion regarding he characerisics of nonsaionariy. The above findings illusrae ha here are dependencies in he observaions and ha he series has a non-saionary dynamics. I seems ha he dependencies and nonsaionariy are caused by he rend, however i may also be caused by he variance. The findings do no reveal any furher informaion regarding he characerisics of he variance. 3.3 Uni Roo ess The Augmened Dickey Fuller (984) and Phillips-Perron (988) (PP) ess are implemened for he U.S GDP series. Under he null hypoheses in boh ess, he series inheris a sochasic rend componen, and under he alernaive hypohesis he series inheris a deerminisic rend componen. The PP es yields a -value of -.49 hrough equaions (.6.3). Tha corresponds o a p-value of.336 represening moderae evidence o accep he null. All calculaions are carried ou in S-Plus. 3

31 Thus, he series conains a uni roo and firs differencing is appropriae o ransform he series ino a saionary series. As menioned earlier, he specificaion of lag p in he Dickey Fuller (984) es requires recursive seps. Based on equaion (.6.), p max is chosen o be 5 for a sample size of 8 and he degree of significance of i is calculaed by (.6.4) which resuls from regression (.6.). The resuls when p is 5 are repored in Table. A - value of.34 indicaes ha he las lag is insignifican since i is lower han he criical value of absolue value of.6. Thus, lag 5 should be reduced by o 4 and he process should be repeaed. The resuls when p is 4 are repored in Table. A -value of -.67 indicaes ha he las lag is insignifican since i is lower han he criical value of absolue value of.6. Thus, lag 4 should be reduced by o 3 and he process should be repeaed. The resuls when p is 3 are repored in Table 3. A -value of -.87 indicaes ha he las lag is significan since i is higher han he criical value of absolue value of.6. Therefore, lag p should be se o 3. As shown in Table 3, he Dickey Fuller (984) es, when p=3 yields a -value of -.37 hrough equaion (.6.4). This corresponds o a p-value of.336 represening moderae evidence o accep he null. Thus, he series conains a uni roo and requires firs differencing o exhibi saionariy. 3

32 3.4 Possible Model Specificaions 3.4. Inspecion of ACF and PACF The auocorrelaion funcion of he firs differenced saionary U.S GDP series is shown on op of Figure 4. I exhibis damping cosine and sine waves. This is a characerisic of an AR () process. Thus, he series may conain an auoregressive componen of order. The parial auocorrelaion funcion of he firs differenced saionary U.S GDP series is shown in he boom panel of Figure 4. The parial auocorrelaion funcion of he underlying GDP series is similar o is auocorrelaion funcion in exhibiing damping waves like cosine and sine. The dualiy beween he moving average process and he auoregressive process indicaes ha a moving average componen of order may exis. The parial auocorrelaion funcion of he G.D.P series also indicaes ha he s and he h lags are saisically significan. The significance of he h lag may be capured by eiher MA () or MA () added o AR () or AR () keeping in mind ha eiher AR () or MA () should exis in he process as indicaed by ACF and PACF. In brief, he inspecion of ACF and PACF of he firs differenced U.S GDP series may sugges ha he series is possibly an ARMA (, ), ARMA (, ) or ARMA (, ). For he original U.S GDP series; i is possibly an ARIMA (,, ), ARIMA (,, ) or ARIMA (,, ). 3

33 3.4. Residual Diagnosics Residual diagnosic indicaes wheher he model is correcly specified or misspecified. If he residuals are independenly and idenically disribued, hen he model is correcly specified. Oherwise, here are auocorrelaions in he residuals and he model is misspecified. In his analysis, hree ypes of echniques are discussed o es wheher residuals behave as an independenly and idenically disribued process. The firs echnique involves deecing ACF for auocorrelaions amongs residuals; he second echnique involves inspecing PACF for he degree significance of he lags; he hird echnique involves conducing he Lung-Box es for esing auocorrelaions amongs residuals. In order o conclude ha he residuals are independenly and idenically disribued, boh echniques should yield similar resuls. The residual diagnosic is conduced for he ARIMA (,, ) and ARIMA (,, ) models. Figure 6 illusraes he resuls for ARIMA (,, ) and Figure 5 illusraes he resuls for ARIMA (,, ). On he second panel of Figure 6, ACF of he residuals indicaes ha he residuals are independenly and idenically disribued for ARIMA (,, ) since all he correlaion coefficiens lie wihin confidence inervals. On he conrary, on he second panel of Figure 5, ACF of he residuals indicaes ha he residuals may no be independenly and idenically disribued for ARIMA (,, ) since he correlaion coefficien for lag is ouside of he confidence inerval. On he hird panel of Figure 6, he PACF of residuals indicaes he significance of 4 h, 5 h and h lag for ARIMA (,, ). On he hird panel of Figure 5, he PACF of residuals indicae he The residual Diagnosic for ARIMA (,, ) is omied because when esimaed, he coefficiens becomes greaer han one suggesing ha he process becomes explosive. 33

34 significance of he h lag and h only for ARIMA (,, ). On he las panel of Figure 6, he Lung Box es indicaes ha he residuals of ARIMA (,, ) have no auocorrelaions as suppored by a p-value lower han.5 for he h lag. On he las panel of Figure 5, he Lung Box es indicaes ha he residuals of ARIMA (,, ) have no auocorrelaions as suppored by a p-value ha is almos zero for consecuive lags. The findings so far illusrae ha each model has is srenghs and weaknesses in comparison o one anoher. ARIMA (,, ) has is srengh in he PACF and p-values for he Lung Box es bu is weakness in he ACF of auocorrelaions. On he oher hand, ARIMA (,, ) has is srengh in he ACF of residuals bu is weakness in he PACF. Thus, he findings of residual analysis fail o disinguish one model from he oher and furher analysis is required for model selecion. 3.5 Forecass A useful mehod of comparing differen models is o forecas he acual daa from he in-sample series. The comparison is finalized in hree seps. Firs, an in-sample daa is generaed from he original series by excluding he las few observaions. Second, he ou-of sample observaions are esimaed by he original models proposed. Third, a model ha has he mos accurae predicion is chosen Forecas for ARIMA (,, ) and ARIMA (,, ) A comparison of ARIMA (,, ) and ARIMA (,, ) is also conduced in hree seps. Firs, he las observaions, which correspond o he las 5 quarers, are 34

35 excluded from he series and a new se of observaions is generaed ending in he s quarer of 999. Second, he quarers ha are beween he firs of 999 and he las of 3 are forecased. Third, he model ha has he mos accurae predicion is chosen. Figure 7 represens a predicion of ARIMA (,, ). Even hough he acual daa appears o lie wihin he disribuion confined by he sandard deviaion, he model fails o capure he general dynamics of he series. I seems ha he rend in he acual daa is larger han he forecased rend. Thus, he model fails o forecas he quarers accuraely. Figure 8 represens a predicion of ARIMA (,, ). The prediced values are around he prediced rend indicaing ha here are no significan deviaions apar from he daa. Thus, he model succeeds in forecasing quarers accuraely and is chosen over ARIMA (,, ) Forecas for rend-saionary and sochasic rend models In his analysis, he model proposed for he U.S GDP series is ARIMA (,, ), in addiion o he ARIMA (,, ) model proposed by Nelson and Murray (997) and he random walk model proposed by Nelson and Plosser (98). Boh of hese models have a uni roo in heir represenaion and require firs differeniaion o exhibi saionary dynamics. However, here are also oher models proposed by Perron (989) and Zivo (99) for U.S GNP which conain a deerminisic rend in heir represenaion and require de-rending o exhibi saionary dynamics. To furher explore which model The dynamics of gross domesic naional produc series should be almos idenical o ha of gross naional produc because gross naional produc differs from he gross domesic produc in separaing naion income from foreign income. 35

36 correcly accommodaes he dynamics of he paricular series, a mehod of forecas as described earlier may be implemened 3. Figure 9 represens a Perron (989) ype of model which conains a deerminisic rend wih an auoregressive componen of order. Even hough he model seems o capure he rend in he daa, here are some deviaions visible apar from he acual daa. Thus, he model fails o forecas he GNP series accuraely. Figure represens a Nelson and Plosser (98) ype of a random walk model. Even hough he acual daa appears o lie wihin he disribuion confined by he sandard deviaion, he model fails o capure he general dynamics of he series. I appears ha he model esimaes he same value for each quarer. Thus, he model fails o forecas he quarers precisely. The model ha has he only precise esimaes is ARIMA (,, ). Thus, he resuls of he findings of he analysis are correc. The U.S GDP series has a sochasic rend in is represenaion and is bes described ARIMA (,, ). 3 A mehod of forecas is no implemened for ARIMA (,, ) because when esimaed, he coefficiens becomes greaer han one suggesing ha he process becomes explosive. 36

37 4. Conclusions The purpose of his analysis is o specify an ARIMA (p, d, q) model o examine he impac of shor-run variaions on he long-run rend for he U.S GDP series. The analysis is divided ino wo pars. In he firs par, ime series conceps are briefly inroduced o sudy an ARIMA (p, d, q) model. In he second par, a lieraure review on U.S GDP series is provided ogeher wih he empirical findings. The models ha are proposed in he lieraure 4 are compared agains he ARIMA (,, ) model by using a forecas mehod which involves creaing an insample daa by excluding he las few observaions and forecasing he ou-of-sample daa. As a resul, i is found ha amongs all he models, ARIMA (,, ) yields he bes forecas accuracy and hence he dynamics of he U.S GDP series is bes described by an ARIMA (,, ) model. This specific model poins ou ha in every period he growh rae is affeced by he previous even and he las wo shocks, and more imporanly i poins ou ha he U.S economy undergoes shor-run flucuaions ha have permanen effecs on he long-run rend of he economy. There can be hree limiaions of he findings of his analysis. One limiaion may be due o he breviy of he sample making i difficul o fully idenify he long-run dynamics of he series. Usually an infinie sample is needed o characerize he long-run dynamics of he series (Romer, ). Thus, an aemp o claim ha he long-run rend is sochasic under a sample size of 8 may be oo asserive. The second limiaion is ha here may be a long memory process in he U.S GDP as indicaed by he ACF in he op panel of Figure. Thus, a parially inegraed process may have been even more 4 Excep for ARIMA (,, ), AR () coefficien is greaer han one. 37

38 appropriae o capure he dynamics of he underlying GDP series. The hird limiaion is ha he sudied models here are linear. As Poer (995) suggesed, a nonlinear model ouperform he sandard linear models for he U.S GNP. Furher research could be conduced by using longer ime spans; he dynamics of he long memory could be furher examined and also a non-linear model could be designed. Overall, hese possible models can be compared by he forecas echnique described and he one ha has he bes predicive can be sudied. 38

39 5. References BLACK, FISHER.(98). General Equilibrium and Business Cycles. Naional Bureau of Economic Research Working Paper No. 95. BOX, G.E.P. and PIERCE, D. (97). Disribuion of Residual Auocorrelaions in Auoregressive-inegraed Moving Average Time Series Models. Journal of he American Saisical Associaion, 65, BOX, G.E.P., JENKINS, G.M., and REINSEL, G.C. (994). Time Series Analysis: Forecasing and conrol, 3 rd ediion, Prenice Hall: Englewood Cliffs, New Jersey. CRYER, J.D. (985). Time Series Analysis. Duxbury Press, Boson. DICKEY, D.A. and FULLER, W.A. (979). Disribuion of he Esimaes for Auoregressive Time Series wih a Uni Roo. Journal of he American Saisical Associaion, HAMILTON, J.D. (994). Time Series Analysis. Jersey. Princeon Universiy Press, New KEYNES, JOHN MAYNARD (936). The General Theory of Employmen, Ineres Rae, and Money. London:Macmillian. KYDLAND, FINN E., and PRESCOTT, EDWARD C. (98). Time o Build and Aggregae Flucuaions. Economerica 5, LEYBOURNE S.J. and MCCABE, B.P.M. (994). A Consisen Tes for a Uni Roo. Journal of Business and Economics Saisics,, LJUNG, G., BOX and BOX, G.E.P. (978). On a Measure of Lack of Fi in Time Series Models. Biomerica, 66, LONG, JOHN B., and PLOSSER, CHARLES I. (983). Real Business Cycles. Journal of Poliical Economy,9, PERRON, PIERRE (989). The grea crash, he oil shock, and uni roo hypohesis. Economerica, 57, PHILLIPS, P.C.B. (987). Time Series Regression wih a Uni Roo. Economerica, 55, 7-3. PHILLIPS, P.S.B. and PERRON (988). Tesing for Uni Roos in Time Series Regression. Biomerica, 75,

40 POTTER, SIMON M. (995). A Nonlinear Approach o US GNP. Journal of Applied Economerics,, 9-5. PRESCOTT, EDWARD C. (986). Theory Ahead of Business-Cycle Measuremen. Carneige-Rocheser Conference Series on Public Policy 5, -4. ROMER, DAVID (). Advanced Macroeconomics, Second ediion McGraw- Hill,New York. SAID, S.E. and D.DICKEY (984). Tesing for Uni Roos in Auoregressive Moving- Average Models wih Unknown Order, Biomerica, 7, SCHWERT, W. (989). Tes for Uni Roos: A Mone Carlo invesigaion, Journal of Business and Economic Saisics, 7, NELSON, CHARLES R. and PLOSSER CHARLES I. (98). Trends and Random walks in macroeconomic ime series: Some Evidence and Implicaions, Journal of Moneary Economics,, NELSON, CHARLES R. and MURRAY, CHRISTIAN J. (997). The Uncerain rend Trend in U.S GDP, Universiy of Washingon Working Paper. TSAY, R.S. (). Analysis of Financial Time Series, John Willey & Sons, New York. ZIVOT, ERIC and DONALD W.K. ANDREWS (99). Furher Evidence on he grea crash, he oil price shock, and he uni roo hypohesis, Journal of Business and Economics Saisics,, 5-7. ZIVOT, E. and WANG, JIAHUI (). Modeling Financial Time Series wih S-PLUS. Springer-Verlag. 4

41 QUARTERLY AND SEASONALLY ADJUSTED U.S.GDP (log scale) GDP(log scale) Year FIGURE : Quarerly and seasonally adused Gross Domesic Produc in log scale. The sample period is from January, 947 o Ocober, 3. x-axis represens he years of he sample and y-axis represens he sample in log scale. The presence of rend indicaes non-saionariy. Source: U.S. Bureau of Economic Analysis, Naional Income and Produc Accouns. 4

42 ACF OF U.S. GDP (log scale) ACF Lag PARTIAL ACF OF U.S. GDP (log scale) Parial ACF Lag FIGURE : Top: Auocorrelaion funcion of quarerly and seasonally adused GDP series. x-axis corresponds o lags and y-axis corresponds o correlaion coefficiens of he series. If he series is idenically and independenly disribued, he correlaion coefficiens should be wihin confidence inerval bands. The confidence inerval bands are represened by he doed line. The figure indicaes he significance of he auocorrelaions in he series. Boom: Parial auocorrelaion funcion of quarerly and seasonally adused U.S. GDP series. x-axis corresponds o lags and y- axis corresponds o correlaion coefficiens of he series. If he lag is insignifican, he correlaion coefficien lies wihin confidence inerval bands. The confidence inerval bands are represened by he doed line. The figure indicaes he s lag is saisically significanly. 4

43 FIRST DIFFERENCE OF QUARTERLY AND SEASONALY ADJUSTED U.S.GDP(log scale) Firs Difference Year FIGURE 3: The firs difference of quarerly and seasonally adused U.S. GDP in log scale. x-axis represens he years of he sample and y-axis represens he firs difference of he series. The sample period is from January, 947 o Ocober, 3. The mean revering dynamics is an indicaion of saionariy. 43

44 ACF OF THE FIRST DIFFERENCE OF U.S. GDP (log scale) ACF Lag PARTIAL ACF OF THE FIRST DIFFERENCE OF U.S. GDP (log scale) Parial ACF Lag FIGURE 4: Top: Auocorrelaion funcion of he firs difference of quarerly and seasonally adused U.S. GDP series. x-axis corresponds o lags and y-axis corresponds o correlaion coefficiens of he series. The presence of damping and cosine waves may be an indicaive of AR() process. Boom: Parial auocorrelaion funcion of quarerly and seasonally adused U.S. GDP series. x-axis corresponds o lags and y-axis corresponds o correlaion coefficiens of he series. If he lag is insignifican, he correlaion coefficien lies wihin confidence inerval bands. The confidence inerval bands are represened by he doed line. The figure indicaes he s and he h lags are saisically significan. MA() process is suspeced due o he dualiy beween AR and MA processes. 44