2 Univariate Stationary Processes

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1 Univariae Saionary Processes As menioned in he inroducion, he publicaion of he exbook by GEORGE E.P. BOX and GWILYM M. JENKINS in 97 opened a new road o he analysis of economic ime series. This chaper presens he Box-Jenkins Approach, is differen models and heir basic properies in a raher elemenary and heurisic way. These models have become an indispensable ool for shor-run forecass. We firs presen he mos imporan approaches for saisical modelling of ime series. These are auoregressive (AR) processes (Secion.) and moving average (MA) processes (Secion.), as well as a combinaion of boh ypes, he so-called ARMA processes (Secion.3). In Secion.4 we show how his class of models can be used for predicing he fuure developmen of a ime series in an opimal way. Finally, we conclude his chaper wih some remarks on he relaion beween he univariae ime series models described in his chaper and he simulaneous equaions sysems of radiional economerics (Secion.5).. Auoregressive Processes We know auoregressive processes from radiional economerics: Already in 949, DONALD COCHRANE and GUY H. ORCUTT used he firs order auoregressive process for modelling he residuals of a regression equaion. We will sar wih his process, hen rea he second order auoregressive process and finally show some properies of auoregressive processes of an arbirary bu finie order... Firs Order Auoregressive Processes Derivaion of Wold s Represenaion A firs order auoregressive process, an AR() process, can be wrien as an inhomogeneous sochasic firs order difference equaion, (.) x = + x - + u, G. Kirchgässner e al., Inroducion o Modern Time Series Analysis, Springer Texs in Business 7 and Economics, DOI.7/ _, Springer-Verlag Berlin Heidelberg 3

2 8 Univariae Saionary Processes where he inhomogeneous par + u consiss of a consan erm and a pure random process u. Le us assume ha for = he iniial value x is given. By successive subsiuion in (.) we ge x = + x + u x = + x + u = + ( + x + u ) + u = + + x + u + u x 3 = + x + u 3 or x 3 = x + u + u + x = ( ) + x u u + - u + + u + u, x = x + j u j. j For = +, we ge (.) x = + x j u j. j The developmen and hus he properies of his process are mainly deermined by he assumpions on he iniial condiion x. The case of a fixed (deerminisic) iniial condiion is given if x is assumed o be a fixed (real) number, for example for =, i.e. no random variable. Then we can wrie: x = x + j u j. j This process consiss of ime dependen deerminisic and sochasic pars. Thus, i can never be weakly saionary, since firs and second order mo-

3 . Auoregressive Processes 9 mens are ime dependen. I is, however, asympoically saionary because he ime dependence vanishes for -. We can imagine he case of sochasic iniial condiions as (.) being generaed along he whole ime axis, i.e. - < <. If we observe he process only for posiive values of, he iniial value x is a random variable which is generaed by his process. Formally, he process wih sochasic iniial condiions resuls from (.) if he soluion of he homogeneous difference equaion has disappeared. This is only possible if <. Therefore, in he following, we resric o he inerval < <. If lim x is bounded, (.) for - converges o (.3) x = j u j. j The ime dependence has disappeared. According o Secion.5, he AR() process (.) has he Wold represenaion (.3) wih j = j and <. This resuls in he convergence of j j = =. j j Thus, assuming sochasic iniial condiions, he process (.) is weakly saionary. The Lag Operaor Equaion (.3) can also be derived from relaion (.) by using he lag operaor defined in Secion.3: (.') ( L)x = + u. If we solve for x we ge (.4) x = + L L u. The expression /( L) can formally be expanded o a geomeric series, L = + L + L + 3 L 3 +. Thus, we ge x = ( + L + L + ) + ( + L + L + )u = ( ) + u + u - + u - +,

4 3 Univariae Saionary Processes and because of < x = j u j. j The firs erm could have been derived immediaely if we subsiued he value for L in he firs erm of (.4). (See also relaion (.8) on p. ). Calculaion of Momens Due o represenaion (.3), he firs and second order momens can be calculaed. As E[u ] = holds for all, we ge for he mean E[x ] = E j j u j E[x ] = = j E u j j = i.e. he mean is consan. I is differen from zero if and only if. Because of >, he sign of he mean is deermined by he sign of. For he variance we ge V[x ] = Ex = E = E[(u + u - + u ) ] j j u j = E[ u + u + 4 u + + u u - + u u - + ] = ( ), because E[u u s ] = for s and E[u u s ] = for = s. Applying he summaion formula for he geomeric series, and because of <, we ge he consan variance V[x ] =. The covariances can be calculaed as follows: Cov [x,x - ] = Ex x

5 . Auoregressive Processes 3 = E[(u + u u ) (u - + u -- + u )] = E[(u + u u -+ + (u - + u -- + u )) (u - + u -- + u )] Thus, we ge = E[(u - + u -- + u ) ]. Cov [x,x - ] = V[x - ] =. The auocovariances are only a funcion of he ime difference and no of ime, and we can wrie: (.5) () =, =,,,.... Therefore, he AR() process wih < and sochasic iniial condiions is weakly saionary. An Alernaive Mehod for he Calculaion of Momens Under he condiion of weak saionariy, i.e. for < and sochasic iniial condiions, he mean of x is consan. If we apply he expecaion operaor on equaion (.), we ge: E[x ] = E[ + x - + u ] = + E[x - ] + E[u ]. Because of E[u ] = and E[x ] = E[x - ] = for all we can wrie E[x ] = =. If we consider he deviaions from he mean, x = x and subsiue his in relaion (.), we ge: x + = + x + + u. From his i follows ha

6 3 Univariae Saionary Processes x = + ( ) + x + u = + ( ) + x + u (.6) x = x + u. This is he AR() process belonging o (.) wih E[ x ] =. If we muliply equaion (.6) wih x for and ake expecaions we can wrie: (.7) E[ x x ] = E[ x x ] + E[ x u ]. Because of (.3) we ge x = u - + u -- + u This leads o (.8) E[ x u ] = for. for Because of he saionariy assumpion and because of he (even) symmery of he auocovariances, () = (-), equaion (.7) resuls in or or = : E[ x ] = E[ xx ] +, () = () +, = : E[ xx ] = E[ ], x () = (). This leads o he variance of he AR() process () =. For (.7) implies () = () () = () = ()

7 . Auoregressive Processes 33 (3) = () = 3 () () = (-) = (). Thus, he covariances can be calculaed from he linear homogeneous firs order difference equaion wih he iniial value () = /( ). The Auocorrelogram () (-) = Because of () = ()/(), he auocorrelaion funcion (he auocorrelogram) of he AR() process is (.9) () =, =,,.... This funcion converges geomerically o zero for, and is infinie sum equals /( ) since <. This convergence is monoone for posiive and oscillaing for negaive values of. Example. For = and {.9,.5, -.9}, Figures. o.3 each presen one realisaion of he corresponding AR() process wih T = 4 observaions. To generae hese series, we used realisaions of normally disribued pure random processes wih mean zero and variance one. We always dropped he firs 6 observaions o eliminae he dependence of he iniial values. The realisaion for =.9, presened in Figure., is relaively smooh. This is o be expeced given he heoreical auocorrelaion funcion because random variables wih a considerable disance beween each oher sill have high posiive correlaions. The developmen of he realisaion in Figure. wih =.5 is much less sysemaic. The geomeric decrease of he heoreical auocorrelaion funcion is raher fas. The fourh order auocorrelaion coefficien is only.65. Conrary o his, he realisaion of he AR() process wih = -.9, presened in Figure.3, follows a well pronounced zigzag course wih, however, alernaing posiive and negaive ampliudes. This is consisen wih he heoreical auocorrelaion funcion indicaing ha all random variables wih even-numbered disance are posiively correlaed and hose wih odd-numbered disance negaively correlaed.

8 34 Univariae Saionary Processes 7.5 x a) Realisaion b) Theoreical auocorrelaion funcion ˆ c) Esimaed auocorrelaion funcion wih confidence inervals Figure.: AR() process wih =.9

9 . Auoregressive Processes 35 x a) Realisaion b) Theoreical auocorrelaion funcion ˆ c) Esimaed auocorrelaion funcion wih confidence inervals Figure.: AR() process wih =.5

10 36 Univariae Saionary Processes x a) Realisaion 5 5 b) Theoreical auocorrelaion funcion ˆ c) Esimaed auocorrelaion funcion wih confidence inervals Figure.3: AR() process wih = -.9

11 . Auoregressive Processes 37 I generally holds ha he closer he parameer is o +, he smooher he realisaions will be. For negaive values of we ge zigzag developmens which are he more pronounced he closer is o -. For = we ge a pure random process. The auocorrelaion funcions esimaed by means of relaion (.) wih he given realisaions are also presened in Figures. o.3. The doed parallel lines show approximae 95 percen confidence inervals for he null hypohesis assuming ha he rue process is a pure random process. In all hree cases, he esimaed funcions reflec quie well he ypical developmen of he heoreical auocorrelaions. Example. In a paper on he effec of economic developmen on he elecoral chances of he German poliical paries during he period of he social-liberal coaliion from 969 o 98, GEBHARD KIRCHGÄSSNER (985) invesigaed (besides oher issues) he ime series properies of he populariy series of he paries consruced by monhly surveys of he Insiue of Demoscopy in Allensbach (Germany). For he period from January 97 o April 98, he populariy series of he Chrisian Democraic Union (CDU), i.e. he share of voers who answered ha hey would voe for his pary (or is Bavarian siser pary, he CSU) if here were a general elecion by he following Sunday, is given in Figure.4. The auocorrelaion and he parial auocorrelaion funcion (which is discussed in Secion..4) are also presened in his figure. While he auocorrelaion funcion goes slowly owards zero, he parial auocorrelaion funcion breaks off afer =. This argues for an AR() process. The model has been esimaed wih Ordinary Leas Squares (OLS), he mehod proposed in Secion..5 for he esimaion of auoregressive models. Thus, we ge: CDU = CDU - + û, (3.43) (7.) R =.683, SE =.586, Q() =.56 (p =.36). The esimaed values are given in parenheses, SE denoes he sandard error of he residuals. The auocorrelogram, which is also given in Figure.4, does no indicae any higher-order process. Moreover, given he high p-value, he Ljung-Box Q saisic wih correlaion coefficiens (i.e. wih degrees of freedom) gives no reason o rejec his model. The mean is calculaed as 8.53 ˆ I shows ha abou 48.5 percen of he voers voed on average for he CDU during his period.

12 38 Univariae Saionary Processes Percen year ˆ ( ) ˆ ( ) a) Populariy of he CDU/CSU, b) Auocorrelaion ( ) and parial ( ) auocorrelaion funcions wih confidence inervals 5 5 c) Esimaed auocorrelaion funcion of he residuals of he esimaed AR()-process wih confidence inervals Figure.4: Populariy of he CDU/CSU, 97 98

13 . Auoregressive Processes 39 Sabiliy Condiions Along wih he sochasic iniial value, he condiion <, he so-called sabiliy condiion, is crucial for he saionariy of he AR() process. We can also derive he sabiliy condiion from he linear homogeneous difference equaion, which is given for he process iself by x x - =, for is auocovariances by () (-) = and for he auocorrelaions by () (-) =. These difference equaions have sable soluions, i.e. lim ( ) =, if and only if heir characerisic equaion (.) = has a soluion (roo) wih an absolue value smaller han one, i.e. if < holds. We ge an equivalen condiion if we do no consider he characerisic equaion bu he lag polynomial of he corresponding difference equaions, (.) L =. This implies ha he soluion has o be larger han one in absolue value. (Sricly speaking, L, which denoes an operaor, has o be subsiued by a variable, which is ofen denoed by z. To keep he noaion simple, we use L in boh meanings.) Example.3 Le us consider he sochasic process (E.) y = x + v. In his equaion, x is a saionary AR() process, x = x - + u, wih < ; v is a pure random process wih mean zero and consan variance v which is uncorrelaed wih he oher pure random process u wih mean zero and consan variance. u We can inerpre he sochasic process y as an addiive decomposiion of wo saionary componens. Then y iself is saionary. In he sense of MILTON FRIEDMAN (957) we can inerpre x as he permanen (sysemaic) and v as he ransiory componen.

14 4 Univariae Saionary Processes Wha does he correlogram of y look like? As boh x and v have zero mean, E[y ] =. Muliplying (E.) wih y - and aking expecaions resuls in E[y - y ] = E[y - x ] + E[y - v ]. Due o y - = x - + v -, we ge E[y - y ] = E[x - x ] + E[v - x ] + E[x - v ] + E[v - v ]. As u and v are uncorrelaed, i holds ha E[v - x ] = E[x - v ] =, and because of he saionariy of he wo processes, we can wrie (E.) y () = x () + v (). For = we ge he variance of y as u y () = x () + v = + v. For >, because of v () = for, we ge from (E.) Thus, we finally ge y () = x () = u. y () = ( ) / v u, =,,..., for he correlogram of y. The overlay of he sysemaic componen by he ransiory componen reduces he auocorrelaion generaed by he sysemaic componen. The larger he variance of he ransiory componen, he sronger is his effec... Second Order Auoregressive Processes Generalising (.), he second order auoregressive process (AR()) can be wrien as (.) x = + x - + x - + u, wih u denoing a pure random process wih variance and. Wih he lag operaor L we ge (.3) ( L L ) x = + u. Wih (L) = L L we can wrie (.4) (L) x = + u.

15 . Auoregressive Processes 4 As for he AR() process, we ge he Wold represenaion from (.4) if we inver (L); i.e. under he assumpion ha - (L) exiss and has he propery (.5) (L) - (L) = we can solve for x in (.4): (.6) x = - (L) + - (L) u. If we use he series expansion wih undeermined coefficiens for - (L) = + L + L +... i has o hold ha = ( L L )( + L + L + 3 L ) because of (.5). This relaion is an ideniy only if he coefficiens of L j, j =,,,..., are equal on boh he righ and he lef hand side. We ge L L L... L L L... L L Comparing he coefficiens of he lag polynomials on he righ- and lefhand side finally leads o L : = L : = =. L : = = +. L 3 3 : 3 = 3 = +. By applying his so-called mehod of undeermined coefficiens, we ge he values j, j =, 3,..., from he linear homogeneous difference equaion j j- j- = wih he iniial condiions = and =. The sabiliy condiion for he AR() process requires ha, for j, he j converge o zero, i.e. ha he characerisic equaion of (.), (.7) =, has only roos wih absolue values smaller han one, or ha all soluions of he lag polynomial in (.3),.

16 4 Univariae Saionary Processes (.8) L L = are larger han one in modulus. Togeher wih sochasic iniial condiions, his guaranees he saionariy of he process. The sabiliy condiions are fulfilled if he following parameer resricions hold joinly for (.7) and (.8): + (- ) + (- ) >, (- ) + (- ) >, (- ) >. As a consan is no changed by he applicaion of he lag operaor, he number can subsiue he lag operaor in he corresponding erms. Thus, due o (.6), he Wold represenaion of he AR() process is given by (.9) x = ju j, =. Under he assumpion of saionariy, he expeced value of he sochasic process can be calculaed direcly from (.) since E[x ] = E[x - ] = E[x - ] =. We ge = + + or (.) E[x ] = = j. As he sabiliy condiions are fulfilled, > holds, i.e. he sign of also deermines he sign of. In order o calculae he second order momens, we can assume wihou loss of generaliy ha =, which is equivalen o =. Muliplying (.) wih x -,, and aking expecaions leads o (.) E[x - x ] = E[x - x - ] + E[x - x - ] + E[x - u ]. Because of represenaion (.9), relaion (.8) holds here as well. This leads o he following equaions (.) : () () () : () () () : () () (),

17 . Auoregressive Processes 43 and, more generally, he following difference equaion holds for he auocovariances (),, (.3) () (-) (-) =. As he sabiliy condiions hold, he auocovariances which can be recursively calculaed wih (.3) are converging o zero for. The relaions (.) resul in (.4) V[x ] = () = for he variance of he AR() process, and in and () = () = ( ) [( ) ], ( ) [( ) ] ( ) [( ) ] for he auocovariances of order one and wo. The auocorrelaions can be calculaed accordingly. If we divide (.3) by he variance () we ge he linear homogeneous second order difference equaion, (.5) () (-) (-) = wih he iniial condiions () = and () = /( ) for he auocorrelaion funcion. Depending on he values of and, AR() processes can generae quie differen developmens, and, herefore, hese processes can show considerably differen characerisics. Example.4 Le us consider he AR() process (E.3) x = +.5 x -.56 x - + u wih a variance of u of. Because he characerisic equaion = has he wo roos =.8 and =.7, (E.3) is saionary, given ha we have sochasic iniial condiions. The expeced value of his process is,

18 44 Univariae Saionary Processes =.5.56 = 6.6. The variance of (E.3) can be calculaed from (.4) as () = 9.3. A realisaion of his process (wih 8 observaions) is given in Figure.5 in which he (esimaed) mean was subraced. Thus, he realisaions flucuae around zero, and he process always ends o go back o he mean. This mean-revering behaviour is a ypical propery of saionary processes. Due o (.5) we ge ().5 (-) +.56 (-) =, =, 3,..., wih () =, () =.96 for he auocorrelaion funcion. The general soluion of his homogeneous difference equaion is () = C (.8) + C (.7), where C and C are wo arbirary consans. Taking ino accoun he wo iniial condiions we ge () =.6 (.8).6 (.7) for he auocorrelaion coefficiens. This developmen is also expressed in Figure.5. The coefficiens are always posiive bu sricly monoonically decreasing. Iniially, he esimaed auocorrelogram using he given realisaion is also monoonically decreasing, bu, conrary o he heoreical developmen, he values begin o flucuae from he enh lag onwards. However, excep for he coefficien for = 6, he esimaes are no significanly differen from zero; hey are all inside he approximae 95 percen confidence inerval indicaed by he doed lines. The characerisic equaions of sable auoregressive processes of second or higher order can resul in conjugae complex roos. In his case, he ime series exhibi dampened oscillaions, which are shocked again and again by he pure random process. The soluion of he homogeneous par of (.) for conjugae complex roos can be represened by x = d (C cos (f ) + C sin (f )) wih C and C again being arbirary consans ha can be deermined by using he iniial condiions. The dampening facor d = corresponds o he modulus of he wo roos, and f = arccos

19 . Auoregressive Processes 45 x a) Realisaion b) Theoreical auocorrelaion funcion ˆ c) Esimaed auocorrelaion funcion wih confidence inervals Figure.5: AR() process wih =.5, = -.56

20 46 Univariae Saionary Processes x a) Realisaion b) Theoreical auocorrelaion funcion ˆ c) Esimaed auocorrelaion funcion wih confidence inervals Figure.6: AR() process wih =.4 and = -.85

21 . Auoregressive Processes 47 is he frequency of he oscillaion. The period of he cycles is P = /f. Processes wih conjugae complex roos are well-suied o describe business cycle flucuaions. Example.5 Consider he AR() process (E.4) x =.4 x -.85 x - + u, wih a variance of u of. The characerisic equaion = has he wo soluions =.7 +.6i and =.7-.6i. ( i sands for he imaginary uni: i = -.) The modulus (dampening facor) is d =.9. Thus, (E.4) wih sochasic iniial condiions and a mean of zero is saionary. According o (.4) he variance is given by () = A realisaion of his process wih 8 observaions is given in Figure.6. Is developmen is cyclical around is zero mean. For he auocorrelaion funcion we ge because of (.5). The general soluion is ().4 (-) +.85 (-) =, =, 3,..., () =, () =.76, () =.9 (C cos (.79 ) + C sin (.79 )). Taking ino accoun he wo iniial condiions, we ge for he auocorrelaion coefficiens () =.9 (cos (.79 ) +. sin (.79 )), wih a frequency of f =.79. In case of quarerly daa, his corresponds o a period lengh of abou 9 quarers. Boh he heoreical and he esimaed auocorrelaions in Figure.6 show his kind of dampened periodical behaviour. Example.6 Figure.7 shows he developmen of he hree monh money marke rae in Frankfur (GSR) from he firs quarer of 97 o he las quarer of 998 as well as he auocorrelaion and he parial auocorrelaion funcions explained in Secion..4. Whereas he auocorrelaion funcion ends only slowly owards zero, he parial auocorrelaion funcion breaks off afer wo lags. As will be shown below, his indicaes an AR() process. For he period from 97 o 998, esimaion wih OLS resuls in he following:

22 48 Univariae Saionary Processes Percen year ˆ ( ) ˆ ( ) a) Three monh money marke rae in Frankfur b) Esimaed auocorrelaion ( ) and parial auocorrelaion ( ) funcions wih confidence inervals 5 5 c) Esimaed auocorrelaion funcion of he residuals of he esimaed AR()-process wih confidence inervals Figure.7: Three monh money marke rae in Frankfur,

23 . Auoregressive Processes 49 GSR = GSR GSR - + û,. (.8) (7.5) (-6.6) R =.9, SE =.8, Q(6) = (p =.37), wih values being again given in parenheses. On he. percen level, boh esimaed coefficiens of he lagged ineres raes are significanly differen from zero. The auocorrelogram of he esimaed residuals (given in Figure.7c) as well as he Ljung-Box Q saisic which is calculaed wih 8 correlaion coefficiens (and 6 degrees of freedom) does no indicae any higher order process. The wo roos of he process are.7 ±.6i, i.e. hey indicae dampened cycles. The modulus (dampening facor) is d =.76; he frequency f =.79 corresponds o a period of 79.7 quarers and herefore of nearly years. Correspondingly, his oscillaion canno be deeced in he esimaed auocorrelogram presened in Figure.7b...3 Higher Order Auoregressive Processes An AR(p) process can be described by he following sochasic difference equaion, (.6) x = + x - + x p x -p + u, wih p, where u is again a pure random process wih zero mean and variance. Using he lag operaor we can also wrie: (.6') ( L L... p L p ) x = + u. If we assume sochasic iniial condiions, he AR(p) process in (.6) is saionary if he sabiliy condiions are saisfied, i.e. if he characerisic equaion (.7) p p- p-... p = only has roos wih absolue values smaller han one, or if he soluions of he lag polynomial (.8) L L... p L p = only have roos wih absolue values larger han one. If he sabiliy condiions are saisfied, we ge he Wold represenaion of he AR(p) process by he series expansion of he inverse lag polynomial, as L... L p p = + L + L +...

24 5 Univariae Saionary Processes (.9) x = ju j.... p Generalising he approach ha was used o calculae he coefficiens of he AR() process, he series expansion can again be calculaed by he mehod of undeermined coefficiens. From (.9) we ge he consan (uncondiional) expecaion as E[x ] = =.... p Again, similarly o he AR() and AR() cases, a necessary condiion for sabiliy is... p >. Wihou loss of generaliy we can se =, i.e. =, in order o calculae he auocovariances. Because of () = E[x - x ], we ge according o (.6) (.3) () = E[x - ( x - + x p x -p + u )]. For =,,..., p, i holds ha j (.3) () () () (p) p () () () (p ) p (p) (p ) (p ) () p because of he symmery of he auocovariances and because of E[x - u ] = for = and zero for >. This is a linear inhomogeneous equaion sysem for given i and o derive he p + unknowns (), (),..., (p). For > p we ge he linear homogeneous difference equaion o calculae he auocovariances of order > p: (.3) () (-)... p (-p) =. If we divide (.3) by (), we ge he corresponding difference equaion o calculae he auocorrelaions: (.33) () (-)... p (-p) =. The iniial condiions (), (),..., (p) can be derived from he so-called Yule-Walker equaions. We ge hose if we successively inser =,,..., p in (.33), or, if he las p equaions in (.3) are divided by (),

25 . Auoregressive Processes 5 () = + () + 3 () p (p-) () = () () p (p-) (.34) (p) = (p-) + (p-) + 3 (p-3) p If we define ' = ((), (),..., (p)), ' = (,,..., p ) and () () (p) () () (p ) R pp (p ) (p ) (p 3) we can wrie he Yule-Walker equaions (.34) in marix form, (.35) = R. If he firs p auocorrelaion coefficiens are given, he coefficiens of he AR(p) process can be calculaed according o (.35) as (.36) = R -. Equaions (.35) and (.36) show ha here is a one-o-one mapping beween he p coefficiens and he firs p auocorrelaion coefficiens of an AR(p) process. If here is a generaing pure random process, i is sufficien o know eiher or o idenify he AR(p) process. Thus, here are wo possibiliies o describe he srucure of an auoregressive process of order p: he parameric represenaion ha uses he parameers,,..., p, and he non-parameric represenaion wih he firs p auocorrelaion coefficiens (), (),..., (p). Boh represenaions conain exacly he same informaion. Which represenaion is used depends on he specific siuaion. We usually use he parameric represenaion o describe finie order auoregressive processes (wih known order). Example.7 Le he fourh order auoregressive process x = 4 x -4 + u, < 4 <, be given, where u is again whie noise wih zero mean and variance. Applying (.3) we ge: () = 4 (4) +, () = 4 (3), () = 4 (),

26 5 Univariae Saionary Processes From hese relaions we ge (3) = 4 (), (4) = 4 (). () =, 4 () = () = (3) =, (4) = 4. As can easily be seen, only he auocovariances wih lag = 4j, j =,,... are differen from zero, while all oher auocovariances are zero. Thus, for > we ge he auocorrelaion funcion () = j 4 4 for 4j, j,,.... elsewhere. Only every fourh auocorrelaion coefficien is differen from zero; he sequence of hese auocorrelaion coefficiens decreases monoonically like a geomeric series. Employing such a model for quarerly daa, his AR(4) process capures he correlaion beween random variables ha are disan from each oher by a mulipliciy of four periods, i.e. he srucure of he correlaions of all variables which belong o he i-h quarer of a year, i =,, 3, 4, follows an AR() process while he correlaions beween variables ha belong o differen quarers are always zero. Such an AR(4) process provides a simple possibiliy of modelling seasonal effecs which ypically influence he same quarers of differen years. For empirical applicaions, i is advisable o firs eliminae he deerminisic componen of a seasonal variaion by employing seasonal dummies and hen o model he remaining seasonal effecs by such an AR(4) process...4 The Parial Auocorrelaion Funcion Due o he sabiliy condiions, auocorrelaion funcions of saionary finie order auoregressive processes are always sequences ha converge o zero bu do no break off. This makes i difficul o disinguish beween processes of differen orders when using he auocorrelaion funcion. To cope wih his problem, we inroduce a new concep, he parial auocorrelaion funcion. The parial correlaion beween wo random variables is he correlaion ha remains if he possible impac of all oher random variables has been eliminaed. To define he parial auocorrelaion coefficien, we use he new noaion,

27 . Auoregressive Processes 53 x = k x - + k x kk x -k + u, where ki is he coefficien of he variable wih lag i if he process has order k. (According o he former noaion i holds ha i = ki i =,,,k.) The coefficiens kk are he parial auocorrelaion coefficiens (of order k), k =,,. The parial auocorrelaion measures he correlaion beween x and x -k which remains when he influences of x -, x -,..., x -k+ on x and x -k have been eliminaed. Due o he Yule-Walker equaions (.35), we can derive he parial auocorrelaion coefficiens kk from he auocorrelaion coefficiens if we calculae he coefficiens kk, which belong o x -k, for k =,,... from he corresponding linear equaion sysems () () (k) k () () () (k ) k (), k =,,.... (k ) (k ) (k 3) kk (k) Wih Cramer s rule we ge (.37) kk () () () () (k ) (k ) (k) () (k), k =,,.... () (k ) (k ) (k ) Thus, if he daa generaing process (DGP) is an AR() process, we ge for he parial auocorrelaion funcion: = () = () () () () () = () () () =,

28 54 Univariae Saionary Processes because of () = (). Generally, he parial auocorrelaion coefficiens kk = for k > in an AR() process. If he DGP is an AR() process, we ge = (), = () () (), kk = for k >. The same is rue for an AR(p) process: all parial auocorrelaion coefficiens of order higher han p are zero. Thus, for finie order auoregressive processes, he parial auocorrelaion funcion provides he possibiliy of idenifying he order of he process by he order of he las non-zero parial auocorrelaion coefficien. We can esimae he parial auocorrelaion coefficiens consisenly by subsiuing he heoreical values in (.37) by heir consisen esimaes (.). For he parial auocorrelaion coefficiens which have a heoreical value of zero, i.e. he order of which is larger han he order of he process, we ge asympoically ha hey are normally disribued wih E[ ˆ kk ] = and V[ ˆ kk ] = /T for k > p. Example.8 The AR() process of Example. has he following heoreical parial auocorrelaion funcion: = () = and zero elsewhere. In his example, akes on he values.9,.5 and -.9. The esimaes of he parial auocorrelaion funcions for he realisaions in Figures. and.3 are presened in Figure.8. I is obvious for boh processes ha hese are AR() processes. The esimaed value for he process wih =.9 is ˆ =.9, while all oher parial auocorrelaion coefficiens are no significanly differen from zero. We ge ˆ = -.9 for he process wih = -.9, while all esimaed higher order parial auocorrelaion coefficiens do no deviae significanly from zero. The AR() process of Example.4 has he following heoreical parial auocorrelaion funcion: =.96, = -.56 and zero elsewhere. The realisaion of his process, which is given in Figure.5, leads o he empirical parial auocorrelaion funcion in Figure.8. I corresponds quie closely o he heoreical funcion; we ge ˆ =.95 and ˆ = -.6 and all higher order parial auocorrelaion coefficiens are no significanly differen from zero. The same holds for he AR() process wih he heoreical non-zero parial auocorrelaions =.76 and = -.85 given in Example.5. We ge he esimaes ˆ =.76 and ˆ = -.78, whereas all higher order parial correlaion coefficiens are no significanly differen from zero.

29 . Auoregressive Processes 55 kk kk kk kk k 5 5 AR() process wih =.9 k 5 5 AR() process wih = -.9 k 5 5 AR() process wih =.5, = -.56 k 5 5 AR() process wih =.4, = -.85 Figure.8: Esimaed parial auocorrelaion funcions

30 56 Univariae Saionary Processes..5 Esimaing Auoregressive Processes Under he assumpion of a known order p we have differen possibiliies o esimae he parameers: (i) If we know he disribuion of he whie noise process ha generaes he AR(p) process, he parameers can be esimaed by using maximum likelihood (ML) mehods. (ii) The parameers can also be esimaed wih he mehod of momens by using he Yule-Walker equaions. (iii) A furher possibiliy is o rea (.6) x = + x - + x p x -p + u, as a regression equaion and apply he ordinary leas squares (OLS) mehod for esimaion. OLS provides consisen esimaes. Moreover, if (.6) fulfils he sabiliy condiions, T( ˆ ) as well as T( ˆ ), i =,,..., p, are asympoically normally disribued. i i If he order of he AR process is unknown, i can be esimaed wih he help of informaion crieria. For his purpose, AR processes wih successively increasing orders p =,,..., p max are esimaed. Finally, he order p* is chosen which minimises he respecive crierion. The following crieria are ofen used: (i) The final predicion error which goes back o HIROTUGU AKAIKE (969) FPE = T m T m T T (p) (u ˆ ). (ii) Closely relaed o his is he Akaike informaion crierion (HIROTUGU AKAIKE (974)) AIC = T (p) ˆ T T ln (u ) m. (iii) Alernaives are he Bayesian crierion of GIDEON SCHWARZ (978) SC = T (p) lnt ˆ T T ln (u ) m (iv) as well as he crierion developed by EDWARD J. HANNAN and BARRY G. QUINN (979)

31 . Auoregressive Processes 57 HQ = T (p) ln(lnt) ˆ T T ln (u ) m. (p) û are he esimaed residuals of he AR(p) process, while m is he number of esimaed parameers. If he consan erm is esimaed, oo, m = p + for an AR(p) process. These crieria are always based on he same principle: They consis of one par, he sum of squared residuals (or is logarihm), which decreases when he number of esimaed parameers increases, and of a penaly erm, which increases when he number of esimaed parameers increases. Whereas he firs wo crieria overesimae he rue (finie) order asympoically, he wo oher crieria esimae he rue order of he process consisenly. For T 6, he penaly erm of SC is larger han he one of HQ which iself is larger han he one of AIC. This leads o he following ordering of he esimaed AR orders: SC order HQ order AIC order. Please noe ha choosing such an order does no always imply ha we have whie noise residuals. This has o be checked independenly. Many compuer programmes like, for example, EViews, do no exacly repor he crieria given in (ii) hrough (iv). Relying on he log-likelihood funcion insead of on he sum of squared residuals direcly, hey add + ln().8379, which does, of course, neiher affec he order nor which value of p minimises he informaion crieria. Example.9 As in Example.6, we ake a look a he developmen of he hree monh money marke ineres rae in Frankfur am Main. If, for his series, we esimae AR processes up o he order p = 4, we ge he following resuls (for T = 6): p = : AIC = , HQ = 4.843, SC = 4.857; p = : AIC =.78, HQ =.7373, SC =.7655; p = : AIC =.4457, HQ =.4746, SC =.569; p = 3: AIC =.469, HQ =.4995, SC =.5559; p = 4: AIC =.4778, HQ =.56, SC = Wih all hree crieria we ge he minimum for p =. Thus, he opimal number of lags is p* =, as used in Example.6.

32 58 Univariae Saionary Processes. Moving Average Processes Moving average processes of an infinie order have already occurred when we presened he Wold decomposiion heorem. They are, above all, of heoreical imporance as, in pracice, only a finie number of (differen) parameers can be esimaed. In he following, we consider finie order moving average processes. We sar wih he firs order moving average process and hen discuss general properies of finie order moving average processes... Firs Order Moving Average Processes The firs order moving average process (MA()) is given by he following equaion: (.38) x = + u u -, or (.38') x = (l L)u, wih u again being a pure random process. The Wold represenaion of an MA() process (as of any finie order MA process) has a finie number of erms. In his special case, he Wold coefficiens are =, = - and j = for j. Thus, is finie for all finie values of, i.e. an MA() j j process is always saionary. Taking expecaions of (.38) leads o E[x ] = + E[u ] E[u - ] =. The variance can also be calculaed direcly, V[x ] = E[(x ) ] = E[(u u - ) ] = E[( u u u - + u )] = ( + ) = (). Therefore, he variance is consan a any poin of ime. For he covariances of he process we ge E[(x )(x + )] = E[(u u - )(u + u + - )] = E[(u u + u u + u - u + + u - u + - )].

33 . Moving Average Processes 59 The covariances are differen from zero only for = ±, i.e. for adjoining random variables. In his case () = -. Thus, for an MA() process, all auocovariances and herefore all auocorrelaions wih an order higher han one disappear, i.e. () = () = for. The correlogram of an MA() process is () =, () =, () = for. If we consider () as a funcion of, () = f(), i holds ha f() = and f() = -f(-), i.e. ha f() is poin symmeric o he origin, and ha f().5. f() has is maximum a = - and is minimum a =. Thus, an MA() process canno have a firs order auocorrelaion above.5 or below -.5. If we know he auocorrelaion coefficien () =, for example, by esimaion, we can derive (esimae) he corresponding parameer by using he equaion for he firs order auocorrelaion coefficien, ( + ) + =. The quadraic equaion can also be wrien as (.39) + and i has he wo soluions + =,, = 4 Thus, he parameers of he MA() process can be esimaed non-linearly wih he mehod of momens: he heoreical momens are subsiued by heir consisen esimaes and he resuling equaion is used for esimaing he parameers consisenly. Because of.5, he quadraic equaion always resuls in real roos. They also have he propery ha =. This gives us he possibiliy o model he same auocorrelaion srucure wih wo differen parameers, where one is he inverse of he oher. In order o ge a unique parameerisaion, we require a furher propery of he MA() process. We ask under which condiions he MA() process (.38) can have an auoregressive represenaion. By using he lag operaor represenaion (.38') we ge.

34 6 Univariae Saionary Processes u = + L x. An expansion of he series /( L) is only possible for < and resuls in he following AR() process u = + x + x - + x or x + x - + x = + u. This represenaion requires he condiion of inveribiliy ( < ). In his case, we ge a unique parameerisaion of he MA() process. Applying he lag polynomial in (.38'), we can formulae he inveribiliy condiion in he following way: An MA() process is inverible if and only if he roo of he lag polynomial L = is larger han one in modulus. Example. The following MA() process is given: (E.5) x = -, ~ N(, ), wih = -.5. For his process we ge E[x ] =, V[x ] = ( +.5 ) 4 = 5,.5 () = =.4,.5 () = for. Solving he corresponding quadraic equaion (.39) for his value of () leads o he wo roos = -. and = -.5. If we now consider he process (E.5a) y = + -, ~ N(, ), we obain he following resuls: E[y ] =, V[y ] = ( +. ) = 5,

35 . Moving Average Processes 6. () = =.4,. () = for, i.e. he variances and he auocorrelogram of he wo processes (E.5) and (E.5a) are idenical. The only difference beween hem is ha (E.5) is inverible, because he inveribiliy condiion < holds, whereas (E.5a) is no inverible. Thus, given he srucure of he correlaions, we can choose he one of he wo processes ha fulfils he inveribiliy condiion wihou imposing any resricions on he srucure of he process. Wih equaion (.37), he parial auocorrelaion funcion of he MA() process can be calculaed in he following way: = (), = () () () () = () <, () 33 = () () () () () () () () = 3 () () for, 44 = () () () () () () () () () () () () = 4 () ( () ) () <, ec.

36 6 Univariae Saionary Processes If is posiive, () is negaive and vice versa. This leads o he wo possible paerns of parial auocorrelaion funcions, exemplified by = ±.8: =.8, ii {-.49,-.3,-., -.7,... }, = -.8, ii {.49,-.3,., -.7,... }. Thus, conrary o he AR() process, he auocorrelaion funcion of he MA() process breaks off, while he parial auocorrelaion funcion does no. These properies hold generally, since inverible finie order MA processes are equivalen o infinie order AR processes... MA() and Temporal Aggregaion The ime series which are discussed in his book are measured in discree ime, wih inervals of equal lengh. Exchange raes, for example, are normally quoed a he end of each rading day. For economeric analyses, however, monhly, quarerly, or even annual daa are used, raher han hese daily values. Usually, averages or end-of-period daa are used for emporal aggregaion. Thus, wo aggregaion schemes have o be disinguished. The firs one is skip sampling (or: sysemaic sampling) where only every m h daa poin is recorded. If x is he basic series a =,, 3,, he skip sampled series y s wih new ime scale s is end-of-period daa, y = x m, y = x m, y 3 = x 3m,, y s = x sm. Such an aggregaion is ypical for sock variables. However, he second scheme of averaging over m non-overlapping periods is also widely used, in paricular for raes or indices: y x x... x m m m y x x... x m m m m y s xsm x sm... x(s )m. m

37 . Moving Average Processes 63 In he following, we do no presen a general heory of emporal aggregaion bu jus discuss a special case of paricular applied ineres, he random walk, wih x = x - + u, where an arificial MA() srucure arises due o aggregaion by averaging. I is sraighforward o see ha sysemaic sampling does no affec he random walk propery, since in his case we can wrie y s = x + sm u. From his represenaion we ge y s = y s- + s, wih s being whie noise: s = u sm + u sm u (s-)m+, wih E[ s ] = and mu for E( s s ) =. elsewhere Hence, he random walk propery is inheried by y s, only he variance of he differences y s y s- is inflaed in he obvious way. In case of averaging, y s, maers ge more complicaed. I can, however, be shown ha he differences y s ys s follow no longer a whie noise process bu an MA() scheme hidden behind u u s sm sm... mu s... u m s u m 3 s. m m We omi deails bu refer o HOLBROOK WORKING (96) who showed ha wih increasing aggregaion level, m, one obains he auocorrelaion funcion

38 64 Univariae Saionary Processes () =,. V s 4, elsewhere E s s, Noe ha he above auocorrelaion funcion corresponds o he following MA()-process s us us where u s is whie noise, and he limiing value (for m ) of he MA parameer is GEORGE C. TIAO (97) generalised his resul he following way: If x x - is no generaed by whie noise bu by an inverible MA() process, hen y s y s behaves wih growing m like he MA() process us u s, where is independen of he underlying MA() srucure of x x -. This resul even coninues o hold when he assumpion ha x x - is MA() is replaced by a more general moving average process of higher order as inroduced in subsecion..3. Example. Consider averaging over m = periods, y s xs xs. For he random walk x = x - + u, i holds ha s ys ys = (x s + x s- x s- x s-3 ) = ( u s + u s- + u s- ). This process can be described as u s s us wih = 3.7, and

39 . Moving Average Processes 65 E( s s ) = 3 u for u for, 4 elsewhere such ha for m = he auocorrelaion coefficien a lag one becomes () = /6. Example. Example.3 as well as Figure.8 presen he end-of-monh exchange rae beween he Swiss Franc and he U.S. Dollar over he period from January 974 o December. The auocorrelogram of he firs differences of he logarihms of his ime series indicaes ha hey follow a pure random process. The ess we applied did no rejec his null hypohesis. If we use monhly averages insead of end-of-monh daa, he following MA() process can be esimaed for he firs difference of he logarihms of his exchange rae: ln(e ) = û +.38 û -, (-.53) (6.9) R =.8, SE =.8, Q() = 8.6 (p =.694), JB =.94 (p =.), wih he values again given in parenheses. ln( ) denoes he naural logarihm. The esimaed coefficien of he MA() erm is highly significanly differen from zero. The Ljung-Box Q-saisic indicaes ha here is no longer any significan auocorrelaion in he residuals. As m is relaively large (in his conex), he esimaed values of he MA() erm should no be oo differen from he heoreical value given by GEORGE C. TIAO (97). The heoreical value -.68 lies in he wo-sigma confidence inerval of he esimaed parameer Higher Order Moving Average Processes In general, he moving average process of order q (MA(q)) can be wrien as (.4) x = + u u - u -... q u -q wih q and u as a pure random process. Using he lag operaor we ge (.4') x = ( L L... q L q )u = (L)u.

40 66 Univariae Saionary Processes From (.4) we see ha we already have a finie order Wold represenaion wih k = for k > q. Thus, here are no problems of convergence, and every finie MA(q) process is saionary, no maer wha values are used for j, j =,,..., q. For he expecaion of (.4) we immediaely ge E[x ] =. Thus, he variance can be calculaed as: V[x ] = E[(x ) ] = E[(u u -... q u -q ) ] = E[( u + u qu q u u -... q- q u -q+ u -q )]. From his we obain V[x ] = ( q ). For he covariances of order we can wrie Cov[x, x + ] = E[(x )(x + )] = E[(u u -... q u -q ) (u + u q u +-q )] = E[u (u + u q u +-q ) u - (u + u q u +-q ) q u -q (u + u q u +-q )]. Thus, for =,,..., q we ge = : () = ( q- q ), (.4) = : () = ( q- q ), = q: (q) = q, while we have () = for > q. Consequenly, all auocovariances and auocorrelaions wih orders higher han he order of he process are zero. I is a leas heoreically possible o idenify he order of an MA(q) process by using he auocorrelogram. I can be seen from (.4) ha here exiss a sysem of non-linear equaions for given (or esimaed) second order momens ha deermines (makes i possible o esimae) he parameers,..., q. As we have al-

41 . Moving Average Processes 67 ready seen in he case of he MA() process, such non-linear equaion sysems have muliple soluions, i.e. here exis differen values for,,... and q ha all lead o he same auocorrelaion srucure. To ge a unique parameerisaion, he inveribiliy condiion is again required, i.e. i mus be possible o represen he MA(q) process as a saionary AR() process. Saring from (.4'), his implies ha he inverse operaor - (L) can be represened as an infinie series in he lag operaor, where he sum of he coefficiens has o be bounded. Thus, he represenaion we ge is an AR() process u = + - (L) x () = () + cx j j j, where = ( L... q L q )( + c L + c L +... ), and he parameers c i, i =,,... are calculaed by using again he mehod of undeermined coefficiens. Such a represenaion exiss if all roos of L... q L q = are larger han one in absolue value. Example.3 Le he following MA() process x = u +.6 u -. u - be given, wih a variance of given for he pure random process u. For he variance of x we ge V[x ] = ( ) =.37. Corresponding o (.4) he covariances are () = =.54 () =.. () = for > This leads o he auocorrelaion coefficiens () =.39 and () = -.7. To check wheher he process is inverible, he quadraic equaion +.6 L. L =

42 68 Univariae Saionary Processes has o be solved. As he wo roos -.36 and 7.36 are larger han in absolue value, he inveribiliy condiion is fulfilled, i.e. he MA() process can be wrien as an AR() process x = ( +.6 L. L ) u, u = x.6l.l = ( + c L + c L + c 3 L 3 + ) x. The unknowns c i, i =,,..., can be deermined by comparing he coefficiens of he polynomials in he following way: = ( +.6 L. L )( + c L + c L + c 3 L 3 + ) = + c L + c L + c 3 L L +.6 c L +.6 c L 3 +. L. c L 3 I holds ha c +.6 = c =.6, c +.6 c. = c =.46, c c. c = c 3 =.34, c c 3. c = c 4 =.5,. Thus, we ge he following AR() represenaion x.6 x x -.34 x x -4 = u. Similarly o he MA() process, he parial auocorrelaion funcion of he MA(q) process does no break off. As long as he order q is finie, he MA(q) process is saionary whaever is parameers are. If he order ends owards infiniy, however, for he process o be saionary he series of he coefficiens has o converge jus like in he Wold represenaion..3 Mixed Processes If we ake a look a he wo differen funcions ha can be used o idenify auoregressive and moving average processes, we see from Table. ha he siuaion in which neiher of hem breaks off can only arise if here is an MA() process ha can be invered o an AR() process, i.e. if he Wold represenaion of an AR() process corresponds o an MA() process. However, as pure AR or MA represenaions, hese processes canno

43 .3 Mixed Processes 69 be used for empirical modelling because hey can only be characerised by means of infiniely many parameers. Afer all, according o he principle of parsimony, he number of esimaed parameers should be as small as possible when applying ime series mehods. In he following, we inroduce processes which conain boh an auoregressive (AR) erm of finie order p and a moving average (MA) erm of finie order q. Hence, hese mixed processes are denoed as ARMA(p,q) processes. They enable us o describe processes in which neiher he auocorrelaion nor he parial auocorrelaion funcion breaks off afer a finie number of lags. Again, we sar wih he simples case, he ARMA(,) process, and consider he general case aferwards. Table.: Characerisics of he Auocorrelaion and he Parial Auocorrelaion Funcions of AR and MA Processes Auocorrelaion Funcion Parial Auocorrelaion Funcion MA(q) breaks off wih q does no break off AR(p) does no break off breaks off wih p.3. ARMA(,) Processes An ARMA(,) process can be wrien as follows, (.4) x = + x - + u u -, or, by using he lag operaor (.4') ( L) x = + ( L) u, where u is a pure random process. To ge he Wold represenaion of an ARMA(,) process, we solve (.4') for x, x = + L L u. I is obvious ha mus hold, because oherwise x would be a pure random process flucuaing around he mean = /( ). The j, j =,,,..., can be deermined as follows:

44 7 Univariae Saionary Processes L = + L + L + 3 L 3 + L L = ( L)( + L + L + 3 L 3 + ) L = + L + L + 3 L 3 + L L L 3. Comparing he coefficiens of he wo lag polynomials we ge L : = L : = = L : = L 3 : 3 = = ( ) 3 = ( ) L j : j j- = j = j- ( ). The j, j can be deermined from he linear homogeneous difference equaion j j- = wih = as iniial condiion. The j converge owards zero if and only if <. This corresponds o he sabiliy condiion of he AR erm. Thus, he ARMA(,) process is saionary if, wih sochasic iniial condiions, i has a sable AR() erm. The Wold represenaion is (.43) x = + u + ( ) u - + ( ) u - + ( ) u Thus, he ARMA(,) process can be wrien as an MA() process. To inver he MA() par, < mus hold. Saring from (.4') leads o u = + L x. L If /( L) is developed ino a geomeric series we ge u = + ( L)( + L + L +... ) x = + x + ( ) x - + ( ) x - + ( ) x

45 .3 Mixed Processes 7 This proves o be an AR() represenaion. I shows ha he combinaion of an AR() and an MA() erm leads o a process wih boh MA() and AR() represenaion if he AR erm is sable and he MA erm inverible. We obain he firs and second order momens of he saionary process in (.4) as follows: E[x ] = E[ + x - + u u - ] = + E[x - ]. Due o E[x ] = E[x - ] =, we ge =, i.e. he expecaion is he same as in an AR() process. If we se = wihou loss of generaliy, he expecaion is zero. The auocovariance of order can hen be wrien as (.44) E[x - x ] = E[x - ( x - + u u - )], which leads o () = () + E[x u ] E[x u - ] for =. Due o (.43), E[x u ] = and E[x u - ] = ( ). Thus, we can wrie (.45) () = () + ( ( )). (.44) leads o () = () + E[x - u ] E[x - u - ] for =. Because of (.43) his can be wrien as (.46) () = (). If we inser (.46) in (.45) and solve for (), he resuling variance of he ARMA(,) process is (.47) () =. Insering his ino (.46), we ge ( )( ) (.48) () =

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