STATIONARY LINEAR VECTOR TIME SERIES PROCESSES Richard T Baillie, ( ).

Transcription

1 STATIONARY LINEAR VECTOR TIME SERIES PROCESSES Richard T Baillie, (.4.4). Inroducion In he case of univariae ime series processes, i was seen ha he Wold decomposiion provides a rich class of linear models o describe a scalar, saionar ime series process. I urns ou ha exending linear ARMA processes o he linear mulivariae ARMA case is relaivel sraighforward. The populaion characerisics of Vecor Auoregressions (VARs) and Vecor Auoregressive Moving Average (VARMA) processes appear o have been firs derived b Quenouille (957), alhough he were no rouinel applied unil he earl 98s. An imporan issue concerns he relaionship beween a mulivariae ime series model and a dnamic economeric model esimaed from ime series daa. In man was he wo are esseniall he same. In paricular, mulivariae ime series models ma ofen have an excessive number of parameers, and economic heor is ideall necessar in order o impose resricions on he parameer space. Also he pure VAR and VARMA mulivariae ime series models are esseniall linear, and economic heor ma provide was of formulaing non-linear relaionships. However, in man cases, a dnamic economeric model ma be approximaed b a linear mulivariae ime series model. Hence, he wo fields of mulivariae ime series analsis and dnamic economeric models are ver closel linked and iner-relaed. The undersanding and appreciaion of dnamic economeric models ineviabl requires a good working knowledge of ime series analsis. A maor problem area o be addressed laer concerns how o bes deal wih he forms of non saionari frequenl encounered in much economic ime series daa. Iniiall, i is necessar o deal wih he class of linear, saionar mulivariae ime series processes ha forms he basis for all subsequen work.

2 Vecor Form of Wold s Decomposiion Consider a ime series vecor of g random variables denoed b (,,... ) = / g so ha / represens a g dimensional vecor of random variables measured a discree inervals of ime. I is assumed ha / is covariance saionar, or inegraed of order zero, i.e. I(), so ha each elemen of he g dimensional vecor is saionar. The mulivariae form of Wold s decomposiion hen implies ha apar from fixed period harmonics ha can be represened as an infinie order moving average process given b () = B ε, = where he B are gxg dimensional coefficien marices and ε is a g dimensional vecor whie noise process wih he properies ha E( ε ) = and E εε =Ω, s= and E εε =, s. Hence Ω is a smmeric, posiive definie / ( s) / ( s) covariance marix of he innovaions. Auocovariance Funcion for Vecor Processes The dependence beween and a value a a fuure or previous ime period is represened b he auocovariance funcion, which for a saionar process will be ime invarian. The auocovariance funcion for a vecor process becomes a marix. In paricular, he auocovariance funcion beween and kis given b Γk and is defined as

3 () Γ = ( ) E k k = E g k k g k k k g k k k g k = E g k g k g g k so ha a pical elemen of Γ is γ ( k), k i which is he scalar auocovariance beween i and γ. Noe ha in general γ ( k) γ ( k). However, γ ( k) = γ ( k), so ha k i i i i ( ) Γ = E k + k and Γ = = =Γ (3) E( ) E( ) k + k + k k The covariance marix of he general vecor process in Wold s decomposiion is 3

4 / / / / ε iε i ε ε = i= = Γ = E( ) = E B B = E B B (4) Γ = = B ΩB / and he auocovariance marix a lag k is / Γ k= E( k) = E Bε Biε k i = i= / Γ = E( ε + Bε + B ε B ε + B ε +...)( ε + Bε + Bε +...) k k k k+ k k k k (5) / Γ k = B+ kωb, = which can be a useful formula for calculaing auocovariance funcions. The VAR() Process Before considering he general higher order VAR(p) model, i is ver convenien o sar wih he VAR() model, which can also be modified o deal wih man higher order mulivariae ime series models. The VAR() process is (6) = A + ε B successive subsiuion, = ε + Aε + A ε +... A ε + A 4

5 and from he same argumens as wih he scalar case, a necessar condiion for saionari is ha lim( A ) =, which is he null marix. An alernaive derivaion is o wrie (6) as, (7) AL ( ) = ( I AL ) = ε, and o be inverible for a valid Wold decomposiion, i is necessar for all he roos of he deerminenal polnomial AL ( ) o lie ouside he uni circle. (See he appendix for furher deails). If he marix AL ( ) is inverible, hen he Wold decomposiion of he VAR() is (8) = ( ) ε = ε. = I AL A The auocovariance funcion of he VAR() can also be found b direc analog wih he scalar AR() process. On ransposing, = A + ε and on premulipling b ε (9) ε = ε A + εε Since () E( ε ) =, k k i follows from aking expecaions hrough equaion (9) ha E( ε ) =Ω. On posmulipling hrough (6) b / and aking expecaions: = A + ε () Γ = A Γ +Ω 5

6 Also on pos-mulipling hrough (6) b k and aking expecaions: ( ε ) Γ = A Γ + E k k k and hence () Γ = AΓ, for k k k Subsiuing for Γ from () ino () gives (3) Γ = AΓ A +Ω On recalling ha he column sacking operaor, s(.), possesses he proper ha ( ) ( ) ( ) s ABC = C A s B, where s() = vec (), hen from (3) ( Γ )( ) ( Γ ) + ( Ω) s A A s s (4) s( ) ( I A A) s( ) Γ = Ω, which gives a direc expression for he covariance marix of of in erms of A and he covariance marix of ε, which is Ω. Also, b successive subsiuion from (), k Γ = A Γ k, and hence 6

7 k (5) s( Γ ) = ( I A ) s( Γ ) k An alernaive expression for Γ is available from he infinie moving average represenaion, or Wold decomposiion in (), (6) Γ = E A ε ε A = A ΩA = = = Example of a Saionar VAR() Process The following example is based on he bivariae g = case wih one lag, so ha p =. Then, (7) (/ 3) (/ 6) ε = + (/ 3) (/ ) ε, which can be expressed in lag operaor form as (/3) L (/6) L ε = (/ 3) L (/ ) L ε and (8) ( ) [ ][ ] A L = (/3) L (/) L (/8) L = (5/6) L+ (/9) L [ (/6) L][ (/3) L] = Hence he deerminenal polnomial has roos of 6 and 3/; boh of which lie ouside he uni circle. Hence he process is saionar. The infinie moving average represenaion, or 7

8 Wold decomposiion is obained from (7) as A ε = =, and in his case he firs four erms in he Wold decomposiion are ε ε ε = ε ε ε..97 ε ε ε ε 4... I can be seen ha each elemen of A is ending o zero for increasing powers of A, which is enirel consisen wih he process being saionar. The auocovariance marix of is Γ and from (3), Γ = AΓ A +Ω ( Γ ) = ( ) ( Ω) s I A A s Given Ω= 5, hen, s ( ) ( ) ( ) ( ) ( ) E E E E Γ = s ( Ω ) = 5 8

9 (/ 9) (/8) (/8) (/ 36) (/ 3) (/ 6) (/ 3) (/ 6) (/ 9) (/ 6) (/8) (/) A A= = (/ 3) (/ ) (/ 3) (/ ) (/ 9) (/8) (/ 6) (/) (/ 9) (/ 6) (/ 6) (/ 4) Then ( ) ( ) ( ) ( ) E E s( Γ ) = = E E Γ = ,.5.43 Γ = , ec. Saionar Vecor Auoregression (VAR) The mos widel used vecor model used in ime series economeric work is he Vecor Auoregression of order p, or VAR(p), which is () p = A + ε, = where A are gxg dimensional coefficien marices and ε is he vecor whie noise process which was defined previousl. The i h equaion of he VAR (p) will be () p g, = a ( k) + ε i i k i k= = 9

10 where ai ( k) is he ( i, ) elemen of A k. Hence all he las p lagged values of each variable, including iself, explain he curren value of i plus he random innovaionε i. In lag operaor form he VAR (p) can be expressed as (3) A ( L) = ε where (4) AL ( ) I AL p = = Analsis of he VAR(p) From Companion Form One grea aracion wih he VAR() model in (6) is ha i can be easil used o derive he properies of higher order vecor ime series models. The increase in he number of equaions in he VAR() poses lile problems for he calculaions required; while he increase in he order (i.e. value of p) is far more awkward. This fac was also apparen for he univariae AR(p) model compared wih he univariae AR() process. For his reason, i is paricularl convenien o express he VAR(p) in companion form as a VAR() model b sacking he ssem o allow for he original model o occup he firs g rows of he VAR() and all he oher rows o be ideniies. Then, (5) A A Ap ε I... = I I p+ p which can be expressed as

11 (6) Y = AY + ξ where Y and ξ are gp in dimension and A is gp gp. Wold Decomposiion of he VAR(p) On using he resuls for he pure VAR() model, he companion form VAR(p) can be wrien b successive subsiuion as (7) Y = A ξ = ' On defining a g gpdimensional selecion marix, N as N = [ I ] he ssem of equaions can hen be pre-muliplied b N' o obain NY = N AY + Nξ / / / hence (8) = N AY + ε / which is an alernaive form of expressing he VAR(p) in () wih N'A being he firs g rows of A in (5). On pre-mulipling (7) b he selecion marix, = ξ = ( ) ε = =, NA NA N so ha he h MAR coefficien marix is he op lef corner g gmarix of A. Hence for he VAR (p) model he Wold decomposiion marices, or infinie moving average represenaion coefficien marices B are defined as

12 (9) B = NA N SinceY = A ξ + AY, or equivalenl for he g equaion VAR(p) process, = ( ) ε ( ) = NA N + NA Y = i again follows ha for o be saionar i mus be ime invarian, so ha a necessar condiion is for ( A) lim =. A formal es for saionari of he VAR(p) in () is ha all he roos of AL ( ) in (3) mus lie ouside he uni circle, which is an analogous resul for he pure VAR(). A more formal proof is in he appendix. Granger Causali: A variable x Granger causes anoher variable, if x has predicable conen in he formaion of predicions of fuure. Tha is if knowledge of pas x significanl improves predicion of fuure, hen x Granger Causes. More formall, MSE( E,,,...) σ + = MSE( E,,,... x, x, x,...) σ + =, x If σ < σ, hen x Granger Causes. x, Noe ha he concep of Granger Causali is designed for he siuaion in engineering erminolog of a black box, ha is where he inpu and oupu are observed, bu he innermos workings of he black box are unknown.

13 Tes for Linear Granger Causali: : H o + = φ + φ + φ φp p+ + ε+ H : + = φ + φ + φ φp p+ + βx + βx βkx k+ + ε+ Noes: (i) A es for β = β =... = β k = can be done b eiher he Wald, Likelihood Raio or LM ess. (ii) The above equaions can also be inerpreed as one equaion of a VAR, wih and wihou resricions. For example, consider he bivariae VAR(p) model, p = a ( k) + a ( k) + ε k k k= k= p p = a ( k) + a ( k) + ε k k k= k= p In he general unresriced case, here is bi-direcional Granger causali beween boh variables. If a () = a () =... = a ( p) =, hen onl depends on is own pas hisor and no on ha of. However, boh he pas of and are informaive, (or have predicive conen) for modeling he behavior of. Hence is exogenous, bu Granger causes. causes. Conversel, if a() = a() =... = a( p) =, hen is exogenous, bu Granger (iii) Bi-direcional causali is alwas a disinc possibili. So he es mus be run in boh direcions. (iv) Conemporaneous causali can occur hrough he off diagonal elemen of he error covariance marix Ω. Genuine Granger causali mus occur wih pas informaion reducing he predicion MSE raher han curren informaion. 3

14 (v) Sims (97) provided a differen compuaional mehod for Granger Causali, which is harder and less inuiive. Sims applied he concep o he mone and income debae a ha ime. However, condiioning on omied, missing inermediae variables is a poenial problem wih all of hese ess. Auoregressive Final Form (ARFF) Given a mulivariae VARMA or VAR model, one ineresing quesion concerns he univariae ime series represenaion of a componen of he vecor process. The A L VAR(p) model, ( ) = ε, can be invered o give he Wold decomposiion, ( ) = A L ε which can be expressed as (3) = + ( ) ( ) A L A L ε where A( L ) + represens he adoin marix of A( L ) and A( L ) is he deerminenal polnomial associaed wih A( L) and will generall be a polnomial in he lag operaor of order gp. On mulipling boh sides of (3) b A( L ), he ARFF equaions are obained. A L A L ε (3) ( ) = ( ) + From his represenaion, i can be seen ha a pical elemen of, sa i will follow a univariae ARMA(pg, p) process. This can be seen since he sum of g moving average processes, each of order p, will also be a moving average process of order a mos p, which is one of he resuls found b Granger and Morris (976) and was referenced in chaper 3. 4

15 Anoher aspec of he ARFF represenaion is ha he auoregressive process will be he same for each univariae process in he vecor. To illusrae his, reurn o he previous example where (/ 3) (/ 6) ε = + (/ 3) (/ ) ε Then from (3), (/ ) L (/ 6) L ε (5/ 6) L (/ 9) L + = (/ ) L (/ 3) L ε The univariae process for will be (5/ 6) L+ (/ 9) L = ε (/ ) ε (/ 6) ε The righ hand side is he sum of wo componen MA() processes, which will also be anoher MA() process, denoed b u. Then u = ε (/ ) ε (/ 6) ε = ξ + θξ u ( ) = ( + (/ 4) ) + (/ 36) + (/ 6) = ( + ) γ σ σ σ θ σ ξ u () = (/) (/6) = ξ γ σ σ θσ To make progress wih he link beween he wo componen MA() processes and he aggregae one, ξ, is difficul since i involves solving non linear equaions. However, in order o provide a simple illusraion, suppose 5

16 E ( εε ) =Ω and 4 Ω= 36 γ = = + θ σ and γ () = = θσ, which gives he firs order Then u ( ) 6 ( ) ξ auocorrelaion coefficien on he aggregae MA() process o be u ξ ρ θ = =, 3 + θ () u and on solving (/ 3) θ + θ + (/ 3) = gives θ=.68 and θ=.38. On aking he inverible roo of.38, leads o a soluion for he variance of he whie noise process of he componen MA() o be σ = ( ) /(.38) = Hence ARMA(,) process ξ = (5 / 6) (/ 9) + ξ.38ξ and σ = 5.36 ξ Noe ha he variance of he innovaion (i.e. one sep ahead predicion error) is 5.36 which exceeds ha of he corresponding innovaion variance of 4 in he 'srucural' model. This is an illusraion of he fac ha going o he reduced form univariae model for one variable will generall be inefficien, since i will have larger disurbance variance, which is he same as he one sep ahead predicion MSE. A similar siuaion also arises for. Inerpreaion of Impulse Response Weighs The sandard VAR(p) model is se up so ha onl lagged values of oher variables ener each equaion and explain he lef hand side variable. An conemporaneous relaionship beween he elemens of will hen be presen in he off diagonal elemens of he disurbance covariance marix Ω. In he analsis of MARs and impulse response weighs from an esimaed VAR(p), he convenional MAR of 6

17 = B ε = has he normalizaion of B = I and E ( εε ) =Ω. Sims (98) and ohers generall renormalize b premulipling hrough () b a lower riangular marix R o obain p ( ) R = RA + ξ = where ξ = Rε and since R R I ( ) E ξξ = I. I also follows ha RR Ω =, i follows ha E ( ξ ) =, ( ) E ξξ =, s and =Ω. The advanage wih his represenaion is ha he effec of differen innovaions a differen lags can be more clearl deeced. Alernaivel i is also possible o use anoher lower riangular marix R - which is defined so ha s RDR =Ω and D is a diagonal marix, alhough in mos applicaions R is usuall chosen so ha he covariance marix of he disurbances will have an ideni marix. On reurning o he previous example (/ 3) (/ 6) ε = + (/ 3) (/ ) ε where, E ( εε ) =Ω= 5. We choose a marix R such ha R =, hen RΩ R = I and 5 RR = =Ω 7

18 The VAR() can hen be expressed as = (/ 3) (/ 6) + ξ = (/3) + (/) + ξ, where he new innovaions ξ and ξ are no onl uncorrelaed a all lags, bu conemporaneousl also. The conemporaneous par of he relaionship beween and is now apparen in he second equaion raher han in he marix Ω. 8

19 Appendix: Condiions for Saionari of he VAR(p) The VAR() in companion form is Y = A ξ + AY =. However, onl he firs g rows conaining he VAR(p) are of direc ineres and can be seen o be ( ) ε ( ) = NA N + NA Y = In order for o be saionar i mus be ime invarian, so ha a necessar condiion is for ( A) lim =. If A has g disinc eigenvalues λ, λ,..., λ g, which are sacked ino he diagonal marix, λ λ Λ=. λ g and on defining H as he corresponding marix of eigenvecors, so ha = Λ, hen A ( H H)( H H)...( H H) ( H H) A H H H = H, and = Λ Λ Λ = Λ. In order o deermine he eigenvalues of A, i is herefore necessar o solve he deerminenal equaion A λi =, and requiring all he eigenvalues o be inside he uni circle. This is equivalen o all he roos of AL ( ) o lie ouside he uni circle. For example, consider he VAR(3), hen 9

20 A A A3 I A λi A A3 A λi = I λ I = I λi I I I λi = = A + λa + λ A λ I = λ I λ A = In he general case for he VAR(p): A λ I A A3 A p I λi I λi A λi = I λi I λi I λi I λi p p p p p p p = = λ I λ A λ A A = λ I λ A = The condiion ha all he roos of he above lie inside he uni circle is hen equivalen o he condiion for he VAR(p) model o be covariance saionar is for all he roos of p AL ( ) = I AL o lie ouside he uni circle. =

21