Γ(h)=0 h 0. Γ(h)=cov(X 0,X 0-h ). A stationary process is called white noise if its autocovariance

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1 A family, Z,of random vecors : Ω R k defined on a probabiliy space Ω, A,P) is called a saionary process if he mean vecors E E =E M = M k E and he auocovariance marices are independen of. k cov, -h )=E E ) -h E -h ) he auocovariance funcion of a saionary process is defined by Γh)=cov, -h ). Exercise: Show ha Γh)=Γ h). MS A saionary process is called whie noise if is auocovariance funcion saisfies Γh)= h. Since Γ) does no have o be a diagonal marix, any wo componens of whie noise can be correlaed wih each oher conemporaneously. A saionary process is called a firs order auoregressive process or AR) process) if i can be expressed as φ φk ) M = M O M M + M, k 3 k 4 φ φkk 443 ) k 443 k 3 Φ where is whie noise wih mean vecor. Each componen of an AR) process depends no only on lagged values of iself bu also on lagged values of he oher componens.

2 Subsiuing in an AR) equaion =Φ - + firs Φ for - and hen Φ for -, gives =ΦΦ )+ =Φ - +Φ - + =Φ Φ )+Φ - + =Φ 3-3 +Φ - +Φ - + M m =Φ m -m + Φ j j Suppose ha Φ is diagonalizable, i.e., here is an inverible marix C such ha Λ=C - ΦC is a diagonal marix. I hen follows from Λ=C - ΦC Φ=CΛC - For example, any Hermiian marix a complex square marix ha is equal o is own conjugae ranspose) is diagonalizable. hus, real symmeric marices are diagonalizable. ha Φ =ΦΦ=CΛC - CΛC - =CΛ C - Φ 3 =Φ Φ=CΛ C - CΛC - =CΛ 3 C - hus, M Φ m =CΛ m C -. λ Φ m =C M m λ =C M λ λ m O O m M λ k M m λ k C - C - will vanish as m only if he no necessarily real numbers λ,,λ k have modulus less han. MR

3 A k k marix Φ is diagonalizable if and only if i has k linearly independen eigenvecors diagonalizaion heorem). Proof: Suppose ha c,,c k are linearly independen eigenvecors wih eigenvalues λ,,λ k. hen If hen λ Φ = c,,c k ) M O M c,,c k ) -, λ k Φ = Φc,,c k )c,,c k ) - Φc,,Φc k ) = Φc,,c k ) = Φc,,Φc k )c,,c k ) - = λ c,,λ k c k )c,,c k ) - λ = c,,c k ) M O M c,,c k ) -. λk = c,,c k ) λ M = λ c,,λ k c k ). O λ M k MD 3

4 Exercise: Show ha if a marix Φ has differen eigenvalues λ and λ, he corresponding eigenvecors c and c will be linearly independen. Soluion: Since λ and λ are differen, a leas one of hem, say λ, is no equal o zero. Assuming ha he anihesis is valid, we obain c =νc for some ν Φc =νφc, λ c =νλ c, and c = λ λ νc, Since λ and c, c, his is in conradicion wih he λ anihesis. ME Exercise: Find he eigenvalues λ and λ of he marix Φ=. Hin: If c is an eigenvecor wih eigenvalue λ, i.e., or, equivalenly, Φc=λc λiφ)c=, hen he inveribiliy of he marix λiφ would imply ha c=λiφ) - =, which is inconsisen wih he requiremen ha c mus be a non-zero vecor. he eigenvalues can herefore be found by solving he equaion deλiφ)=. MV 4

5 If Φ is a k k marix, he characerisic polynomial deλiφ) has degree k. According o he fundamenal heorem of algebra i has herefore k complex) roos, if each roo is couned wih is algebraic mulipliciy. Since eigenvecors corresponding o differen eigenvalues are independen, Φ can only be non-diagonalizable if here exiss an eigenvalue wih algebraic mulipliciy m a > and geomeric mulipliciy m g <m a. he geomeric mulipliciy of an eigenvalue is he number of linearly independen eigenvecors wih ha eigenvalue. Exercise: Show ha he marix MN is non-diagonalizable. Φ= 3 Hin: he eigenvecors corresponding o an eigenvalue λ can be found by solving he equaion λiφ)c= for c. he condiion ha all he eigenvalues of Φ are less han in absolue value, i.e., is equivalen o z deφzi), z de z ΦzI)), z dei z Φ), and z deizφ). Remark: If all roos of he polynomial dei-zφ) lie ouside of he uni circle, he sequence Φ, Φ, Φ 3, is absoluely summable and j j converges componenwise) in mean square o. Φ 5

6 sing lag-operaor noaion he equaion can also be wrien as Φ - = IΦ) =, where IΦ is a marix-valued polynomial. For example, in he bivariae case we have I Φ) = φ = φ φ φ φ φ φ φ φ) φ = φ + φ) A saionary process is called an auoregressive process of order p or ARp) process) if i can be expressed as or, equivalenly, as =Φ - + +Φ p -p + Φ - Φ p -p =, where is whie noise wih mean vecor. sing lag-operaor noaion, he laer equaion can also be wrien as where Φ) =, Φ)=IΦ Φ p p is a marix-valued polynomial. = φ φ ) ) φ φ ) ). 6

7 e be a general linear process represened by = Ψ j j, where is whie noise wih E = and var )=Σ. We have and E = Ψ je j = Γ k)=cov, -k )=E Ψ j = Ψ r= r j E = Ψ j+ kσψ j. j k)j Ψ -r -j+ k) Since neiher E nor cov, -k ) depend on, he process is weakly saionary. P Ψ j he specral densiies of and are given by f ω)= π f ω)= π e k= iωk iωk e = π k= Γ k)= Σ, π Ψ i k e ω = π k= j+ k Γ k) ΣΨ j Ψ j ΣΨ iωj e Ψ jσ) k= = π Ψ j e -iωj Σ k= = π Ψ jk Ψ jk e iωj-k) j+ k e iωj+k) Ψ j e -iωj Σ Ψ -iωk k e * ) k=. PD 7

8 A represenaion IΦ) = of an AR) process is called causal if can be expressed in erms of presen and pas shocks, i.e., = j j Is specral densiy is given by f ω)= π j j Φ ) = Φ. j Φ e -iωj Σ iω = π Φe ) I Σ k= k e -iωk Φ ) * iω ) * I Φe ). Exercise: Derive he sum formula PG n Φ j =IΦ n+ )IΦ) for a geomeric series of marices. Hin: Muliply each side of he equaion by IΦ. Remark: Moreover, if all eigenvalues of Φ have modulus less han, we have = j j Φ =IΦ). Exercise: Show ha PV = j j Φ e -iωj =IΦe -iω ), if all eigenvalues of Φ have modulus less han. Analogously, f ω)= π IΦ e -iω Φ p e -iωp ) - Σ IΦ e -iω Φ p e -iωp ) - ) * is he specral densiy of an ARp) process wih causal represenaion IΦ Φ p p ) =. 8

9 Exercise: Reexamine he relaionship beween changes in he indusrial producion and changes in he duraion of unemploymen wih parameric mehods. Wrie an R funcion for he calculaion of he specral densiy of a vecor auoregressive process. var.spec <- funcionfr,ar.p) { # fr vecor of frequencies # AR.p ARp) model esimaed by R funcion ar nf <- lenghfr); p <- AR.p$order sigma <- AR.p$var.pred; k <- lenghsigma[,]) Id <- diag,nrow=k,ncol=k) # ideniy marix sp <- arraydim=cnf,k,k)) for w in :nf) { A <- Id for l in :p) A <- A-AR.p$ar[l,,]*exp-i*fr[w]*l) A <- solvea) # inverse of A sp[w,,] <- A%*%sigma%*%ConjA)) } reurnsp/*pi)) } Esimae AR model of order p=3. AR.3 <- arxy,order.max=3,aic=f,demean=) # aic=f order is fixed, no seleced auomaically # AR.3$ar: array of dim 3,,) wih AR coefficiens AR.3$ar[,,] # lag Series Series Series Series AR.3$ar[,,] # lag Series Series Series Series AR.3$ar[3,,] # lag 3 Series Series Series Series AR.3$var.pred # variance no explained by AR model Series Series Series 4.437e e-6 Series e e-3 9

10 Esimae he univariae specral densiies. parmfrow=c,),mar=c,,,)) p <- spec.pgramxy[,],aper=,der=f,fas=f,plo=f) f <- p$freq**pi; plof,p$spec/*pi),ype="l") sp.3 <- var.specf,ar.3); linesf,sp.3[,,],col="red") p <- spec.pgramxy[,],aper=,der=f,fas=f,plo=f) plof,p$spec/*pi),ype="l") linesf,sp.3[,,],col="red") Esimae he cospecrum. parmfrow=c,)) plof,resp.3[,,]),ype="l") ablineh=,ly="dashed") # dashed horizonal line he overall negaive relaionship beween he wo variables is mainly due o he low frequencies.

11 Esimae he squared coherency and he phase specrum. parmfrow=c,)) plof,modsp.3[,,])^/sp.3[,,]*sp.3[,,]),ype="l") plof,argsp.3[,,]),ype="l") he squared coherency is large a he low frequencies. here he slope of he phase specrum is approximaely, which indicaes ha changes in he duraion of unemploymen lag wo monhs behind changes in indusrial producion.

12 A saionary process is called an auoregressive moving average process of order p,q) or ARMAp,q) process) if i can be expressed as =Φ - + +Φ p -p + +Θ - + +Θ q -q or, equivalenly, as Φ - Φ p -p = +Θ - + +Θ q -q, where is whie noise wih mean vecor. sing lag-operaor noaion, he laer equaion can also be wrien as where and Φ) =Θ), Φ)=IΦ Φ p p Θ)=I+Θ + +Θ q q are marix-valued polynomials. An ARMAp,) process is an ARp) process. An ARMA,q) process is also called a moving average process of order q or MAq) process). he ARMAp,q) equaion IΦ Φ p p ) = I+Θ + +Θ q q ) is said o be causal if z deizφ z p Φ p ). I is said o be inverible if z dei+zθ+ +z q Θ q ). Exercise: Show ha he bivariae AR) process 4 ) ) = is causal and inverible. PC

13 3 Exercise: Show ha he bivariae MA) process = + 3 ) ) + ) ) is causal and inverible. PI Exercise: Show ha he bivariae ARMA,) process ) ) = ) ) is causal and inverible. PA I does no make sense o esimae he parameer marices Φ,,Φ p,θ,,θ q, and Σ of an ARMAp,q) process if hey are no unique. o ensure idenifiabiliy in he univariae case, where Φ) and Θ) are jus scalar polynomials, we mus require, in addiion o causaliy and inveribiliy, ha Φz) and Θz) have no common zeros. For example, he equaion 4 ) =+ ) can be wrien more parsimoniously as ) =, because he polynomials 4 z =+ z) z) and + z have a common zero.

14 In he mulivariae case, he marix-valued polynomials Φz) and Θz) can have a common lef facor even if deφz)) and deθz)) have no common zero. o avoid he difficulies involved in he idenificaion of mulivariae ARMA processes, many ime series analyss use only mulivariae AR models for he modeling of mulivariae ime series. Exercise: Show ha he equaion P φ + θ ) ) can be wrien more parsimoniously as φ = θ + ) ) = ) ) alhough he polynomials and have no common zero. φ + θ deφz))=dei z) θ deθz))=dei+ z) Hin: Muliply boh Φz) and Θz) by Θ - θz z)=. Remark: he inverse of he marix-valued polynomial Θz) is also a marix-valued polynomial. Is deerminan is a consan unequal o zero. Such a marix-valued polynomial is called unimodular. 4