An introduction to ARMA and GARCH processes 402KB (by Fulvio

46 Pages·2010·0.39 MB·English

by  Fulvio Corsi

Introduction to ARMA and GARCH processes
FulvioCorsi
SNSPisa
3March2010
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 1/24
Stationarity
Strictstationarity:
(X1,X2,...,Xn)=d (X1+k,X2+k,...,Xn+k) foranyinteger n>1,k
Weak/second-order/covariancestationarity:
E[x]=µ
t
E[X −µ]2 =σ2 <+∞(i.e. constantandindipendentoft)
t
E[(X −µ)(X −µ)]=γ(|k|)(i.e. indipendentoftforeachk)
t t+k
Interpretation:
meanandvarianceareconstant
meanreversion
shocksaretransient
covariancebetweenX andX tendsto0ask→∞
t t−k
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 2/24
Stationarity
Strictstationarity:
(X1,X2,...,Xn)=d (X1+k,X2+k,...,Xn+k) foranyinteger n>1,k
Weak/second-order/covariancestationarity:
E[x]=µ
t
E[X −µ]2 =σ2 <+∞(i.e. constantandindipendentoft)
t
E[(X −µ)(X −µ)]=γ(|k|)(i.e. indipendentoftforeachk)
t t+k
Interpretation:
meanandvarianceareconstant
meanreversion
shocksaretransient
covariancebetweenX andX tendsto0ask→∞
t t−k
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 2/24
Stationarity
Strictstationarity:
(X1,X2,...,Xn)=d (X1+k,X2+k,...,Xn+k) foranyinteger n>1,k
Weak/second-order/covariancestationarity:
E[x]=µ
t
E[X −µ]2 =σ2 <+∞(i.e. constantandindipendentoft)
t
E[(X −µ)(X −µ)]=γ(|k|)(i.e. indipendentoftforeachk)
t t+k
Interpretation:
meanandvarianceareconstant
meanreversion
shocksaretransient
covariancebetweenX andX tendsto0ask→∞
t t−k
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 2/24
Withe noise
weak(uncorrelated)
E(ǫ)=0 ∀t
t
V(ǫ)=σ2 ∀t
t
ρ(ǫ,ǫ)=0 ∀s6=t where ρ≡ γ(|t−s|)
t s γ(0)
strong(independence)
ǫ ∼I.I.D.(0,σ2)
t
Gaussian(weak=strong)
ǫ ∼N.I.D.(0,σ2)
t
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 3/24
Lag operator
theLagoperatorisdefinedas:
LXt≡Xt−1
isalinearoperator:
L(βXt) = β·LXt=βXt−1
L(Xt+Yt) = LXt+LYt=Xt−1+Yt−1
andadmitspowerexponent,forinstance:
L2Xt = L(LXt)=LXt−1=Xt−2
LkXt = Xt−k
L−1Xt = Xt+1
Someexamples:
∆Xt = Xt−Xt−1=Xt−LXt=(1−L)Xt
yt = (θ1+θ2L)LXt=(θ1L+θ2L2)Xt=θ1Xt−1+θ2Xt−2
Expressionlike
(θ0+θ1L+θ2L2+...+θnLn)
withpossiblyn= ,arecalledlagpolynomialandareindicatedasθ(L)
∞
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 4/24
Lag operator
theLagoperatorisdefinedas:
LXt≡Xt−1
isalinearoperator:
L(βXt) = β·LXt=βXt−1
L(Xt+Yt) = LXt+LYt=Xt−1+Yt−1
andadmitspowerexponent,forinstance:
L2Xt = L(LXt)=LXt−1=Xt−2
LkXt = Xt−k
L−1Xt = Xt+1
Someexamples:
∆Xt = Xt−Xt−1=Xt−LXt=(1−L)Xt
yt = (θ1+θ2L)LXt=(θ1L+θ2L2)Xt=θ1Xt−1+θ2Xt−2
Expressionlike
(θ0+θ1L+θ2L2+...+θnLn)
withpossiblyn= ,arecalledlagpolynomialandareindicatedasθ(L)
∞
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 4/24
Lag operator
theLagoperatorisdefinedas:
LXt≡Xt−1
isalinearoperator:
L(βXt) = β·LXt=βXt−1
L(Xt+Yt) = LXt+LYt=Xt−1+Yt−1
andadmitspowerexponent,forinstance:
L2Xt = L(LXt)=LXt−1=Xt−2
LkXt = Xt−k
L−1Xt = Xt+1
Someexamples:
∆Xt = Xt−Xt−1=Xt−LXt=(1−L)Xt
yt = (θ1+θ2L)LXt=(θ1L+θ2L2)Xt=θ1Xt−1+θ2Xt−2
Expressionlike
(θ0+θ1L+θ2L2+...+θnLn)
withpossiblyn= ,arecalledlagpolynomialandareindicatedasθ(L)
∞
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 4/24
Lag operator
theLagoperatorisdefinedas:
LXt≡Xt−1
isalinearoperator:
L(βXt) = β·LXt=βXt−1
L(Xt+Yt) = LXt+LYt=Xt−1+Yt−1
andadmitspowerexponent,forinstance:
L2Xt = L(LXt)=LXt−1=Xt−2
LkXt = Xt−k
L−1Xt = Xt+1
Someexamples:
∆Xt = Xt−Xt−1=Xt−LXt=(1−L)Xt
yt = (θ1+θ2L)LXt=(θ1L+θ2L2)Xt=θ1Xt−1+θ2Xt−2
Expressionlike
(θ0+θ1L+θ2L2+...+θnLn)
withpossiblyn= ,arecalledlagpolynomialandareindicatedasθ(L)
∞
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 4/24
Moving Average (MA) process
Thesimplestwaytoconstructastationaryprocessistousealagpolynomialθ(L)with
∞j=0θj2<∞toconstructasortof“weightedmovingaverage”ofwithenoisesǫt,i.e.
P
MA(q)
Yt=θ(L)ǫt=ǫt+θ1ǫt−1+θ2ǫt−2+...++θqǫt−q
Example,MA(1)
Yt=ǫt+θǫt−1=(1+θL)ǫt
beingE[Yt]=0
γ(0) = E[YtYt]=E[(ǫt+θǫt−1)(ǫt+θǫt−1)]=σ2(1+θ2);
γ(1) = E[YtYt−1]=E[(ǫt+θǫt−1)(ǫt−1+θǫt−2)]=σ2θ;
γ(k) = E[YtYt−k]=E[(ǫt+θǫt−1)(ǫt−k+θǫt−k−1)]=0 ∀k>1
and,
γ(1) θ
ρ(1) = =
γ(0) 1+θ2
γ(k)
ρ(k) = =0 k>1
γ(0) ∀
hence,whileawithenoiseis“0-correlated”,MA(1)is1-correlated
(i.e.ithasonlythefirstcorrelationρ(1)differentfromzero)
FulvioCorsi IntroductiontoARM(A)andGARCHprocesses SNSPisa3March2010 5/24