Chapter 63: autoregression in time series

張翔老師. 2015. “ARMA Part1.” https://www.youtube.com/watch?v=G-0dR57W-fo.

張翔老師. 2015. “ARMA Part2.” https://www.youtube.com/watch?v=fQaZzO7E6FE.

張翔老師. 2015. “ARMA Part3.” https://www.youtube.com/watch?v=Ocw4NXoO8Xo.

time series [data]

,Yt2,Yt1,Yt,Yt+1,Yt+2,

  • lag
    • 1st lag = lag 1 Yt1
    • kth lag = lag k Ytk

1st difference

ΔYt=YtYt1 approximation for RoR = rate of return

ΔlnYt=lnYtlnYt1=lnYtYt1=ln(1+YtYt1Yt1)=YtYt1Yt1+O((YtYt1Yt1)2)YtYt1Yt1=RoR

{ln(1+x)=n=1(1)n+1xnn=xx22+x33ln(1x)=n=1xnn=xx22x33

11+t=1t+t2t3+ln(1+x)=x011+tdt=xx22+x33x44+

Definition 40.1 autocorrelation = serial correlation

t1t2[V(Yt1,Yt2)0]

{E(Yt)μtV(Yt)σ2t

Definition 18.1 kth order autocovariance

V(Yt,Ytk)γ(t,k)

σ2tV(Yt)=V(Yt,Yt)=V(Yt,Yt0)=γ(t,k=0)=γ(t,0)

Definition 18.2 kth order autocorrelation

ρt,tk=σt,tkσttσtk,tk=σt,tkσ2tσ2tk=σt,tkσtσtk=V(Yt,Ytk)V(Yt)V(Ytk)R(Yt,Ytk)ρ(t,k)

Definition 18.3 stationary time series

{E(Yt)μt=μ(1)independent of tV(Yt)σ2t=σ2<(2)independent of tV(Yt,Ytk)γ(t,k)=γ(k)(3)independent of t

properties

{V(Yt)σ2t=γ(t,k=0)=γ(k=0)(3)=γ(0)γ0(2)=σ2(4)V(Ytk)=γ(0)R(Yt,Ytk)ρ(t,k)V(Yt,Ytk)V(Yt)V(Ytk)=γ(t,k)γ(0)γ(0)(3)=(4)γ(k)γ(0)γkγ0ρ(k)ρk(5)γ(k)=γ(k)(6)ρ(k)=ρ(k)

γ(k)(3)=V(Yt,Ytk)=V(Ytk,Yt)=V(Yt,Yt+k)=V(Yt,Yt(k))(3)=γ(k)(6)

point estimation

{ˆE(Yt)¯Yˆμ=1TTt=1YtE(Yt)=μˆV(Yt)ˆγ0=1TTt=1(Yt¯Y)2V(Yt)=σ2=γ0ˆV(Yt,Ytk)ˆγk=1TTt=k+1(Yt¯Y)(Ytk¯Y)=1TTkt=1(Yt¯Y)(Yt+k¯Y)V(Yt,Ytk)=γkˆV(Yt,Ytk)ˆV(Yt)ˆV(Ytk)ˆρk=ˆγkˆγ0=1TTt=k+1(Yt¯Y)(Ytk¯Y)1TTt=1(Yt¯Y)2R(Yt,Ytk)=ρk

Y1Y1+k+Y2Y2+k++YtYt+k++YTkYT

1st-order autocorrelation estimation

ˆρ1=ˆγ1ˆγ0=1TTt=1+1=2(Yt¯Y)(Yt1¯Y)1TTt=1(Yt¯Y)2

63.1 AR(1) = 1st-order autoregressive model = first-order autoregressive model

張翔老師. 2015. “ARMA Part3.” https://www.youtube.com/watch?v=Ocw4NXoO8Xo.

Definition 18.4 AR(1) = 1st-order autoregressive model$

Yt=β0+β1Yt1+Et

EtN(0,σ2E) Yt=β0+β1Yt1+Et=β0+β1(β0+β1Yt2+Et1)+Et=β0(1+β1)+β21Yt2+Et+β1Et1=β0(1+β1)+β21(β0+β1Yt3+Et2)+Et+β1Et1=β0(1+β1+β21)+β31Yt3+Et+β1Et1+β21Et2=β0(1+β1+β21+)+Et+β1Et1+β21Et2+|β1|<1=β01β1+k=0βk1Etk

{μ=E(Yt)=E(β01β1+i=0βi1Eti)=β01β1+i=0βi1E(Eti)EtN(0,σ2E)=β01β1μ=β01β1γ0=σ2=V(Yt)=V(μ+i=0βi1Eti)=V(i=0βi1Eti)=i=0(βi1)2V(Eti)EtN(0,σ2E)=i=0(β21)iσ2E=σ2E1β21γ0=σ2=σ2E1β21γk=V(Yt,Ytk)=V(μ+i=0βi1Eti,μ+i=0βi1Etki)=V(i=0βi1Eti,i=0βi1Etki)=γk=βk1σ2E1β21ρk=R(Yt,Ytk)=γkγ0=βk1σ2E1β21σ2E1β21=βk1ρk=γkγ0=βk1

ρ1=γ1γ0=β11=β1

γk=V(Yt,Ytk)=V(μ+i=0βi1Eti,μ+i=0βi1Etki)=V(i=0βi1Eti,i=0βi1Etki)=

Y=Y1,Y2,,YT=[Y1Y2YT]T=[Y1Y2YT]

AR(1) covariance matrix

by (59.3)

V(Y)=E[[YE(Y)][YE(Y)]T]=[σ21σ12σ1Tσ21σ22σ2TσT1σT2σ2T]=[σ11σ12σ1Tσ21σ22σ2TσT1σT2σTT]=[σij]T×T=Σ=[γ0γ1γT1γ1γ0γT2γT1γT2γ0]=[σ2E1β21β11σ2E1β21βT11σ2E1β21β11σ2E1β21σ2E1β21βT21σ2E1β21βT11σ2E1β21βT21σ2E1β21σ2E1β21]=σ2E1β21[1β1βT11β11βT21βT11βT211]=σ2[1β1β21βT11β11β1βT21β21β11βT31βT11βT21βT311]=σ2[1ρ1ρ2ρT1ρ11ρ1ρT2ρ2ρ11ρT3ρT1ρT2ρT31]=σ2[1ρρ2ρT1ρ1ρρT2ρ2ρ1ρT3ρT1ρT2ρT31]

YD(μ,Σ)=d(μY,ΣY)=d(E[Y],V[Y])

where

Σ=[γ0γ1γT1γ1γ0γT2γT1γT2γ0]=σ2[1ρρ2ρT1ρ1ρρT2ρ2ρ1ρT3ρT1ρT2ρT31]

63.2 AR(2) = 2nd-order autoregressive model = second-order autoregressive model

Definition 63.1 AR(2) = 2nd-order autoregressive model

Yt=β0+β1Yt1+β2Yt2+Et

EtN(0,σ2E)

63.3 AR(p) = pth-order autoregressive model

Yt=β0+pi=1βiYti+Et

63.4 MA(q) = qth-order moving-average model

Yt=μ+Et+qi=1αiEti

63.5 ARMA(p,q) = pth,qth-order autoregressive-moving-average model

Yt=Et+qi=1αiEti+pi=1βiYti