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]
⋯,Yt−2,Yt−1,Yt,Yt+1,Yt+2,⋯
- lag
- 1st lag = lag 1 Yt−1
- kth lag = lag k Yt−k
1st difference
ΔYt=Yt−Yt−1 approximation for RoR = rate of return
ΔlnYt=lnYt−lnYt−1=lnYtYt−1=ln(1+Yt−Yt−1Yt−1)=Yt−Yt−1Yt−1+O((Yt−Yt−1Yt−1)2)≈Yt−Yt−1Yt−1=RoR
{ln(1+x)=∞∑n=1(−1)n+1xnn=x−x22+x33−⋯ln(1−x)=−∞∑n=1xnn=−x−x22−x33−⋯
11+t=1−t+t2−t3+⋯ln(1+x)=∫x011+tdt=x−x22+x33−x44+⋯
Definition 40.1 autocorrelation = serial correlation
∃t1≠t2[V(Yt1,Yt2)≠0]
{E(Yt)≜μtV(Yt)≜σ2t
Definition 18.1 kth order autocovariance
V(Yt,Yt−k)≜γ(t,k)
σ2t≜V(Yt)=V(Yt,Yt)=V(Yt,Yt−0)=γ(t,k=0)=γ(t,0)
Definition 18.2 kth order autocorrelation
ρt,t−k=σt,t−k√σtt√σt−k,t−k=σt,t−k√σ2t√σ2t−k=σt,t−kσtσt−k=V(Yt,Yt−k)√V(Yt)√V(Yt−k)≜R(Yt,Yt−k)≜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,Yt−k)≜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(Yt−k)=γ(0)R(Yt,Yt−k)≜ρ(t,k)≜V(Yt,Yt−k)√V(Yt)√V(Yt−k)=γ(t,k)√γ(0)√γ(0)(3)=(4)γ(k)γ(0)≜γkγ0≜ρ(k)≜ρk(5)γ(k)=γ(−k)(6)⇒ρ(k)=ρ(−k)
γ(k)(3)=V(Yt,Yt−k)=V(Yt−k,Yt)=V(Yt′,Yt′+k)=V(Yt′,Yt′−(−k))(3)=γ(−k)⇒(6)
point estimation
{ˆE(Yt)≜¯Y≜ˆμ=1TT∑t=1Yt→E(Yt)=μˆV(Yt)≜ˆγ0=1TT∑t=1(Yt−¯Y)2→V(Yt)=σ2=γ0ˆV(Yt,Yt−k)≜ˆγk=1TT∑t=k+1(Yt−¯Y)(Yt−k−¯Y)=1TT−k∑t=1(Yt−¯Y)(Yt+k−¯Y)→V(Yt,Yt−k)=γkˆV(Yt,Yt−k)√ˆV(Yt)√ˆV(Yt−k)≜ˆρk=ˆγkˆγ0=1TT∑t=k+1(Yt−¯Y)(Yt−k−¯Y)1TT∑t=1(Yt−¯Y)2→R(Yt,Yt−k)=ρk
Y1Y1+k+Y2Y2+k+⋅⋯+YtYt+k+⋅⋯+YT−kYT
1st-order autocorrelation estimation
ˆρ1=ˆγ1ˆγ0=1TT∑t=1+1=2(Yt−¯Y)(Yt−1−¯Y)1TT∑t=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+β1Yt−1+Et
Et∼N(0,σ2E) Yt=β0+β1Yt−1+Et=β0+β1(β0+β1Yt−2+Et−1)+Et=β0(1+β1)+β21Yt−2+Et+β1Et−1=β0(1+β1)+β21(β0+β1Yt−3+Et−2)+Et+β1Et−1=β0(1+β1+β21)+β31Yt−3+Et+β1Et−1+β21Et−2⋮=β0(1+β1+β21+⋯)+Et+β1Et−1+β21Et−2+⋯|β1|<1=β01−β1+∞∑k=0βk1Et−k
{μ=E(Yt)=E(β01−β1+∞∑i=0βi1Et−i)=β01−β1+∞∑i=0βi1E(Et−i)Et∼N(0,σ2E)=β01−β1μ=β01−β1γ0=σ2=V(Yt)=V(μ+∞∑i=0βi1Et−i)=V(∞∑i=0βi1Et−i)=∑∞i=0(βi1)2V(Et−i)Et∼N(0,σ2E)=∞∑i=0(β21)iσ2E=σ2E1−β21γ0=σ2=σ2E1−β21γk=V(Yt,Yt−k)=V(μ+∞∑i=0βi1Et−i,μ+∞∑i=0βi1Et−k−i)=V(∞∑i=0βi1Et−i,∞∑i=0βi1Et−k−i)=⋯γk=βk1σ2E1−β21ρk=R(Yt,Yt−k)=γkγ0=βk1σ2E1−β21σ2E1−β21=βk1ρk=γkγ0=βk1
ρ1=γ1γ0=β11=β1
γk=V(Yt,Yt−k)=V(μ+∞∑i=0βi1Et−i,μ+∞∑i=0βi1Et−k−i)=V(∞∑i=0βi1Et−i,∞∑i=0βi1Et−k−i)=⋯
Y=⟨Y1,Y2,⋯,YT⟩=[Y1Y2⋯YT]T=[Y1Y2⋮YT]
AR(1) covariance matrix
by (59.3)
V(Y)=E[[Y−E(Y)][Y−E(Y)]T]=[σ21σ12⋯σ1Tσ21σ22⋯σ2T⋮⋮⋱⋮σT1σT2⋯σ2T]=[σ11σ12⋯σ1Tσ21σ22⋯σ2T⋮⋮⋱⋮σT1σT2⋯σTT]=[σij]T×T=Σ=[γ0γ1⋯γT−1γ1γ0⋯γT−2⋮⋮⋱⋮γT−1γT−2⋯γ0]=[σ2E1−β21β11σ2E1−β21⋯βT−11σ2E1−β21β11σ2E1−β21σ2E1−β21⋯βT−21σ2E1−β21⋮⋮⋱⋮βT−11σ2E1−β21βT−21σ2E1−β21⋯σ2E1−β21]=σ2E1−β21[1β1⋯βT−11β11⋯βT−21⋮⋮⋱⋮βT−11βT−21⋯1]=σ2[1β1β21⋯βT−11β11β1⋯βT−21β21β11⋯βT−31⋮⋮⋮⋱⋮βT−11βT−21βT−31⋯1]=σ2[1ρ1ρ2⋯ρT−1ρ11ρ1⋯ρT−2ρ2ρ11⋯ρT−3⋮⋮⋮⋱⋮ρT−1ρT−2ρT−3⋯1]=σ2[1ρρ2⋯ρT−1ρ1ρ⋯ρT−2ρ2ρ1⋯ρT−3⋮⋮⋮⋱⋮ρT−1ρT−2ρT−3⋯1]
Y∼D(μ,Σ)=d(μY,ΣY)=d(E[Y],V[Y])
where
Σ=[γ0γ1⋯γT−1γ1γ0⋯γT−2⋮⋮⋱⋮γT−1γT−2⋯γ0]=σ2[1ρρ2⋯ρT−1ρ1ρ⋯ρT−2ρ2ρ1⋯ρT−3⋮⋮⋮⋱⋮ρT−1ρT−2ρT−3⋯1]