January 21, 2014
{X}tis weakly stationary
µ = E[X]
γ (h) = Cov[X, X ]
X t t+h
correlation function: ρ (h) = CXrr[X, X ] = Cov[X,tX ]t+har(X)Var(X ) =tγ(ht+hγ(0)γ(0) = t t+h
γ(h)/γ(0)
Property:
1. |γ(h)| ≤ γ(0) for any h
2. γ(h) = γ(-h) for any h
Proof: Cov[X, X ] t γ(ht+h γ(-h) = Cov[X, X ] t t-h
Ex. White noise: WN(0, σ ) 2
{X}tis called a white noise if
E[X] = 0 for any i
i
Cov[X, X]i= 0jfor any i≠j
2
Var[X] =iσ < ∞
White noise sequence is the building block for many more complicated time series models.
Inputs -> filter -> outputs
White noise -> linear time series model Ex. Moving average process of order 1: MA(1) model
2
Suppose Z , Z , Z0, …,1Z is 2 WN(0, σ n
Let X = i + aZ i i-1
X is called a MA(1) process.
E[X] =iE[Z i + aE[Z ] = i-1
2
Cov[X, X ]i= Ci+hZ+aZ , Z +aZ i i-1 i+h i+h-1 = Cov[Z, Z ] i aCi+hZ, Z i i+h-1] + aCov[Z , Z ]i-1a Coi+h i-
1, Z i+h-1] = A+B+C+D
1. If h=0, A=σ , B=0, C=0, D=σ => Cov[X, X] = σ + a σ = σ (a +1) 2 2 2 2 2
i i
2. If h=1, A=0, B=σ , C=0, D=0 => Cov[X, X ] = aσ i i+1 2
3. If h>1, A=0, B=0, C=0, D=0 => Cov[X, X ] = 0 i i+h
To summarize, X is a statiInary process with covariance function
γ(h) = {σ (a +1) if h=0
aσ if h=1
0 if h>1
Ex. Auto Regressive Process of order 1: AR(1)
Let {Z} bi a WN(0, σ ), iϵZ 2
Let X = ij:0-∞ a Z j i-j
X i AR(1)
Need |a|<1
Observation: X = aX + Z
i i-1 I
Proof: RHS = a ∑j:0-∞ a Z j i-1-j Z =i∑j:1-∞ a Z + Z = ∑ji-j∞ a Zi= LHS j i-j
HW: Check whether or not X = aX + Z is statioiary if i-1 0 I 0
{X} ts weakly stationary.
Proof:
E[X] =i∑j:0-∞ a E[Z ] = 0 i-j
For all h≥0,
j r j r
Cov[X, X ]i= Ci+h∑ a Z , ∑ a Z i-j i+h-r] = ∑∑ a a Cov[Z , Z i-j i+h-r
(Cov ≠ 0 iff i-j = i+h-r -> j = r-h)
= ∑ a a j j+h σ = σ a ∑a = σ a /(1-a ) 2 h 2
γ Xh) = σ a /(1-a ) -> absolute value for h≥0
Suppose a>0
ACVF & A

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