Module 5: Time series Flashcards
Weakly stationary requ.
Mean contant
Cov(Xt,Xt-h) only depends on h. (Ie time difference is all that matters).
Neither depend on time t.
White noise Zt has
Zero mean, constant variance.
All observations are uncorrelated.
Important result for
Cov(aX+bY,cZ)
=Cov(aX+cZ) + Cov(bY,cZ)
=ac Cov(X,Z) + bc Cov(Y,Z)
What does B do
takes Zt back to Zt-1
AR(p) is
Auto regressive model of order p.
Xt=£1Xt-1+£2Xt-2…+Zt
until p terms
MA(q) is
Moving average order q
Xt=Zt+£1*Zt-1…….£qZt-q
£ are constants.
ARMA(p,q) in words is
A model that is influenced by both past results and noise.
Auto-covariance function γ(τ)
Cov( X(t+τ) , X(t) )
Auto Correlation Function ρ(τ)
γ(τ) / γ(0 = Corr( X(t+τ) , X(t) )
▽^j *Xt (j is superscript) Difference operator
(1-B)^j * Xt
▽^d *Xt (d is subscript) Lag-d
(1-B^d)* Xt
White noise is
A RV with mean 0, var σ^2
IID noise
RV with mean 0, var σ^2
IID is/is not white noise
Yes it is. IID is in the subset of white noise.
What is a linear process
if Xt can be represented as
∑ νj * Z(t-j)
V are constants, with finite sum.
So this is a linear combo of past white noises.
Compact form of Linear process
ν(B)*Zt
where v(B) = ∑v*B^j ie the polynomial of B that moves them all back.
An ARMA is causal if
X is expressable in terms of current and past Zt
Or, if roots to φ(B)=0 are outside unit circle.
ARMA is invertable if
Zt is expressable in terms of current and past Xt
or if roots to θ(B)=0 are outside unit circle
ACF of AR(p)
tails off
ACF of MA(q)
cuts off after q
2 methods of estimating parameters
MLE, MOM
Easiest model selection method
Plot the SACF and SPACF
What is Portmanteau stat
Function of the SACF of the residuals…distributed as chi squ.
What is box-jenkins method of forcasting
Iteration method of getting forward results.
