Time series re-run Flashcards
The autocovariance function (ACVF) is defined by
γX (τ ) = Cov(Xt+τ , Xt). gamma
The autocorrelation function (ACF) is defined by
ρX (τ ) = γX (τ )/γX (0) ro = corr(Xt+τ , Xt).
{Xt} is said to be weakly stationary (second-order stationary)
E [Xt] = µ for all t Cov(Xt+τ , Xt) = γ (τ ) for all t and τ Constant Mean
∇^j Xt =
(1 − B)^j Xt power outside the bracket
Lag operator ∇dXt =
= (1 − B^d) Xt power inside the bracket
{Xt} is IID noise if Xt and Xt+h are…
independently and identically distributed and with mean zero. Xt ∼ IID(0, σ^2)
{Xt} is white noise with zero mean if
µX = 0, γX (0) = σ^2 γX (h) = (σ^2 if h = 0, otherwise 0)
IID is white noise…
But the converse is not true.
{Xt} is a linear process if
Can be represented by the sum of constants times past Z terms.
MA is in terms of
past Z terms. (Maz mate!)
All linear processes are
stationary
AR is in terms of
past X terms.
AR model condition for stationarity
Modulus of roots not on unit circle.
φ(B) factorises the
X terms
θ(B) factorises the
Z terms
For an ARMA to be stationary
φ(z) has no roots on the unit circle
An ARMA model is casual iff
roots of the equation φ(z) = 0 are outside the unit circle
ARMA model is invertible iff
roots θ(z) = 0 are outside the unit circle.
2 Methods of calculating ACF of ARMA process
1) Express Xt as ψ(B)Zt 2) Yule-Walker equations - Multiply by Xt-h and then take expectations. This finds ACVF, then divide to get ACF).
PACF idea
use PACF to measure the direct correlation only (by controlling or removing the indirect correlation caused by intermediate terms).
Γh matrix =
γ(0) across to, and down to γ(h − 1).
γh =
γ(1) down to γ(h)
Ch matrix
C1 down to Ch
Γh, γh and Ch eqn
γp − Γp φ(1 to p) = Cp
hth partial autocorrelation function (PACF) notation
α(h), is 1 if h=0, otherwise the last element in ψhh vector
ψh vector
= [Γh]-1 γh
2 Methods of Model Parameter Selection
MoM, MLE
Common criteria to evaluate model fit
AIC, AICC, BIC.
We will use the
AICC=-2ln(L(B)) + 2n(p+q+1)/(n-p-q-2)
Another way of model checking
Residual analysis
Under the null hypothesis that residuals Zt ∼ WN(0, 1),
it is can be shown that the ACF of {Zt} follows
ρˆZ (h) ∼ N (0, 1/n)
Ie ACF of residuals Z, follows normal.
So the confidence intervals at significance 95% of the sample autocorrelations are
0 ± 1.96/ sqrt(n) .
So if its outside this, consider it non-zero.
What is Portmanteau stat used for
Checking model fit
When using Port stat, Under the null hypothesis that Zt ∼ WN(0, 1), the port stat is distributed via
Chi^2 (h-p-q)
so reject if Q > χ21−α(h − p − q).
2 methods of forcasting
Box-Jenkins
Best linear predictor
Solution for coeficients of best linear predictor
an = Γn −1 γn(h).
Where Γn = γ(0) across and down to γ(n-1)