Time series re-run

Question 1

The autocovariance function (ACVF) is defined by

Answer 1

γX (τ ) = Cov(Xt+τ , Xt). gamma

Question 2

The autocorrelation function (ACF) is defined by

Answer 2

ρX (τ ) = γX (τ )/γX (0) ro = corr(Xt+τ , Xt).

Question 3

{Xt} is said to be weakly stationary (second-order stationary)

Answer 3

E [Xt] = µ for all t Cov(Xt+τ , Xt) = γ (τ ) for all t and τ Constant Mean

Question 4

∇^j Xt =

Answer 4

(1 − B)^j Xt power outside the bracket

Question 5

Lag operator ∇dXt =

Answer 5

= (1 − B^d) Xt power inside the bracket

Question 6

{Xt} is IID noise if Xt and Xt+h are…

Answer 6

independently and identically distributed and with mean zero. Xt ∼ IID(0, σ^2)

Question 7

{Xt} is white noise with zero mean if

Answer 7

µX = 0, γX (0) = σ^2 γX (h) = (σ^2 if h = 0, otherwise 0)

Question 8

IID is white noise…

Answer 8

But the converse is not true.

Question 9

{Xt} is a linear process if

Answer 9

Can be represented by the sum of constants times past Z terms.

Question 10

MA is in terms of

Answer 10

past Z terms. (Maz mate!)

Question 11

All linear processes are

Answer 11

stationary

Question 12

AR is in terms of

Answer 12

past X terms.

Question 13

AR model condition for stationarity

Answer 13

Modulus of roots not on unit circle.

Question 14

φ(B) factorises the

Answer 14

X terms

Question 15

θ(B) factorises the

Answer 15

Z terms

Question 16

For an ARMA to be stationary

Answer 16

φ(z) has no roots on the unit circle

Question 17

An ARMA model is casual iff

Answer 17

roots of the equation φ(z) = 0 are outside the unit circle

Question 18

ARMA model is invertible iff

Answer 18

roots θ(z) = 0 are outside the unit circle.

Question 19

2 Methods of calculating ACF of ARMA process

Answer 19

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).

Question 20

PACF idea

Answer 20

use PACF to measure the direct correlation only (by controlling or removing the indirect correlation caused by intermediate terms).

Question 21

Γh matrix =

Answer 21

γ(0) across to, and down to γ(h − 1).

Question 22

γh =

Answer 22

γ(1) down to γ(h)

Question 23

Ch matrix

Answer 23

C1 down to Ch

Question 24

Γh, γh and Ch eqn

Answer 24

γp − Γp φ(1 to p) = Cp

Question 25

hth partial autocorrelation function (PACF) notation

Answer 25

α(h), is 1 if h=0, otherwise the last element in ψhh vector

Question 26

ψh vector

Answer 26

= [Γ_h]^-1 γ_h

Question 27

2 Methods of Model Parameter Selection

Answer 27

MoM, MLE

Question 28

Common criteria to evaluate model fit

Answer 28

AIC, AICC, BIC.

We will use the

AICC=-2ln(L(B)) + 2n(p+q+1)/(n-p-q-2)

Question 29

Another way of model checking

Answer 29

Residual analysis

Question 30

Under the null hypothesis that residuals Zt ∼ WN(0, 1),

it is can be shown that the ACF of {Zt} follows

Answer 30

ρˆZ (h) ∼ N (0, 1/n)

Ie ACF of residuals Z, follows normal.

Question 31

So the confidence intervals at significance 95% of the sample autocorrelations are

Answer 31

0 ± 1.96/ sqrt(n) .

So if its outside this, consider it non-zero.

Question 32

What is Portmanteau stat used for

Answer 32

Checking model fit

Question 33

When using Port stat, Under the null hypothesis that Zt ∼ WN(0, 1), the port stat is distributed via

Answer 33

Chi^2 (h-p-q)

so reject if Q > χ²_1−α(h − p − q).

Question 34

2 methods of forcasting

Answer 34

Box-Jenkins

Best linear predictor

Question 35

Solution for coeficients of best linear predictor

Answer 35

a_n = Γ_n ⁻¹ γ_n(h).

Where Γ_n = γ(0) across and down to γ(n-1)