Module 5: Time series

Question 1

Weakly stationary requ.

Answer 1

Mean contant
Cov(Xt,Xt-h) only depends on h. (Ie time difference is all that matters).

Neither depend on time t.

Question 2

White noise Zt has

Answer 2

Zero mean, constant variance.

All observations are uncorrelated.

Question 3

Important result for

Cov(aX+bY,cZ)

Answer 3

=Cov(aX+cZ) + Cov(bY,cZ)

=ac Cov(X,Z) + bc Cov(Y,Z)

Question 4

What does B do

Answer 4

takes Zt back to Zt-1

Question 5

AR(p) is

Answer 5

Auto regressive model of order p.
Xt=£1Xt-1+£2Xt-2…+Zt
until p terms

Question 6

MA(q) is

Answer 6

Moving average order q

Xt=Zt+£1*Zt-1…….£qZt-q

£ are constants.

Question 7

ARMA(p,q) in words is

Answer 7

A model that is influenced by both past results and noise.

Question 8

Auto-covariance function γ(τ)

Answer 8

Cov( X(t+τ) , X(t) )

Question 9

Auto Correlation Function ρ(τ)

Answer 9

γ(τ) / γ(0 = Corr( X(t+τ) , X(t) )

Question 10

▽^j *Xt (j is superscript) Difference operator

Answer 10

(1-B)^j * Xt

Question 11

▽^d *Xt (d is subscript) Lag-d

Answer 11

(1-B^d)* Xt

Question 12

White noise is

Answer 12

A RV with mean 0, var σ^2

Question 13

IID noise

Answer 13

RV with mean 0, var σ^2

Question 14

IID is/is not white noise

Answer 14

Yes it is. IID is in the subset of white noise.

Question 15

What is a linear process

Answer 15

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.

Question 16

Compact form of Linear process

Answer 16

ν(B)*Zt

where v(B) = ∑v*B^j ie the polynomial of B that moves them all back.

Question 17

An ARMA is causal if

Answer 17

X is expressable in terms of current and past Zt

Or, if roots to φ(B)=0 are outside unit circle.

Question 18

ARMA is invertable if

Answer 18

Zt is expressable in terms of current and past Xt

or if roots to θ(B)=0 are outside unit circle

Question 19

ACF of AR(p)

Answer 19

tails off

Question 20

ACF of MA(q)

Answer 20

cuts off after q

Question 21

2 methods of estimating parameters

Answer 21

MLE, MOM

Question 22

Easiest model selection method

Answer 22

Plot the SACF and SPACF

Question 23

What is Portmanteau stat

Answer 23

Function of the SACF of the residuals…distributed as chi squ.

Question 24

What is box-jenkins method of forcasting

Answer 24

Iteration method of getting forward results.