Flashcards in L6 TS models Deck (43)

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1

## What is a strictly stationary process?

### A stochastic process whose unconditional probability distribution does not change when shifted in time (see notes)

2

## What is a weakly stationary (ie. covariance stationary) process? What 3 conditions must it satisfy?

###
A strictly stationary process whereby the covariance can change over time

It must satisfy the following 3 equations:

1) E(yt) = μ for t=1 to infinity

2) E(yt-μ)(yt-μ)=σ^2 (ie. is constant and less than infinite)

3) E(yt1-μ)(yt2-μ)=γ(t2-t1) (covariance) for all t1 and t2

3

## What is γ(s)?

### Autocovariance; the covariance between period t and period s

4

## What is the autocorrelation function (/correlogram)?

###
Plot of τs against s=0,1,2...

ie. shows the autocorrelation between the current period and period 's' as you go further into the past

5

## What is a white noise process?

### A process with virtually no discernible structure

6

## What equations define a WNP?

###
1) E(yt)=μ

2) var(yt)=σ^2

3) γ(t-r) = 0 for all t is not equal to r (σ^2 otherwise)

7

## What will a WNP ACF look like?

### It will be 0 at all points apart from a single peak at of 1 (ie. correlation=1)

8

## What does the Box-Piece test test?

### It tests the joint hypothesis that all m of the τk correlation coefficients are SIMULTANEOUSLY EQUAL TO ZERO

9

## How is the Q-statistics distributed?

### Asymptotically as a chi-squared(m)

10

## What is the condition for stationarity for an AR(p) model?

### Condition for stationarity in an AR(p) model is that the roots of 1-Ø1z-Ø2z(2)-...-Øpz(p)=0 all lie outside the unit circle (see notes P1S2)

11

## See

### Notes P1S2 'testing for stationarity of an AR(p) model

12

## What is Wold's decomposition theorem?

### Any stationary AR(p) series can be decomposed into the sum of two uncorrelated processes; a purely deterministic part and a purely stochastic part, which will be a MA(infinity)

13

## If an AR model is stationary, what will its ACF (autocorrelation function) do?

### Decay exponentially to zero

14

## See and learn

### Examples 3i, ii and iii in my notes

15

## See and learn

### recursive structure of an AR(1) process (in notes P2S1)

16

## What does the PACF measure? How is it denoted?

### Denoted τkk, it measures the correlation between an observation k periods ago, and the current observation, after controlling for observations at intermediate lags (ie. all lags less than k)

17

## When will the PACF=ACF?

### At lag 1

18

## What is the PACF useful for?

###
Telling the difference between an AR process and an ARMA process

In the case of an AR(p) there are direct connections between yt and y(t-s) only for s is less than or equal to p; therefore AFTER lag p, the theoretical PACF will be zero

19

## How can an MA(q) be written and why?

### As a AR(infinity) because there are direct connections between yt and all its previous values tf for MA(q) its theoretical PACF will be geometrically declining (see notes)

20

## What is an ARMA model?

### ARMA (p,q) is made by combining the AR(p) and MA(q) models

21

## 3 conditions an ARMA model satisfies?

###
E(ut)=0

E(ut^2)=σ^2

E(ut,us)=0 for all t not equal to s

22

## What is the invertibility condition?

### The invertibility condition requires the MA(q) part of the model to have roots of θ(z)=0 greater than one (ie. outside the unit circle again)

23

## How will the ACF look for an ARMA model?

### It will display combos of behaviour derived from both the AR and MA parts, but for lags beyond q, the ACF will simply be identical to the AR(p) model

24

## How does the ACF look for an AR(p) process? How do you tell the AR order?

###
Geometrically decaying

Number of spikes in PACF=AR order (p)

25

## How does the PACF look for an MA(q) process? How do you tell the MA order?

###
Geometrically decaying PACF

Number of spikes in the ACF=MA order (q) (see slides 39-45 for examples)

26

## What are the three steps to building ARMA models via the Box-Jenkins approach?

###
1) Identification

2) Estimation

3) Model diagnostic checking

27

## What is involved in the identification step of the Box-Jenkins method?

### Need to determine the order of the model using graphical procedures (note: now are better methods of doing this)

28

## What is involved in the estimation step of the Box-Jenkins method?

### Here we estimate the parameters of the model using either least squares of MLE (depending on the model)

29

## What is involved in the model diagnostic checking step of the Box-Jenkins method?

### 2 methods of doing this: 1) deliberate overfitting and 2) residual diagnostics (learn what this actually means!)

30