L6 TS models

Question 1

What is a strictly stationary process?

Answer 1

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

Question 2

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

Answer 2

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

Question 3

What is γ(s)?

Answer 3

Autocovariance; the covariance between period t and period s

Question 4

What is the autocorrelation function (/correlogram)?

Answer 4

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

Question 5

What is a white noise process?

Answer 5

A process with virtually no discernible structure

Question 6

What equations define a WNP?

Answer 6

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)

Question 7

What will a WNP ACF look like?

Answer 7

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

Question 8

What does the Box-Piece test test?

Answer 8

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

Question 9

How is the Q-statistics distributed?

Answer 9

Asymptotically as a chi-squared(m)

Question 10

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

Answer 10

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)

Question 11

See

Answer 11

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

Question 12

What is Wold’s decomposition theorem?

Answer 12

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)

Question 13

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

Answer 13

Decay exponentially to zero

Question 14

See and learn

Answer 14

Examples 3i, ii and iii in my notes

Question 15

See and learn

Answer 15

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

Question 16

What does the PACF measure? How is it denoted?

Answer 16

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)

Question 17

When will the PACF=ACF?

Answer 17

At lag 1

Question 18

What is the PACF useful for?

Answer 18

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

Question 19

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

Answer 19

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)

Question 20

What is an ARMA model?

Answer 20

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

Question 21

3 conditions an ARMA model satisfies?

Answer 21

E(ut)=0
E(ut^2)=σ^2
E(ut,us)=0 for all t not equal to s

Question 22

What is the invertibility condition?

Answer 22

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)

Question 23

How will the ACF look for an ARMA model?

Answer 23

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

Question 24

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

Answer 24

Geometrically decaying

Number of spikes in PACF=AR order (p)

Question 25

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

Answer 25

Geometrically decaying PACF

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

Question 26

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

Answer 26

1) Identification
2) Estimation
3) Model diagnostic checking

Question 27

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

Answer 27

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

Question 28

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

Answer 28

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

Question 29

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

Answer 29

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

Question 30

What is an updated way of doing the identification step of Box-Jenkins?

Answer 30

Identification would not typically be done using ACFs since we want to form a parsimonious model
Since the variance of estimators is inversely proportional to the number of DofF, this means that excessive models may be inclined to fit the features of the data -> MOTOVATION FOR INFORMATION CRITERIA!

Question 31

What are the two key features of an information criteria?

Answer 31

1) It must have a term that is a function of the RSS

2) There must be a penalty for adding extra parameters

Question 32

How should we use an information criteria?

Answer 32

We should choose the criteria that minimises the information criteria

Question 33

What are the three most popular information criteria?

Answer 33

1) Akaike’s IC
2) Schwarz’s Bayesian IC
3) Hannan-Quin IC

Question 34

Comparison between SBIC and AIC?

Answer 34

SBIC embodies a stiffer penalty

Question 35

2 characteristics of the SBIC? (pro and con)

Answer 35

Pro: strongly consistent
Con: inefficient

Question 36

2 cons of the AIC?

Answer 36

Not consistent, and will typically choose ‘larger’ models than the AIC

Question 37

What is an ARIMA model?

Answer 37

I stands for integrated; an integrated AR process is one with a characteristic root on the unit circle

Question 38

How will researchers typically deal with ARIMA models? What is the relationship between ARMA and ARIMA models?

Answer 38

They will difference the variable as necessary then build an ARMA model on the differenced variables - an ARMA (p,q) model in the variable differenced ‘d’ times is equivalent to an ARIMA (p,q,d) model on the original data?

Question 39

What is exponential smoothing?

Answer 39

Exponential smoothing helps us to determine the weight we attach to previous observations when forecasting future ones - we expect recent observations to carry the most power in helping forecast future values of a series

Question 40

See

Answer 40

Exponential smoothing in notes

Question 41

3 reasons single/simple exponential smoothing does not work well with financial data?

Answer 41

1) There is little structure to smooth
2) It cannot allow for seasonality
3) Forecasts do not converge on the LT mean as s tends to infinity

Question 42

How would we modify single exponential smoothing to allow for: a) trends? or b) seasonality?

Answer 42

Trend - Holt’s method

Seasonality - Winter’s method

Question 43

2 advantages of exponential smoothing?

Answer 43

simple to use

easy to update model if new realisation becomes available (finish)