# L6 TS models Flashcards Preview

## advanced econometric theory > L6 TS models > Flashcards

Flashcards in L6 TS models Deck (43)
1
Q

What is a strictly stationary process?

A

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

2
Q

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

A

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
Q

What is γ(s)?

A

Autocovariance; the covariance between period t and period s

4
Q

What is the autocorrelation function (/correlogram)?

A

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
Q

What is a white noise process?

A

A process with virtually no discernible structure

6
Q

What equations define a WNP?

A

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
Q

What will a WNP ACF look like?

A

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

8
Q

What does the Box-Piece test test?

A

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

9
Q

How is the Q-statistics distributed?

A

Asymptotically as a chi-squared(m)

10
Q

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

A

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
Q

See

A

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

12
Q

What is Wold’s decomposition theorem?

A

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
Q

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

A

Decay exponentially to zero

14
Q

See and learn

A

Examples 3i, ii and iii in my notes

15
Q

See and learn

A

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

16
Q

What does the PACF measure? How is it denoted?

A

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
Q

When will the PACF=ACF?

A

At lag 1

18
Q

What is the PACF useful for?

A

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
Q

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

A

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
Q

What is an ARMA model?

A

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

21
Q

3 conditions an ARMA model satisfies?

A

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

22
Q

What is the invertibility condition?

A

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
Q

How will the ACF look for an ARMA model?

A

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
Q

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

A

Geometrically decaying

Number of spikes in PACF=AR order (p)

25
Q

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

A

Geometrically decaying PACF

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

26
Q

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

A

1) Identification
2) Estimation
3) Model diagnostic checking

27
Q

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

A

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

28
Q

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

A

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

29
Q

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

A

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

30
Q

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

A

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!

31
Q

What are the two key features of an information criteria?

A

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

2) There must be a penalty for adding extra parameters

32
Q

How should we use an information criteria?

A

We should choose the criteria that minimises the information criteria

33
Q

What are the three most popular information criteria?

A

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

34
Q

Comparison between SBIC and AIC?

A

SBIC embodies a stiffer penalty

35
Q

2 characteristics of the SBIC? (pro and con)

A

Pro: strongly consistent
Con: inefficient

36
Q

2 cons of the AIC?

A

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

37
Q

What is an ARIMA model?

A

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

38
Q

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

A

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?

39
Q

What is exponential smoothing?

A

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

40
Q

See

A

Exponential smoothing in notes

41
Q

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

A

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

42
Q

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

A

Trend - Holt’s method

Seasonality - Winter’s method

43
Q