Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

EC226 > Term 2 lecture notes 2 autocorrelaton > Flashcards

Term 2 lecture notes 2 autocorrelaton Flashcards

1
Q

What is Autocorrelation?

A

It measures how today’s value of something is compared to t-1 or t-2 etc

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What does the ACF do?

A

It plots correlation p(zt, zt-1) = cov(zt, zt-1) / sqrt v(zt) . v(zt-j)

gamma j / gamma 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What is gamma j?
What is gamma 0

A

Cov(zt, zt-j) = gamma j
V(zt) = gamma 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

What allows p(zt, zt-j) = gamma j/ gamma 0

A

In reality this = cov(zt, zt-j) / sqrt v(zt) . v(zt-j)

as stationarity means v(zt) = sigma squared for all t

V(zt-j) = V(zt) so then denominator = V(Zt)^2

if this is squarerooted it is V(zt)

therefore it because gamma 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What can p(zt, zt-j) also be written as?

what happens as j gets bigger?

What is p0?

A

= pj

as j gets bigger pj tends to 0

p0 = 1 as cov(zt,zt)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What does the autocorrelation function plot graphically?

Explain what this graph is showing?

A

x axis = j
y axis row j

At period 0, x axis is at 0 and y axis is at 1 as it is the correlation with zt and itself

At period 1 x axis is 1 and then next value

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

In words how can the autocorrelation function be remembered?

A

after a shock in period 0 how much of it is remembered j periods later

Gives a pictorial representation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What does it mean if in a graph a row j is negative?

A

it means it has a negative correlation with the shock but could be persistent

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What is a white noise process?

What is the functional form and what are the assumptions?

A

A perfectly unforcastable process that has no information in it whatsoever.

zt = epsilont

assumptions
E(zt) = 0
V(epsilont) = sigma^2
Cov(zt, zt-j) = 0 j is not equaled to 0
Varies normally (0, sigma^2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

How do you graphically see a white noise process?

A

x axis j
y axis row j

cross at rowj = 1

Then all the j are 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What can zt be?

A

Some series eg inflation
or some residuals

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What is the set up of an AR(1) model?

What is the most important condition?

A
  1. Zt = phi . Zt-1 + epsilon t

assumptions
E(epsilon t) = 0
V(epsilon t) = sigma squared
Cov(et,et-j) = 0 for j not equal to 0

  1. abs value of phi less than 1 for it to be stationary (roots must lie within unit circle)
    this implies zt is stationary so:

E(zt) = mew
V(zt) = sigma squared
Cov(zt, zt-j) = gamma j

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Graphically what does the AR(1) model look like?
What does a negative phi pattern look like specifcially

What is pj in the AR(1)

A

with phi > 0 Smooth geometric decay to 0
with phi < 0 zig zag decay to 0

pj = phi^j

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What does phi measure in AR(1)

What does it mean if phi = 0.9
0.5

What does it mean if phi is negative?

A

The persistence of a shock

shock is very persistent

effect of shock halves every period less persistent

if phi is negative it means the shock has a negative impact (negative relationship with lagged zt

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

What is a way in words to describe a AR(1)

A

When you have been shocked /shoved off equilibrium path how long does it take to get back on equilibrium path.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What is a set up of an AR(2) model?

A

Zt = phi1 . Zt-1 + phi2 . Zt-2 + epsilon
E(epsilont) = 0
V(epsilont) = sigma squared
Cov(epsilont, epsilont-1) = 0
epsilon varies N (0, epsilon t)

abs value of phi +1 + phi 2 less than one for stationarity which is:
E(zt) = mew for all t
V(zt) = sigma squared for all t
Cov(zt,zt-j) = gamma j

17
Q

How is the shape of the AR(2) ACF plot what does it depend on?

A

Depends on if phi 1 is pos or neg
phi 2 is pos or neg
relative size of phi 1 to phi 2

18
Q

How could an AR(2) look like an
AR(1) smooth decay
AR(1) zig zag

How could an oscilatory AR(2) look like?

A

AR(1) smooth decay if phi 1 and phi 2 are positive but phi 1 is bigger than phi 2
Phi 1 is negative , phi 2 is negative , phi 1 must be bigger in absolute sense.

If it has complex roots.

19
Q

What is an AR(3) model and therefore AR(P)

A

Zt = phi1 . zt-1 + phi2 . zt-2 + phi3 . zt-3

Zt = phi1 . zt-1 + phi2 . zt-2 + phi3 . zt-3 +phi p . zt-p

20
Q

What is the set up of an MA(1) model?

What does it do?

What is the set up of an MA(2) model?

What makes it different?

A

Zt = theta . epsilon t-1 + epsilon

it remembers the shock for one period

Zt = theta1 . epsilon t-1 + theta2 . epsilon t-2 + epsilon

21
Q

What is an ARMA(1,1) model?

What is an ARMA (0,1)

A

It has an autoregressive part and a MA part

zt = phi1 . zt-1 + theta1 . et-1

ARMA(0,1) is a MA(1)

22
Q

What are the roots of an AR model?

A

AR(1) has one root
AR(2) has two roots
AR(3) has three roots

23
Q

How does stationarity link to roots?

A

If all roots lie within the unit circle the process is said to be stationary

24
Q

Why do we calculate roots?

A

Tells us if process that generated that process is stable or explosive

25
Q

How do we calculate root of AR(1)
First give working then give short cut

how do we know if its stationary?

A

Zt = phi1 . Zt-1
LZt = Zt-1

Zt = Phi1 . Lzt + epsilont
Zt(1-phi1L) = epsilon t

let w = 1/L
(1-phi1/w)

w - phi1 = 0

w = phi1

if abs value of w is less than 1 it is stationary

26
Q

How do we calculate the root of AR(2)

How do we know if it is stationary?

What are the two conditions of stationarity in AR(2)?

Whaha roots in AR(2)?

A

Same method as subbing in lagged operator collecting like Zt then plugging in w=1/L then timsing through w^2

w^2 - phi1 . w - phi2

if w1 and w2 are both abs value less than 1 it is stationary

phi 1 + phi2 < 0
Phi 1- phi2 <0

the part under root sign is negative

27
Q

What is the roots situation in an MA process?

Is MA stationary?

A

We do not talk about roots but talk instead about invertability

It is a linear sum of white noise processes so must be stationary

28
Q

What does it mean if a process is invertible?

What does this mean?

A

if it can be written as an infinite AR

It means it can be written

29
Q

How do you solve for if a MA(1) process is invertible?

How do you solve if MA(2) is invertible?

A

-You use the same lagged method as AR and if abs value of w is less than one MA is invertible

-You use lagged operator model and then sub in l equation for w then solve for roots.

30
Q

Why are PACF functions useful?

A

Sometimes it is difficult to differentiate between ACF so PACF gives more information:

PACF shows what is the contribution of lag zt-j in explaining zt holding constant all the previous lags.

31
Q

How do you derive the PACF of an MA?

A

You use backward substituion of lags to make it an infinite AR

31
Q

How do you derive a PACF for AR model?

what does p33 show

A

Zt = p11 .zt-1 + epsilon t then save p11
Zt = p21 .zt-1 + p22 . zt-2 + epsilon save p22
Zt = p31 . Zt-1 + p32 . Zt-2 + p33 . Zt-3 + epsilon

The ffect of the lag zt-3 on Zt holding all other lags constant

32
Q
A
32
Q
A

EC226 (23 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions