Term 2 lecture notes 2 autocorrelaton

Question 1

What is Autocorrelation?

Answer 1

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

Question 2

What does the ACF do?

Answer 2

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

gamma j / gamma 0

Question 3

What is gamma j?
What is gamma 0

Answer 3

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

Question 4

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

Answer 4

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

Question 5

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

what happens as j gets bigger?

What is p0?

Answer 5

= pj

as j gets bigger pj tends to 0

p0 = 1 as cov(zt,zt)

Question 6

What does the autocorrelation function plot graphically?

Explain what this graph is showing?

Answer 6

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

Question 7

In words how can the autocorrelation function be remembered?

Answer 7

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

Gives a pictorial representation

Question 8

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

Answer 8

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

Question 9

What is a white noise process?

What is the functional form and what are the assumptions?

Answer 9

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)

Question 10

How do you graphically see a white noise process?

Answer 10

x axis j
y axis row j

cross at rowj = 1

Then all the j are 0

Question 11

What can zt be?

Answer 11

Some series eg inflation
or some residuals

Question 12

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

What is the most important condition?

Answer 12

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

2. 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

Question 13

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)

Answer 13

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

pj = phi^j

Question 14

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?

Answer 14

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

Question 15

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

Answer 15

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

Question 16

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

Answer 16

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

Question 17

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

Answer 17

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

Question 18

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

How could an oscilatory AR(2) look like?

Answer 18

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.

Question 19

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

Answer 19

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

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

Question 20

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?

Answer 20

Zt = theta . epsilon t-1 + epsilon

it remembers the shock for one period

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

Question 21

What is an ARMA(1,1) model?

What is an ARMA (0,1)

Answer 21

It has an autoregressive part and a MA part

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

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

Question 22

What are the roots of an AR model?

Answer 22

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

Question 23

How does stationarity link to roots?

Answer 23

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

Question 24

Why do we calculate roots?

Answer 24

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

Question 25

How do we calculate root of AR(1) First give working then give short cut how do we know if its stationary?

Answer 25

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

Question 26

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)?

Answer 26

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

Question 27

What is the roots situation in an MA process? Is MA stationary?

Answer 27

We do not talk about roots but talk instead about invertability It is a linear sum of white noise processes so must be stationary

Question 28

What does it mean if a process is invertible? What does this mean?

Answer 28

if it can be written as an infinite AR It means it can be written

Question 29

How do you solve for if a MA(1) process is invertible? How do you solve if MA(2) is invertible?

Answer 29

-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.

Question 30

Why are PACF functions useful?

Answer 30

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.

Question 31

How do you derive the PACF of an MA?

Answer 31

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

Question 31

How do you derive a PACF for AR model? what does p33 show

Answer 31

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