Term 2 lecture notes 2 autocorrelaton Flashcards
What is Autocorrelation?
It measures how today’s value of something is compared to t-1 or t-2 etc
What does the ACF do?
It plots correlation p(zt, zt-1) = cov(zt, zt-1) / sqrt v(zt) . v(zt-j)
gamma j / gamma 0
What is gamma j?
What is gamma 0
Cov(zt, zt-j) = gamma j
V(zt) = gamma 0
What allows p(zt, zt-j) = gamma j/ gamma 0
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
What can p(zt, zt-j) also be written as?
what happens as j gets bigger?
What is p0?
= pj
as j gets bigger pj tends to 0
p0 = 1 as cov(zt,zt)
What does the autocorrelation function plot graphically?
Explain what this graph is showing?
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
In words how can the autocorrelation function be remembered?
after a shock in period 0 how much of it is remembered j periods later
Gives a pictorial representation
What does it mean if in a graph a row j is negative?
it means it has a negative correlation with the shock but could be persistent
What is a white noise process?
What is the functional form and what are the assumptions?
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 do you graphically see a white noise process?
x axis j
y axis row j
cross at rowj = 1
Then all the j are 0
What can zt be?
Some series eg inflation
or some residuals
What is the set up of an AR(1) model?
What is the most important condition?
- 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
- 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
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)
with phi > 0 Smooth geometric decay to 0
with phi < 0 zig zag decay to 0
pj = phi^j
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?
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
What is a way in words to describe a AR(1)
When you have been shocked /shoved off equilibrium path how long does it take to get back on equilibrium path.
What is a set up of an AR(2) model?
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
How is the shape of the AR(2) ACF plot what does it depend on?
Depends on if phi 1 is pos or neg
phi 2 is pos or neg
relative size of phi 1 to phi 2
How could an AR(2) look like an
AR(1) smooth decay
AR(1) zig zag
How could an oscilatory AR(2) look like?
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.
What is an AR(3) model and therefore AR(P)
Zt = phi1 . zt-1 + phi2 . zt-2 + phi3 . zt-3
Zt = phi1 . zt-1 + phi2 . zt-2 + phi3 . zt-3 +phi p . zt-p
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?
Zt = theta . epsilon t-1 + epsilon
it remembers the shock for one period
Zt = theta1 . epsilon t-1 + theta2 . epsilon t-2 + epsilon
What is an ARMA(1,1) model?
What is an ARMA (0,1)
It has an autoregressive part and a MA part
zt = phi1 . zt-1 + theta1 . et-1
ARMA(0,1) is a MA(1)
What are the roots of an AR model?
AR(1) has one root
AR(2) has two roots
AR(3) has three roots
How does stationarity link to roots?
If all roots lie within the unit circle the process is said to be stationary
Why do we calculate roots?
Tells us if process that generated that process is stable or explosive
How do we calculate root of AR(1)
First give working then give short cut
how do we know if its stationary?
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
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)?
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
What is the roots situation in an MA process?
Is MA stationary?
We do not talk about roots but talk instead about invertability
It is a linear sum of white noise processes so must be stationary
What does it mean if a process is invertible?
What does this mean?
if it can be written as an infinite AR
It means it can be written
How do you solve for if a MA(1) process is invertible?
How do you solve if MA(2) is invertible?
-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.
Why are PACF functions useful?
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.
How do you derive the PACF of an MA?
You use backward substituion of lags to make it an infinite AR
How do you derive a PACF for AR model?
what does p33 show
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
