Term 2 H2 Serial Correlation

Question 1

What is serial correlation?

Answer 1

Some form of linear dependence in a series.

Question 2

how do you measure an ACF?

What is particular if the model has no lags?

Answer 2

corr(zt, zt−k) = =
cov(zt, zt−k)/
V (zt)
=
γk/
γ0
= pk

where ρ0 = 1 and |ρj | < 1, j ≥ 1.

Question 3

What is the autocorrelation function?

What is this telling us?

Answer 3

This is a pictorial representation of the linear dependence

Plots values of row j against j

-Telling us proportion of intial shock that is remembered in time j
memory of shocks.

Question 4

What are the 4 types of ACF models?

Answer 4

White noise
AR (Auto regressive)
MA (moving average)
ARMA (auto reg moving average)

Question 5

What is the more simple way to present the formula of ACF?

What is the assumption under which this holds?

Answer 5

-gamma j = cov(zt,zt-1)
gamma 0 = v(zt)

i) E(zt) = m for all time
ii) V(zt) =sigma ^2
iii) cov(zt, zt-h) = gamma h

Question 6

1. What is a white noise process

Answer 6

zt = εt
E(εt) = 0 (constant mean)
V(εt) = σ^2 (constant variance)
cov(zt, zt−j ) = 0 j not equaled to 0
εt is normally distributed (perfectly unforcastable)

Question 7

What would the plot of a white noise process look like?

Answer 7

there is only 1 point in the j=0.
If you shock the process today the 100% of the shock remains today but then dissipates out of the system.

Question 8

What is the graphical representation of an AR(1):

a) with ϕ > 0

b) with ϕ < 0

c)What does phi change?

Answer 8

a) smooth decay to zero

b) zig zaggy decay to 0

c) the higher the phi the more persistent the shock is.

Question 9

What is the basic construction of an AR(1) model?

Answer 9

ρj = ϕ^j

Question 10

What are the yule walker equations?

Answer 10

• Yule-Walker Equations:
  method to estimate the coefficients of an AR model using the autocorrelations of the time series.

-Gammas (γ) in the Yule-Walker Equations:
The ‘gammas’ (γ) represent the autocovariance values of the time series at different lags.

phi is the coefficient

by solving this system of equations if gives you the value of phi

Question 11

What is an analogy to think about the ACF

Answer 11

After being shoved of a path how fast and with what method can you be moved back to equilbrium

Question 12

What is the form of an AR(1) ?

Answer 12

zt = ϕzt−1 + εt

if zt is stationary absolute phi1<1

E(εt) = 0
v(εt) = sigma^2
cov(zt, zt−j ) = 0 j not equaled to 0

Question 13

what is the form of the AR(2)?

Answer 13

zt = ϕ1zt−1 + ϕ2zt−2 + εt

if zt is stationary absolute phi1+phi2<1
E(εt) = 0
v(εt) = sigma^2
cov(zt, zt−j ) = 0 j not equaled to 0

Question 14

What does gamma stand for in the AR(1)

Answer 14

gamma 0 = variance of t as it is the covariance of zt and zt

gamma 1 = covariance of zt and zt-1

Question 15

What is the recursive sequence that can be used for the AR(2) model?

1.do it for gamma
2.do it for row

Answer 15

gamma j = phi 1 gamma j-1+ phi 2 gamma j-2

pj= gamma j/ gamma 0

row j = phi 1 row j -1 + phi 2 row j -2

Question 16

What does the AR(2) shape and style of ACF depend on?

Answer 16

1. phi1 is positive / negative
2. phi 2 is positive / negative
3. relative of phi 1 to phi 2

Question 17

For the AR(2) what are the different possibilities in shape and what do these depend on?

Answer 17

1. Smooth decay to 0, depends phi 1 >0 and phi 2 > 0 but phi 1 is much bigger than phi 2.
2. Zig Zag to 0 phi 1 < 0 and phi 2 < 0 but phi 1 is much smaller than phi 2
3. Oscillatory nature to 0, phi 1 and phi 2 have complex roots

Question 18

What is the format for an AR(3) model
and what are the assumptions

Answer 18

zt = phi1zt-1 c+ phi2zt-2 + phi3zt-3 + epsilon

Constant mean
Constant variance

Question 19

What is the AR(P) format?

What are the yule walker equations for an AR(P)?

Answer 19

zt = ϕ1zt−1 + ϕ2zt−2 + . . . + ϕpzt−p + εt

γ1 = ϕ1γ0 + ϕ2γ1 + ϕ3γ2 + . . . ϕpγp−1

γ2 = ϕ1γ1 + ϕ2γ0 + ϕ3γ1 + . . . ϕpγp−2

γj = ϕ1γj−1 + ϕ2γj−2 + . . . + ϕpγj−p j ≥ p

Question 20

What is an MA(1) model?

Give me the format and what it means?

Answer 20

zt = θεt−1 + εt

zt is a weighted average of theta of epsilon - 1 and 100% of epsilon t.

Question 21

What feature does an MA(1) have?

How do you calculate the shock?

Answer 21

It only remembers the shock for 1 period. Then the influence of the shock is less than 0

theta / 1 + theta squared

Question 22

What is the format of a MA(2) model

give me what the special feature about this is?

Answer 22

zt = θ1εt−1 + θ2εt−2 + εt

That the shock dissipates after two periods

Question 23

What will an MA(2) look like graphically?

Answer 23

Remembers shock for 2 periods before returning to eq path (0).

Question 24

Why will an MA(p) only remember the shocks for p periods?

Answer 24

This is because after j>p gamma j = 0

Question 25

What is the arma model?

How do you set up the model?

Answer 25

Arma is a combination of AR and MA model.

Set up: has one AR parameter and one MA parameter

zt = ϕzt−1 + θ1εt−1 + εt

Question 26

Why are we interested in the roots of an AR process?

How do we investigate for this?

Answer 26

To see if it is a stable/ stationary process

if all roots are within the unit circle this means it is a stationary/ stable process.

Question 27

How does the number of roots link to the AR model?

Answer 27

An AR(p) model has p roots.H

Question 28

How do you work out the roots of an AR (1)

How would this vary for an AR(2)

Answer 28

You find the characteristic equation then you solve for W

define 1/L = w

if abs value of W is less than 1 the process is stable and zt is stationary

AR(2) you would need to check w1 and w2 are both less than 1 to be stationary.

Question 29

What are the conditions for stationarity?

Answer 29

phi 1 + phi 2 < 1
phi 2 - phi 1 < 1

Question 29

What are complex roots

and what does this imply?

Answer 29

root of phi 1 + 4 phi 2 is less than 0

Oscilatory patters is then implied.

Question 30

What are the roots in MA processes?

Answer 30

In MA processes we do not talk about roots we talk about invertability

-This is because they have to be stationary as it is the sum of white noise processes.

Question 31

What is the format of solving a AR(1) equation?

What is the format of solving an AR(2) equation

Answer 31

w - phi 1 = 0

w^2 - phi 1 (w) - phi 2 = 0

Question 32

What does it mean if an MA process is invertible?

Answer 32

it means it can be written as an infinite AR.
Can move from MA(1) or MA(2) to infinite AR

Question 33

What is the format of solving for invertability of

AR(1)

AR(2)

Answer 33

• AR(1)

w - phi1 = 0
if abs value of w < 1 then it is invertible

-AR(2)
W^2- phi1w- phi 2 = 0
if w1 and w2 abs value are less than 1 it is invertible

Question 34

What is a PACF and why is it useful?

Answer 34

-Contribution of zt-j to zt having held constant the rest of the lags.

-complement tool Another tools which can be used to distinguish the type of serial correlation present in some series

Question 35

How do you construct a PACF?

Answer 35

regress zt = p11zt-1 + εt save p11
regress zt = p21zt-1 + p22zt-2 + ε save p22
regress zt = p31zt-1+p32zt-2+p33zt-3 + ε save p33

for j terms
zt = pj1zt-1+pj2zt-2+pj3zt-3+pjjzt-j ε save pjj

Question 35

How does a AR(1) vary with a PACF graphically?

and why is this?

Answer 35

AR(1) would have smooth decay
Whilst PACF would have only one significant term and the rest 0 crosses.

This is because you are comparing a AR(1) to a PACF and the AR(1) is the true model and this causes many of the terms to drop out as zero.

Question 36

How does a AR(2) vary with an PACF?

Answer 36

AR(2) should typically have smooth decay
PACF would have two signifiant terms

Question 37

How can you get the PACF from an MA?

Answer 37

You write it as an infinite sum of a geometric series which is just an infinite AR.

Question 38

What is the important step do go from ACF to PACF

What is an issue that could arise?

Answer 38

See it as the normal ACF is the true model and that the PACF is the estimated model and then compare the two.

There is omitted variable bias.

Question 40

How does an MA(1) and PACF for it vary graphically?

Answer 40

MA(1) would be non-zero for one point
PACF would be non-zero for all points increasingly getting closer to zero zig zagging up and down.

Question 41

What is the link between serial correlation and unbiasedness in a non dynamic model?

Answer 41

• serial correlation does not impact biasedness if there is no lag of the dependent variable.

Question 42

What is the link between serial correlation and variance? Non-dynamic

Answer 42

• in the presence of serial correlation the OLS estimate of the variance is no longer suitable as it is the variance + something.

Question 43

How do you deal with the variance in the presence of serial correlation?

Answer 43

You use the Newey-West Heteroscadastic AC consistent standard error

Question 44

What is the link between serial correlation and unbiasedness in mode with lagged dependent variable.

Answer 44

as covariance between y and error term is not zero
and covariance between error term and error -1 is not equal to zero

This means that error term is correlated to yt-1

Therefore, even as T tends to infinity OLS is both biased and inconsistent.

Question 45

How do you combat the issue with dependent lagged variables with serial correlation.

What method is it?

Answer 45

You transform the equation to have a serially uncorrelated error term to at least yield consistent estimates.

generalised least squares method.