HANDOUT 3

Question 1

3 conditions for weak stationarity

Answer 1

1. E(Yt) = M
2. V(Yt) = sigma^2 Y
3. COV(Yt, Yt-h) = gamma h

Question 2

Does AR1 satisfy condition 1 for weak stationarity? Give E(Yt) when we include a mean.

Answer 2

YES

E(Yt) = M / (1 - phi)

Question 3

Does AR1 satisfy condition 2 for weak stationarity? Give V(Yt) when we include a mean.

Answer 3

V(Yt) = sigma^2 / (1 - phi^2)

Question 4

Does AR1 satisfy condition 3 for weak stationarity?

Answer 4

YES
Pj = phi ^j
As long as phi<1, also satisfies weak dependence.

Question 5

Therefore, what condition do we need for an AR1 process to be stationary?

Answer 5

Phi<1 in absolute magnitude

Question 6

A process which is stationary is integrate of order…

Answer 6

I(0)

Question 7

I(1) means

Answer 7

Integrated of order 1

The first difference of the series is stationary

Question 8

How many unit roots does as I(1) series have?

Answer 8

1

Question 9

How many unit roots does an I(2) series have?

Answer 9

2

Question 10

Random walk model formula (with drift)

Answer 10

Yt = M + Yt-1 + €t
Phi = 1

Question 11

Yt = when we back substitute a random walk model

Answer 11

Yt = Mt + Y0 + sum €j

Question 12

E(Yt) for a random walk model. Does this satisfy stationarity?

Answer 12

E(Yt) = Mt + Y0

E(Yt) depends on t = not stationary

Question 13

sum €j refers to

Answer 13

Stochastic trend

Question 14

Mt =

Answer 14

deterministic trend

Question 15

V(Yt) for a random walk model. Does this satisfy stationarity?

Answer 15

V(Yt) = t sigma^2

NO - uncertainty increases over time

Question 16

Ph = for a random walk model

Answer 16

Ph = (t - h) / t

Question 17

In theory, ACF and PACF for random walk

Answer 17

ACF = horizontal line at p=1
PACF = spike for 1 at p=1, others 0

Question 18

In practice, ACF for random walk

Answer 18

Finite t

Ph decays at a LINEAR rate

Question 19

random walk without drift

Answer 19

M=0

Stochastic trend only

Question 20

random walk with drift

Answer 20

M≠0
Stochastic trend AND deterministic trend
ACF shows even less decay to zero.

Question 21

The test for unit roots / non-stationarity is called

Answer 21

Augmented Dickey-Fuller (ADF) test

Question 22

Easier method for ADF test (for p*=0)

Answer 22

change Yt = gamma Yt-1 + €t

gamma = phi - 1

Question 23

H0 and H1 for ADF test

Answer 23

H0: gamma = 0 - non-stationary
H1: gamma < 0 - stationary

Question 24

ADF test is one/two sided?

Answer 24

ONE sided - the root is either unity or less than.

Question 25

Test stat for ADF test

Answer 25

Do a t-test

Question 26

Problem with CVs for ADF test

Answer 26

We test under H0 for non-stationarity so the distribution of the test stat is shifted left compared to t/z test = more likely to reject H0.

Question 27

Solution to CVs for ADF test

Answer 27

Use Mackinnon's CVs table 10 | CVT = phi infinity + (phi1/T) + (phi2/T^2)

Question 28

When do we reject H0 for ADF test?

Answer 28

1 sided test so only reject if test stat < CV.

Question 29

What is the biggest root of an AR?

Answer 29

Biggest root = sum of all parameters

Question 30

gamma for AR2 =

Answer 30

gamma = phi 1 + phi 2 - 1

Question 31

For an AR(P), test equation =

Answer 31

change Yt = gamma Yt-1 + sum j=1,...,p* | delta j change Yt-j + €t

Question 32

state 3 ways of choosing P* for ADF test

Answer 32

1. Look at number spikes on PACF 2. Minimise a selection criteria 3. Test between different models

Question 33

Max P* =

Answer 33

T^1/3

Question 34

How do we test between P*=3 and P*=2?

Answer 34

Test H0: delta 3 = 0 i.e. is the coefficient on | change Yt-3 significant? If we do not reject H0, then test between P*=2 and P*=1.

Question 35

As well as choosing P*, what else do we have to decide on for the ADF test?

Answer 35

Deterministic elements e.g. do we include an intercept and/or trend?

Question 36

Model A =

Answer 36

no intercept, no trend

Question 37

Model B =

Answer 37

intercept, but no trend

Question 38

Model C=

Answer 38

both an intercept and a trend

Question 39

When should we use Model A? explain.

Answer 39

NEVER Under H0, E(Yt) = Y0 Under H1, E(Yt) = 0 Only reconcile by assuming Y0=0 under H0 but unrealistic to assume the initial value of any variable is zero.

Question 40

When should we use Model B? explain.

Answer 40

Under H0, E(Yt) = Mt + Y0 - time trend Under H1, E(Yt) = M / (1 - phi) - constant To reconcile assume M = 0 under H0 Only good if realistic to assume zero average growth rate e.g. ER, IR, inflation

Question 41

When should we use Model C? explain.

Answer 41

Most of the time Under H0, E(Yt) = Mt + Y0 + alpha t(t+1)/2 Under H1, E(Yt) = M/1-phi + alpha/1-phi t Assume alpha=0 under H0 Realistic - the growth rate of a variable doesn't usually following a trend.

Question 42

Power of ADF test

Answer 42

LOW POWER = we often do not reject H0 = often find non-stationarity when actually series is stationary.

Question 43

Perron's result

Answer 43

A stationary series with a structural break (in intercept/trend) will be found to be non-stationary in an ADF test.

Question 44

If we do not reject H0, how can we test to see if the series is I(1) or actually I(2)?

Answer 44

change^2 Yt = M + gamma changeYt-1 + sum delta j change^2 Yt-j + €t H0: gamma=0 - series is I(2) H1: gamma < 0 - series is I(1) as 1st difference is stationary.

Question 45

Are many economic series I(2)?

Answer 45

Not many - maybe Venezuelan prices | NO economic series > I(2)