HANDOUT 3 Flashcards
3 conditions for weak stationarity
- E(Yt) = M
- V(Yt) = sigma^2 Y
- COV(Yt, Yt-h) = gamma h
Does AR1 satisfy condition 1 for weak stationarity? Give E(Yt) when we include a mean.
YES
E(Yt) = M / (1 - phi)
Does AR1 satisfy condition 2 for weak stationarity? Give V(Yt) when we include a mean.
V(Yt) = sigma^2 / (1 - phi^2)
Does AR1 satisfy condition 3 for weak stationarity?
YES
Pj = phi ^j
As long as phi<1, also satisfies weak dependence.
Therefore, what condition do we need for an AR1 process to be stationary?
Phi<1 in absolute magnitude
A process which is stationary is integrate of order…
I(0)
I(1) means
Integrated of order 1
The first difference of the series is stationary
How many unit roots does as I(1) series have?
1
How many unit roots does an I(2) series have?
2
Random walk model formula (with drift)
Yt = M + Yt-1 + €t Phi = 1
Yt = when we back substitute a random walk model
Yt = Mt + Y0 + sum €j
E(Yt) for a random walk model. Does this satisfy stationarity?
E(Yt) = Mt + Y0
E(Yt) depends on t = not stationary
sum €j refers to
Stochastic trend
Mt =
deterministic trend
V(Yt) for a random walk model. Does this satisfy stationarity?
V(Yt) = t sigma^2
NO - uncertainty increases over time
Ph = for a random walk model
Ph = (t - h) / t
In theory, ACF and PACF for random walk
ACF = horizontal line at p=1 PACF = spike for 1 at p=1, others 0
In practice, ACF for random walk
Finite t
Ph decays at a LINEAR rate
random walk without drift
M=0
Stochastic trend only
random walk with drift
M≠0
Stochastic trend AND deterministic trend
ACF shows even less decay to zero.
The test for unit roots / non-stationarity is called
Augmented Dickey-Fuller (ADF) test
Easier method for ADF test (for p*=0)
change Yt = gamma Yt-1 + €t
gamma = phi - 1
H0 and H1 for ADF test
H0: gamma = 0 - non-stationary
H1: gamma < 0 - stationary
ADF test is one/two sided?
ONE sided - the root is either unity or less than.
Test stat for ADF test
Do a t-test
Problem with CVs for ADF test
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.
Solution to CVs for ADF test
Use Mackinnon’s CVs table 10
CVT = phi infinity + (phi1/T) + (phi2/T^2)
When do we reject H0 for ADF test?
1 sided test so only reject if test stat < CV.
What is the biggest root of an AR?
Biggest root = sum of all parameters
gamma for AR2 =
gamma = phi 1 + phi 2 - 1
For an AR(P), test equation =
change Yt = gamma Yt-1 + sum j=1,…,p*
delta j change Yt-j + €t
state 3 ways of choosing P* for ADF test
- Look at number spikes on PACF
- Minimise a selection criteria
- Test between different models
Max P* =
T^1/3
How do we test between P=3 and P=2?
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.
As well as choosing P*, what else do we have to decide on for the ADF test?
Deterministic elements e.g. do we include an intercept and/or trend?
Model A =
no intercept, no trend
Model B =
intercept, but no trend
Model C=
both an intercept and a trend
When should we use Model A? explain.
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.
When should we use Model B? explain.
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
When should we use Model C? explain.
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.
Power of ADF test
LOW POWER = we often do not reject H0 = often find non-stationarity when actually series is stationary.
Perron’s result
A stationary series with a structural break (in intercept/trend) will be found to be non-stationary in an ADF test.
If we do not reject H0, how can we test to see if the series is I(1) or actually I(2)?
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.
Are many economic series I(2)?
Not many - maybe Venezuelan prices
NO economic series > I(2)
