Term 2 lecture notes stationarity Flashcards
How could you show why a
yt = mew + phi . yt-1 + epsilont
is statonary
Backward substitute it and show that it becomes an infinite geometric sequence that can be written as M/ 1- phi + phit . y0 + sum of j0 to t-1 . et-j
phi^t tends to zero this term tends to zero and at et-j is 0 it tends to zero
so it ends up being mew / 1- phi
This shows that mean of stationary process does not depend on time and is constant
Doing the same for variance it collapses to something that does not depend on time
Cov eventually goes back to mean of process
How could you show why yt = mew + yt-1 + epsilon is non-stationary
In which direction is it not stationary?
Backward substitute
Then get tmew + y0 + sum of et-j
Then take expectations
you are left with t mew + y0 which shows the series depends on time
if mew is positive it is increasing if mew is negative it is decreasing.
The variance depends on t
The cov shows that if you get pushed off equilibrium path you will never go back. The correlation depends upon t
Can a non-stationary process satisfy weak stationarity?
If a non stationary process has a mean zero it can satisfy weak stationarity.
What can we say about uncertainty in a non-stationary model?
Uncertainty increases in a inear rate through time as the mean and variance are multiplied by t
Graphically what is the difference between a stationary and non-stationary model?
What is another major difference with the interpretation of stationary and non stationary models
Jumps around constant mean
Spread whilst jumping around will be constant
Non-stationary is accumulating past shocks without discounting them so is increasing the level of uncertainty
In stationary models you will go back to equilibrium path whilst when shoved off path in non-stationary you will never go back
Graphically how can you show how Non stationary impacts ACF
- in infinite it is a horizontal line
-in finite it is a smooth linear decay
How can you breakdown the format of a non stationary model
yt = tmew (drift) + y0 (initial value) + Sigmaet-j (stochastic strend)
Why is it that a non-stationary model has such so much uncertainty?
Due to the stochastic trend the variance is getting increasingly larger which decreases the chance of being
As if you have an inifinite variance the chance you go back to the point you started at is approaching 0
What is the ADF
Augmented Dicky Fuller Test
and how do you do? for AR 1
-to test for stationarity
H0: gamma = 0 unit root non stationary
H1: gamma < 0 stationary
Then use a classic T test but follow the DF distribution
- look at if it has constant and trend then do
CV = the CV - C1/ T - C2/T^2\
Only reject if test statistic is more negative than CV
When rewriting an equation for ADF test how many lags do you include?
1 less than the original equation.
How can a non stationary series be described?
-A series when shocked does not return to equilibrium path
What is perron’s result?
What are the scenarios in which this can happen?
If you get a stationary series with a structural break an ADF test will identify non-stationarity
Can also happen if there is a structural break in time trend.
What is the power of ADF test?
Probability of reject H0 when H0 is truly false.
What happens under the null with a series with no constant and no trend?
Model A with no constant and no trend as under the null of (Non-stationary) it means that the series must start at 0 but this is unlikely for all macro time series.
What happens under the null hypothesis when a model has a constant but no time trend
Under alternative?
When is it used?
The expectation is tmew + y0
so it causes a trend but with increasing uncertainty .
Under alternative (stationary) it is a constant)
You use it if you think the series does not exhibit a time trend
