Stationarity, AR(1) Models & Unit Root Tests Flashcards
What is stationary?
It is where statistical properties don’t change over time
This means:
1) the mean is constant over time
2) the variance is constant over time
) the covariance (correlation) between values depends only on the time gap, not the time itself
Why does stationary matter?
OLS only works properly on stationary data
Non-stationary data can lead to spurious regression (i.e false relationships)
What are the conditions for weak stationary
Yt is weak stationary if:
1) Mean does not depend on time
E(Yt) = u (constant mean)
2) Variance does not depend on time
Var(Yt) = σ2 (constant variance)
3) Covariance depends only on time difference
Cov(Yt,Yt-h) depends only on h (gap) not t
What is the AR(1) model?
The Autoregressive (AR) Model uses past values of Y to predict the present:
Yt=pYt-1 +ut
where:
p = autocorrelation coefficient
ut = white noise error term
What are the stationary conditions for AR(1)?
The process is stationary if ∣ρ∣<1.
if p = 1, the process is a random walk (non-stationary)
How can a time series have a unit root?
when p=1 it is a unit root meaning
Yt= Yt-1 +ut
What does a unit root cause?
It makes the series non stationary and randomly drift over time
How do we test for a unit root?
Use the Dickey-Fuller (DF) test, which tests:
H0: p=1 (unit root -> non stationary)
HA: P < 1 (stationary)
What is the dickey-Fuller test regression?
ΔYt=α+γYt−1+ut
(unit root is represented by the gamma symbol)
what do the results of the Dickey-Fuller test mean?
If the unit root is significantly negative then reject H0 = stationary
If the Unit root is not significant, fail to reject H0 = series has a unit root
How to interpret the DF test?
if the test statistic is more negative than the critical value, the series is stationary
if it is less negative than the critical value, the series has a unit root
What does an AR(1) model look like?
Yt=α+ρYt−1+ut
When is an AR(1) process stationary?
When ∣ρ∣<1
What is the interpretation of ρ=1?
The series has a unit root and is non-stationary (random walk)
What does 𝜌 represent in an AR(1) model?
It’s the autoregressive coefficient — how much the current value depends on the lagged value.
What does it mean if ∣ρ∣<1?
The series is stationary (its mean, variance, and autocovariance are constant over time)
What are the consequences of running OLS on a non-stationary series?
It can lead to spurious regression — high
𝑅^2, significant t-stats, but no real relationship
