Download Trend Stationarity

Cointegration, Stationarity and Error Correction Models.
From the notes you need to know:
The definition of weak or covariance stationarity on page 2 of the notes plus an
intuitive understanding of the term.
Testing for unit roots, the Dickey Fuller test and the Augmented Dickey Fuller test
The concept of cointegration (pages 2 & 3).
Testing for cointegration (Page 5-6)
The Engle Granger representation theorem
Error Correction Models – including superconsistency of the long run equilibrium
parameters (page 4-5)
The Engle Granger two step method
The Role of cointegration in economics
To supplement these we have:
Trend Stationarity
The following is an AR(1) (autoregressive with lag depth or order 1) with a
deterministic linear trend term
Yt = θYt-1 + δ + γt + εt
(1)
Where |θ| < 1
The moving average (MA) representation of this on past error terms is
Yt = θtY0 + μ0 + μ1t + εt + θεt-1 + θ2εt-2 + θ3εt-3 + …..
(2)
E(Yt) = θtY0 + μ0 + μ1t → μ0 + μ1t as t → ∞
(3)
This has a finite, unchanging variance, but no constant mean (because of μ1t). Thus
the process is not stationary.
However, the deviation from the mean
Yt = Yt – E[Yt] = Yt - μ0 - μ1t
(4)
is stationary and hence Yt is called a trend stationary process. Thus the shocks to the
process are transitory and the process is ‘mean reverting’, with the mean μ0 + μ1t
being the ‘attractor’.
UNIT ROOT PROCESSES
The following is an AR(1) model with a unit root θ=1
Yt = θYt-1 + δ + εt = Yt-1 + δ + εt
(5)
It is clearly non-staionary with no specified mean. But the process ∆Yt is stationary
and we term Yt a difference stationary process (in this case I(1))
Yt = Yt-1 + δ + εt
(6)
Yt = Yt-2 + δ + εt + δ + εt-1 (substituting for Yt-1)
(7)
Yt = Yt-3 + δ + εt + δ + εt-1 + δ + ε-2 (substituting for Yt-2)
Yt-3 + δ + δ + δ + εt + εt-1 + ε-2
(7)
And so on until:
Yt = Y0 + δt + Σεi
(8)
Hence E[Yt] = Y0 + δt
(9)
The effect of the initial value Y0 stays in the process. We can also see from (8)
that shocks have permanent effects and the figure below shows the impact of a
large shock. The process has no attractor.
DETERMINISTIC TRENDS AND TREND STATIONARITY
A time series that is stationary around a deterministic trend is called a trend
stationary process.
Examples of Different Processes
To test for trend stationarity we include a trend term and typically a constant term
in the Dickey Fuller/ADF regressions. However this changes the asymptotic
distribution relating to the test shifting the distribution to the left, the inclusion of
a time trend shifts it still further to the left which is why we need different critical
values for the DF and ADF tests depending on whether a constant and/or a time
trend is included in the specification.
But apart from this the test is the same and relates to the t statistic on Yt-1. We are
also interested in whether the coefficient on the constant term is significant.