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StationarityDefinition, meaning and consequences
Matthieu StiglerMatthieu.Stigler@gmail.com
November 14, 2008
Version 1.1
This document is released under the Creative Commons Attribution-Noncommercial 2.5 India
license.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 1 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 2 / 56
Economies of scale and increasing returns
So-called Nobel Prize has been attributed to Krugman:
Economies of scale combined with reduced transport costsalso help to explain why an increasingly larger share of the worldpopulation lives in cities and why similar economic activities areconcentrated in the same locations. [This], in turn, stimulatesfurther migration to cities.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 3 / 56
The increasing returns are:
Self production: Learning curve
Common production: adoption/network externalities
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 4 / 56
How is a keyboard organised?
0 10 20 30 40 50
01
23
45
Number of people
Ave
rage
cos
t
QWERTZDVORAK
QWERTZ vs DVORAK costs
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 5 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 6 / 56
1 Stationarity
2 ARMA models for stationary variables
3 Some extensions of the ARMA model
4 Non-stationarity
5 Seasonality
6 Non-linearities
7 Multivariate models
8 Structural VAR models
9 Cointegration the Engle and Granger approach
10 Cointegration 2: The Johansen Methodology
11 Multivariate Nonlinearities in VAR models
12 Multivariate Nonlinearities in VECM models
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 7 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 8 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 9 / 56
What is a time serie?
In econometrics, we deal with three types of data:
Cross individual data: Xi i : individu
Times series data: Xt t : time
Panel data: Xit t : time
Definition
A time series is a variable X indexed by the time t: Xt t = 1, 2, . . . ,T .
Examples
The time t can be annual, monthly, daily...
Let be Xt the annual GDP.
Let Yt be the monthly temperature
Zt be the daily stocks
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 10 / 56
Some distinctions
The time series can:
Take discrete or continuous values.
Be measured at discrete (monthly, daily) or continous time (signalprocessing, finance).
Be measured at regular or irregular intervals.
In this lecture, usually we will refer to time series that take continousvalues and are measured at discrete and regular intervalls.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 11 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 12 / 56
Description of a time serie
To describe a time serie, we will concentrate on:
Its Data Generating Process (DGP)
The joint distribution of its elements
Its “moments”:I Its expected value, E[Xt ] ≡ µt .I Its variance, Var[Xt ] ≡ σ2 ≡ γ0.I Its autocovariance or autocorrelation of order k, Cov[Xt ,Xt−k ] ≡ γk .
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 13 / 56
Variance
The “second” moments play an important role in time series and have aslight different definition:
Definition
The variance of Xt , denoted by γ0(t), is given by:Var(Xt) ≡ E
[(Xt − E(Xt))2
]Definition
The autocovariance of Xt of order k, denoted by γk(t), is given by:Cov(Xt ,Xt−k) ≡ E [(Xt − E(Xt))(Xt−k − E(Xt−k))]
Definition
The autocorrelation of Xt of order k, denoted by ρk(t), is given by:
Corr(Xt ,Xt−k) ≡ Cov(Xt ,Xt−k )√Var(Xt)
√Var(Xt−k)
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 14 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 15 / 56
The stationarity is an essential property to define a time series process:
Definition
A process is said to be covariance-stationary, or weakly stationary, ifits first and second moments are time invariant.
E(Yt) = E[Yt−1] = µ ∀ tVar(Yt) = γ0 <∞ ∀ tCov(Yt ,Yt−k) = γk ∀ t, ∀ k
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 16 / 56
Weak stationarity
The third condition states that the autocovariances only depend onthe decay in the time but not in the time itself.
Hence, the structure of the serie does not change with the time.
If a process is stationary, γk = γ−k
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 17 / 56
Mean reverting propriety
A stationary process has the propriety to be mean reverting :
it will fluctuate around its mean.
This mean will act as an attractor.
It will cross the mean line an infinite number of ways.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 18 / 56
Strong stationarity
Definition
f (Z1,Z2, ...,Zt) = f (Z1+k ,Z2+k , ...,Zt+k)
.
Implies weak stationarity
Definition not very useful as informations about the density functionare difficult to obtain
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 19 / 56
ergodicity
Not absolutely useful...
Definition
limn→∞ E [|Y (yi , . . . , yi+k)Z (yi+n, . . . , yi+n+l)|] =E [Y (yi , . . . , yi+k)] · E [Z (yi+n, . . . , yi+n+l)]
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 20 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 21 / 56
Example 1
We define now the simplest stationary process:
Definition
An independent white noise process is a sequence of independantelements with same expected value and variance.
We will denote it εt ∼ iid(0, σ2)
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 22 / 56
Example 2
Consider the first order auto-regressive process AR(1) with a constant:
Yt = c + ϕYt−1 + εt εt ∼ iid(0, σ2)
Theorem
An AR(1) process is asymptotically stationary if |ϕ| < 1
We will need fot its proof to remember the properties of a geometricprogression:
Lemma (Infinite geometric progression)
|α| < 1⇔∑∞
i=0 αi = 1 + α + α2 + α3 + . . . = 1
1−α
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 23 / 56
Proof.
The AR(1) can be written as:
Yt = ct−1∑i=0
ϕi + ϕtY0 +t−1∑i=0
ϕiεt−i
If |ϕ| < 1, it can be simplified into:
Yt =c
1− ϕ+
t−1∑i=0
ϕiεt−i
We have then:
E(Xt) = c1−ϕ
Var(Xt) = σ2
1−ϕ2
Cov(Xt ,Xt−j) = ϕj
1−ϕ2σ2
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 24 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 25 / 56
Corollary
An AR(1) process is nonstationary if |ϕ| ≥ 1. Furthemore, if:
|ϕ| = 1 it is difference stationary
|ϕ| > 1 it is explosive.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 26 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 27 / 56
Proof.
The AR(1) process without constant Yt = ϕYt−1 + εt εt ∼ iid(0, σ2)with |ϕ| = 1 can be rewritten as:
Yt = Y0 +t−1∑i=0
εt−i
We have then:
E(Xt) = Y0
Var(Xt) = tσ2
Cov(Xt ,Xt−j) = (t − j)σ2
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 28 / 56
The AR(1) process with |ϕ| = 1 is called a random walk. It is said to bedifference stationary.
Definition
The difference operator takes the difference between a value of a time serieand its lagged value. ∆Xt ≡ Xt − Xt−1
Definition
A process is said to be difference stationary if it becomes stationary afterbeing differenced once.
Note: a difference stationary process is also called integrated of order 1and denoted by Xt ∼ I (1)
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 29 / 56
Theorem
A random walk is difference stationary.
Proof.
∆Yt = Yt − Yt−1 = Yt−1 + εt − Yt−1 = εt
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 30 / 56
Stationary AR(1) Random Walk
Yt = ϕYt−1 + εt Yt = Yt−1 + εt
|ϕ| < 1, εt ∼ iid(0, σ2) εt ∼ iid(0, σ2)
Yt = Y0 +∑t−1
i=0 ϕiεt−1 Yt = Y0 +
∑t−1i=0 εt−1
E(Yt) = 0( if Y0 = 0) E(Yt) = 0( if Y0 = 0)
Var(Yt) = σ2
1−ϕ2 Var(Yt) = σ2t
Cov(Yt ,Yt−j) = ϕj
1−ϕ2σ2 Cov(Yt ,Yt−j) = (t − j)σ2
Corr(Yt ,Yt−j) = ϕj Corr(Yt ,Yt−j) =√
t−st∞−→ 1
δYtδεt−i
∞−→ 0 δYtδεt−i
= 1
Is mean reverting Tends to move away from the mean
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 31 / 56
Simulation of AR(1) and RW
0 20 40 60 80 100
−15
−10
−5
0
Index
RW
Random Walk and AR
RWAR((1)) :: ρρ == 0.4
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 32 / 56
Simulation of AR(1) and RW
0 20 40 60 80 100
−6
−4
−2
02
Index
RW
Random Walk and AR
RWAR((1)) :: ρρ == 0.4
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 33 / 56
Simulation of AR(1) and RW
0 20 40 60 80 100
−10
−8
−6
−4
−2
02
Index
RW
Random Walk and AR
RWAR((1)) :: ρρ == 0.9
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 34 / 56
Stationary AR(1) Random Walk
Yt = ϕYt−1 + εt Yt = Yt−1 + εt
|ϕ| < 1, εt ∼ iid(0, σ2) εt ∼ iid(0, σ2)
Yt = Y0 +∑t−1
i=0 ϕiεt−1 Yt = Y0 +
∑t−1i=0 εt−1
E(Yt) = 0( if Y0 = 0) E(Yt) = 0( if Y0 = 0)
Var(Yt) = σ2
1−ϕ2 Var(Yt) = σ2t
Cov(Yt ,Yt−j) = ϕj
1−ϕ2σ2 Cov(Yt ,Yt−j) = (t − j)σ2
Corr(Yt ,Yt−j) = ϕj Corr(Yt ,Yt−j) =√
t−st∞−→ 1
δYtδεt−i
∞−→ 0 δYtδεt−i
= 1
Is mean reverting Tends to move away from the mean
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 35 / 56
0 20 40 60 80 100
−5
05
1020
t
YYt == 0 ++ 0t ++ 0.3Yt −− 1
0 20 40 60 80 100
−5
515
t
Y
Yt == 0 ++ 0t ++ 0.5Yt −− 1
0 20 40 60 80 100
−10
010
20
t
Y
Yt == 0 ++ 0t ++ 0.95Yt −− 1
0 20 40 60 80 100
010
3050
t
YYt == 0 ++ 0t ++ 1Yt −− 1
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 36 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 37 / 56
The random walk with drift
Xt = a + xt−1 + εt εt ∼ iid(0, σ2)Can be rewritten as:
Yt = at + Y0 +t−1∑i=0
εt−i
Expectation is also time-varying:
E(Xt) = Y0 + at
Var(Xt) = . . . = f (t)
Cov(Xt ,Xt−j) = . . . = f (t)
But it is still difference stationary:
Proof.
∆Yt = Yt − Yt−1 = a + Yt−1 + εt − Yt−1 = a + εt
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 38 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 39 / 56
Trend stationarity
Yt = a + bt + ϕYt−1 + εt−1
Yt = at−1∑i=0
ϕi + at−1∑i=0
ϕi + ϕtY0 +t−1∑i=0
ϕiεt−i
If |ϕ| < 1, it can be simplified into:
Yt =a
1− ϕ+ b
t−1∑i=0
ϕi (t − i) +t−1∑i=0
ϕiεt−i
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 40 / 56
Proof.
Yt = a + bt + ϕYt−1 + εt−1
= a + bt + ϕ(a + b(t − 1) + ϕYt−2 + εt−2) + εt−i
= a(1 + ϕ) + bt + ϕb(t − 1) + ϕ2Yt−2 + ϕεt−2 + εt−1
= a(1 + ϕ) + bt + ϕb(t − 1) + ϕ2(a + b(t − 2) + ϕYt−3 + εt−3)
+ ϕεt−2 + εt−1
= a(1 + ϕ+ ϕ2) + b(t + ϕ(t − 1) + ϕ2(t − 2)) + ϕ3Yt−3
+ ϕεt−3 + ϕεt−2 + εt−1
= . . .
= ϕtY0 + at−1∑i=0
ϕi + bt−1∑i=0
ϕi (t − i) +t−1∑i=0
ϕiεt−i
=a
1− ϕ+ b
t−1∑i=0
ϕi (t − i) +t−1∑i=0
ϕiεt−i
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 41 / 56
Thus the model Yt = a + bt + ϕYt−1 + εt−1 has:
E(Xt) = f (t)
Var(Xt) 6= f (t)
Cov(Xt ,Xt−j) 6= f (t)
It is non-stationary as its expectation is time varying. However, itsvariance does not vary with time!
This process is called trend-stationary: if one detrends it, the series isstationary:
Proposition
Yt ≡ a + bt + ϕYt−1 + εt−1 is not stationaryYt − bt = a + ϕYt−1 + εt−1 is stationary
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 42 / 56
0 100 200 300 400 500
050
100
150
200
250
TS: y=0.5 t + eDS: y(t)=a+y(t−1)+e
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 43 / 56
0 100 200 300 400 500
050
100
150
200
250
TS: y=0.5 t + eDS: y(t)=a+y(t−1)+e
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 44 / 56
Comparisons between Trend and difference stationarity:
TS DS
DGP: Yt = α + βt + εt Yt = c + Yt−1 + εt
DGP ′ Yt = Y0 + ct +∑t−1
i=0 εt−1
E(Yt) α + βt Y0 + ct
Var(Yt) σ2 tσ2
Cov(Yt ,Yt−j) 0 (t − j)σ2
δYtδεt−i
∞−→ 0 = a
E[yt+s − yt+s|t ]2 6= f (s) = f (s)
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 45 / 56
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 46 / 56
So both TS and DS exhibit a trend tendency but with stable or increasingvariance.
This trend is said:
Deterministic: TS process
Stochastic: DS process
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 47 / 56
0 20 40 60 80 100
020
4060
t
YYt == 0 ++ 0.5t ++ 0.3Yt −− 1
AR(1)AR(1)+e
0 20 40 60 80 100
020
6010
0
t
Y
Yt == 0 ++ 0.5t ++ 0.5Yt −− 1
AR(1)AR(1)+e
0 20 40 60 80 100
020
6010
0
t
Y
Yt == 0.5 ++ 0t ++ 1Yt −− 1
AR(1)RW
0 20 40 60 80 100
020
4060
t
YYt == 0.5 ++ 0t ++ 1Yt −− 1
AR(1)RWAR(1)+eRW+e
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 48 / 56
Outline
1 Preliminary
2 Lectures
3 DefinitionsTime seriesDescription of a time seriesStationarity
4 Stationary processes
5 Nonstationary processesThe random-walkThe random-walk with driftTrend stationarity
6 Economic meaning and examples
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 49 / 56
Nelson and Plosser (1982) studyNelson and Plosser (1982) investigate 14 time series:
Real GNP
Nominal GNP
Real Per Capita GNP
Industrial Production Index
Total Employment
Total Unemployment Rate
GNP Deflator
Consumer Price Index
Nominal Wages
Real Wages
Money Stock (M2)
Velocity of money
Bond Yield (30-year corporate bonds)
Stock PricesMatthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 50 / 56
Nelson and Plosser (1982) study
1860 1900 1940 1980
5.0
6.0
7.0
Real GNP
1860 1900 1940 1980
1112
1314
15
Nominal GNP
1860 1900 1940 1980
7.0
7.5
8.0
8.5
Real Per Capita GNP
1860 1900 1940 1980
01
23
45
Industrial Production Index
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 51 / 56
1860 1900 1940 1980
10.0
10.5
11.0
11.5
Total Employment
1860 1900 1940 1980
0.5
1.5
2.5
Total Unemployment Rate
1860 1900 1940 1980
3.0
4.0
5.0
6.0
GNP Deflator
1860 1900 1940 1980
3.5
4.5
5.5
Consumer Price Index
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 52 / 56
Nelson and Plosser (1982) study 3
1860 1900 1940 1980
78
910
Nominal Wages
1860 1900 1940 1980
3.0
3.4
3.8
4.2
Real Wages
1860 1900 1940 1980
23
45
67
Money Stock (M2)
1860 1900 1940 1980
0.5
1.0
1.5
Velocity of money
1860 1900 1940 1980
24
68
1012
Bond Yield (30−year corporate bonds)
1860 1900 1940 1980
12
34
5
Stock Prices
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 53 / 56
Results of the test of Trend stationary vs Difference stationary:
Results
13 series can be viewed as DS, one (unemployment) as TS.
The distinction between the two classes of processes isfundamental and aceptance of the purely stochastic view ofnon-stationarity has broad implications for our understanding ofthe nature of economic phenomena
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 54 / 56
We conclude that macroeconomic models that focus onmonetary disturbances as a source of purely transitory(stationary) fluctuations may never be successful in explaining avery large fraction of output fluctuations and that stochasticvariation due to real factors is an essential element of any modelof economic fluctuations.
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 55 / 56
Fill var and cov page 33
Please check 35 and 36
Give title plots 38 39
Page 4: The axes should really begin with 0;0 so that only 2 lines ofthe rect should be visible
Matthieu Stigler Matthieu.Stigler@gmail.com () Stationarity November 14, 2008 56 / 56
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