stationarity - definition, meaning and...

StationarityDefinition, meaning and consequences

Matthieu [email protected]

November 14, 2008

Version 1.1

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Matthieu Stigler [email protected] () Stationarity November 14, 2008 1 / 56

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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


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.


The increasing returns are:

Self production: Learning curve

Common production: adoption/network externalities


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


Outline

1 Preliminary

2 Lectures






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


Outline

1 Preliminary

2 Lectures






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


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.


Outline

1 Preliminary

2 Lectures






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 .


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)


Outline

1 Preliminary

2 Lectures






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


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


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.


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


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)]


Outline

1 Preliminary

2 Lectures






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)


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


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


Outline

1 Preliminary

2 Lectures






Corollary

An AR(1) process is nonstationary if |ϕ| ≥ 1. Furthemore, if:

|ϕ| = 1 it is difference stationary

|ϕ| > 1 it is explosive.


Outline

1 Preliminary

2 Lectures






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


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)


Theorem

A random walk is difference stationary.

Proof.

∆Yt = Yt − Yt−1 = Yt−1 + εt − Yt−1 = εt


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


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



0 20 40 60 80 100

−6

−4

−2

02

Index

RW

Random Walk and AR

RWAR((1)) :: ρρ == 0.4



0 20 40 60 80 100

−10

−8

−6

−4

−2

02

Index

RW

Random Walk and AR

RWAR((1)) :: ρρ == 0.9


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


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


Outline

1 Preliminary

2 Lectures






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


Outline

1 Preliminary

2 Lectures






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


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


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


0 100 200 300 400 500

050

100

150

200

250

TS: y=0.5 t + eDS: y(t)=a+y(t−1)+e


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)


So both TS and DS exhibit a trend tendency but with stable or increasingvariance.

This trend is said:

Deterministic: TS process

Stochastic: DS process


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


Outline

1 Preliminary

2 Lectures






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 [email protected] () 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


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


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


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


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.


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


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