Stationarity, Non Stationarity, Unit Roots and Spurious Regression

7/27/2019 Stationarity, Non Stationarity, Unit Roots and Spurious Regression

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Stationarity, Non Stationarity, Unit

Roots and Spurious Regression

Roger Perman

Applied Econometrics Lecture 11


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Stationary Time Series

Exhibits mean reversion in that it fluctuates around a constant

long run mean

Has a finite variance that is time invariant

Has a theoretical covariance between values ofytthat dependsonly on the difference apart in time

)( tyE)0()y(E)y(Var 2tt

)()y)(y(E)y,y(Cov tttt


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WHITE NOISE PROCESS

Xt= ut ut~ I I D(0, 2)

Stationary time series

White Noise

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


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Xt= 0.5*Xt-1+ ut ut~ I I D(0, 2)

Stationary without drift

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


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0 50 100 150 200 250 300 350 400 450

.25

.5

.75

1

1.25

1.5

1.75

2

2.25

2.5

Many Economic Series Do not Conform to the

Assumptions of Classical Econometric Theory

Share Prices

0 50 100 150 200 250

.35

.4

.45

.5

.55

.6

.65

.7 Exchange Rate

1960 1965 1970 1975

8.7

8.8

8.9

9

9.1

9.2

9.3 Income


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Non Stationary Time Series

There is no long-run mean to which the series returns

and/or

The variance is time dependent and goes to infinity

as time approaches to infinity

Theoretical autocorrelations do not decay but, in finite

samples, the sample correlogram dies out slowly

The results of classical econometric theory

are derived under the assumption that variables of concern are stationary.

Standard techniques are largely invalid where data is non-stationary


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Non-stationary time series

UK GDP (Yt)

The level of GDP (Y) is not constant and the mean increases over time. Hence the level of

GDP is an example of a non-stationary time series.

GDP Level

0

20

40

60

80

100

120

1992Q

3

1993Q

2

1994Q

1

1994Q

4

1995Q

3

1996Q

2

1997Q

1

1997Q

4

1998Q

3

1999Q

2

2000Q

1

2000Q

4

2001Q

3

2002Q

2

2003Q

1


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RANDOM WALK

Xt= Xt-1+ ut ut~ I I D(0, 2)

Mean: E(Xt) = E(Xt-1) (mean is constant in t)

X1= X0+ u1 (take initial valueX0)X2= X1+ u2= (X0+ u1) + u2

Xt= X0+ u1+ u2++ ut

E(Xt) = E(X0+ u1+ u2++ ut)(take expectations)

= E(X0)= constant


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RANDOM WALK

Xt= Xt-1+ ut ut~ I I D(0, 2)

Xt= X0+ u1+ u2++ ut

Variance:Var(Xt) = Var(X0) + Var(u1) ++ Var(ut)= 0 + 2++ 2

=t 2

(variance is not constant through time)


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Non-stationary time series: Random Walk

Xt= Xt-1+ ut ut~ I I D(0, 2)

Random Walk

0

0.5

1

1.5

2

2.5


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Constant covariance - use of correlogram

Covariance between two values ofXtdepends only on the

difference apart in time for stationary series.

Cov(Xt,Xt+k) = (k) (covariance is constant in t)

(A) Correlation for 1980 and 1985 is the same as for 1990

and 1995. (i.e. t = 1980 and 1990, k = 5)

(B) Correlation for 1980 and 1987 is the same as for 1990 and1997. (i.e. t = 1980 and 1990, k = 7)


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UK GDP (Yt) - correlogram

For non-stationary series the Autocorrelation Function (ACF) declines towards zero at a

slow rate as kincreases.

0 1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ACF-Y


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First difference of GDP is stationary

Yt=Yt-Yt-1- Growth rate is reasonably constant through time.Variance is also reasonably constant through time


GDP Growth (YBEZ)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1992Q3

1993Q2

1994Q1

1994Q4

1995Q3

1996Q2

1997Q1

1997Q4

1998Q3

1999Q2

2000Q1

2000Q4

2001Q3

2002Q2

2003Q1


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UK GDP Growth ( Yt) - correlogram

Sample autocorrelations decline towards zero as kincreases. Decline is rapid for stationary

series.

0 1 2 3 4 5 6 7

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

ACF-DY


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Relationship between stationary and non-stationary process

AutoRegressive AR(1) process

Xt= +Xt-1+ ut ut~ I I D(0, 2)

< 1 stationary process

- process forgets past = 1 non-stationary process

- process does not forget past

= 0 without drift

0 with drift

Non-stationary Time Series: summary


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Stationary time series with drift

Xt= + 0.5*Xt-1+ ut ut~ I I D(0, 2)

Stationary with Drift

-0.2

0

0.2

0.4

0.6

0.8

1

1.2


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Non-stationary time series: Random Walk with Drift

Xt= + Xt-1+ ut ut~ I I D(0, 2)

Random Walk with Drift

0

2

4

6

8

10

12

Ti S i M d l


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General Models

AutoRegressive AR(1) process without dr if t

Xt= Xt-1+ ut

< 1 stationary process- process forgets past

= 1 non-stationary process

- process does not forget past

AutoRegressive AR(k) process without dr if t

Xt= 1Xt-1+ 2Xt-2+ 3Xt-3+ 4Xt-4++ kXt-k+ ut

Time Series Models: summary

S i R i P bl


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Spurious Regression Problem

yt= yt-1+ ut ut~ iid(0,2)

xt

= xt-1

+ vt

vt

~ iid(0,2)

utand vtare serially and mutually uncorrelated

yt=0+1xt+ t

since ytand xtare uncorrelated random walks we should expect R2

to tend to zero. However this is not the case.

Yule (1926): spurious correlation can persist in large samples withnon-stationary time series.

- if two series are growing over time, they can be correlated

even if the increments in each series are uncorrelated

S i R i P bl


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Two random walks generated from Excel using RAND() command

hence independent


xt= xt-1+ vt vt~ iid(0,2)

Two Random Walks

-4-2

0

2

4

6

8

10

12

14

1 40 79 118 157 196 235 274 313 352 391 430 469

RW1 RW2

S i R i P bl


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Plot Correlogram using PcGive

(Tools, Graphics, choose graph, Time series ACF, Autocorrelation

Function)



0 5 10

0.25

0.50

0.75

1.00

ACF-RW1

0 5 10

0.25

0.50

0.75

1.00 ACF-RW2

S i R i P bl


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Estimate regression using OLS in PcGive

yt=0+1xt+ t

based on two random walks



EQ( 1) Modelling RW1 by OLS (using lecture 2a.in7)The estimation sample is: 1 to 498

Coefficient t-value

Constant 3.147 25.8

RW2 -0.302 -15.5

sigma 1.522 RSS 1148.534

R^2 0.325 F(1,496) = 239.3 [0.000]**

log-likelihood -914.706 DW 0.0411

no. of observations 498 no. of parameters 2


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Trend: Deterministic or Stochastic?

0 50 100 150 200 250 300 350 400 450 500

100

200

0 50 100 150 200 250 300 350 400 450 500

100

200

0 5 10 15 20 25

.25

.5

.75

1

0 5 10 15 20 25

.25

.5

.75

1

The First

The Second

Y a Yt t t

1 1

Y a a Y a t t t

1 2 1 3

(with a2 0)


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Y a a Y a t t t

1 2 1 3

This series has a deterministic trend (if a3 > 0)

Classical inference is valid

(provided that a2is less than 1).

The series is transformed to a stationary series by

subtracting the deterministic trend from the left side

(and so the right side).

Y a t a a Y t t

3 1 2 1


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Y a Yt t t

1 1

This series is non-stationary - the trend is stochastic

Classical inference is not valid

The series is called difference stationary

Y Y at t t

1 1

Random Walk With Drift


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Parameter Set Description Properties

1 f b

Stationarity, Non Stationarity, Unit Roots and Spurious Regression

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