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The Land of the Unit RootsThe Land of the Unit Roots

Gloria González-RiveraUniversity of California, RiversideandJesús Gonzalo U. Carlos III de Madrid

Why should we care about the existence of unit roots?Why should we care about the existence of unit roots?

• Growth

• Forecast

• The effect of a shock

• Spurious Regression

•Asymptotic Results

• Testing for unit roots

• Problems of testing for unit roots

• Structural Breaks

Some plots: InflationSome plots: Inflation

Some Plots: Production

Some Plots: Dutch stock market indexSome Plots: Dutch stock market index

How do we model Growth?How do we model Growth?

Most of the macro-economic variables: GNP, Consumption, Investment, ...etc show a growing pattern through time. This pattern is impossible to be captured with our stationary ARMA models:

)(ˆlim)|()(ˆ

)()(

)(

0

22

lZZElZ

ZVarZE

aLZ

nl

nlnn

iiatt

tt

tZ

tt

How do we describe trends as the following?

How do we model Growth? (cont)How do we model Growth? (cont)

Two options:

• An ARMA model with a deterministic trend component

(TS=Trend Stationary)

• A Unit Root process with a drift term (DS=difference stationary)

t

tcZEaLtcZ ttt )(

ttt aZZ 1

1. Deterministic time trend

2. Stochastic trend. Unit root processes

tcZEaLtcZ ttt )(

ttt aZZ 1

21

0

1

0

21

0

2

1

0

121

)(

))()()((),cov(

)(var)(

)(

.....2

:onsubstituti backward with; :conditions initial

a

Nt

jjt

Nt

jjt

tttt

a

Nt

jjttt

Nt

jjtt

tttttt

N

NtaaE

ANtZANtZEZZ

NtaNtA

aNtAZ

aaZaZZ

AZNt

Random walk with drift

1: tocompared large for

)()(

)(22

2

Nt

Nt

Nt

NtNt

Nt

aa

a

tt

How can we relate processes 1. and 2. ?

ttt

tt

aZZ

aLtcZ

1.2

)(.1

Consider the process:

)1)....(1)(1()......1(

:roots in polynomial AR thedecomposed

)....1()......1(

),(where.3

112

111

11

LrLrLrLL

aLLuLL

qpARMAuutcZ

pp

p

tq

qtp

p

ttt

)1(Z2.) (model)()1()1(

)1(

difference and 3. modelGet

)()1)....(1(

)......1()1(

then321and1 Suppose

1.) (model)(

1 then,stationary is If

)()1)....(1)(1(

)......1(

t

11

112

1

111

1

112

11

1

IaLuLZL

utcZ

utcZ

aLaLrLr

LLuL

...p ,irr

aLtcZ

iru

aLaLrLrLr

LLu

ttt

tt

tt

tt

p

qq

t

i

tt

it

tt

p

qq

t

Differencing this type of non-stationary process makes it stationary

Warning

tt

tt

aLLZL

aLtcZ

)()1()1(

)(

Non-invertible MA

)2()()1(

...311 that Suppose

2

112

11

IZaLZL

pirrr

ttt

i

Two differences are needed to achieved stationarity

* In general, ),,( qdpARIMAZt

d-differences are necessary for stationarity

tqtd

p aLZLL )()1)((

ForecastsForecasts

Deterministic time trend:

t

tal

alnln

lnlnnlnln

nlnln

lnln

tt

Z

aLMSE

lZZEMSE

aaaale

aalnclZ

aLlncZ

aLtcZ

ofpart stationary

)(var....)1(lim

)...1())(ˆ(

....)(

....)()(ˆ

)()(

)(

222

21

221

22

21

2

112211

11

Forecasts (cont)Forecasts (cont)Unit Root:

........

....)|(

.........

......)|(

)|(......)|()|()|(

......

).....()()(

...)(ˆ

)()1(

121

11

121

1111

11

11

1211

11

nl

nllnnln

nnn

nlnlnlnlnln

nnnnlnnlnnln

nnlnlnln

nnnlnlnlnlnln

nlnln

t

Z

t

a

aZlZE

Zaa

aaaaZE

ZZEZEZEZE

ZZZZZ

ZZZZZZZZ

aalZ

aLZL

t

Forecasts: ExamplesForecasts: Examples

nnnln

ttt

nnln

tt

aZlZE

aaZL

ZlZE

aZL

)|(

0...;)1(.2

)|(

0....)1(.1

3211

21

Forecast error

2

2221

21

2

112122111

1122111111

11

)(ˆlim

.....)1()1(1)(ˆ

)...1(......)1()1(

.....)....()....(

)())1(ˆ(....))1(ˆ())(ˆ(

)(ˆ)(

lZZE

lZZE

aaaa

aaaaaaa

ZZZZlZZlZZ

lZZle

nlnl

anln

nllnlnln

nnllnlnnllnln

nnnnnlnnln

nlnn

(l elements in sum )

Forecasts: Examples (cont)Forecasts: Examples (cont)Example

ARIMA(0,1,1)

22

221

21

2

11

)1)(1(1

)1(....)1(1)(ˆ

)1(

a

anln

ttt

l

lZZE

aaZL

The Effect of a ShockThe Effect of a Shock t ah t Z

Transitory shock:

Permanent shock: 0tahtZ

h

0tahtZ

h

Examples:

(1)

h as 0h

tahtZ

....tah...2hta2

1htahtahtZ

...2ta21tatatZ

1|| ta1tZtZ

The effect of a shock (cont)The effect of a shock (cont)

0

0,2

0,4

0,6

0,8

1

1,2

T-1 TT+1

T+2T+3

T+4T+5

T+6T+7

T+8T+9

T+10T+11

T+12

e

0

0,2

0,4

0,6

0,8

1

1,2

T-1 TT+1

T+2T+3

T+4T+5

T+6T+7

T+8T+9

T+10T+11

T+12

y

The unit impulse response function for the AR(1) process yt = 0.8 yt-1 + et

The effect of a shock (cont)The effect of a shock (cont)

The Effect of a Shock (cont)The Effect of a Shock (cont)

t ah t Z

(2)

h as 0h1tahtZ

....ta...2hta1htahtahtZ...2ta1tatatZ

| ta1tZtZ

(3)

0)1(tahtZ

hta)L(~

L1hta

)1(htZ

ta)L(~

L1ta

)1(tZ

ta)L(~

)L1(ta)1(ta)L(tZ)L1(

The Effect of a Shock (cont)The Effect of a Shock (cont)

t ah t Z

Q1: Calculate the effect of a perturbation on at in the following TS model:

ta)L(tu wheretutctZ

Spurious RegresionsSpurious Regresions

Consider two independent random walks:

0,01

1

kttkttstttt

ttt

vEvuEustvEuvxx

uyy

By construction, there is no relation between x and y.

Consider the regression

ttt xy e

Q2: Which values do you expect the estimates of and will take?What about the R2? You will find the right answer in two more lectures.

Some Asymptotic ResultsSome Asymptotic Results

Consider the stationary case

)21(T

12

0T

12

1

t

1t2yˆvarˆE

t

1t2y

t

1tyty

ˆ:OLS

e

e

11 e ttt yy

Asymptotically (CLT) from the previous lecture:

)1,0()ˆ( 2 NT

Some asymptotic results (cont)Some asymptotic results (cont)

When the asymptotic result is not valid to perform inferences because

1

ondistributi ddegeneratea has ˆ

0)1ˆ(0)ˆvar(

T

What to do when ?1

ondistributiFuller -Dickey: ondistributi standard-nona has

e

e

t

1t2y

t

t1ty

1ˆ:1under ;

t

1t2y

t

t1ty

ˆ

tt

ttt

yT

yT

T2

12

1

1

1

)1ˆ(e

t

ttyT

?1

1e)1,0(),0(

0.....

2

011

Nt

ytNy

yy

tt

ttt

eee

12

11

1

2

1

2

11

2

1

2

11

2

1)(

2

1)(

2

1

)(2

1

2)(

2112

(by LLN)

2

2

2

12

221

220

221

221

21

221

211

221

2

221

e

e

e

ee

eee

ee

eee

D

ttt

tt

T

ttt

ttT

ttt

ttT

tt

t

t

tt

ttt

ttttt

ttttttt

yT

TT

yy

T

Ty

Ty

T

yyyyy

yyy

yyyy

?1

12

2

t

tyT

zero.not islimit in the variance thebecause variable

random a toeconvergencget that wedoingBy . 2Tby sum thisdivide tohave weso

2T Order2)1t2t)(1t(t)3/1()

t

1t2y(Var

t t2

T)1T(2)1t(21t

2Ey

t

1t2yE

2)1t()1t2yvar()2)1t(,0(N1ty

In summary, the statistic

t

t

ttt

yT

yT

T1

22

1

1

1

)1ˆ(e

has a non-standard distribution known as the Dickey-Fuller distributionthat is dominated by the chi-squared behavior of the numerator.

We can construct a pseudo-t test as

1

1ˆ

ˆ1ˆ

2

2

21

22ˆ

ˆ

T

sy

st tt

tt

e

ee

This pseudo-t test does not have the usual limiting Gaussian distribution

-1.95

5%

%5)65.1(

%595.1ˆ

1ˆ

ZP

P

Dickey-Fuller distribution

Normal distributionReject the unit root too often

Some Asymptotic Results (cont)Some Asymptotic Results (cont)

The asymptotic distributions can be written in a more compact way

e

1

0

2W

)12)1(W)(2/1(1

0

2W

1

0

WdW

t

21ty

2T

1t

t1tyT

1

)1ˆ(T

2)dr2)r(W(

)12)1(W)(2/1(

2)dr2)r(W(

WdW

ˆˆ1ˆ

t

Some Asymptotic Results (cont)Some Asymptotic Results (cont)

where W(r) is a Brownian Motion (see the applets from this lecture).A Brownian Motion is defined by the following properties:•W(0)=0•W(t) has stationary and independent increments and for all t and ssuch that for t>s we have W(t)-W(s) is N(0, (t-s))•W(t) is N(0,t) for every t•W(t) is sample path continuous.

Testing for Unit Roots (DF test)Testing for Unit Roots (DF test)

Problem: Tests for unit root are conditional on the presence of deterministic regressors; and tests for the presence of deterministic regressors are

conditional on the presence of unit root.

Reparametrization of the model

ity)(stationar0:

root)(unit 0:

)1(

1

0

11

111

eee

H

H

yyy

yyyy

ttttt

ttttt

Dickey-Fuller consider three models with deterministic regressors:

ttt

ttt

ttt

yty

yy

yy

ee

e

11

11

11

)3(

)2(

)1(

Pseudo-t statistics for :

(3) model

(2) model

(1) model

0:

0:

1

0

H

H

Pseudo-F statistic)/(

/)(

kTRSS

rRSSRSSF

U

UR

for :

(3) model 0:

(3) model 0:

(2) model 0:

30

20

10

FH

FH

FH

ttt

ttt

ttt

yty

yy

yy

ee

e

11

11

11

)3(

)2(

)1(

05.0)41.3t(P

05.0)86.2(P

05.0)95.1(P

05.0)64.1e(P

Testing for Unit Roots: A procedure for the DF testTesting for Unit Roots: A procedure for the DF test

1. Start with a general model

2.) to(go

root)unit (no

0:

0:

1

0

accept

reject

H

H

2. Test for trend

3.) to(go

Normal usemay 0)(0: 30 accept

rejectFH

3. Estimate

4.) to(go

root)unit (no

0:

0:

1

0

1

accept

reject

H

H

yy ttt

e

4. Test for drift

5.) to(go

Normal usemay 0)(0: 10 accept

rejectFH

5. Estimate

(stop)

root)unit (no

0:

0:

1

0

1

accept

reject

H

H

yy ttt

e

ttt yty e 1

Augmented Dickey-Fuller testAugmented Dickey-Fuller test

The previous results are only valid when the error term et is iid.

If this is not the case, for instance if et follows a linear process:

then it can be proved that it can be proved that we can re-write the DF regression

by adding lags of the increments of yt-1 until the error term becomes

iid. This solves the problem. The strategy is the same and the A.D

are the same as before.

ta)L(t e

ta

p

1i

ityi1tytty

Q3: Think on two different ways to choose the right order “p”.

Q4: Discuss briefly two reasons why we deal with the null of unit root instead of the null of stationarity.

Q5: I am sure you have read and heard many many many times that unit root tests do not have power. What about other tests? Any comments?

Structural Breaks versus unit RootsStructural Breaks versus unit Roots

See notes and discussion in class.

See Part IV of “Unit Roots, Cointegration and Unit Roots, Cointegration and Structural ChangeStructural Change” by Maddala and Kim. Cambridge University Press 1998.

Applications in FinanceApplications in Finance

Asset price levels (logs) US marketAsset price levels (logs) US market

US stock market returns US stock market returns 1980-20001980-2000

Looking at information in past prices: Looking at information in past prices: returns and past returnsreturns and past returns

Histogarm US returnsHistogarm US returns

Interest rate USAInterest rate USA

Histogram of Interest rates in the USAHistogram of Interest rates in the USA

Scatter plot: unveiling information of past interest Scatter plot: unveiling information of past interest rate levelrate level

ACF and PAC for interest ratesACF and PAC for interest rates

AC and PAC of interest rate changes

Regression for the DF testRegression for the DF test

Regression for the DF testRegression for the DF test

ADF test for interest ratesADF test for interest rates

Model fitModel fit