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Stationarity, cointegration Arnaud Chevalier

University College DublinJanuary 2004



STATIONARITY

Typically, we only observe one set of realisations for any particular series.However if y t is stationary, the mean, variance and autocorrelations canusually be well approximated by sufficient long time averages based on asingle set of realisations.A stochastic process having a finite mean and variance is co-variancestationary if for all t and t-s:

? A ? A E

y E y E

y E y E

y st t

st t

222

)()(

W Q Q

Q

!!

!!

? A ? A s s jt jt st t y y E y y E K Q Q Q Q !!

A series is covariance stationary if its mean and all autocovariances areunaffected by a change of time origin.For a covariance stationary series, autocorrelation between y t and y s is:

0/ K K V s s | The autocorrelation is independent of time



* stati a it c iti s a A (1) cess

t t t yaa y I ! 110 Suppo se t e proc ess sta r te i per iod zero, so 0 is a determi isti c i itial cond ition.

§§!!

!1

0

101

1

0

10

t

iit

it t

i

it a yaaa y I

§!

!1

11

t

i

t it y Ey

So t e mean is ti me de pendent, and t is seq uence is t e r e or e notstati onary.

|a1|<1, 01 yt conv erges to 0 as t .

Also as t e com es la rge, we a ve: )1( 11

1aaaa

i

i

i }§!

us §g

!

!i

it i

t aaa y I

And or large t, Ye t = a 0 (1-a1), w i ch is i nite a nd time inde pendent.he limit of the var iance is:

? A2

22

1112 ...)(! t t t t aa E Y E I I I Q

? A ? A4

aaa !! W W Which is als o f inite a nd time-inde pendent.



Similarly , limiting alues f all autocovariances are f inite and ime-independent: ? A ? A ? A _ a! st st st t t t st t E y y E I I I I I I Q Q

? A ? A211

241

211

2 1/...1 aaaaa s s !!

f a1|<1 and t large , y t is a stationary process



4 STATIONARY RESTRICTION S FOR AN ARMA p

First consider an ARMA : t t t t t aa I I F ! 112211 (16)

Yt can also be written as an infinite MA pro cess:

§g

!

!0i

it it I E (17)

For the 2 expressions to be e ual we must have: t t t t t t t t I I FI EI EI EI EI EI E ! 113120221101110 ...)(...)(...

This means:

2i

1

2211

1111011

0

u!

!!

!

iii aa

aa

EEE

FE FEE

E

From (17) it is easy to see that E(y t)= and var (yt)= §g

!iiEW are finite and time invariant .

The covarian ce between y t and yt-s is constant and independent of t. ..,cov ! EEEEEW s s s st t y y

These results can be generalised to an ARMA ( p )

A finite MA pro cess will alwa ys be stationar y An infinite MA pro cess is stationar y if is finite



5 U L I FU I

* (1) pro ess t t t y y I ! 11

? A ? A2

112

21

2

1/

1/

aa

a s

s !

!

K

K

hu , h e au o orr e a on ar e s s K K V /! , hen e 1, 1=a1, =(a1)s

For an (1) pro ess a nece ssar y ond on or stationar ity is a1 |<1



* (2) process: t t t t aa I 2211

For an (2) to be stationar , the roo ts o (1-a1L-a2L2

) need to beoutside the un it cir cle.e use the u le- a lk er technique to ind the au tocorre lations:

Multip l ing t b t-s or s= ,« and tak ing ex pectations, e or :

)()()(...

)()()(

2211

2211

st t st t st t st t

t t t t t t t t

y E y y E a y y E a y Ey

y E y y E a y y E a y Ey

!

!

I

I

hus,110 W K K K ! aa

2211! s s s aa K K K Dividing s b ields:

2211! s s s aa V V V (19 ) and e k no that =1.he roo ts o (19) lie inside the un it cir cle.



* u to orr elati on un tion o an (1) pro e . 1! t t t y FI I B the ule- alk er equation , e get:

? A W F FI I FI I K !!! t t t t t t E y y E

? A 221111 )( FW FI I FI I K !!! t t t t t t E y y E

nd ? A 10)( 11 "!!! s E y y E st st t t st t s FI I FI I K

hu , e have 1,0),1/(,1 10 "!!! s s V F F V V



* u tocorre lation o f an M (1,1) process : 111! t t t t ya y FI I

By the u le- a lk er equations, e f ind that: W F FW K K ! aa (2 ) 2

11 FW K K ! a (21 )

112 K K a!

11!

s sa K K

olving (2 ) and (21 ) simultaneous ly, e ge t:

221

111

221

12

11

121

W F F

K

W F F

K

a

aa

a

a

!

!

ence V

12

111 21

1

a

aa! .

nd ! s s V V f or a ll u s



6 PART A A TOCORR AT ON

In an AR (1) process yt and yt-2 ar e corr elate d even though yt-2 does not dir ectly a pp ear in the model. In contr ast the par tial autocorr elati on betwee n yt and yt-s eliminates the eff ect of the intervening values. So for an AR (1)

process the par tial autocorr elati on betwee n yt and yt-2 is 0. The most dir ect way to o btain par tial autocorr elati on is to for m the ser ies :

Q| t t y y*

The coeff icient J in the fo llowing AR (1) is the par tial autocorr elati on betwee n yt and yt-1. Since ther e is no intervening value this is als o the autocorr elati on.

t t t e! *

111

* J

Simila r ly 22J gives the par tial autocorr elati on betwee n yt and yt-2.

t t t t e! *

222

*

121

* J J



P ar tial autocorre lation can also be found f r om the Yule Wa l er equations:

2121222

111

1/ V V VJ

VJ

!!

§

§

!

!!1

1,1

1

1,1

1 s

j j j s

s

j j s j s s

ss

VJ

VJ VJ

For an AR (p) pr ocess, there is no direct corre lation betwee n yt and yt-s for s> p. An MA (1) pr ocess can be wr itten as an AR ( ), so always have par tial autocorre lation, decaying slowly over time.

Fea tures of autocorre lation and par tial autocorre lation, can be used to determine the ty pe of you r pr ocess.

For and ARMA (p,q), the P ACF decay star t af ter lag p



T o summarise:

Process AC F PACF hite noise All All

AR ( p) eca towards S pik es though lag p, a f ter A(1) S pik e at lag 1, af ter e ca

AR A (1,1) eca a f ter lag 1 eca af ter lag 1 AR A ( p,q) eca beginning at lag q eca beginning at lag p



7 .6 eter mining the or der of an autoregressi on

ore la gs means more i nf or mation is used, but at t he cost of

additional esti mation uncertai nt (esti mating too man

coeffi cients).

- F-statisti csStart wit h a model wit h a lar ge numbers of lags, test w hether

the coeffi cient on the last la g is si gnificant, if not reduce the

number of la gs and start t he pr ocess a gain. W hen the tr ue value

of the model is p, the test will still esti mate t he model t o be > p,

5% of the time.



- In or mation cr iter ia

In or mation cr iter ia tr ade o the improvement in the f it of the model with th e

number of estimated coeff icients. The most po pular inf or mation cr iter ia ar e the

Bayes Inf or mation Cr iter ia, also called Schwar z inf or mation cr iter ia and the Akaike

Inf or mation Cr iter ia.

T p

T pSSR

p AIC

T T

pT

pSSR p BIC

2)1(

)(ln)(

ln)1(

)(ln)(

¹ º ¸©

ª¨

!

¹ º ¸©

ª¨

!

,

You choo se the model minimizing the inf or mation cr iter ia. The diff er ence betwee n the AIC and BICis that the ter m in ln T in the BIC is r e placed by 2 in the AIC, so the second ter m ( penalty f or number of lags) is not as large. A smalle r decr ease in the SSR is needed in the AIC to justi f y including an additional r egr essor. In large sample, AIC will over esti mate p with a non- zero proba bility.



Simi lar ly the optim al umb er of lags in the add itional r egr essor s need

to be estim ated. he same methods can be used, if the r egr essions has

K coe ff icients including the int er cep t, then the I is:

T

T K

T

K SSR K B I

ln)(ln)( ¹

º ¸©

ª¨!

Important: all models should be estimated using the samesample:

So make sur e to star t ith the mode l ith the most lags, and keep thi

as you r or k ing sam le f or this test.

In r actice a convenient shor tcut is to im ose that all the r egr essor s

have the same numb er of lags to r educe the numb er of mode ls that

need com ar ing.



7.7 Nonstationar ity I: tr ends

If the e pendent var ia ble and/or r egr essor s ar e non stationar y, then y po thesis testing and f or ecast will be unr elia ble. T her e ar e two

comm on ty pe of non stationar ity, with their own solutions.

7.7.1 Wh at is a tr end?

A tr end is a per sist ent long -term movement of a var ia ble over time.

A eterministi c tr end is a non -r ando m f unction of time (linear

in time f or exam ple).

- A stochastic tr end is r ando m and var ies over time. For

exam ple, a stochastic tr end in inf lation may exhi bit a

pr olonge per iod of incr ease f ollowe by a per iod of ecr ease.

Since econo mic ser ies ar e the consequences of com plex

econo mic f or ces, tr ends ar e usef ully though t of as having a

lar ge unp r e icta ble, r ando m com pon ent.



T he r ndom lk mo del of tr end .T he simpl est model of v r i ble ith sto has ti tr end is the r andom

alk . tim e ser ies t is said to f ollow a r andom walk if the han ge in

t is iid:

t t t uY Y ! 1

wher e ut has onditio nal mean er o: 0,...),|( 1 !t t t u

T he basi idea of a r and om walk is that the value of the ser ies

tomo rr ow is its value today lus an un r edi table ompo nen t.



W he series have a te e cy to in cr ease or d ecr ease, th e r a om

walk ca includ e a drift compon e t.

§!

!!t

j jt t t ut uY Y

0

010 F F



If t f ollows a r a om walk the it is not stationar y: the var ia ce of the r a om walk i cr eases over time so the dist r i bu tion o f t cha ges over time.

Var (Y t )=var (Y t-1 ) + var (u t )

or Y t to b e stationar y, we must have var (Y t)=var (Y t-1 ) which impo ses that

var (ut )=0 .lternatively, say Y0 =0, the Y1=u1 , Y2 =u1 + u 2 a mor e ge er ally, Y t =

u1 +u2 + +u t . Because, the ut ar e uncorr elate , var (Y t ) =t u2

T he var ia ce of Y t de pe s on t a i cr eases as t i cr eases. Because the

var ia ce of a r a om walk i cr eases withou t bound, its popu lation

autocorr elations ar e not def i e .



The r andom walk can b e seen as a s pecial case of the AR(1 )

odel in which =1 , th en t cont ains a stochastic tr end and a no

stochastic tr end. I f | |<1 , th en t is station ar y as long as ut is

station ar y.

For an AR( p) to b e station ar y involv es the root s of the f ollo win

olynomi al to b e all gr eater th an 1. Th e root s ar e f ound b y

solving:

0...1 221 !p

p z z z F F F

In th e s pecial case of an AR(1 ), th e pol ynomi al is simpl y:

11

11 F

F !! z z .

The condition th at th e root s ar e less than unit y is equiv alent t



If an AR ( p) has a r oot equals one, the serie s is said to have a unit (autoregressive) r oot. If t has a unit r oot, it con tains a stoch astic trend and is not stationar y. (the two

ter ms can be used inter- changea bly).

If a serie s has a unit r oot, the estimator of the autoregressive coeff icient in an AR ( p) is

biased towar ds 0, t-stat have a non -nor mal distri bution, two inde pendent serie s may

a ppear relate d.

1) bias towar ds 0

Suppo se that the tr ue mod el is a random walk: ( t t t uY Y ! 1 ) but the econom etri cian

estimates an AR (1) ( t t t uY Y !

11 F ). Since the serie s is non stationar y, the OLS assump tions are not satisf ied and it can be

shown that:

T E /3Ö ! . So with 20 year s of quarterl y data, you would e pect 3.0Ö !



Mon te car lo with 100 r e plications gives: ar ia ble | O bs Mea St . D ev. M i Max

------------- -----------------------------------------------------

R ES1 | 100 .9270481 .057000 9 .7792 42 1.010915

1) on norm al distr i bu tion

If a r egr essor h as a stochastic tr e , the OL S t-statisti cs have a

nonnorm al distr i bu tion und er the nu ll hy po thesis . One impor tant

case in which it is possi ble to ta bu late this distr i bu tion is in the

context of an AR with unit roo t; we will go b ack to this.



1) s pur ious r egr essi on

S inf lation was r ising f rom the mid-6 0s through the ea r ly 80¶s, s o was t he

Ja panese GD P over the sa me per iod.

reg inflation gdp_jp if daten>=19651 & daten<=19814,robustRegression with robust standard errors Number of obs = 68

F( 1, 66) = 113.38Prob > F = 0.0000R-squared = 0.5605Root MSE = 2.2989

------------------------------------------------------------------------------| Robust

inflation | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

gdp_jp | .1871328 .0175741 10.65 0.000 .152045 .2222207_cons | -2.938637 .7660354 -3.84 0.000 -4.468076 -1.409198

------------------------------------------------------------------------------

. reg inflation gdp_jp if daten>=19821 & daten<=19994,robustRegression with robust standard errors Number of obs = 70

F( 1, 68) = 5.49Prob > F = 0.0221R -squared = 0.0797Root MSE = 1.5262

------------------------------------------------------------------------------| Robust

inflation | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

gdp_jp | -.0304821 .0130145 -2.34 0.022 -.0564521 -.0045121_cons | 6.30274 1.378873 4.57 0.000 3.551242 9.054239

------------------------------------------------------------------------------



da t196 197 198 199

-5

5

1

15



da t



7.7.2 T esting f or u nit roo t

T he most comm only use test in pr actice is the Dick ey an uller test.

* D ick ey uller in the AR(1 ) model

In the AR(1 ) case, we want to test whether 11 ! F , if we cannot r ejec t

the null hy po thesis then t contains a unit roo t an is not stationar y

(contains a stochastic tr en ).

owever, the test is best im plemente by substr acting t-1 to bo th si es,

it then becomes:

0: 0!H vs 1: 0H

t t t uY Y !( 10 H F wher e 11! FH

T he OL t-stat testi ng 0!H is calle the Dick ey- uller statisti cs.

Note: the test is one si e because the r elevant alter native is that the

ser ies is stationar y.



Regression with robust standard errors Number of obs = 68F( 1, 66) = 4.47Prob > F = 0.0383R-squared = 0.0852Root MSE = 1.7954

------------------------------------------------------------------------------| Robust

dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

inf_1 | -.1559304 .0737577 -2.11 0.038 -.3031924 -.0086683_cons | 1.07776 .4075892 2.64 0.010 .2639821 1.891538

------------------------------------------------------------------------------

The DF statisti cs does not have a nor mal distr i bution, so the cr itical values ar e s pecif ic

to the test.Ta ble 7.1 Critic a l v a lues f r Au mented Dickey a nd Fuller test

10% 5% 1%

Inter ce pt only -2.57 -2.86 - .4

Inter ce pt an time tr en - .12 - .41 - .96

So in the pr evious r egr ession we cannot r eject at any level of statisti cal conf i ence that

0!H , so the ser ies has a unit root, an is not stationar y.



* ick ey-Fu ller test in the ( p) model

For an ( p), the ick ey Fuller test is base on the f ollowing r egr essi on:

t pt pt t t t uY Y Y Y Y (((!( K K K H ...221110

(7.7)

0: 0!H vs 1: 0H The F statistics is the LS-t-statistics testi ng 0!H . If 0 is r e jecte , Yt

is stati onar y.

The num ber of p-lags nee e is unk nown. Studies sugg est that f or the F

it is bette r to h ave too many lags r ather than too f ew, so it is r ecomme n e to use the I to dete r mine the num ber of lags f or the F.



* Di ck ey Full er allowing f or a linear tr en

Som e series have an obviou s linear tr en (Ja panese GD P ) so it will b e

uninf orm ative to t est th eir stationarity without accounting f or th e tr en .

Alter natively, if t is stationar y aroun a determi nistic linear tr en , th e

tr en must be a e to (7.7) which becom es:

t pt pt t t t uY Y Y Y t Y (((!( K K K H E F ...220

If 0 is r e jecte , t is stationar y aroun a determi nistic tim e tr en .

If the series is f oun to h ave a unit root, th en the f ir st di ff er ence of the

series does not h ave a tr en . For exampl e: t t t u! 10 F then

t t uY !( 0 F is stationar y.



Rq: T he power of a test is equal to the pro ba bilit y of r e jecting a f alse null

hypo thesis (1-pro b T ype II). Monte Ca r lo have shown that UR test have

low power , they canno t distingu ish betwee n a unit roo t and a stati onary near un it roo t process . T hus the test will of ten ind icate that a ser ies con tains

a UR .

t t t t

t t t t

z z z y y y

I I !! 21

21

15.01.11.01.1

Checking f or UR ,

Wit h the f ir st process , we have: 10y,1

0)11.0)(1(

01.01.11 2

!! !

!

y y y

y y

Wit h the second process , we have the f ollowing roo ts: z=0.940 5, z=0.1 595.

So the f ir st process has a UR and the second on e is stati onary.



t

y z

0 00 200 300 400

-20

- 0

0

0



Similar ly, it can be diff icult to disti ngu ish bet ee n a tr end stati onary and a unit roo t proc ess ith

dr if t.

/02.0

02.01

1 t t t

t t

x x

t w

I I

!!

t

w x

1

-5

5

1



In the shor t run , the for ecast from stati onary and non- stati onary mo dels

ill be close , how ever the long term for ecast will be quite diff er ent.

lso, the power of the un it roo t test is dr asti cally aff ected b y the dat

gener ating proc ess . If we ina ppropr iatel y om it the interce pt or time

tr end, the power of the test can go to 0. or e amp le om itting the

tr end leads to an upw ar d b ias in the esti mate d value of K in:

t pt pt t t t uY Y Y Y t Y (((!( K K K H E F ...221110

(7 .8)



T hu s a procedur e for testi ng can take the fo llow ing form :

1- se the least r est r ictive mo del (7 .8) to test for .

test have low pow er to r e ject Ho, so if Ho is r e jected ther e i

no need to proceed fur ther. If no t go to ste p 2.

2- T est 0!E , if no t use (7 .8) to test for ste p 1 If yes , use (7 .7 ) to test for , if Ho is r e jected conc lude no un it roo t, i

no t, go to ste p .

- T est 0! F , if no t go back to ste p 2,

If yes , use §!

(!% p

j jt t y y y y

11H to test for .



7.8 N on station ar y: Br eak s

A second ty pe of non station ar y arises when the popul ation r egr ession

f unction changes over the cour se of the sampl e

A br eak can arise either f rom a iscr ete change in the popul ation

r egr ession coeff icients at a istinct ate ( poli cy change) or f rom

gr a ual evolution of the coeff icients over a long er period of tim e (change in the structur e of the econom y).

If the br eak is not noti ce , estim ates will be base on the aver age

behaviour of the series over the period of tim e and not the true

r elation ship at the end of the period , thu s f or ecast will be poor .



7.8.1 testing f or br eak s at a k nown date

To k ee p it simpl e, l et¶s consid er th e ADL(1 ,1) mod el. Let¶s denote X the period

at which th e br eak is suppos e to h ave ha pp ene .

r eate a dumm y varia bl e (D t)tak ing v alues 0 bef or e X and 1 af ter X . D is also

inter acte to t-1 and X t-1.

t t t t t t t t t X DY D D X Y Y ! )()( 12111111 K K K H F F

nder th e hy poth esis o f no br eak , 21 !!! K K K can b e teste using a F-test.

nder th e alternative of a br eak , at least on e of these coeff icients will b e

diff er ent f rom . This is usu ally r ef err e as a how test.

This a ppro ach can b e modi f ie to check f or a br eak in a subs et of the

coeff icients b y including onl y the bin ar y varia ble inter actions f or th at subs et of

r egr essions o f inter est.



7.8.2 Testing f or br eak at an un k nown d ate

Of ten the date of a poss i ble br eak is un k nown, bu t you m ay susp ec t the r ange

dur ing wh ich the br eak took place, say between 10 and X X . Th e how test is us e

to test f or br eak s at all dates between 10 and X X ., then us ing the lar gest of the

r esulting F-statistics to test f or a br eak at an un k nown d ate. Th is is o f ten r ef err e

as Qu andt Lik elihood R atio. Since, Q LR is the lar gest of a ser ies of F-statistics,

its d istr i bution is sp ecial and d e pends on the numb er of r estr ictions teste q (nbr

of coeff icients, including the inter ce pt allowe to b r eak), 10 and X X , expr esse as

a f r action o f the total sample size. For the lar ge sample a ppr oximation to the

distr i bution o f the QLR to be a good on e, 10 and X X cannot be too close to the end

of the sample, For this r eason, the QLR is compu te over a tr imm e r ange so that

T 15.!X and T 85.0!X .

The QLR test can detect a single discr ete br eak , mu lti ple discr ete br eak s and/or

slow evolution o f the r egr ession f unction. If ther e is a distinct br eak in the

r egr ession f unction, the date at which the lar gest how s tatistics occur s is an

estimator of the br eak date.



Say, we want to check that our estimates o f the determ inants o f inflation in the

S ov er the 1962:I and 1999 :4 per iod. Mor e specif ically, we ar e concerne

that the inter ce pt and un emp loyment may have change over time. Th e f irs t

per iod we can check f or s tru ctur al br eak is 0.15T is 1967:4. So we cr ea te a

dumm y var ia ble f or obs ervations af ter 1967:4 and inter act it with

unemp loyment var ia bles: Source | SS df MS Number of obs = 15

-------------+------------------ ------------ F( 13, 138) = 7.41Model | 184.330595 13 14.1792765 Prob > F = 0.000

Residual | 283.045198 138 1.91246756 R -squared = 0.3944-------------+------------------ ------------ Adj R-squared = 0.3412

Total | 467.375793 151 2.90295524 Root MSE = 1.382

-----------------------------------------------------------------------------dinf | Coef. Std. Err. t P>|t| [95% Conf. Interv al]

-------------+---------------------------------------------------------------dinf_1 | -.4009554 .0824812 -4.86 0.000 -.5639484 -.2379 623dinf_2 | -.3433158 .0892349 -3.85 0.000 -.5196549 -.1669 767dinf_3 | .0545284 .0850863 0.64 0.523 -.1136126 .2226693dinf_4 | -.038809 .0754606 -0.51 0.608 -.1879284 .1103105

unemp_1 | -1.719641 1.254766 -1.37 0.173 -4.199214 .7599307unemp_2 | 3.46834 2.364168 1.47 0.144 -1.203546 8.140225unemp_3 | -3.370699 2.164944 -1.56 0.122 -7.648893 .9074963unemp_4 | 1.666702 1.155521 1.44 0.151 -.6167486 3.950152

D | 1.775541 1.839904 0.97 0.336 -1.860335 5.411417D_unemp_1 | -1.225527 1.351754 -0.91 0.366 -3.896758 1.445703D_unemp_2 | .2032217 2.560099 0.08 0.937 -4.855847 5.26229D_unemp_3 | 2.394236 2.370403 1.01 0.314 -2.28997 7.078442D_unemp_4 | -1.668078 1.255425 -1.33 0.186 -4.148952 .8127955

_cons | -.2276938 1.757672 -0.13 0.897 -3.701068 3.245681-----------------------------------------------------------------------------

. testparm D-D_unemp_4F( 5, 148) = 0.85

Prob > F = 0.5135



F=0 .85, we now r e-esti mate this mod el wit h =1 if t>=19 6 8:1, and until

1993 :I.

For e amp le, a br eak at 1981 :4 leads to Regression with robust standard errors Number of obs = 152

F( 13, 138) = 8.42Prob > F = 0.0000R-squared = 0.4223Root MSE = 1.367

------------------------------------------------------------------------------| Robust

dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

dinf_1 | -.4075559 .0932063 -4.37 0.000 -.591853 -.2232587dinf_2 | -.3777853 .0977229 -3.87 0.000 -.5710131 -.1845574dinf_3 | .0515292 .0798247 0.65 0.520 -.1063085 .2093669dinf_4 | -.0260024 .0826179 -0.31 0.753 -.1893631 .1373584

unemp_1 | -2.705181 .6911244 -3.91 0.000 -4.071744 -1.338618unemp_2 | 3.54704 1.300035 2.73 0.007 .9764752 6.117605unemp_3 | -2.025859 1.188034 -1.71 0.090 -4.374964 .3232453unemp_4 | .9846463 .5641419 1.75 0.083 -.1308334 2.100126

D | -.0729984 .9544203 -0.08 0.939 -1.960177 1.81418D_unemp_1 | -.5718067 .8773241 -0.65 0.516 -2.306543 1.162929

D_unemp_2 | .1754026 1.576346 0.11 0.912 -2.941512 3.292317D_unemp_3 | 2.79729 1.599601 1.75 0.083 -.3656069 5.960186D_unemp_4 | -2.432152 .8388761 -2.90 0.004 -4.090865 -.7734395

_cons | 1.350888 .733964 1.84 0.068 -.100382 2.802157------------------------------------------------------------------------------

. testparm D-D_unemp_4F( 5, 138) = 3.31

Prob > F = 0.0074





7.8. P seudo ou t of sample f or ecast

1) choose the numb er o f observations P f or which you will gener ate pseudo ou t of

sample f or ecast, say P =10 . Let¶s def ine s=T-P

2) Estimate the r egr ession on the sho r tene sample: t=1,..,s

) Compu te the f or ecast f or the f ir st per iod b eyond the shor tene samp le: s sY |1~

4) The f or ecast error : s s s s Y Y |111 ~~ !

5) R e peat ste ps 2-4 f or each date f rom T-p 1 to T- 1 (r eestimating the r egr ession

each time) .

6) The pseudo f or ecast error s can be examine to see if they ar e consist ent with a

stationar y r elationshi p



For exampl e, going back to our pr e iction of inf lation , using ata up to

199 : 4, we can pr e ict inf lation f or 1994:1, oing so until 1999: 4, we have 24

pseudo f or ecasts. Regression with robust standard errors Number of obs = 128

F( 13, 114) = 7.37Prob > F = 0.0000R -squared = 0.4210Root MSE = 1.4729

------------------------------------------------------------------------------| Robust

dinf | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------dinf_1 | -.4190169 .0998416 -4.20 0.000 -.6168024 -.2212315dinf_2 | -.3961329 .1031673 -3.84 0.000 -.6005065 -.1917593dinf_3 | .039491 .0844715 0.47 0.641 -.1278463 .2068283dinf_4 | -.0449508 .0860523 -0.52 0.602 -.2154198 .1255181

unemp_1 | -2.679112 .6980463 -3.84 0.000 -4.061936 -1.296288unemp_2 | 3.465039 1.325757 2.61 0.010 .8387247 6.091353unemp_3 | -1.987951 1.22184 -1.63 0.106 -4.408407 .4325056unemp_4 | .9924426 .5769953 1.72 0.088 -.1505805 2.135466

D | .4808356 1.389741 0.35 0.730 -2.27223 3.233901

D_unemp_1 | -.9707623 .9465191 -1.03 0.307 -2.845809 .9042847D_unemp_2 | .6794326 1.700203 0.40 0.690 -2.688656 4.047521D_unemp_3 | 2.716406 1.821819 1.49 0.139 -.8926028 6.325415D_unemp_4 | -2.525234 .9671997 -2.61 0.010 -4.441249 -.6092183

_cons | 1.414308 .7407146 1.91 0.059 -.0530417 2.881658

The inf lation r ate is pr e icte to rise by 1.9 per centage point s. But the true

value is 0.9, so our f or ecast erro is ±1 per centage point s.

D i hi 24 i fi d h h f i 0 37 hi h i i ifi l



Doing this 24 times, w e f ind that the aver age for ecast err or is 0.37 which is s ignif icantl

diff er ent f r om 0 (t=-2.7 1). T his sugg ests that the for ecasts wer e biased over the per iod,

systematically for ecasting higher inflation. T his would suggest that the mod el has been

unsta ble (br eak ).



7.9 Co integr ation

7.9.1 Co integr ation and error corr ection

Ser ies can move together so closely over the long run that they a pp ear to h ave the same tr end

com pon ent. For exam ple, the months and 12months S inter est r ate.

daten

FYFF FYGM3

20004

. 3

.



mor eover , the s pr ead between the two ser ies does not a ppear to have a tr end.

daten00 00 00 00 20000

2

0

2

4

T he two ser ies have a common stochastic tr end, they ar e said to be cointegr ated. .



Suppo se X t and Y t ar e integr ated of ord er 1. If ther e e ist a coeff icient U such

that t t X Y U is integr ated of ord er 0 (stati onary) , then the 2 ser ies ar e said to be

cointegr ated with a cointegr ating coeff icient U .

Unit roo t testi ng can be e tended to test f or cointegr ation. If X t and Yt ar e

cointegr ated, then t t X Y U is I(0) (the null hypo thesis of a unit roo t is r e jected)

other wise t t X Y U isI(1).

* T esting f or co integr ation when U is k nown.

In som e cases , econom ic theory sugg ests a value of U . In this case a F test on

the ser ies X Y z t t U! is conduc ted.



In our example, let¶ s assume that theor y sugge st that U =1. There i s no trend in d s pread, so we

simpl y estimate:. reg dspread spread_1 dspread_1 dspread_2 dspread_3 dspread_4

Source | SS df MS Number of obs = 163-------------+------------------------------ F( 5, 157) = 11.73

Model | 20.0646226 5 4.01292452 Prob > F = 0.0000Residual | 53.706531 157 .342079815 R-squared = 0.2720

-------------+------------------------------ Adj R-squared = 0.2488Total | 73.7711536 162 .455377491 Root MSE = .58488

------------------------------------------------------------------------------dspread | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------spread_1 | -.2506278 .0719562 -3.48 0.001 -.3927548 -.1085007

dspread_1 | -.283247 .091436 -3.10 0.002 -.4638504 -.1026437dspread_2 | .0230289 .0910197 0.25 0.801 -.1567521 .20281dspread_3 | -.0599991 .0895151 -0.67 0.504 -.2368085 .1168102dspread_4 | .048277 .0791148 0.61 0.543 -.1079897 .2045436

_cons | .1548892 .063015 2.46 0.015 .0304227 .2793557------------------------------------------------------------------------------

Lags AI

4 -1.049

-1.059

2 -1.063

1 -1.072

The t- stat on s pread_ 1 = -3.48, which is greater than the critical value (1% of the ADF) so we re ject

the null h y pothe sis that 0!H , the serie s does not have a unit root, and i s there f ore I (0). The 2

intere st rate serie s are cointegrated.



* testing f or cointegr ation whe n U is unk nown.

In gener al U is unk nown, the cointegr ation coeff icient must be estim ate pr ior to testing f or

unit root. Th is pr eliminar y ste p mak es it necessar y to u se diff er ent cr itical values f or the

subsequent unit root test.

Ste p 1: estim ate t t t X Y LE ! (7.12)

Ste p2: a Dick ey Fuller t -test is use to test f or u nit roo t in the r esi uals f rom (1): t LÖ

This proce ur e is calle the Engle -Gr anger Augmente Dick ey Fuller Test. C r itical values f or

the EGAD F ar e:

N br o f X in (7.12):

Cointegr ate var ia bles

10% 5% 1%

1 -3.12 -3.41 -3.96

2 -3.52 -3.80 -4.36

3 -3.84 -4.16 -4.73

4 -4.20 -4.49 -5.07



. reg dnu nu_1 dnu_1 dnu_2 dnu_3 dnu_4


Model | 31.2052888 5 6.24105775 Prob > F = 0.0000Residual | 45.5212134 157 .289944035 R -squared = 0.4067

-------------+------------------------------ Adj R-squared = 0.3878Total | 76.7265022 162 .473620384 Root MSE = .53846

------------------------------------------------------------------------------dnu | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

nu_1 | -.5739985 .1150186 -4.99 0.000 -.8011821 -.3468149dnu_1 | -.1574595 .1139771 -1.38 0.169 -.3825858 .0676667dnu_2 | .0752181 .1052652 0.71 0.476 -.1327006 .2831369dnu_3 | .0053021 .0974368 0.05 0.957 -.1871541 .1977583dnu_4 | .1237554 .0782992 1.58 0.116 -.0309003 .278411_cons | .0016953 .0421806 0.04 0.968 -.0816193 .0850099

R eject the null hy po thesis of a unit root, the two ser ies ar e cointegr ate .



*E rror corr ection mod el

If 2 ser ies a r e cointegr ated, then the f or ecast of tand (( t Y can be improv ed by including an

error corr ection term.

If Xt and Yt ar e cointegr ated, one wa y to eliminate t he stochasti c tr end is to compu te t he se r ies

t t X Y U which is stati onary and can be used f or analysis. The te rm t t X Y U is calle d the e rror

corr ection term t t t qt qt ¡ t ¡ t t X X X ((((!( )(....... 1111110 UEK K F F F

similar ly, we als o have:

t t t qt qt pt pt t u X Y X X Y Y X ((((!( )(....... 1111110 UEK K F F F

if U is unk nown, then the E rror Corr ection Models can be esti mated using 1Öt L .



Inter est r ate change accor ding to stoch asti c shoc ks and pr evious per iod

deviati on from the long- term equili br ium t t X Y U =0. l phas can be

interpr eted as the s peed of ad justment.

T he a bsence of Gr anger causalit y for cointegr ate d var ia bles r equir es

that the s peed of ad justment is 0 as well as all gamm as (r es p, all betas )

to be 0. f cour se at least one of the al phas has to be non- zero for the 2

ser ies to be cointegr ated.

or t y( to be I(0) , t t X Y U needs to be I(0) since the error term andall f ir st diff er ence terms ar e I(0) , hence the 2 ser ies ar e cointegr ated

(1,1).



reg dfyff dfyff_1-dfyff_4 dfy3m_1-dfy3m_4 spread_1


Model | 69.5220297 9 7.72466996 Prob > F = 0.0000Residual | 249.54597 153 1.63101941 R -squared = 0.2179

-------------+------------------------------ Adj R-squared = 0.1719Total | 319.067999 162 1.96955555 Root MSE = 1.2771

------------------------------------------------------------------------------dfyff | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------dfyff_1 | -.0014132 .2136881 -0.01 0.995 -.4235733 .4207468dfyff_2 | -.0264828 .2208415 -0.12 0.905 -.4627751 .4098095dfyff_3 | .1002626 .2129522 0.47 0.638 -.3204438 .5209689dfyff_4 | .1444413 .1802188 0.80 0.424 -.2115972 .5004798dfy3m_1 | .0068489 .2541142 0.03 0.979 -.4951767 .5088745dfy3m_2 | -.1758844 .275382 -0.64 0.524 -.7199263 .3681576dfy3m_3 | .2220654 .2653096 0.84 0.404 -.3020777 .7462086dfy3m_4 | -.3159166 .2272404 -1.39 0.166 -.7648506 .1330174

spread_1 | -.4598352 .1585354 -2.90 0.004 -.7730361 -.1466342_cons | .2955998 .1381308 2.14 0.034 .0227098 .5684897

------------------------------------------------------------------------------

the lag s pr ea does help to pr e ict change in int er est r ate in th e one year tr easur e bond r ate.

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