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Stationarity, cointegration Arnaud Chevalier
University College DublinJanuary 2004
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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
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* 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.
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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
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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
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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
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* (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.
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* 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
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* 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
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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
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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
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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
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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.
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- 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.
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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.
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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.
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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.
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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
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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 .
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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
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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 !
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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.
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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
------------------------------------------------------------------------------
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da t196 197 198 199
-5
5
1
15
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da t
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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.
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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.
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* 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.
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* 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.
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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.
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t
y z
0 00 200 300 400
-20
- 0
0
0
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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
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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)
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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 .
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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 .
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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.
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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.
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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
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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
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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
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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
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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 ).
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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
.
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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. .
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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.
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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.
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* 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
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. reg dnu nu_1 dnu_1 dnu_2 dnu_3 dnu_4
Source | SS df MS Number of obs = 163-------------+------------------------------ F( 5, 157) = 21.53
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 .
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*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 .
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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).
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reg dfyff dfyff_1-dfyff_4 dfy3m_1-dfy3m_4 spread_1
Source | SS df MS Number of obs = 163-------------+------------------------------ F( 9, 153) = 4.74
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.