modelling lon run relationship in finance

Brooks chapter 7

1. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
1
Chapter 7
Modelling long-run relationship in finance

2. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Stationarity and Unit Root Testing
Why do we need to test for Non-Stationarity?
•Thestationarityorotherwiseofaseriescanstronglyinfluenceits
behaviourandproperties-e.g.persistenceofshockswillbeinfinitefor
nonstationaryseries
•Spuriousregressions.Iftwovariablesaretrendingovertime,a
regressionofoneontheothercouldhaveahighR
2
evenifthetwoare
totallyunrelated
•Ifthevariablesintheregressionmodelarenotstationary,thenitcan
beprovedthatthestandardassumptionsforasymptoticanalysiswill
notbevalid.Inotherwords,theusual“t-ratios”willnotfollowat-
distribution,sowecannotvalidlyundertakehypothesistestsaboutthe
regressionparameters.

3. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Value of R
2
for 1000 Sets of Regressions of a
Non-stationary Variable on another Independent
Non-stationary Variable

4. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Value of t-ratio on Slope Coefficient for 1000 Sets of
Regressions of a Non-stationary Variable on another
Independent Non-stationary Variable

5. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Two types of Non-Stationarity
•Variousdefinitionsofnon-stationarityexist
•Inthischapter,wearereallyreferringtotheweakformorcovariance
stationarity
•Therearetwomodelswhichhavebeenfrequentlyusedtocharacterise
non-stationarity:therandomwalkmodelwithdrift:
y
t=+y
t-1+u
t (1)
andthedeterministictrendprocess:
y
t=+t+u
t (2)
whereu
tisiidinbothcases.

6. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Stochastic Non-Stationarity
•Notethatthemodel(1)couldbegeneralisedtothecasewherey
tisan
explosiveprocess:
y
t=+y
t-1+u
t
where>1.
•Typically,theexplosivecaseisignoredandweuse=1to
characterisethenon-stationaritybecause
–>1doesnotdescribemanydataseriesineconomicsandfinance.
–>1hasanintuitivelyunappealingproperty:shockstothesystem
arenotonlypersistentthroughtime,theyarepropagatedsothata
givenshockwillhaveanincreasinglylargeinfluence.

7. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Stochastic Non-stationarity: The Impact of Shocks
•Toseethis,considerthegeneralcaseofanAR(1)withnodrift:
y
t=y
t-1+u
t (3)
Lettakeanyvaluefornow.
•Wecanwrite: y
t-1=y
t-2+u
t-1
y
t-2=y
t-3+u
t-2
•Substitutinginto(3)yields:y
t=(y
t-2+u
t-1)+u
t
=
2
y
t-2+u
t-1+u
t
•Substitutingagainfory
t-2: y
t=
2
(y
t-3+u
t-2)+u
t-1+u
t
=
3
y
t-3+
2
u
t-2+u
t-1+u
t
•Successivesubstitutionsofthistypeleadto:
y
t=
T
y
0+u
t-1+
2
u
t-2+
3
u
t-3+...+
T
u
0+u
t

8. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
The Impact of Shocks for
Stationary and Non-stationary Series
•Wehave3cases:
1.<1
T
0asT
Sotheshockstothesystemgraduallydieaway.
2.=1
T
=1T
Soshockspersistinthesystemandneverdieaway.Weobtain:
asT
Sojustaninfinitesumofpastshocksplussomestartingvalueofy
0.
3.>1.Nowgivenshocksbecomemoreinfluentialastimegoeson,
sinceif>1,
3
>
2
>etc.



0
0
i
tt uyy

9. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Detrending a Stochastically Non-stationary Series
•Goingbacktoour2characterisationsofnon-stationarity,ther.w.withdrift:
y
t=+y
t-1+u
t (1)
andthetrend-stationaryprocess
y
t=+t+u
t (2)
•Thetwowillrequiredifferenttreatmentstoinducestationarity.Thesecondcase
isknownasdeterministicnon-stationarityandwhatisrequiredisdetrending.
•Thefirstcaseisknownasstochasticnon-stationarity.Ifwelet
y
t=y
t-y
t-1
and Ly
t=y
t-1
so (1-L)y
t=y
t-Ly
t=y
t-y
t-1
Ifwetake(1)andsubtracty
t-1frombothsides:
y
t-y
t-1=+u
t
y
t =+u
t
Wesaythatwehaveinducedstationarityby“differencingonce”.

10. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Detrending a Series: Using the Right Method
•Althoughtrend-stationaryanddifference-stationaryseriesareboth
“trending”overtime,thecorrectapproachneedstobeusedineachcase.
•Ifwefirstdifferencethetrend-stationaryseries,itwould“remove”the
non-stationarity,butattheexpenseonintroducinganMA(1)structureinto
theerrors.
•Converselyifwetrytodetrendaserieswhichhasstochastictrend,thenwe
willnotremovethenon-stationarity.
•Wewillnowconcentrateonthestochasticnon-stationaritymodelsince
deterministicnon-stationaritydoesnotadequatelydescribemostseriesin
economicsorfinance.

11. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Sample Plots for various Stochastic Processes:
A White Noise Process-4
-3
-2
-1
0
1
2
3
4
14079118157196235274313352391430469

12. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Sample Plots for various Stochastic Processes:
A Random Walk and a Random Walk with Drift-20
-10
0
10
20
30
40
50
60
70
11937557391109127145163181199217235253271289307325343361379397415433451469487
Random Walk
Random Walk with Drift

13. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Sample Plots for various Stochastic Processes:
A Deterministic Trend Process-5
0
5
10
15
20
25
30
14079118157196235274313352391430469

14. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Autoregressive Processes with
differing values of (0, 0.8, 1)-20
-15
-10
-5
0
5
10
15
153105157209261313365417469521573625677729781833885937989
Phi=1
Phi=0.8
Phi=0

15. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Definition of Non-Stationarity
•Consideragainthesimpleststochastictrendmodel:
y
t =y
t-1+u
t
or y
t =u
t
•Wecangeneralisethisconcepttoconsiderthecasewheretheseries
containsmorethanone“unitroot”.Thatis,wewouldneedtoapplythe
firstdifferenceoperator,,morethanoncetoinducestationarity.
Definition
Ifanon-stationaryseries,y
tmustbedifferenceddtimesbeforeitbecomes
stationary,thenitissaidtobeintegratedoforderd.Wewritey
tI(d).
Soify
tI(d)then
d
y
tI(0).
AnI(0)seriesisastationaryseries
AnI(1)seriescontainsoneunitroot,
e.g.y
t=y
t-1+u
t

16. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Characteristics of I(0), I(1) and I(2) Series
•An I(2) series contains two unit roots and so would require differencing
twice to induce stationarity.
•I(1) and I(2) series can wander a long way from their mean value and
cross this mean value rarely.
•I(0) series should cross the mean frequently.
•The majority of economic and financial series contain a single unit root,
although some are stationary and consumer prices have been argued to
have 2 unit roots.

17. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
How do we test for a unit root?
•The early and pioneering work on testing for a unit root in time series
was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976).
The basic objective of the test is to test the null hypothesis that =1 in:
y
t= y
t-1+ u
t
against the one-sided alternative <1. So we have
H
0: series contains a unit root
vs. H
1: series is stationary.
•We usually use the regression:
y
t= y
t-1+ u
t
so that a test of =1 is equivalent to a test of =0 (since -1=).

18. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Different forms for the DF Test Regressions
•DickeyFullertestsarealsoknownastests:,
,
.
•Thenull(H
0)andalternative(H
1)modelsineachcaseare
i)H
0:y
t=y
t-1+u
t
H
1:y
t=y
t-1+u
t,<1
Thisisatestforarandomwalkagainstastationaryautoregressiveprocessof
orderone(AR(1))
ii)H
0:y
t=y
t-1+u
t
H
1:y
t=y
t-1++u
t,<1
ThisisatestforarandomwalkagainstastationaryAR(1)withdrift.
iii)H
0:y
t=y
t-1+u
t
H
1:y
t=y
t-1++t+u
t,<1
This is a test for a random walk against a stationary AR(1) with drift and a
time trend.

19. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Computing the DF Test Statistic
•Wecanwrite
y
t=u
t
wherey
t=y
t-y
t-1,andthealternativesmaybeexpressedas
y
t=y
t-1++t+u
t
with==0incasei),and=0incaseii)and=-1.Ineachcase,the
testsarebasedonthet-ratioonthey
t-1termintheestimatedregressionof
y
tony
t-1,plusaconstantincaseii)andaconstantandtrendincaseiii).
Theteststatisticsaredefinedas
teststatistic=
•Theteststatisticdoesnotfollowtheusualt-distributionunderthenull,
sincethenullisoneofnon-stationarity,butratherfollowsanon-standard
distribution.CriticalvaluesarederivedfromMonteCarloexperimentsin,
forexample,Fuller(1976).Relevantexamplesofthedistributionare
shownintable4.1below




SE()

20. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Critical Values for the DF Test
Thenullhypothesisofaunitrootisrejectedinfavourofthestationaryalternative
ineachcaseiftheteststatisticismorenegativethanthecriticalvalue.Significance level10% 5% 1%
C.V. for constant
but no trend
-2.57-2.86-3.43
C.V. for constant
and trend
-3.12-3.41-3.96
Table 4.1: Critical Values for DF and ADF Tests (Fuller,
1976, p373).

21. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
The Augmented Dickey Fuller (ADF) Test
•Thetestsaboveareonlyvalidifu
tiswhitenoise.Inparticular,u
twillbe
autocorrelatediftherewasautocorrelationinthedependentvariableofthe
regression(y
t)whichwehavenotmodelled.Thesolutionisto“augment”
thetestusingplagsofthedependentvariable.Thealternativemodelin
case(i)isnowwritten:
•ThesamecriticalvaluesfromtheDFtablesareusedasbefore.Aproblem
nowarisesindeterminingtheoptimalnumberoflagsofthedependent
variable.
Thereare2ways
-usethefrequencyofthedatatodecide
-useinformationcriteria

 
p
i
tititt uyyy
1
1

22. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Testing for Higher Orders of Integration
•Considerthesimpleregression:
y
t=y
t-1+u
t
WetestH
0:=0vs.H
1:<0.
•IfH
0isrejectedwesimplyconcludethaty
tdoesnotcontainaunitroot.
•ButwhatdoweconcludeifH
0isnotrejected?Theseriescontainsaunit
root,butisthatit?No!Whatify
tI(2)?Wewouldstillnothaverejected.So
wenowneedtotest
H
0:y
tI(2)vs.H
1:y
tI(1)
WewouldcontinuetotestforafurtherunitrootuntilwerejectedH
0.
•Wenowregress
2
y
tony
t-1(pluslagsof
2
y
tifnecessary).
•NowwetestH
0:y
tI(1)whichisequivalenttoH
0:y
tI(2).
•Sointhiscase,ifwedonotreject(unlikely),weconcludethaty
tisatleast
I(2).

23. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
The Phillips-Perron Test
•Phillips and Perron have developed a more comprehensive theory of
unit root nonstationarity. The tests are similar to ADF tests, but they
incorporate an automatic correction to the DF procedure to allow for
autocorrelated residuals.
•The tests usually give the same conclusions as the ADF tests, and the
calculation of the test statistics is complex.

24. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Criticism of Dickey-Fuller and
Phillips-Perron-type tests
•Main criticism is that the power of the tests is low if the process is
stationary but with a root close to the non-stationary boundary.
e.g. the tests are poor at deciding if
=1 or =0.95,
especially with small sample sizes.
•If the true data generating process (dgp) is
y
t
= 0.95y
t-1
+ u
t
then the null hypothesis of a unit root should be rejected.
•One way to get around this is to use a stationarity test as well as the
unit root tests we have looked at.

25. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Stationarity tests
•Stationarity tests have
H
0
: y
t
is stationary
versus H
1
: y
t
is non-stationary
So that by default under the null the data will appear stationary.
•One such stationarity test is the KPSS test (Kwaitowski, Phillips,
Schmidt and Shin, 1992).
•Thus we can compare the results of these tests with the ADF/PP
procedure to see if we obtain the same conclusion.

26. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Stationarity tests (cont’d)
•A Comparison
ADF / PP KPSS
H
0
: y
t
I(1) H
0
: y
t
I(0)
H
1
: y
t
I(0) H
1
: y
t
I(1)
•4 possible outcomes
Reject H
0
and Do not reject H
0
Do not reject H
0
and Reject H
0
Reject H
0
and Reject H
0
Do not reject H
0
and Do not reject H
0

27. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Cointegration: An Introduction
•Inmostcases,ifwecombinetwovariableswhichareI(1),thenthe
combinationwillalsobeI(1).
•Moregenerally,ifwecombinevariableswithdifferingordersof
integration,thecombinationwillhaveanorderofintegrationequaltothe
largest.i.e.,
ifX
i,tI(d
i)fori=1,2,3,...,k
sowehavekvariableseachintegratedoforderd
i.
Let (1)
Thenz
tI(maxd
i)z Xt iit
i
k


,
1

28. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Linear Combinations of Non-stationary Variables
•Rearranging (1), we can write
where
•This is just a regression equation.
•But the disturbances would have some very undesirable properties: z
t´is
not stationary and is autocorrelated if all of the X
iare I(1).
•We want to ensure that the disturbances are I(0). Under what circumstances
will this be the case?

 
i
i
t
t
z
z
i k  
1 1
2,' ,,..., X Xz
t iit t
i
k
1
2
, ,
' 



29. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Definition of Cointegration (Engle & Granger, 1987)
•Letz
tbeak1vectorofvariables,thenthecomponentsofz
tarecointegrated
oforder(d,b)if
i)Allcomponentsofz
tareI(d)
ii)Thereisatleastonevectorofcoefficientssuchthatz
tI(d-b)
•Manytimeseriesarenon-stationarybut“movetogether”overtime.
•Ifvariablesarecointegrated,itmeansthatalinearcombinationofthemwill
bestationary.
•Theremaybeuptorlinearlyindependentcointegratingrelationships(where
rk-1),alsoknownascointegratingvectors.risalsoknownasthe
cointegratingrankofz
t.
•Acointegratingrelationshipmayalsobeseenasalongtermrelationship.

30. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Cointegration and Equilibrium
•Examples of possible Cointegrating Relationships in finance:
–spot and futures prices
–ratio of relative prices and an exchange rate
–equity prices and dividends
•Market forces arising from no arbitrage conditions should ensure an
equilibrium relationship.
•No cointegration implies that series could wander apart without bound
in the long run.

31. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Equilibrium Correction or Error Correction Models
•Whentheconceptofnon-stationaritywasfirstconsidered,ausual
responsewastoindependentlytakethefirstdifferencesofaseriesofI(1)
variables.
•Theproblemwiththisapproachisthatpurefirstdifferencemodelshaveno
longrunsolution.
e.g.Considery
tandx
tbothI(1).
Themodelwemaywanttoestimateis
y
t=x
t+u
t
Butthiscollapsestonothinginthelongrun.
•Thedefinitionofthelongrunthatweuseiswhere
y
t=y
t-1=y;x
t=x
t-1=x.
•Henceallthedifferencetermswillbezero,i.e.y
t=0;x
t=0.

32. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Specifying an ECM
•One way to get around this problem is to use both first difference and
levels terms, e.g.
y
t= 
1x
t+ 
2(y
t-1-x
t-1)+u
t (2)
•y
t-1-x
t-1is known as the error correction term.
•Providing that y
tandx
tare cointegrated with cointegrating coefficient
, then (y
t-1-x
t-1) will be I(0) even though the constituents are I(1).
•We can thus validly use OLS on (2).
•The Granger representation theorem shows that any cointegrating
relationship can be expressed as an equilibrium correction model.

33. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Testing for Cointegration in Regression
•Themodelfortheequilibriumcorrectiontermcanbegeneralisedto
includemorethantwovariables:
y
t
=
1
+
2
x
2t
+
3
x
3t
+…+
k
x
kt
+u
t
(3)
•u
tshouldbeI(0)ifthevariablesy
t,x
2t,...x
ktarecointegrated.
•Sowhatwewanttotestistheresidualsofequation(3)toseeifthey
arenon-stationaryorstationary.WecanusetheDF/ADFtestonu
t.
Sowehavetheregression
withv
tiid.
•However,sincethisisatestontheresidualsofanactualmodel,,
thenthecriticalvaluesarechanged.uuvt t t 1 ut

34. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Testing for Cointegration in Regression:
Conclusions
•EngleandGranger(1987)havetabulatedanewsetofcriticalvalues
andhencethetestisknownastheEngleGranger(E.G.)test.
•WecanalsousetheDurbinWatsonteststatisticorthePhillipsPerron
approachtotestfornon-stationarityof.
•Whatarethenullandalternativehypothesesforatestontheresiduals
ofapotentiallycointegratingregression?
H
0:unitrootincointegratingregression’sresiduals
H
1:residualsfromcointegratingregressionarestationaryut

35. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Methods of Parameter Estimation in
Cointegrated Systems:
The Engle-Granger Approach
•Thereare(atleast)3methodswecoulduse:EngleGranger,EngleandYoo,
andJohansen.
•TheEngleGranger2StepMethod
Thisisasingleequationtechniquewhichisconductedasfollows:
Step1:
-MakesurethatalltheindividualvariablesareI(1).
-ThenestimatethecointegratingregressionusingOLS.
-Savetheresidualsofthecointegratingregression,.
-TesttheseresidualstoensurethattheyareI(0).
Step 2:
-Use the step 1 residuals as one variable in the error correction model e.g.
y
t= 
1x
t+ 
2( )+u
t
where =y
t-1-x
t-11
ˆ
t
u 1
ˆ
t
u ut ˆ

36. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
An Example of a Model for Non-stationary Variables:
Lead-Lag Relationships between Spot
and Futures Prices
Background
•Weexpectchangesinthespotpriceofafinancialassetanditscorresponding
futurespricetobeperfectlycontemporaneouslycorrelatedandnottobe
cross-autocorrelated.
i.e.expectCorr(ln(F
t),ln(S
t))1
Corr(ln(F
t),ln(S
t-k))0k
Corr(ln(F
t-j),ln(S
t))0j
•Wecantestthisideabymodellingthelead-lagrelationshipbetweenthetwo.
•WewillconsidertwopapersTse(1995)andBrooksetal(2001).

37. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Futures & Spot Data
•Tse (1995): 1055 daily observations on NSA stock index and stock
index futures values from December 1988 -April 1993.
•Brooks et al(2001): 13,035 10-minutely observations on the FTSE
100 stock index and stock index futures prices for all trading days in
the period June 1996 –1997.

38. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Methodology
•Thefairfuturespriceisgivenby
whereF
t
*
isthefairfuturesprice,S
tisthespotprice,risa
continuouslycompoundedrisk-freerateofinterest,disthe
continuouslycompoundedyieldintermsofdividendsderivedfromthe
stockindexuntilthefuturescontractmatures,and(T-t)isthetimeto
maturityofthefuturescontract.Takinglogarithmsofbothsidesof
equationabovegives
•First,testf
tands
tfornonstationarity.t
*
t
(r-d)(T-t)
F = Se t)-d)(T-(r s f
tt 
*

39. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Dickey-Fuller Tests on Log-Prices and Returns for High
Frequency FTSE Data Futures Spot
Dickey-Fuller Statistics
for Log-Price Data
-0.1329 -0.7335
Dickey Fuller Statistics
for Returns Data
-84.9968 -114.1803

40. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Cointegration Test Regression and Test on Residuals
•Conclusion:logF
tandlogS
tarenotstationary,butlogF
tandlogS
tare
stationary.
•Butamodelcontainingonlyfirstdifferenceshasnolongrunrelationship.
•Solutionistoseeifthereexistsacointegratingrelationshipbetweenf
tand
s
twhichwouldmeanthatwecanvalidlyincludelevelstermsinthis
framework.
•Potentialcointegratingregression:
wherez
tisadisturbanceterm.
•Estimatetheregression,collecttheresiduals,,andtestwhethertheyare
stationary.z
t ttt zfs 
10

41. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Estimated Equation and Test for Cointegration for
High Frequency FTSE DataCointegrating Regression
Coefficient

0

1
Estimated Value
0.1345
0.9834
DF Test on residuals
tzˆ
Test Statistic
-14.7303

42. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Conclusions from Unit Root and Cointegration Tests
•Conclusion:arestationaryandthereforewehaveacointegrating
relationshipbetweenlogF
tandlogS
t.
•FinalstageinEngle-Granger2-stepmethodistousethefirststage
residuals,astheequilibriumcorrectionterminthegeneralequation.
•Theoverallmodelisz
t z
t ttttt vFSzS 
 111110 lnlnˆln 

43. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Estimated Error Correction Model for
High Frequency FTSE Data
Look at the signs and significances of the coefficients:
• is positive and highly significant
• is positive and highly significant
• is negative and highly significantCoefficient Estimated Value t-ratio


0
9.6713E-06 1.6083

 -8.3388E-01 -5.1298


1
0.1799 19.2886

1 0.1312 20.4946 1
ˆ 1
ˆ
 


44. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Forecasting High Frequency FTSE Returns
•Isitpossibletousetheerrorcorrectionmodeltoproducesuperior
forecaststoothermodels?
Comparison of Out of Sample Forecasting AccuracyECM ECM-COC ARIMA VAR
RMSE 0.00043820.00043500.00045310.0004510
MAE 0.4259 0.4255 0.4382 0.4378
% Correct
Direction
67.69% 68.75% 64.36% 66.80%

45. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Can Profitable Trading Rules be Derived from the
ECM-COC Forecasts?
•The trading strategy involves analysing the forecast for the spot return, and
incorporating the decision dictated by the trading rules described below. It is assumed
that the original investment is £1000, and if the holding in the stock index is zero, the
investment earns the risk free rate.
–Liquid Trading Strategy -making a round trip trade (i.e. a purchase and sale of
the FTSE100 stocks) every ten minutes that the return is predicted to be positive
by the model.
–Buy-&-Hold while Forecast Positive Strategy -allows the trader to continue
holding the index if the return at the next predicted investment period is positive.
–Filter Strategy: Better Predicted Return Than Average-involves purchasing the
index only if the predicted returns are greater than the average positive return.
–Filter Strategy: Better Predicted Return Than First Decile -only the returns
predicted to be in the top 10% of all returns are traded on
–Filter Strategy: High Arbitrary Cut Off-An arbitrary filter of 0.0075% is
imposed,

46. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008Trading StrategyTerminal
Wealth
( £ )
Return ( % )
{Annualised}
Terminal
Wealth (£)
with slippage
Return ( % )
{Annualised}
with slippage
Number
of trades
Passive
Investment
1040.92 4.09
{49.08}
1040.92 4.09
{49.08}
1
Liquid Trading1156.21 15.62
{187.44}
1056.38 5.64
{67.68}
583
Buy-&-Hold while
Forecast Positive
1156.21 15.62
{187.44}
1055.77 5.58
{66.96}
383
Filter I 1144.51 14.45
{173.40}
1123.57 12.36
{148.32}
135
Filter II 1100.01 10.00
{120.00}
1046.17 4.62
{55.44}
65
Filter III 1019.82 1.98
{23.76}
1003.23 0.32
{3.84}
8
Spot Trading Strategy Results for Error Correction
Model Incorporating the Cost of Carry

47. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Conclusions
•Thefuturesmarket“leads”thespotmarketbecause:
•thestockindexisnotasingleentity,so
•somecomponentsoftheindexareinfrequentlytraded
•itismoreexpensivetotransactinthespotmarket
•stockmarketindicesareonlyrecalculatedeveryminute
•Spot&futuresmarketsdoindeedhavealongrunrelationship.
•Sinceitappearsimpossibletoprofitfromlead/lagrelationships,their
existenceisentirelyconsistentwiththeabsenceofarbitrage
opportunitiesandinaccordancewithmoderndefinitionsofthe
efficientmarketshypothesis.

48. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
The Engle-Granger Approach: Some Drawbacks
This method suffers from a number of problems:
1. Unit root and cointegration tests have low power in finite samples
2. We are forced to treat the variables asymmetrically and to specify one as
the dependent and the other as independent variables.
3. Cannot perform any hypothesis tests about the actual cointegrating
relationship estimated at stage 1.
-Problem 1 is a small sample problem that should disappear
asymptotically.
-Problem 2 is addressed by the Johansen approach.
-Problem 3 is addressed by the Engle and Yoo approach or the Johansen
approach.

49. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•OneoftheproblemswiththeEG2-stepmethodisthatwecannotmake
anyinferencesabouttheactualcointegratingregression.
•TheEngle&Yoo(EY)3-stepproceduretakesitsfirsttwostepsfromEG.
•EYaddathirdstepgivingupdatedestimatesofthecointegratingvector
anditsstandarderrors.
•Themostimportantproblemwithboththesetechniquesisthatinthe
generalcaseabove,wherewehavemorethantwovariableswhichmaybe
cointegrated,therecouldbemorethanonecointegratingrelationship.
•Infacttherecanbeuptorlinearlyindependentcointegratingvectors
(whererg-1),wheregisthenumberofvariablesintotal.
The Engle & Yoo 3-Step Method

50. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•So,inthecasewherewejusthadyandx,thenrcanonlybeoneor
zero.
•Butinthegeneralcasetherecouldbemorecointegratingrelationships.
•Andifthereareothers,howdoweknowhowmanythereareor
whetherwehavefoundthe“best”?
•The answer to this is to use a systems approach to cointegration which
will allow determination of all rcointegrating relationships -
Johansen’s method.
The Engle & Yoo 3-Step Method (cont’d)

51. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•TouseJohansen’smethod,weneedtoturntheVARoftheform
y
t
= 
1
y
t-1
+
2
y
t-2
+...+
k
y
t-k
+u
t
g×1 g×gg×1g×gg×1 g×gg×1g×1
intoaVECM,whichcanbewrittenas
y
t
=y
t-k
+
1
y
t-1
+
2
y
t-2
+...+
k-1
y
t-(k-1)
+u
t
where= and
isalongruncoefficientmatrixsinceallthey
t-i
=0.
Testing for and Estimating Cointegrating Systems
Using the Johansen Technique Based on VARs


k
j
giI
1
)( 


i
j
gji I
1
)(

52. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Letdenoteaggsquarematrixandletcdenoteag1non-zerovector,
andletdenoteasetofscalars.
•iscalledacharacteristicrootorsetofrootsofifwecanwrite
c=c
ggg1g1
•Wecanalsowrite
c=I
p
c
andhence
(-I
g
)c=0
whereI
g
isanidentitymatrix.
Review of Matrix Algebra
necessary for the Johansen Test

53. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Sincec0bydefinition,thenforthissystemtohavezerosolution,we
requirethematrix(-I
g)tobesingular(i.e.tohavezerodeterminant).
-I
g=0
•Forexample,letbethe22matrix
•Thenthecharacteristicequationis
-I
g
Review of Matrix Algebra (cont’d)






51
24 

















 
51
24
10
01
0
5 1
24
5 4 2 918
2



 ()()

54. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Thisgivesthesolutions=6and=3.
•ThecharacteristicrootsarealsoknownasEigenvalues.
•Therankofamatrixisequaltothenumberoflinearlyindependentrowsor
columnsinthematrix.
•WewriteRank()=r
•Therankofamatrixisequaltotheorderofthelargestsquarematrixwe
canobtainfromwhichhasanon-zerodeterminant.
•Forexample,thedeterminantofabove0,thereforeithasrank2.
Review of Matrix Algebra (cont’d)

55. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•SomepropertiesoftheeigenvaluesofanysquarematrixA:
1.thesumoftheeigenvaluesisthetrace
2.theproductoftheeigenvaluesisthedeterminant
3.thenumberofnon-zeroeigenvaluesistherank
•ReturningtoJohansen’stest,theVECMrepresentationoftheVARwas
y
t
=y
t-1
+
1
y
t-1
+
2
y
t-2
+...+
k-1
y
t-(k-1)
+u
t
•Thetestforcointegrationbetweenthey’siscalculatedbylookingatthe
rankofthematrixviaitseigenvalues.(Toprovethisrequiressome
technicalintermediatesteps).
•Therankofamatrixisequaltothenumberofitscharacteristicroots
(eigenvalues)thataredifferentfromzero.
The Johansen Test and Eigenvalues

56. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Theeigenvaluesdenoted
i
areputinorder:

1

2
...
g
•Ifthevariablesarenotcointegrated,therankofwillnotbe
significantlydifferentfromzero,so
i
=0i.
Thenif
i
=0,ln(1-
i
)=0
Ifthe’sareroots,theymustbelessthan1inabsolutevalue.
•Sayrank()=1,thenln(1-
1
)willbenegativeandln(1-
i
)=0
•Iftheeigenvalueiisnon-zero,thenln(1-
i
)<0i>1.
The Johansen Test and Eigenvalues (cont’d)

57. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Theteststatisticsforcointegrationareformulatedas
and
whereistheestimatedvaluefortheithorderedeigenvaluefromthe
matrix.

trace
teststhenullthatthenumberofcointegratingvectorsislessthan
equaltoragainstanunspecifiedalternative.

trace
=0whenallthe
i
=0,soitisajointtest.

max
teststhenullthatthenumberofcointegratingvectorsisragainst
analternativeofr+1.
The Johansen Test Statistics 
max(,) ln(

)rr T
r 
1 1
1 


g
ri
itrace
Tr
1
)
ˆ
1ln()( 

58. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Decomposition of the Matrix
•For any 1 < r < g, is defined as the product of two matrices:
= 
gg gr rg
•contains the cointegrating vectors while gives the “loadings” of
each cointegrating vector in each equation.
•For example, if g=4 and r=1, and will be 41, and y
t-k
will be
given by:
or  
kt
y
y
y
y






























4
3
2
1
14131211
14
13
12
11




  
kt
yyyy
















414313212111
14
13
12
11






59. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Johansen&Juselius(1990)providecriticalvaluesforthe2statistics.
Thedistributionoftheteststatisticsisnon-standard.Thecritical
valuesdependon:
1.thevalueofg-r,thenumberofnon-stationarycomponents
2.whetheraconstantand/ortrendareincludedintheregressions.
•IftheteststatisticisgreaterthanthecriticalvaluefromJohansen’s
tables,rejectthenullhypothesisthattherearercointegratingvectors
infavourofthealternativethattherearemorethanr.
Johansen Critical Values

60. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Thetestingsequenceunderthenullisr=0,1,...,g-1
sothatthehypothesesfor
trace
are
H
0
:r=0 vs H
1
:0<rg
H
0
:r=1 vs H
1
:1<rg
H
0
:r=2 vs H
1
:2<rg
... ... ...
H
0
:r=g-1 vs H
1
:r=g
•Wekeepincreasingthevalueofruntilwenolongerrejectthenull.
The Johansen Testing Sequence

61. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Buthowdoesthiscorrespondtoatestoftherankofthematrix?
•ristherankof.
•cannotbeoffullrank(g)sincethiswouldcorrespondtotheoriginal
y
t
beingstationary.
•Ifhaszerorank,thenbyanalogytotheunivariatecase,y
t
depends
onlyony
t-j
andnotony
t-1
,sothatthereisnolongrunrelationship
betweentheelementsofy
t-1
.Hencethereisnocointegration.
•For1<rank()<g,therearemultiplecointegratingvectors.
Interpretation of Johansen Test Results

62. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Hypothesis Testing Using Johansen
•EGdidnotallowustodohypothesistestsonthecointegratingrelationship
itself,buttheJohansenapproachdoes.
•Ifthereexistrcointegratingvectors,onlytheselinearcombinationswillbe
stationary.
•Youcantestahypothesisaboutoneormorecoefficientsinthe
cointegratingrelationshipbyviewingthehypothesisasarestrictiononthe
matrix.
•Alllinearcombinationsofthecointegratingvectorsarealsocointegrating
vectors.
•Ifthenumberofcointegratingvectorsislarge,andthehypothesisunder
considerationissimple,itmaybepossibletorecombinethecointegrating
vectorstosatisfytherestrictionsexactly.

63. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Hypothesis Testing Using Johansen (cont’d)
•Astherestrictionsbecomemorecomplexormorenumerous,itwill
eventuallybecomeimpossibletosatisfythembyrenormalisation.
•Afterthispoint,iftherestrictionisnotsevere,thenthecointegrating
vectorswillnotchangemuchuponimposingtherestriction.
•Ateststatistictotestthishypothesisisgivenby

2
(m)
where,
arethecharacteristicrootsoftherestrictedmodel
arethecharacteristicrootsoftheunrestrictedmodel
risthenumberofnon-zerocharacteristicrootsintheunrestrictedmodel,
andmisthenumberofrestrictions.
i
* 
i 


r
i
iiT
1
*)]1ln()1[ln( 

64. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Cointegration Tests using Johansen:
Three Examples
Example1:Hamilton(1994,pp.647)
•DoesthePPPrelationshipholdfortheUS/Italianexchangerate-
pricesystem?
•AVARwasestimatedwith12lagson189observations.TheJohansen
teststatisticswere
r 
max criticalvalue
0 22.12 20.8
1 10.19 14.0
•Conclusion:thereisonecointegratingrelationship.

65. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Example 2: Purchasing Power Parity (PPP)
•PPP states that the equilibrium exchange rate between 2 countries is
equal to the ratio of relative prices
•A necessary and sufficient condition for PPP is that the log of the
exchange rate between countries A and B, and the logs of the price
levels in countries A and B be cointegrated with cointegrating vector
[ 1 –1 1] .
•Chen (1995) uses monthly data for April 1973-December 1990 to test
the PPP hypothesis using the Johansen approach.

66. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Cointegration Tests of PPP with European Data
Tests for
cointegration between
r = 0 r  1 r  2 1 2
FRF – DEM 34.63* 17.10 6.26 1.33 -2.50
FRF – ITL 52.69* 15.81 5.43 2.65 -2.52
FRF – NLG 68.10* 16.37 6.42 0.58 -0.80
FRF – BEF 52.54* 26.09* 3.63 0.78 -1.15
DEM – ITL 42.59* 20.76* 4.79 5.80 -2.25
DEM – NLG 50.25* 17.79 3.28 0.12 -0.25
DEM – BEF 69.13* 27.13* 4.52 0.87 -0.52
ITL – NLG 37.51* 14.22 5.05 0.55 -0.71
ITL – BEF 69.24* 32.16* 7.15 0.73 -1.28
NLG – BEF 64.52* 21.97* 3.88 1.69 -2.17
Critical values 31.52 17.95 8.18 - -
Notes: FRF- French franc; DEM – German Mark; NLG – Dutch guilder; ITL – Italian lira; BEF –
Belgian franc. Source: Chen (1995). Reprinted with the permission of Taylor and Francis Ltd.
(www.tandf.co.uk).

67. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Example 3: Are International
Bond Markets Cointegrated?
•Mills & Mills (1991)
•Iffinancialmarketsarecointegrated,thisimpliesthattheyhavea
“commonstochastictrend”.
Data:
•Dailyclosingobservationsonredemptionyieldsongovernmentbondsfor
4bondmarkets:US,UK,WestGermany,Japan.
•For cointegration, a necessary but not sufficient condition is that the yields
are nonstationary. All 4 yields series are I(1).

68. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Testing for Cointegration Between the Yields
•TheJohansenprocedureisused.Therecanbeatmost3linearlyindependent
cointegratingvectors.
•Mills&Millsusethetraceteststatistic:
where
i
aretheorderedeigenvalues.


g
ri
itrace Tr
1
)
ˆ
1ln()(  Johansen Tests for Cointegration between International Bond Yields
Test statistic Critical Values r (number of cointegrating
vectors under the null hypothesis) 10% 5%
0 22.06 35.6 38.6
1 10.58 21.2 23.8
2 2.52 10.3 12.0
3 0.12 2.9 4.2
Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.

69. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Testing for Cointegration Between the Yields
(cont’d)
•Conclusion:Nocointegratingvectors.
•ThepaperthengoesontoestimateaVARforthefirstdifferencesofthe
yields,whichisoftheform
where
They set k = 8.X
XUS
XUK
XWG
XJAP
t
t
t
t
t
i
i i i i
i i i i
i i i i
i i i i
t
t
t
t
t







































()
()
()
()
, ,




11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
1
2
3
4




 


k
i
titit XX
1


70. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Variance Decompositions for VAR
of International Bond Yields
Variance Decompositions for VAR of International Bond Yields
Explained by movements in Explaining
movements in
Days
ahead US UK Germany Japan
US 1 95.6 2.4 1.7 0.3
5 94.2 2.8 2.3 0.7
10 92.9 3.1 2.9 1.1
20 92.8 3.2 2.9 1.1

UK 1 0.0 98.3 0.0 1.7
5 1.7 96.2 0.2 1.9
10 2.2 94.6 0.9 2.3
20 2.2 94.6 0.9 2.3

Germany 1 0.0 3.4 94.6 2.0
5 6.6 6.6 84.8 3.0
10 8.3 6.5 82.9 3.6
20 8.4 6.5 82.7 3.7

Japan 1 0.0 0.0 1.4 100.0
5 1.3 1.4 1.1 96.2
10 1.5 2.1 1.8 94.6
20 1.6 2.2 1.9 94.2

Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.

71. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Impulse Responses for VAR of
International Bond Yields
Impulse Responses for VAR of International Bond Yields
Response of US to innovations in
Days after shock US UK Germany Japan
0 0.98 0.00 0.00 0.00
1 0.06 0.01 -0.10 0.05
2 -0.02 0.02 -0.14 0.07
3 0.09 -0.04 0.09 0.08
4 -0.02 -0.03 0.02 0.09
10 -0.03 -0.01 -0.02 -0.01
20 0.00 0.00 -0.10 -0.01

Response of UK to innovations in
Days after shock US UK Germany Japan
0 0.19 0.97 0.00 0.00
1 0.16 0.07 0.01 -0.06
2 -0.01 -0.01 -0.05 0.09
3 0.06 0.04 0.06 0.05
4 0.05 -0.01 0.02 0.07
10 0.01 0.01 -0.04 -0.01
20 0.00 0.00 -0.01 0.00

Response of Germany to innovations in
Days after shock US UK Germany Japan
0 0.07 0.06 0.95 0.00
1 0.13 0.05 0.11 0.02
2 0.04 0.03 0.00 0.00
3 0.02 0.00 0.00 0.01
4 0.01 0.00 0.00 0.09
10 0.01 0.01 -0.01 0.02
20 0.00 0.00 0.00 0.00

Response of Japan to innovations in
Days after shock US UK Germany Japan
0 0.03 0.05 0.12 0.97
1 0.06 0.02 0.07 0.04
2 0.02 0.02 0.00 0.21
3 0.01 0.02 0.06 0.07
4 0.02 0.03 0.07 0.06
10 0.01 0.01 0.01 0.04
20 0.00 0.00 0.00 0.01

Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.