cointegration for market forecast in the spanish stock market
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Cointegration for market forecast in the Spanishstock marketCarmen Ansotegui & María Victoria EstebanPublished online: 05 Oct 2010.
To cite this article: Carmen Ansotegui & Mara Victoria Esteban (2002) Cointegration for market forecast in the Spanishstock market, Applied Economics, 34:7, 843-857, DOI: 10.1080/00036840110058932
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Cointegration for market forecast in the
Spanish stock market
CARMEN ANSOTEGUI and MARIÂ A VICTORIA ESTEBAN*
ESADE. Avda, Pedralbes 60-62, 08034 Barcelona, SpainE-mail: [email protected] .and Departamento de EconometrõÂa y EstadõÂstica, Universidad del PaõÂs Vasco, AvenidaLehendakari Agirre, 83, 48015 Bilbao, SpainE-mail: [email protected].
This study is interested in empirically testing the existence of a long-run relationshipbetween the Spanish stock market and its fundamentals, and in checking to whichextent this relationship helps in forecasting. This study is concerned with the behav-iour of the aggregate Madrid Stock Exchange in a macroeconomic context. It alsoidenti®es as macroeconomic fundamentals : industrial production as a proxy for realactivity, in¯ation and interest rates. This study tests the existence of cointegration byJohansen’s procedure. The long-run relationships among the variables implied by theexistence of cointegration do not allow inference to the interrelations among thevariables. To get some insight into the short-run interactions among the variables, animpulse response analysis was performed. This study compares the forecasting abil-ity of its model with respect to alternative multivariate speci®cations in terms ofRMSE. Also measured is the value of the forecast for the ®nancial agents assessingthe extent to which it helps improve asset allocation.
I . INTRODUCTION
Models of asset price determination, such as Abel (1988),
Balvers et al. (1990) and Cochrane (1991) establish that the
aggregate stock market is determined by macroeconomic
fundamentals. Although Shiller (1981) proves the stock
market is too volatile in respect to its fundamentals, several
empirical studies in the US try to identify the variables at
the macroeconomic level which can predict stock market
returns. More speci®cally, Fama (1981, 1990), Fama and
French (1989), Keim and Stambaugh (1986) , Barro (1990),
Chen (1991) and Chen et al. (1986) among others, identify
the term spread and the default spread as macro-®nancial
variables, which vary with the business cycle and determine
stock market returns. Only Fama (1981, 1990), Barro
(1989) and Chen et al. (1986) explicitly include industrial
production in the analysis.Fama (1981) for the US and Peiro (1995) for three
European countries ®nd that industrial production
explains a greater proportion of variance for longer
return horizons than for shorter ones. In another line of
research, Estrella and Hardouvelis (1991) report that the
term structure of interest rates predicts cumulativechanges in real output up to four years into the future.
These two facts, together with the high variability of
stock returns, lead us to consider that the stock market
behaviour with respect to fundamentals is di� erent in the
long and short run. This fact is recognized by Kandel and
Stambaugh (1988) who model stock returns for short and
long run.
Applied Economics ISSN 0003±6846 print/ISSN 1466±4283 online # 2002 Taylor & Francis Ltd
http://www.tandf.co.uk/journalsDOI: 10.1080/0003684011005893 2
Applied Economics, 2002, 34, 843±857
843
* Corresponding author E-mail: [email protected].
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Starting from the premise that the stock market behavesdi� erently in the long and short run, we are interested intesting whether the Spanish stock market is driven byfundamentals, at least in the long-run, which allow forpossible speculative movements of prices in the short-run.Cointegration analysis is used to test the existence of long-run equilibrium relationships between the stock marketand its fundamentals . The short-run interactions amongthe variables are inferred through the impulse responsefunctions of the system when cointegration restrictionsare imposed.
Ansotegui (1992) ®nds cointegration among stockmarket indexes and fundamentals for the US andJapanese economies. The same elements of predictabilityfound for the American economy are found by Esteban(1997) for the Spanish stock market. This study tests todetermine if, in the Spanish stock market, there is a long-run relationship between these predictable elements andfundamentals. If cointegration exists, we can concludethat the Spanish stock market and its fundamental s evolvetogether in the long-run.
The ability of a model to forecast the stock marketprice for di� erent time horizons is relevant to the marketparticipants. This study will test the extent to which theinclusion of long-run restrictions helps forecast the stockmarket price for di� erent time horizons in relation with analternative multivariate model. For ®nancial analysts, theforecast is relevant insofar as it allows them to perform aproper asset allocation. To test the ®nancial value of aforecast, the probability of a correct forecast conditionalupon the realization of the market is measured as proposedby Merton (1981). Also compared is accumulated wealthunder di� erent investment strategies based on the modelforecast.
II . MODEL SPECIFICATION
For the study of the Spanish stock market, this study pro-poses a vector autoregressive (VAR) Xt which includes theMadrid Stock Exchange Index and the variables consid-ered as fundamentals, listed in Table 1.
The Madrid Stock Exchange Index, (IND), is su� cientlyrepresentative of the Spanish Stock Market, since itaccounts for more than 90% of total trading volume.The Industrial Production Index, (IPI), is included as aproxy for real economic activity. This study is aware of
the limitations of this variable as a proxy for real activity,especially in the Spanish economy, where the weight of thestock market is accounted for by services, utilities andsemi-public ®rms. In¯ation (INF) is calculated on amonthly basis from the Consumer Price Index and givenin percentages. The variable Interest Rate (RFR), includedin the model, corresponds to the monthly percentual rateo� ered by the one-year Letras del Tesoro, Spanish TreasuryBills, in the secondary market. These securities were notissued prior to January 1983, so the pre-1983 rate used isthe preferential rate in the banking system.
Monthly data from February 1980 to December 1992have been used in the study. Data for the IndustrialProduction Index, Consumer Price Index and InterestRates were obtained from the Bank of Spain’s BoletõÂnEstadõÂstico. Data for the Madrid Stock Exchange Indexwere obtained from DATASTREAM.
The time paths of all the variables in the system, in bothlevels and di� erences, are graphed in Fig. 1. Visual check-ing shows the existence of outliers in the system thatrequire some explanation: The variable RFR su� ers achange of scale in January 1983, which coincides with thechange in the series taken as the short-run interest rate bythe Banco de Espa·nna. The variable INF shows irregularbehaviour from January to March 1986, which probablycoincides with the introduction of VAT in Spain and thecountry’s entry in the European Community. Finally, as adependent economy, Spain mimics the world stock marketcrash of October 1987.
In order to account for the above irregularities in thedi� erent series, three structural dummy variables are intro-duced in the VAR: d83, d86 and d87. The dummies d83and d86 take value one in January 1983 and 1986 respect-ively, and zero in the rest. Since we are interested in lookingat the interaction between real and ®nancial economy weintroduced a dummy with value one in October 1987 inorder to isolate the macroeconomic relations from externalin¯uences.
Formally, the proposed VAR speci®cation for theSpanish stock market is the following:
Xt ˆ ¦1Xt¡1 ‡ ¦2Xt¡2 ‡ ¢ ¢ ¢ ‡ ¦kXt¡k ‡ ·
‡ ©Dt ‡ ®83d83 ‡ ®86d86 ‡ ®87d87 ‡ °t …1†
where · is a vector of constants and Dt is a set of centeredseasonal dummy variables that are orthogonal to the con-stant term, and sum to zero over a year. °t, t ˆ 1; . . . ; T isan independent p-dimensional Gaussian process with meanzero and variance matrix ¢.
Except for the structural dummies, this formulation ofthe VAR is similar to the one proposed by Johansen(1991a) and Johansen and Juselius (1990). As Dolado(1993) points out, results should be taken with care sincethe impact of structural variables on the asymptotic distri-bution of Johansen’s statistics is not clear.
844 C. Ansotegui and M. V. Esteban
Table 1. Variables in the VAR
X1 ˆ IND = Madrid Stock Exchange Index, in logs.X2 ˆ IPI = Industrial Production Index, in logs.X3 ˆ INF = In¯ation Rate.X4 ˆ RFR = Short Run Interest Rate, in percentage.
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II I . EMPIRICAL ANALYSIS
Johansen’s procedure for cointegration
In the context of a VAR Xt (p £ 1), integrated of order one,cointegration exists if there exists at least one linear com-
bination of these variables which is stationary. In otherwords, the vector process Xt is said to be cointegrated if
each component of Xt is integrated of order one and there
exists a matrix such that 0 < rank… † ˆ r < p and
zt ˆ 0Xt is stationary.In order to test for the existence of cointegration
Johansen’s procedure is applied. Johansen develops a test-
ing procedure for cointegration in the context of a VAR in
several papers. The basic method and asymptotic distribu-
Cointegration for market forecast in the Spanish stock market 845
Fig. 1. Variables in levels and di� erences.
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tions for the test statistics are given in Johansen (1988,1991a). Empirical application of the procedure and thetabulation of the statistics are given in Johansen andJuselius (1990, 1992). For the determination of cointegra-tion rank in the presence of a linear trend, see Johansen(1991b). Johansen (1991a) gives the test statistics and theirdistributions for linear restrictions in the cointegrationvectors and the VAR. This study borrows Johansen’sterminology in this paper.
The procedure requires that the model de®ned byEquation 1 be reparameterized as,
¢Xt ˆ ¦Xt¡1 ‡XK¡1
jˆ1
¡j¢Xt¡j ‡ ·
‡ ©Dt ‡ ®83d83 ‡ ®86d86 ‡ ®87d87 ‡ °t …2†
where,
¡i ˆ ¡…¦i‡1 ‡ ¢ ¦k†
¦ ˆ ¡…I ¡ ¦1 ¡ ¢ ¢ ¢ ¡ ¦k† …3†
If a unit root exists in the non-stationary VAR, thematrix ¦ is of reduced rank, r < p, and can be written as
¦ ˆ ¬ 0, where ¬ and are p £ r matrices. It can be shownthat if the reduced rank of ¦ holds and rank…¦† > 0, 0Xt
is stationary and Equation 2 can be represented by anError Correction Model (ECM) as in Johansen (1991a).
¢Xt ˆ ¬ Xt¡1 ‡XK¡1
jˆ1
¡j¢Xt¡j ‡ ·
‡ ©Dt ‡ ®83d83 ‡ ®86d86 ‡ ®87d87 ‡ °t …4†
In the ECM representation the value 0Xt¡1 ˆ 0 is inter-preted as the long-run equilibrium to which the variables inthe system converge. The disequilibrium error is given bythe cointegration relations, 0Xt¡1. The extent of theadjustment to the disequilibrium error is measured by theloading matrix ¬. The increment of each variable, ¢Xit,can be expressed as the sum of the adjustment to the dis-equilibrium error of the previous period, 0Xt¡1, and thelagged increments of all the variables in the system,
¢Xt¡k; k ˆ 1; . . . ; K .Johansen’s procedure for cointegration involves testing
the rank of matrix ¦ in Equation 2, and therefore the rankof ¬ and . For cointegration to exist, nonstationarity ofthe system is a necessary condition. Johansen’s procedure isrobust even if the Xt vector is not fully integrated of orderone.
Unit Root Test. It is assumed that the lack of stationarityin the system comes from the existence of a unit root foreach variable in its autoregressive representation. Thisstudy uses the augmented Dickey±Fuller (DF) test inorder to determine the existence of unit roots.
For the determination of the number of lags in eachvariable’s equation in the augmented Dickey-Fuller test,this study applies the Campbell and Perron (1991) criterionwhere a maximum number of lags is ®xed, kmax, and theorder is reduced until the last coe� cient is signi®cant foreach variable, and this study arrives at a minimum, k. Inthe IPI equation, and given that the variable has a strongseasonal component, dummy monthly variables, Dt, asde®ned in Equation 1 have been included.
The estimated values of the statistics for DF test ½· and
½½ and the speci®ed number of lags are given in Table 2.The tabulation of the distribution of these statistics underthe null hypothesis of existence of unit root is given inFuller (1976). At the 5% signi®cant level, it is acceptedthat the null hypothesis of existence of a unit root for allthe variables in the system. For the interest rate variable,the lack of stationarity can not be attributed to the changeof the series from 1983 onwards.
The VAR Order. A complete speci®cation of the VARmodel, de®ned by Equation 1, requires the order’s deter-mination. The Ljung±Box test, for lack of correlation onthe residuals is performed. As order three in the VAR inlevels is enough to guarantee the non-correlation of resi-duals, the results of the Ljung±Box test are presented inTable 3. Following this criterion, the VAR in levels is oforder three.
In order to complete the speci®cation of the system, theJarque±Bera test for normality in the VAR is performed.
846 C. Ansotegui and M. V. Esteban
Table 2. Dickey±Fuller test for unit roots
Variable lags ½· ½½
IND 1 ¡1.350 ¡1.886IPI* 2 ¡1.138 ¡1.785INF 11 ¡1.356 ¡2.016RFR 1 ¡1.740 ¡1.923
* Test includes seasonal dummy variables
Table 3. Ljung±Box test
Ljung-Box Q-Statistics
Variable
(k ˆ 3) IND IPI INF RFR
Q…6† ˆ 10.43 8.66 1.83 3.84p-value (0.10) (0.19) (0.93) (0.69)
Q…12† ˆ 19.24 13.42 4.96 8.04p-value (0.08) (0.34) (0.96) (0.78)
Q…24† ˆ 34.56 26.62 13.53 27.24p-value (0.08) (0.32) (0.96) (0.29)
Q…36† ˆ 43.58 36.56 26.03 42.85p-value (0.18) (0.44) (0.89) (0.20)
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Results are given in Table 4. For all variables except RFR,the null hypothesis of normality is accepted, although asGonzalo (1994) notes, Johansen’s procedure is robust forlack of normality.
Cointegration Test. Finally, once the non-stationarity ofthe Xt system has been tested and the VAR fully speci-®ed, this study uses Johansen’s procedure to examine theexistence of cointegration in the Spanish Stock Exchange.The nested testing procedure used in Johansen (1991b) isreplicated. This procedure tests the existence of cointegra-tion jointly with the number of cointegration vectors andthe null hypothesis of the non-existence of a deterministiclinear trend in the system. Testing the existence of cointe-gration and determining the number of cointegration vec-tors and the non-existence of a deterministic linear trendin the system is tantamount to testing a set of nestedhypotheses. The hypothesis rank… † µ r is nested inrank… † µ r ‡ 1. For a given number of cointegrating vec-tors, rank… † ˆ r, the null hypothesis of the non-existenceof a deterministic linear trend, · ˆ ¬ 0
0, is nested in thenull of rank… † ˆ r.
Let the set of nested hypotheses be formulated asfollows:
H1 : rank ˆ p
H*1 : rank ˆ p and · ˆ ¬ 0
H2…r† : rank µ r
H*2…r† : rank µ r and · ˆ ¬ 0
…5†
This study ®rst tests H*2…r† against H*
1…p†. If this null isrejected then H2…r† is tested against H1…p†. If the latter isrejected H*
2…r ‡ 1† is tested against H*1…r† and so on until
the null is accepted.The null is accepted only if all previous hypotheses have
been rejected. For a given number of cointegrating vectors,rank… † ˆ r, if the null of a non-existence of deterministiclinear trend is accepted, the model can be reformulated toincorporate the restriction. If the restriction in Equation 4is imposed, it would be:
¢Xt ˆ ¬ * 0X*
t¡1 ‡XK¡1
jˆ1
¡j¢Xt¡j
‡ ©Dt ‡ ®83d83 ‡ ®86d86 ‡ ®87d87 ‡ °t …6†
with X*t¡1 ˆ …Xt¡1; 1† and * ˆ … ; 0†.The test statistics and critical values are given in Table 5.
We accept the null of one cointegrating vector and the non-existence of a deterministic linear trend. The cointegrationrelations, Yi, and the cointegrating relations corrected by
short run dynamics, VRi, are graphed in Fig. 2. At ®rstglance it can be said that the ®rst relationship is stationary,whereas the rest of the relations have a non-stationary
pattern, which is more clear when looking at the correctedcointegrating relations. These graphs con®rm the formaltesting of existence of a single cointegrating vector. It
should be noted that IPI has a strong seasonal component,which is mirrored in the cointegrating relations.1
Testing on and ¬
Once it is accepted that a cointegrating vector does exist
this study is interested, on the one hand, in checking the
structure of this vector and, on the other, in testing how it
a� ects short-run movements of the system.To begin with, this study focuses on the examination of the
structure of the cointegrating vector by imposing several
linear restrictions. More speci®cally, for each variable wetest whether it does not enter in the cointegrating vectorand whether it constitutes a cointegrating vector by itself.
The statistics to test the two sets of null hypothesis aregiven in Table 6. The statistic H*
3…r† tests whether i ˆ 0for i ˆ IND, IPI , INF , RFR. The statistic H*
4…r† tests
whether the cointegrating vector is such that i ˆ 1, j ˆ 0,
Cointegration for market forecast in the Spanish stock market 847
Table 5. Johansen test for cointegration
Cointegration and Deterministic Linear Trend Test
H*2…r†=H*1…p† H2…r†=H1…p†
H…r† T*r C*r Tr Cr
r µ 0 89.89 53.34 R 82.96 47.18 Rr µ 1 31.95 35.07 A 25.05 29.51 Ar µ 2 11.39 20.17 A 5.54 15.20 Ar µ 3 3.75 9.09 A 1.25 3.96 A
A: accept the null R: reject the null
1 The use of a deseasonalized IPI time series no. alters the results.
Table 4. Jarque±Bera test for normality
Jarque±Bera Statistics
Variable Statistic P-value
IND 1.43 0.48IPI 0.14 0.92INF 0.39 0.82FR 1366.00 0.00
Table 6. Testing on cointegrating vectors
Variable H*3(1) H*4(1) H*5(1)
IND 13.77 51.66 0.14IPI 4.59 51.11 0.02INF 37.11 49.71 37.05RFR 5.37 51.97 0.40
H*3…1† ¹ À2…1†; À2…1†0:05 ˆ 3:84
H*4…1† ¹ À2…3†; À2…3†0:05 ˆ 7:81
H*5…1† ¹ À2…1†; À2…1†0:05 ˆ 3:84
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i 6ˆ j for i ˆ IND, IPI , INF , RFR. The test statistics and theirdistributions for both hypotheses are given in Johansen
(1991a) and Johansen and Juselius (1992). For all the vari-ables in the system this study rejects the null hypothesis thatit does not enter in the cointegrating vector. For all Xi also
rejected is the null hypothesis that the cointegrating vector iscomposed of a single variable. As a result, there are no trivial
cointegrating vectors. This is a redundant result to the extentthat only one cointegrating vector exists and all the variables
enter in the cointegrating vector.Also tested is the characteristics of the loading matrix ¬.
This study is interested in testing what Johansen refers to
as the weak exogeneity test: whether the ¬i that corre-sponds to the equation’s variable Xi is zero. Accepting
848 C. Ansotegui and M. V. Esteban
Fig. 2. Cointegration relationships.
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the null hypothesis that ¬i is zero leads to the conclusionthat the long-run relationship given by the cointegratingvector does not a� ect the short-run dynamics of the vari-able. It must be the case that one of the variables in thesystem is a� ected in the short run by the cointegratingrelationship. Otherwise cointegration would not exist.
There are two equivalent procedures for testing weakexogeneity: maximum likelihood and a two steps pro-cedure. Maximum likelihood testing involves imposinglinear restrictions on the estimated matrix ¬:2
The statistics for the null are presented in Table 6 undercolumn H*5…r†. The two-step testing procedure developedby Gardeazabal and Regu lez (1992) starts with a ®rst stepthat uses Johansen’s procedure to estimate the cointegrat-ing relationships. In a second step, the cointegratingrelation, YY*
t¡1 ˆ *X*t¡1, is imposed in Equation 2, andthe following equation for each variable is estimatedby OLS,
¢Xi;t ˆ ¬iYYt¡1 ‡XK¡1
jˆ1
®j;i¢Xt¡j ‡ ¿iDt
‡ ®i;83d83 ‡ ®i;86d86 ‡ ®i;87d87 ‡ °i;t …7†
where zz*t¡1 is the previously estimated cointegrating rela-tion and ¿i is the vector of seasonal dummies for the i-th
Equation. The estimated values for these parameters arepresented in Table 7. As Stock (1987) shows, the short-run coe� cient estimators have standard distribution, andtherefore regular t-values can be applied for testingpurposes.
Both testing procedures for weak exogeneity lead us toaccept the null hypothesis of weak exogeneity for all thevariables in the system except INF.3 Therefore, the coin-tegrating vector has a direct impact on the determination ofin¯ation, the rest of the variables being weakly exogenousto the system.4
IV. ECONOMIC INTERPRETATION OFRESULTS
This section looks at the interrelations among the variablesin the system and investigates whether any structural rela-tionship can be inferred from our estimation. First focusedon is the cointegration relations as long-run equilibriumrelations. The concept of cointegration has serious limita-tions for direct interpretation as a ®nal equilibrium since itoverlooks the short-run dynamics of the system. This studytherefore performs the impulse response analysis in orderto get more information about these interrelations amongthe variables.
Cointegration for market forecast in the Spanish stock market 849
Table 7. Short-run estimators in the two steps procedure
Coe� cient IND IPI INF RFR
¬ 0.002 (0.37) ¡0.001 (¡0.16) ¡0.233 (¡7.56) 0.002 (0.61)¢X1;t¡1 0.330 (4.18) 0.017 (0.38) 0.304 (0.58) ¡0.073 (¡1.17)¢X1;t¡2 ¡0.101 (¡1.27) 0.064 (1.44) 0.540 (1.02) 0.022 (0.35)¢X2;t¡1 0.296 (1.85) ¡0.847 (¡9.52) ¡2.462 (¡2.31) ¡0.079 (¡0.62)¢X2;t¡2 0.189 (1.25) 70.410 (74.88) 72.270 (72.26) 0.023 (0.19)¢X3;t¡1 ¡0.012 (¡0.68) 0.007 (0.73) 0.157 (1.32) ¡0.016 (¡1.12)¢X3;t¡2 ¡0.009 (¡0.70) ¡0.004 (¡0.57) 0.121 (1.41) ¡0.005 (¡0.50)¢X4;t¡1 ¡0.065 (¡0.82) 0.035 (0.79) 0.142 (0.27) ¡0.118 (¡1.87)¢X4;t¡2 0.164 (2.03) 0.032 (0.71) ¡0.626 (¡1.17) 0.063 (0.99)¿1 ¡0.039 (¡1.07) 0.079 (3.92) 0.639 (2.64) ¡0.037 (¡1.27)¿2 ¡0.045 (¡1.22) 0.000 (0.01) 0.345 (1.41) ¡0.026 (¡0.89)¿3 0.004 (0.11) 0.042 (2.23) 0.114 (0.51) ¡0.021 (¡0.78)¿4 ¡0.036 (¡1.16) 0.021 (1.21) 0.127 (0.61) ¡0.020 (¡0.80)¿5 ¡0.019 (¡0.54) 0.026 (1.33) 0.886 (3.78) ¡0.044 (¡1.58)¿6 ¡0.029 (¡1.07) ¡0.549 (35.86) 0.225 (1.22) ¡0.019 (¡0.86)¿7 0.106 (1.26) 0.107 (2.29) 1.071 (1.91) ¡0.093 (¡1.38)¿8 ¡0.080 (¡1.09) 0.298 (7.29) 0.472 (0.96) 0.034 (0.578)¿9 ¡0.139 (¡1.34) 0.272 (4.73) 1.234 (1.79) ¡0.030 (¡0.36)¿10 ¡0.047 (¡1.28) ¡0.066 (¡3.22) 0.478 (1.96) ¡0.019 (¡0.68)¿11 0.060 (1.8) ¡0.038 (¡2.06) 1.015 (4.58) ¡0.013 (¡0.49)®83 ¡0.029 (¡0.47) 0.014 (0.41) ¡0.185 (¡0.45) 0.550 (11.17)®86 0.049 (0.79) ¡0.012 (¡0.36) 1.220 (2.99) ¡0.034 (¡0.69)®87 ¡0.340 (¡5.18) ¡0.009 (¡0.27) 0.051 (0.13) 0.044 (0.89)
t-Student values in parenthesis
2 See Johansen (1991a) and Johansen and Juselius (1990).3 This result was also obtained by Ansotegui (1992) for the US and Japanese economies.4 The use of a deseasonalised IPI time series no. alters the results.
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The long-run relationships
The existence of cointegration has several interpretations,
the most intuitive one being that the cointegrated variables
tend to evolve together over time and the cointegrating
relations, 0Xt¡1 act as an attractor for the variables in
the system. In the ECM representation, Equation 4, the
interpretation of Xj ˆ 0 is the equilibrium relationship.
The extent of the adjustment to the disequilibrium error
is given by ¬. This interpretation is the one given by Engle
and Granger (1987) and replicated for the VAR by
Johansen (1991a). The existence of cointegration can also
be interpreted as the existence of common trends. Stock
and Watson (1988) shows that when an r cointegrating
relation exists, the p cointegrated variables in the VAR
share the same (p ¡ r) shocks. There is still another in-
terpretation of cointegration given by Campbell and
Shiller (1988), if one variable is the rational forecast of
another variable, and they are integrated of order one,
then the two variables are cointegrated.
A problem in the interpretation of the ECM formulationarises because the cointegrating vectors , and the loading
matrix, ¬, are identi®able up to a constant. This lack of full
identi®ability prevents the direct interpretation of the coin-
tegrating vectors as long-run equilibrium relationships; this
is particularly true when more than one cointegrating rela-
tionship exists. The Matrix ¦ in Equation 2 is identi®able,
but does not help in the identi®cation of the equilibriumrelations since it incorporates the short-run adjustment.
For estimation purposes a normalization in is imposed.
This normalization allows us to determine the existence of
cointegration. The estimation of the parameters ¬ and ¢is conditional on . The estimation of the short-run par-
ameters can be made following the estimation of ¦ ˆ ¬ 0,
which is identi®ed. Therefore, although the cointegrationrelations are not identi®able, it is possible to determine
their existence. In addition, the deviations from the co-
integrating relations are stationary.
Given that a cointegrating relationship exists, it can be
stated that a long-run relationship between stock market
and fundamentals exists, and therefore they evolve together
over time. Although identi®cation is not possible, thisstudy has a single cointegration relation and can normalize
it for the stock-market index and check the underlying
relations by looking at the signs. The maximum likelihood
estimators for the cointegration vector and the loading
matrix are given in Table 8. Note that the dimension of is 5 £ 1, because the cointegrating vector incorporates the
restriction of non-existence of a deterministic linear trend:
· ˆ ¬ 0. The equilibrium relationship is normalized,
Xt ˆ 0, by IND, which gives us:
IND ˆ 15:04
2:78IPI ¡ 4:68
2:78INF ¡ 1:70
2:78RFR ¡ 49:34
2:78…8†
This study obtains a positive relationship between INDand IPI, which shows a positive relationship between the
stock market and real activity. The relationship between
the stock market and RFR is negative, as corresponds to
its role of discount rate. The relationship between IND andin¯ation is negative.
Impulse response functions
The parameters derived from the cointegrating relations
cannot be identi®ed as long-run elasticities since they
ignore the short-run dynamics of the system. The interrela-
tions among the variables, which take into account the
short-run relations of the stock exchange with funda-
mentals, can be analysed through the impulse response
functions. Given the stationarity of deviations from coin-
tegrating relations, it can be assumed that the variables are
in equilibrium at some time t, and any shock to one of the
variables will result in time paths of the system that will
eventually settle down in equilibrium, provided no further
shocks occur.
Impulse response functions are a standard analysis
instrument in stationary VARs. The functions give the
e� ect of a shock on one of the variables in the rest of thevariables in the system. LuÈ tkepohl and Reimers (1990)
derive the impulse response functions for cointegrated
systems. The impulse response functions can be obtained
by recursive substitution of parameters in expression 3 inEquation 2. This leads to:
Xt ˆX1
nˆ0
©n°t¡n ‡ ·X1
nˆ0
©n ‡ ©X1
nˆ0
©nDt¡n …9†
where:
850 C. Ansotegui and M. V. Esteban
Table 8. Maximum likelihood estimators
ˆ
2:78
¡15:04
4:68
1:70
49:34
0
BBBBBB@
1
CCCCCCA¬¬ ˆ
0:00174
¡4:19e ¡ 04
¡0:23348
0:00223
0
BBB@
1
CCCA
¦¦ ˆ
0:00485 ¡0:02622 0:00816 0:00296 0:08601
¡0:00117 0:00631 ¡0:00196 ¡7:12e ¡ 04 ¡0:02069
¡0:64939 3:51223 ¡1:09277 ¡0:39633 ¡11:52019
0:00621 ¡0:03360 0:01046 0:00379 0:11022
0
BBB@
1
CCCA
¢¢ ˆ
0:00265
¡2:45e ¡ 04 7:68e ¡ 04
¡0:00343 0:00190 0:12280
¡3:08e ¡ 04 ¡4:78e ¡ 05 ¡7:88e ¡ 04 0:00170
0
BBB@
1
CCCA
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©n ˆXn
mˆ1
©n¡m¦m
©0 ˆ Ik …10†
¦m ˆ 0 for m > k
The ij-th element of ©n; ¿ij;n, represents the response ofvariable Xi to a unit shock to variable Xj, n periods before,
¿ij:n with n ˆ 1; 2; . . . ; N is the impulse response function.The expression ¿ij:n is not the Moving Average represen-
tation of the system, and the coe� cients of ©n do not tendto zero. Unlike the stationary case, the e� ect of a shockdoes not have to die out in the long run, and the sum of thecoe� cients will not be ®nite.
The impulse response matrix ©n does incorporate infor-mation on contemporaneous e� ects since it ignores thecontemporaneous correlations of residuals. In order toavoid this pitfall it is common to orthogonalize the innova-tions. The orthogonalized innovations are de®ned as
£n ˆ ©nP, P being the lower triangular Choleski decom-position of ¢. The ij-th component of £n, £ij:n, is againinterpreted as the response of variable Xi to an impulse inXj, n periods before, where the size of the impulse is one
standard deviation of °j . The orthogonalized impulseresponse functions should be interpreted with caution,since orthogonalization imposes a pre-order which implies
a semi-structural interpretation of the model. The variable,which enters ®rst in the system, acts as the most exogenous.The movements in this variable at one point in time pre-cede the movement of the rest of the variables, which comeafterwards in the system.
One of the problems in the impulse response functionanalysis is the order in which the variables are introducedin the VAR model. In order to facilitate presentation, thisstudy presents a subset of the impulse response graphicrepresentations for each order considered. As this studyis mainly interested in the behaviour of the stock marketin relation to the rest of the variables in the system, thesubset of impulse response functions is selected with thisaim and includes some other functions this paper consid-ered interesting.
This study initially imposes two orders in the variables,which respond to a clear semi-structure of the model. Inboth cases the ®nancial variables form group, which isanalysed with respect to industrial production. For theset of ®nancial variables it is assumed that in¯ation pre-cedes movements in interest rates, and interest rates pre-cede movements in the stock market. This order of the®nancial variables seems reasonable for the sample periodthat this study analysed, in which control of in¯ation wasthe declared goal of economic policy, and interest rates
were used as an instrument. On the other hand, Spanishpublic debt increased considerably during the 1980s, and
the stock market can be considered as a follower of thedebt market, and therefore moved by interest rates. Thetwo orders depend on whether industrial production pre-
cedes the set of ®nancial variables or vice-versa.Finally, and to summarize the information above, Fig. 3
represents the impulse response functions when the order inwhich the variables have been included is: IPI, INF, RFR,IND; i.e., real activity precedes ®nancial variables. Figure 4
represents the order: INF, RFR, IND, IPI; i.e., the ®nan-cial variables precede industrial production. The functionsXi¡Xj in the graphs represent the analytical function ³ij:n,
and they are the response of variable Xi to an impulse invariable Xj for the 24 periods represented graphically.5
The response of IND to its own shock is always positiveand remains positive for longer periods, achieving stable
response after two periods. The short time it takes for priceto react indicates that the market is e� cient and pricemovements cannot be forecast solely on the basis of past
performance.The response of IND to RFR is negative, declining in the
period immediately following and increasing in subsequentperiods, though stabilized with negative values. When theorder of IND and RFR is reversed, IND precedes RFR in
the VAR, and the initial response is also negative, butincreases later on, stabilizing in positive values. Thisresponse is consistent with the role of interest rates as a
discount factor.The shock in INF produces an initial decline in IND,
which gets worse after the second period and recovers lateron. The response in longer periods remains negative,though not as much as at the outset. This result is consis-
tent with that obtained by Cutler et al. (1988) and theresults obtained by Fama (1981), which indicate that the
stock market does not compensate for in¯ation.The time path of the responses of IND to IPI is quite
similar for the two orders considered. There is a sharp
increase from the ®rst to the second period, a tendencywhich remains in the third period, though not so strongly.The decrease which occurs in the fourth period does not
cancel the initial increase. Afterwards, the evolution isoscillating, stabilizing around the eighth period in a value
above the initial response. When IPI precedes the ®nancialsubset, Fig. 3, the initial response of IND to IPI is negative.However, when the ®nancial variables come ®rst in the
VAR, as in Fig. 4, the initial response of IND to IPI iszero, due to the order imposed on the system. Moreover,
the response of IPI to a shock in IND is negative, or zero ifIPI precedes IND in the VAR in the ®rst period, but in allcases it increases over time and stabilizes in positive values.
Cointegration for market forecast in the Spanish stock market 851
5 The complete set of impulse response functions for the two orders given in the paper and the rest of the orders mentioned herein isavailable from the authors on request.
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852 C. Ansotegui and M. V. Esteban
Fig. 3. Impulse response functions.
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Cointegration for market forecast in the Spanish stock market 853
Fig. 4. Impulse response functions.
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The scheme of responses holds even when the order of the®nancial variables is reversed. It depends only on the situa-tion of IPI respect to the ®nancial variables.
According to Sims (1981), stability of the response func-tions for di� erent orders of the variables is an indicator ofthe suitability of the model. The shape of the responses ofIND to IPI is similar in both orders, which indicates thestability of the system. Moreover, the di� erences in signfound in the response of IND to IPI would prevent usfrom drawing conclusions about the interrelations betweenthe stock market and real activity. In the interpretation ofthese impulse response functions, there are two conjecturesthat could be valid. The ®rst is that this model has omittedvariables that are embodied in the innovations. This con-jecture would distort the impulse response functions andwould invalidate any attempt to make structural interpre-tations of the model. The second conjecture is a positiveone ± there is no economic reason why the stock marketshould react with a negative response to an improvementin real activity. In the worst case, in a purely speculativecapital market, shocks to production would not a� ect theprice index. Therefore the model speci®cation, which leadsto negative responses of IND to IPI for long periods oftime can be discarded. In this case, this study interpretsthat the stock market acts as a rational forecast of produc-tion, and the forecast is more accurate in the long run. Thispositive interpretation is in the line of Fama (1981, 1990)and Peiro (1995), who found that industrial productionwould explain returns only when future values of the indus-trial production index were included in the regression.
V. FORECAST OF STOCK MARKET PRICEINDEX
This section analyses whether the inclusion of cointegrationhelps in forecasting the stock market price. Two ways oflooking at the value of a forecast are examined: the classicand the ®nancial approaches. In the classic approach toforecasting we compare forecasts based on the cointegrat-ing model and alternative multivariate models. For ®nan-cial analysts, the relevant forecast is the change in tendencythat allows them to perform a better asset allocation. Forthis reason, as a ®nancial approach, the value of the fore-cast is measured as the probability of a correct forecastconditional upon the realization of the market as proposedby Merton (1981). A second ®nancial measure of the valueof the forecast is the comparison of the accumulated wealthobtained under four di� erent investment strategies: buyand hold the index, buy and hold bonds, switching betweenstocks and bonds on the basis of a random walk forecast,and switching between bonds and stocks on the basis of thecointegration model forecast. The best strategy will be theone that yields the greatest accumulated wealth.
Classical forecast
This study follows standard forecasting theory, as pre-
sented by Granger and Newbold (1986). The performance
of a model in terms of its ability to forecast is measured
with the Root Mean Squared Error (RMSE).The estimated model, which includes cointegrating
restrictions, will be speci®ed with and without short-run
restrictions. The alternative multivariate models are a
VAR in levels and a VAR in di� erences, which include
the same variables considered in the speci®cation of ourmodel. All these models will be compared with the per-
formance of the non-forecast model, the random walk.
These four multivariate speci®cations amount to a di� erent
set of restrictions in a general VAR. For the general
stochastic process Xt p £ 1 the general VAR formu-
lation is given by Equation 1 and Equation 2 forthe equivalent ECM formulation. The VAR in Levels
(VAR) imposes the restriction rank…¦† ˆ p. The model
including cointegrating restrictions (CI) imposes the
restriction 0 < rank…¦† ˆ r µ p. When r cointegrating
vectors exist, rank…¦† ˆ r. The model including cointegrat-ing and short-run restrictions (CIR) imposes
0 < rank…¦† ˆ r µ p and zero restrictions in ¡’s. The
VAR in Di� erences (VARD) imposes the restriction
rank…¦† ˆ 0. It is equivalent to estimating the model indi� erences. When long-run restrictions exist, this formula-
tion discards the information derived from the cointegrat-
ing relations.
Forecast: The VARL and VARD have been estimated by
OLS. The number of lags to include in the process have
been determined by the Ljung±Box statistic. The number
of lags in the VARL and VARD is two. The models with
cointegration CI (Table 7) and CIR have been estimated
in a two-step procedure. For all models the forecast is
performed outside the sample. This study estimates for
the sample period February 1980 to December 1992.
Reestimation of the model and forecast is performed
24 periods ahead, i.e. from January 1993 to December
1994.
This study compares the forecast accuracy of the di� erentmultivariate speci®cation by comparing their U’Theil value.
U’Theil gives the ratio between the Root Mean Squared
Error of the model and the random walk. Table 9 gives us
these ratios. The best model for forecasting the stock market
price is the VARL because it gives us the lowest RMSE for allthe forecasting horizons both with respect to the random
walk and with respect to the rest of multivariate speci®ca-
tions. The worst performer is the VARD, which does not
consider long-run relations. The inclusion of cointegratingrestrictions (CI) barely helps in forecasting for longer time
horizons with respect to the random walk. Moreover, it does
improve VARL performance.
854 C. Ansotegui and M. V. Esteban
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Financial forecast
The market timer forecast is whether the di� erence between
stockmarket returns and returns on bonds for the next
period is greater than zero. This forecast has value if it
enables analysts to allocate assets so that they will perform
better.
This study measures the value of the forecast with two
instruments: Merton’s measure of market timing and the
comparison of accumulated wealth obtained by a portfolio
constructed with di� erent investment strategies. The meas-ures apply only for the one-period-ahead forecast for the
time period January 1993 to December 1994. Forecast for
longer time horizons are di� cult because we are working
with a limited sample.
Merton’s measure on market timing. Merton (1981)
shows that a su� cient statistic for measuring the value
of a forecast is the probability of a correct forecast
conditional upon market return. Let FRt be the stock
market return forecast for period t, Rt the stock mar-
ket return at time t, and Rft the return of government
bonds at time t. Let p1 and p2 the probabilities of a
correct forecast, i.e.
p1 ˆ Prob…FR µ Rf=R µ Rf †
p2 ˆ Prob…FR > Rf=R > Rf † …11†
Under some conditions, Merton (1981) shows thatp1 ‡ p2 is a su� cient statistic. The condition for a forecastto have no value is p1 ‡ p2 ˆ 1. Under the null hypothesisof no forecast ability, the contingency table is:
R µ Rf R > Rf
FR µ Rf p1 p1
FR > Rf p2 p2
The null of no forecasting ability amounts to a test ofequal proportions. Let n the number of times we forecastFR µ Rf , let n1 and n ¡ n1 the number of successful andunsucessfull predictions respectively given Rt µ Rft. Let N1
be the number of observations where Rt µ Rft, N2 thenumber of observations where Rt > Rf , andN ˆ N1 ‡ N2 the total number of observations. Underthe null of no forecasting ability, the probability ofFR µ Rf is independent of the market realization,R µ Rf or R > Rf , and p1 can be estimated as n=N andthe statistic
z ˆ
n1
N1
¡ n ¡ n1
N2���������������������������������������������n
N1 ¡ n
N
± ²������������������1
N1
‡ 1
N2
svuut
is distributed as a N…0; 1†.In this case, the contingency table of these estimations is
given in Table 10. The statistic z ˆ 0; 06857 and we acceptthe null of no forecasting value of the model.
Financial Performance. Here, this study compares thewealth obtained at period T when using di� erent strate-gies to invest one peseta at period zero. The ®rst strategyis based on the forecast return obtained with this model.This is called the `switching strategy’. If FRt > Rft, thisstudy invests the accumulated wealth at the beginning ofperiod t, Wt in the stock market. If FRt µ Rft, Wt, isinvested at the risk-free interest rate. Applying this strat-egy, the accumulated wealth at the end of the period isWt…1 ‡ Rt) in the ®rst case and Wt…1 ‡ Rft) in the secondcase. A second strategy is `buy and hold stocks’; thewealth obtained at the end of the period will be then
¦Ttˆ1…1 ‡ Rt). A third alternative is `buy and hold bonds’;
the wealth obtained under this strategy at the end of per-iod T is ¦T
tˆ1…1 ‡ Rft). The advantage of the cointegra-
Cointegration for market forecast in the Spanish stock market 855
Table 9. U’Theil
U’Theil for di� erent speci®cations of the model
period CI CIR VARL VARD
1 0.9834 1.0240 0.9693 1.02752 1.0024 1.0568 0.9470 1.06933 1.0024 1.1148 0.9245 1.12914 1.0007 1.0741 0.8911 1.08685 1.0051 1.0848 0.9192 1.09156 1.0044 1.1133 0.9294 1.11827 1.0041 1.0810 0.8879 1.08738 1.0038 1.0685 0.8736 1.07359 1.0076 1.0681 0.8663 1.0719
10 1.0047 1.0525 0.8551 1.055911 1.0008 1.0401 0.8230 1.045612 0.9994 1.0258 0.7261 1.031613 0.9988 0.9775 0.5990 0.985214 0.9996 0.9759 0.5504 0.986715 0.9961 0.9920 0.4215 1.007116 0.9939 0.9982 0.4047 1.013017 0.9922 1.0205 0.2691 1.035918 0.9905 1.0521 0.2103 1.070119 0.9928 1.0536 0.2226 1.067020 0.9993 1.0836 0.2004 1.093021 1.0093 1.1235 0.1884 1.131822 1.0067 1.1194 0.1339 1.126723 1.0036 1.0522 0.1399 1.053724 0.9926 1.0302 0.0266 1.0238
Table 10. Contingency table
Realizations
Forecast R µ Rf R > Rf
FR µ Rf 6 8 14FR > Rf 4 6 10
10 14 24
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tion model is compared with a more simple forecastingprocedure. This study designed a switching strategy basedon a naive forecast. The forecast return for the next per-iod is equal to the return for this period. This strategy isdenoted as `naive switching’.
The accumulated wealth obtained by each strategy at theend of each period is given in Table 11. The comparison ofthese strategies indicates that the switching strategy isbetter than the buy and hold strategies, but this advantageis not clear when compared with the naive switching strat-egy. The graphic representation shows that the two switch-ing strategies are quite similar in terms of time pathevolution of accumulated wealth.
It should be noted that when this study compares theaccumulated wealth obtained under di� erent strategies,this study is not taking into account the cost associatedwith each strategy. In the switching strategies there is acost associated with the change of portfolio from bondsto stocks or vice versa. This cost does not exist in thebuy and hold strategies. Therefore the advantage of theswitching strategies should be taken with some reserva-tions.
VI. CONCLUSIONS
This study is primarily concerned with stock market ration-ality in Spain. This study understands rationality to mean
that in the long run the market is driven by its fundamen-
tals. This framework for the analysis is a VAR consisting
of the stock market price index, in¯ation, interest rates and
industrial production. This study tests the existence of
long-run relations between stock market and fundamentals
by checking the existence of cointegration in the VAR.
Johansen’s procedure for cointegration is used and it is
found that a cointegrating vector exists.The existence of cointegration in the system leads us to
conclude that the variables tend to evolve together overtime. However, nothing more can be said about the inter-
relations of the variables. The cointegration relation can-not be interpreted directly because the relations are notidenti®able.
In order to get a better insight into the economic inter-relations among the variables, an impulse response analysis
is performed. This study is aware that its analysis is con-tingent on all the relevant variables being included in the
system. These ®ndings are consistent with other studies in®nancial literature. The stock market does not compensatefor in¯ation. As is to be expected, the price index moves
negatively with interest rate. Increases in industrial produc-tion lead to increases in in¯ation and decreases in interest
rates. The relations between the stock market and realactivity depend on the structure imposed on the model
through the order of the variables in the system.Economic consistency would require a positive relation
between stock market and production. If this required con-sistency is to hold, it is inferred that the market acts as arational predictor of production in the Spanish economy.
When the forecast performance of the model is consid-ered in terms of RMSE, the results are worse than those of
the VAR in Levels for any forecasting horizon. The VARin Levels outperforms the random walk forecast for all theforecasting horizons considered. This is not true of the
cointegration model, which outperforms the random walkonly for some time horizons. However, the model with
cointegration performs better than the VAR in Di� erences.For the forecast to be relevant for the market timer, we
forecast the di� erence between the market return and therisk-free rate. Forecasting of this model is of no value whenthis study considers the probability of a correct forecast
conditional to the realization of the market, as proposedby Merton.
ACKNOWLEDGEMENTS
The authors thank Marta Regu lez, Gonzalo Rubio, Jesu s
Palau, Rosa Varela, Javier Gardeazabal and Nacho PenÄ a
for their useful comments. Should any errors remain, they
are the authors’ responsibility. This work was supported by
Direccio n General de Investigacio n Cientõ ®ca y
Tecnolo gica and Universidad del Paõ s Vasco (UPV/EHU)
856 C. Ansotegui and M. V. Esteban
Table 11. Cumulated wealth
Cumulated wealth for di� erent strategies
switching buy and holdcointegrating naive stocks bonds
1.07 1.01 1.07 1.011.08 1.02 1.08 1.021.09 1.06 1.12 1.031.11 1.07 1.14 1.041.17 1.13 1.20 1.051.18 1.14 1.21 1.061.22 1.18 1.25 1.071.23 1.32 1.40 1.081.16 1.25 1.32 1.081.17 1.26 1.42 1.091.18 1.19 1.34 1.101.19 1.20 1.48 1.111.31 1.33 1.63 1.111.24 1.26 1.54 1.121.25 1.26 1.45 1.131.26 1.27 1.47 1.131.26 1.28 1.48 1.141.27 1.29 1.35 1.151.28 1.29 1.41 1.151.27 1.29 1.41 1.161.28 1.30 1.34 1.171.29 1.31 1.33 1.181.30 1.31 1.36 1.191.31 1.24 1.28 1.20
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under research grants DGICYT PB97-0621 and UPV038.321-HA 129/99
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