## Cointegraing relations using the vecm model

## Abstraction

This paper analyzes the long tally co-movement between the UK, German, and Gallic stock markets utilizing the Johansen co-integration technique, that is, the Vector Error Correction Model ( VECM ) with a recursive common stochastic tendencies analysis model. The consequence of the analysis indicates that non until after about 1982 did there exists an indicant of increasing co-integration between the major European stock markets. This paper tries to retroflex to an extent the analysis conducted by Pascual ( 2003 ) in his paper titled, “ Measuring European stock markets ( carbon monoxide ) integrating ” and concentrating on the VECM analysis portion of the paper.

## Introduction

The survey of the long-term common tendencies between macroeconomic and fiscal clip series informations is an imperative econometric analysis, because it assist economic expert to find the correlativity between assorted economic variables, which leads to prediction and rational determinations made by persons, houses and the authorities on issues that affects the economic system of a state and as such the universe economic system. The analysis of the integrating of economic and fiscal clip series informations by Christopher Sims ( 1980 ) suggests the Vector Auto-Regression ( VAR ) theoretical account as a believable methodological analysis for this intent. The VAR is an n-variable additive theoretical account in which each variable is explained by its lagged values, plus current and past values of the staying n-1 variables. This therefore implies that more than one variable can be analysed at the same clip to happen out the relationship that exist between them. Therefore the vector arrested development signifier:

Degree centigrades: Users 01mnn0DesktopCapture.PNG ( 1 )

where I are ( n x N ) coefficient matrices and T is an ( n x 1 ) unobservable zero average white noise vector procedure ( serially uncorrelated or independent ) with clip invariant covariance matrix a?‘ . To work out this, it can be treated like a multivariate least square job:

Degree centigrades: Users 01mnn0DesktopCapture.1PNG.PNG ( 2 )

where Y is the matrix of the dependent variables in the signifier of columns stand foring each variable.

In a VAR analysis, it is of import that the variables are stationary I ( 0 ) -meaning no unit root exist in the model-so as to back up the premise that the statistical features of the informations will act the same manner in the hereafter as it has in the yesteryear. However, it is suggested that differencing to make stationarity should non be encouraged, because it is argued that the purpose of VAR analysis is entirely to analyze the correlativity between the variables, and the differing will extinguish information on any long-term relationships between the variables, ( Brooks, 2008 ) .

Economic and fiscal clip series informations, are normally known to hold a common stochastic tendency, this means they are correlated in the sense that they are known to linearly follow a tendency on the long-run. A set of such series are considered as co-integrated when it contains one unit root I ( 1 ) and a additive combination of them is stationary. It was foremost suggested by Granger ( 1981 ) that a vector of clip series that become a stationary procedure when differenced, can besides hold a additive combination that has a stationary procedure without differencing, it can so be said that such variables are co-integrated, which leads to the inquiry of how much differencing should be carried out on the variables in respects to the combination of the clip series considered. It has been identified that when all the variables are differenced from their univariate belongingss suitably, so the theoretical account no longer has a multi-variate additive clip series representation with an invertible moving norm. In such a instance the theoretical account can be said to hold been over-differenced. Engle and Granger ( 1987 ) pointed out that a co-integrated construction can be represented in an mistake rectification theoretical account which includes both the stationary and non-stationary features of macroeconomic clip series, that is, a set non-stationary series combinations that have a common economic factor that affects them in the same manner, so that they exist a common tendency between them and as such will ever travel linearly together in the long-term even if they drift apart from each other in the short-run. These factors could be rising prices, involvement rates and/or economic policies. The mistake rectification theoretical account provides a methodological analysis that can be used to gauge, prognosis and trial co-integration. The Engle and Granger method besides known as the two-step technique is considered non to be believable plenty due to some jobs that involved in its process. This is apparent in the analysis carried out by Xu ( 2005 ) which was to look into the efficiency of the two-step method used by Lattau and Ludvigson ( 2001 ) and the Vector Error Correction Model ( VECM ) method to look into for the co-movement in both German and US informations. It was concluded that the VECM is more appropriate method to analyze the consequence of consumption-wealth ratio ( key ) on stock return and the extra returns in both informations set significantly.

The purpose of this paper is to utilize the VECM to analyze for co-integration between three European stock markets, viz. ; UK, Germany and Gallic stock markets, in an effort to retroflex to an extent the analysis carried out by Pascual ( 2003 ) in his paper “ Assessing European stock markets ( carbon monoxide ) integrating ” utilizing the Johansen trial. However, although Pascual ( 2003 ) uses the quarterly informations of the European stock market indices from 1960 to 1999, this paper will utilize a sample size of 192 observations from 1963 to 2010 of the same stock market indices, this is due to informations handiness issues. Besides, this paper will concentrate chiefly on utilizing the Johansen trial to mensurate the co-movement of the markets, comparing co-integration consequences at different point in clip to happen out if there exists grounds of an increasing convergence of the European stock markets as the observations increases. The undermentioned subdivision is the reappraisal of literatures on assorted analysis undertaken to look into for co-integration utilizing VECM, following is the description of the methodological analysis that will be used in this documents ‘ analysis, followed by presentation and reading of consequences.

## Literature Reappraisals

Fiscal market integrating has been a topic of extended research in economic literatures for a long clip, with the purpose of look intoing the grounds of the co-integration relationship between national stock indices by analyzing the long-term co-movements of these markets. Harmonizing to Corhay et Al ( 1993 ) , this involvement is spurred from the “ addition in the flow of capital across national boundaries, possible additions from international variegation and the being of lead-lag interrelatednesss among stock exchanges ” . However, different methods have been used and improved upon with clip. Pascual ( 2003 ) efforts to turn out that an addition of convergence between the stock indices of the selected European stock markets should non be considered as an accurate illation from the recursive attack proposed by Rangvid ( 2001 ) . In his sentiment the consequences from the Rangvid ( 2001 ) analysis could be misdirecting because an addition in the convergence of the European markets could be interpreted to be as a consequence of the addition in the power of the Johansen trial as the sample size additions from 20 to 156 observations. So hence, it can be said that exist no grounds of an increasing co-integration. He so suggested an alternate method to look into for increasing stock market integrating. He proposes that the mistake rectification term ( ECT ) should be estimated as it can reflect the velocity of accommodation to divergences from the long-term co-integration relationship. A higher value of the coefficient of the ECT, could be interpreted as a higher degree of the stock market integrating, as the sample additions.

Corhay et Al ( 1993 ) , in their analysis recognises that the best attack to analyze stock monetary values when the variables involved are non-stationary is the usage of the co-integration construct or the common stochastic tendencies, which suggest that assorted non-stationary variables do linearly travel together in the long-run. It is in their sentiment that since it is expected that the stock markets of two or more European states are capable to a common market tendency, so it can be said that the markets are co-integrated. Their analysis involved 389 fortnightly observations, that is, from the 1 March 1975 to 30 September 1991, of stock monetary value indices of five major European stock markets ( Germany, France, Italy, UK and the Netherlands ) . Using the VECM attack that would be used subsequently on in this paper, which was proposed by Johansen ( 1988 ) , and Johansen and Juselius ( 1990 ) which is a maximal likeliness attack to gauge and prove the figure of con-integration in VAR theoretical account. In their decision they found grounds that reveals that they exists some long-term stochastic tendencies between several European stock market indices, although it was besides discovered that the Italian stock monetary values seem non to act upon this long-term tendency.

Pukthuanthong and Roll ( 2009 ) in their survey proposes an alternate step of the integrating of planetary markets. They suggest utilizing through empirical observation the explanatory power of multi-factor theoretical account to look into the increasing integrating of planetary markets as the correlativity of states market indexes is considered a hapless step. They explain that unless the same planetary factors affects for case two states indexes at the same proportion, their correlativity would be imperfect even if the planetary factors explain the return of the indexes in both states 100 % . They observed that they seem to be an increasing co-integration between the 17 big states over clip, indicating out that simple correlativity did non give an efficient consequence, because it failed to uncover the full extent of integrating of the states indexes over the past 30+ old ages.

The ground for the involvement by economic analyst and economic policy shapers in the relationship between stock markets and their convergence could be due to the probe of whether there is a possibility of additions from international variegation, most particularly in the position of an investor, for case, in the instance where there be a long-term additive common tendency between national stock markets, so the possibility of deriving from international variegation in the long tally is less likely. Fraser and Oyefeso ( 2005 ) in their survey look into the long-term convergence between U.S. , UK and seven European stock markets. From the Johansen multivariate co-integration trials conducted which was used on a sample of monthly informations over the period from 1974 to 2001of the stock market monetary value indices of a selected set of European states including the UK and U.S. ; France, Denmark, Belgium, Germany, Italy, Sweden and Spain, shows that they exists a long-term relationship between the stock markets due to the presence of a individual common stochastic tendency. The suggested illation from their analysis confirms that stock markets examined are wholly correlated in the long-run or the hereafter. It was besides noted that the consequences obtained from their probe shows a much more degree of integrating than those obtained by Corhay et Al. ( 1993 ) carried out on a specified set of European markets, in their sentiment, this might be as a consequence of the drawn-out clip period. Other paper that have supported the position that the chief stock markets of the universe have converged over the long-term includes that of Kasa ( 1992 ) , where the observation sample are from the monthly and quarterly informations of equity markets of U.S. , Japan, Germany, England and Canada from 1974 to mid-1990. In Taylor and Tonks ( 1989 ) they investigated the consequence of the abolishment of the UK exchange control on the grade of integrating of the UK and abroad stock markets, utilizing the Engle and Granger ( 1987 ) two stairss technique to look into for co-integration on clip series informations. Their consequences show grounds that conforms to that obtained from the antecedently mentioned co-integration analysis. In this instance, with the abolishment of the exchange control, the UK stock exchange has become co-integrated with that of Japan, Germany and the Netherlands, in their sentiment this might hold be due to the fact that since the capital control was now relaxed and as such the undeveloped arbitrage chances have been utilized.

Syriopoulos ( 2004 ) investigates the being of short and long-term correlativity among selected major developed stock markets ; Germany and the US and emerging European stock markets ; Czech Republic, Hungary, Slovakia and Poland. The VECM technique was used and it was inferred that there exists co-integration relationship between the markets. It was in the writers ‘ sentiment that domestic and external forces, which may be referred to as macroeconomic forces, affects the stock markets behaviours, which in bend leads to the long-term equilibrium, it was besides observed that there exists more grade of correlativity between the single European markets and the developed markets in comparing with their fellow emerging markets. This implies that the investing scheme of international variegation of hazard in order to make an efficient market portfolio return may be limited for investors interested in using this investing scheme.

In Karolyi and Stulz ( 1996 ) they investigate the constituents of cross-country daze return co-movements. U.S and Japan portions returns which are traded in the United States were studied to happen out whether macroeconomic proclamations and involvement rates creates dazes that affects the co-movements between the U.S and Nipponese portion returns. From the consequences obtained from the VECM empirical method, it was inferred that these macroeconomic factors do non impact the co-movements and that covariance and correlativities in the markets are high when they extremely volatile. In their sentiment, which is similar to Syriopoulos ( 2004 ) , this means that international variegation as an investing scheme to distribute out hazard might non be every bit effectual as expected, as their analysis shows that variegation in this instance does non supply plenty screen against big dazes to national indices as one might hold expected. It was besides suggested by Karolyi and Stulz ( 1996 ) the covariances between states are non changeless, because they change over clip and can be forecasted.

The inquiry of what could be the ground for the addition in the co-integration in the stock markets arises. What are the macroeconomic or planetary factors that have led to the co-movement of the stock market indices of emerging and developed states? Yang et Al ( 2003 ) survey of the consequence of the constitution of the Economic and Monetary Union ( EMU ) on the short and long-term integrating among 11 European stock markets and US stock market. Their consequences were similar to that obtained by Taylor and Tonks ( 1989 ) and Corhay et Al ( 1993 ) . It was in that sentiment that modern information engineering and amalgamation of stock exchanges in Europe may be the factor that has increased the co-integration among European stock markets.

Furthermore, Ioannidis et Al ( 2006 ) in utilizing the methodological analysis proposed by Lettau and Ludvigson ( 2001 ) , which is the two-step method, examines three states ; Australia, UK and Canada. They confirmed the consequences from the Lettau and Ludvigson ( 2001 ) analysis that suggest that the lagged co-integration variable ( key ) is a important forecaster of the expected return or extra return of the stock markets of the specified states, merely as the instance in U.S. Although, Xu ( 2005 ) uses the VECM to look into the relationship between the consumption-wealth ratio ( key ) on German stock returns. The intent of Xu ( 2005 ) analysis was to compare the efficiency of the methodological analysis proposed by Lettau and Ludvigson ( 2001 ) and the VECM utilizing German and U.S. informations, and it was concluded that the VECM is a more appropriate method to analyze the consequence of key on stock returns and extra returns in both informations set significantly.

It may so be said that key might be regarded a macroeconomic factor that determines the additive tendency of stock market returns in the long-run, since there are grounds that they exist a correlativity between these variables and the fiscal markets returns. With this grounds, the stock market returns could be predictable by concern rhythm at rotational frequences in the long-run.

## Methodology

The methodological analysis that would be used is the Vector Error Correction Model ( VECM ) which has been used most often in the analysis of economic clip series informations. Engle and Granger ( 1987 ) elaborate on the fundamental of the co-integration facet. In this paper, the co-integration analysis in the model of vector autoregressive theoretical account ( VAR ) as proposed by Johansen ( 1988 ) , and Johansen and Juselius ( 1990 ) would be used.

The followers is a statistical account of the VECM analysis utilizing the Johansen technique as denoted by Brooks ( 2008 ) . In order to utilize the Johansen attack, a VAR with K slowdowns incorporating a set of g variables ( g a‰? 2 ) which are assumed to be I ( 1 ) and cointegrated, would hold to be converted into a vector mistake rectification theoretical account ( VECM ) , such that the set up:

yt = I?1 yta?’1 + I?2 yta?’2 + A· A· A· + I?k yta?’k + Greenwich Mean Time

g A- 1 g A- g g A- 1 g A- g g A- 1 g A- g g A- 1 g A- 1 ( 3 )

is transformed to a vector mistake rectification theoretical account ( VECM ) as below:

a?†yt = a??yta?’k + ?“1a?†yta?’1 + ?“2a?†yta?’2 +A· A· A·+?“ka?’1a?†yta?’ ( ka?’1 ) + Greenwich Mean Time ( 4 )

where a??= ( ) – Immunoglobulin and ?“i = ( – Immunoglobulin. From the above VAR equation the g variables are in the first differenced signifier on the left manus side and on the right manus side the k-1 are the slowdowns of the dependent variables in their differenced signifier, each contains a ?“ coefficient matrix that accompanies them.

The matrix ?Y in the Johansen trial can stand for the long-term coefficient matrix, since all the a?†yta?’i will be zero and the error term. Greenwich Mean Time will be set to their expected value of nothing will go forth a??yta?’k = 0, in equilibrium. The rank of the ?Y matrix from its characteristic root of a square matrix is used to cipher the figure of co-integration between the Y. The characteristic root of a square matrixs, which are the figure of its characteristic roots that are different from nothing peers to the rank of a matrix. The symbol I»i denotes the characteristic root of a square matrixs, which are set in go uping order as therefore ; I»1 a‰? I»2 a‰? . . . a‰? I»g. In the instance where the characteristic root of a square matrixs ( I»s ) are roots they have to be less than 1in absolute value and positive, and I»1 will be closest to 1 which is the largest, while I»g will be closest to 0 which is the smallest. When the analysed variables are non co-integrated, the rank of the matrix ?Y will non be different from nothing well, such that I»i a‰? 0 a?ˆ I.

In a Johansen trial, there are two trial statistics that are used to co-integration analysis, they are in the signifier below:

I»trace ( R ) = a?’ Ti ) ( 4 )

( I»trace = 0 when all the I»i = 0, for one = 1, . . . , g. )

and

I»max ( R, R + 1 ) = a?’ T ln ( 1 a?’r+1 ) ( 5 )

where R is the represents the figure of co-integrating vectors under the void hypothesis and I represents the estimated value for the i-th characteristic root of a square matrix from the matrix ?Y . In the I»trace, which is a joint trial has a void hypothesis where the figure of co-integrating vectors is less than or equal to r against an alternate hypothesis that there are more than r. In the I»max tests a separate trial is conducted on each characteristic root of a square matrix with a void hypothesis that is the figure of co-integrating vectors is R and an alternate hypothesis of R + 1. The hint trial starts with P characteristic root of a square matrixs, and so in sequence the largest is removed. Every characteristic root of a square matrix has with it an affiliated different co-integration vector, which is known as the eigenvectors. A significantly non-zero characteristic root of a square matrix shows a important co-integration vector.

The critical values used for the two trial statistics depends on the value of the g – R, the figure of non-stationary elements and how invariables are included in each of the equations. When the critical value is less than the trial statistics, reject the void hypothesis that there are r co-integrating vectors in support of the alternate hypothesis ( r +1 for the I»trace trial or more than R for the I»max trial ) . The trial is conducted in a sequence and under the nothing, R = 0, 1, . . . , g – 1 so that the hypotheses for I»max can be represented every bit below as:

H0: R = 0 versus H1: 0 & lt ; r a‰¤ g

H0: R = 1 versus H1: 1 & lt ; r a‰¤ g

H0: R = 2 versus H1: 2 & lt ; r a‰¤ g

## . . .

## . . .

H0: R = g a?’ 1 versus H1: R = g

From the above, the first trial means the void hypothesis of no presence of co-integrating vectors, therefore the corresponding ?Y matrix have a 0 rank. In the instance where the void hypothesis ( H0: R = 0 ) is rejected, so the nothing that there is one co-integrating vector ( H0: R = 1 ) is tested and the procedure continues, and as such the value of R is continually increased until the void hypothesis is non rejected. The matrix ?Y can ne’er be at full rank ( g ) as this would intend that yt is stationary. In the instance where the matrix ?Y has 0 rank, so by correspondence to the univariate instance, a?†yt depends merely in a?†yt a?’ J and non on yt – 1, which will ensue to no long-term relationship between the elements of yt – 1, which in bend means no co-integration. For case, in 1 & lt ; rank ( ?Y ) & lt ; g, there are r co-integrating vectors. The matrix ?Y is so characterised as the merchandise of two matrices, I± and I? ‘ , of the dimension ( g A- R ) and ( r A- g ) , severally, that is,

?Y = I±I?’ ( 6 )

where matrix I? denotes the co-integrating vectors, while I± , which is known as the accommodation parametric quantity, gives the sum of each co-integrating vector associated with each equation of the vector mistake rectification theoretical account.

In the undermentioned subdivision the VECM attack utilizing the Johansen technique as explained, will be carried out on the selected three European stock markets ; UK, France and Germany to look into the possibility of an increasing market co-integration, utilizing to an extent the recursive attack done by Pascual ( 2003 ) which is similar to that done by Rangvid ( 2001 ) . The Johansen attack is so applied to the vector mistake rectification theoretical account ;

a?†xt = A + a??0xta?’1 + I a?†xta?’1 + Greenwich Mean Time ( 7 )

here ten represents the vector incorporating the logarithm value of the stock market indices for the selected European states. A larger figure of the important co-integrating vectors will be observed as clip goes on if the markets are meeting.

## Datas

The informations used were used to look into for co-integration are the quarterly informations of the European ( UK, Germany and France ) stock market indices from 1963 to 2010 which consequences to a entire sample size of 192 observations obtained from DATASTREAM. The ground for get downing this analysis from 1963 alternatively of 1960 as carried out by Pascual ( 2003 ) is due to informations handiness jobs. Get downing with a sample of 20 quarters from 1960: Q1 to 1964: Q4 for three European stock indices is estimated recursively by adding one excess observation at a clip up to 2010: Q4. In Appendix 1, it can be observed by eye-balling the information, that as more observations are added the lines stand foring each variable seem to pull closer to each other and have an upward tendency. Harmonizing to Pascual ( 2003 ) , the upward tendency can be attributed to two grounds. First, is the figure of bing stochastic tendencies carry oning the three dimensional systems are diminishing with clip as markets become progressively incorporate. Second, as the observations increase from 20 to 156 the hint statistics merge to the long tally values. This may be interpreted as the being of cointegration between the variables, although the necessary analysis must be undertaken to warrant this premise. In the consequence representation subdivision, four different slowdown Windowss, matching to 20, 60, 100, 140 and 192 observations, are analyzed.

## Consequences Presentation

The first measure in the VECM analysis is to look into for stationarity in the variables. Unit Root trial was carried out on the log of the variables utilizing Augmented Dickey-Fuller ( ADF ) , Philips-Perron ( PP ) and Kwiatkowski-Philips-Schmidt-Shin ( KPSS ) trial. The consequences are presented below:

Table 1: Unit of measurement Root/Stationary Trial

Test Type

## United kingdom

## Gram

## F

Critical Value

-0.804

-0.420

-0.568

## ADF

1 % Degree

( -3.464 )

Fail

Fail

Fail

5 % Degree

( -2.876 )

Fail

Fail

Fail

10 % Degree

( -2.574 )

Fail

Fail

Fail

Critical Value

-0.823

-0.473

-0.434

## PP

1 % Degree

( -3.464 )

Fail

Fail

Fail

5 % Degree

( -2.876 )

Fail

Fail

Fail

10 % Degree

( -2.574 )

Fail

Fail

Fail

Critical Value

1.650

1.604

1.615

## KPSS

1 % Degree

A 0.739

Cull

Cull

Cull

5 % Degree

A 0.463

Cull

Cull

Cull

10 % LevelA

0.347

Cull

Cull

Cull

## Decision

## I ( I )

## I ( I )

## I ( I )

From the above tabular array, UK, G and F represents United Kingdom, Germany and France severally, they denote the log of the European stock market. From the above consequences one can deduce that the variables are I ( 1 ) , intending at that place exist unit roots and therefore the variables are non-stationary. These consequences can be illustrated in a unit root graph as below:

Figure 1: Unit of measurement Root graph

Since, one of the bluish points touch the circle, we can reason that the variables are non-stationary. The following measure will be to stipulate the optimum slowdown. The below tabular array contains the slowdown construction of 20, 60, 100 and 140 observations. The optimum slowdown is obtained when the Akaike standard has minimum value. The Akaike Information Criterion is appropriate for this analysis since the ample size is rather little.

## Akaike Information Criterion for VECM with slowdown 2 to dawdle 10

Slowdown

Number of Observations

20

60

100

140

2

## -7.003961

## -5.743871

## -5.612677

## -6.048935

3

-5.538380

-5.503853

-5.965187

4

-5.393658

-5.421785

-5.922316

5

-5.359694

-5.347622

-5.832997

6

-5.167633

-5.172125

-5.773959

7

-5.206056

-5.169274

-5.730607

8

-5.260565

-5.109051

-5.632572

9

-5.083367

-4.979757

-5.535563

10

-5.136492

-4.869142

-5.514630

Table 2: Akaike Information Criterion

From the above tabular array, comparing the information standard shows that VAR ( 1, 2 ) gives the smallest information standards for all the different classs of observations and so it is the best linear indifferent appraisal. For 20 observations merely VAR ( 1, 2 ) was gettable because it is a really little sample size. Following is the cointegration analysis of the variables. Using the Johansen FIML attack for proving the cointegration, there are two basic trials consequences. The max-eigenvalue and the hint trial as explained earlier in this paper. The consequences of this trial are presented below utilizing the given hypothesis determination regulation:

H0: R=0 H1: R & gt ; 0a†’R & gt ; 0

H0: 0E‚Ra‰¤1 H1: R & gt ; 1

H0: 0E‚Ra‰¤2 H1: R & gt ; 2a†’R & gt ; 2. where Roentgen represents rank and is less than 3.

## Co-integration trial – Johansen FIML for 20 observations

Table 3: Unrestricted Cointegration Rank Test ( Trace )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.740448

A 40.38981

A 29.79707

A 0.0021

At most 1 *

A 0.544995

A 17.46023

A 15.49471

A 0.0250

At most 2 *

A 0.213077

A 4.073633

A 3.841466

A 0.0435

Trace trial consequences shows that there exist 3 co-integrating equations at the 5 % degree

Table 4: Unrestricted Cointegration Rank Test ( Maximum Eigenvalue )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.740448

A 22.92958

A 21.13162

A 0.0276

At most 1 *

A 0.544995

A 13.38660

A 14.26460

A 0.0685

At most 2 *

A 0.213077

A 4.073633

A 3.841466

A 0.0435

Max-eigenvalue trial consequences shows that there exist 1 co-integrating equation at the 5 % degree.

## Co-integration trial – Johansen FIML for 60 observations

Table 5: Unrestricted Cointegration Rank Test ( Trace )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.192783

A 24.73114

A 29.79707

A 0.1713

At most 1 *

A 0.123292

A 12.52387

A 15.49471

A 0.1335

At most 2 *

A 0.084363

A 5.023705

A 3.841466

A 0.0250

Trace trial consequences shows that there exist no co-integrating equations at the 5 % degree

Table 6: Unrestricted Cointegration Rank Test ( Maximum Eigenvalue )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.192783

A 12.20727

A 21.13162

A 0.5273

At most 1 *

A 0.123292

A 7.500164

A 14.26460

A 0.4318

At most 2 *

A 0.084363

A 5.023705

A 3.841466

A 0.0250

Max-eigenvalue trial consequences shows that there exist no co-integrating equation at the 5 % degree.

## Co-integration trial – Johansen FIML for 100 observations

Table 7: Unrestricted Cointegration Rank Test ( Trace )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.188373

A 27.20639

A 29.79707

A 0.0967

At most 1 *

A 0.069223

A 6.961074

A 15.49471

A 0.5822

At most 2 *

A 2.85E-05

A 0.002769

A 3.841466

A 0.9555

Trace trial consequences shows that there exist no co-integrating equations at the 5 % degree

Table 8: Unrestricted Cointegration Rank Test ( Maximum Eigenvalue )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.188373

A 20.24532

A 21.13162

A 0.0662

At most 1 *

A 0.069223

A 6.958305

A 14.26460

A 0.4941

At most 2 *

A 2.85E-05

A 0.002769

A 3.841466

A 0.9555

Max-eigenvalue trial consequences shows that there exist no co-integrating equation at the 5 % degree.

## Co-integration trial – Johansen FIML for 192 observations

Table 9: Unrestricted Cointegration Rank Test ( Trace )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.125102

A 31.81745

A 29.79707

A 0.0289

At most 1 *

A 0.027840

A 6.557989

A 15.49471

A 0.6295

At most 2 *

A 0.006443

A 1.221613

A 3.841466

A 0.2690

Trace trial consequences shows that there exist 1 co-integrating equations at the 5 % degree

Table 10: Unrestricted Cointegration Rank Test ( Maximum Eigenvalue )

Hypothesized No. of CE ( s )

Eigenvalue

Trace Statistic

5 %

Critical Value

Prob. **

None *

A 0.125102

A 25.25946

A 21.13162

A 0.0124

At most 1 *

A 0.027840

A 5.336376

A 14.26460

A 0.6989

At most 2 *

A 0.006443

A 1.221613

A 3.841466

A 0.2690

Max-eigenvalue trial consequences shows that there exist 1 co-integrating equation at the 5 % degree.

## Analysis

The above tabular arraies 3-10 shows the co-integration consequences of the recursive trial. The hint and max-eigenvalue trial for 20 observations shows conflicting consequences, this may be due the little sample size or unidentified structural interruptions in the co-integration system. We rely on the consequence of the max-eigenvalue trial, because the hint trial may be assumed to be weaker as it is prone to false rejection due to informations issues. Thus we adopt the consequence of the max-eigenvalue trial, which indicates that there exists one co-integration relationship in the vector variables. In the instance of 60, 100 and 140 observations, the co-integration trial consequences both for the hint and max-eigenvalue trial indicates that they exist no co-integration relationship among the vector variables. Rangvid ( 2001 ) in his similar analysis recognized this when he stated that “ … until about 1982, the indicants of increased convergence between the major European stock markets are non clear cut ” . As the observation is increased to 192, the co-integration consequences shows that there exist one co-integration for both the hint and max-eigenvalue trial, this indicant is confirmed by Rangvid ( 2001 ) , he noted that it was after 1982 that there seem to be marks of an increasing convergence between the markets, in his analysis the max-eigenvalue became significantly big and exceeded the critical value. He besides noted that this was the period when capital limitations were lifted all over the European country. Harmonizing to Pascual ( 2003 ) the co-integration trial shows no important grounds of fluctuation in the integrating of the analyzed European markets. However, due to the increasing velocity of accommodation coefficient between 1965-1986 it is suggested that it indicates an grounds of increasing integrating for the Gallic market.

The cointegration graphs are presented below:

## Decisions

From the above analysis the consequences from the paper by Pascual ( 2003 ) were compared following the Johansen co-integration trial and it was observed that earlier about 1982 there exists weak marks of integrating between the selected European stock markets, but after 1982 the inclination for the market to be driven by the same common stochastic tendency is observed. Therefore in 2010, the co-integration consequence of the hint and max-eigenvalue trial confirmed the old observation as they both indicated that there exists 1 co-integration relationship in the variables.

However, Pascual ( 2003 ) suggested an option to mensurate convergence due to the fact that utilizing the Johansen trial with a recursive attack may supply misdirecting consequences, as the increasing value of the hint statistics may be interpreted as an addition of co-integration, which in fact may be the due to the matching power of the Johansen trial as the observations increases from 20 to 192. He suggests that a more intuitive step of integrating between the stock market could be done by gauging the time-path followed by the coefficients of the mistake rectification term ( ECT ) , since the coefficient of the ECT reflects the velocity of accommodation to fluctuations from the long tally equilibrium, it can be assumed that the higher the values of the coefficient, the higher the grade of stock market co-integration.