Analysis Of Cointegrating Relations Using The Vecm Model Economics Essay

This paper analyzes the long run co-movement between the UK, German, and French stock markets using the Johansen co-integration technique, that is, the Vector Error Correction Model (VECM) with a recursive common stochastic trends analysis framework. The result of the analysis indicates that not until after approximately 1982 did there exists an indication of increasing co-integration between the major European stock markets. This paper tries to replicate to an extent the analysis conducted by Pascual (2003) in his paper titled, "Assessing European stock markets (co)integration" and concentrating on the VECM analysis part of the paper.

Introduction

The study of the long-run common trends between macroeconomic and financial time series data is an imperative econometric analysis, because it assist economist to determine the correlation between various economic variables, which leads to forecasting and rational decisions made by individuals, firms and the government on issues that affects the economy of a country and as such the world economy. The analysis of the integration of economic and financial time series data by Christopher Sims (1980) suggests the Vector Auto-Regression (VAR) model as a credible methodology for this purpose. The VAR is an n-variable linear model in which each variable is in turn explained by its own lagged values, plus current and past values of the remaining n-1 variables. This means more than one variable can be analysed at the same time to find out the relationship that exist between them. Therefore the vector regression form:

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where i are (n x n) coefficient matrices and t is an (n x 1) unobservable zero mean white noise vector process (serially uncorrelated or independent) with time invariant covariance matrix ∑. To solve this, it can be treated like a multivariate least square problem:

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where Y is the matrix of the dependent variables in the form of columns representing each variable.

In a VAR analysis, it is important that the variables are stationary I(0)-meaning no unit root exist in the model-so as to support the assumption that the statistical characteristics of the data will behave the same way in the future as it has in the past. However, it is suggested that differencing to create stationarity should not be encouraged, because it is argued that the aim of VAR analysis is solely to examine the correlation between the variables, and the differing will eliminate information on any long-run relationships between the variables, (Brooks, 2008).

Economic and financial time series data, are usually known to have a common stochastic trend, this means they are correlated in the sense that they are known to linearly follow a trend on the long-run. A set of such series are considered as co-integrated when it contains one unit root I(1) and a linear combination of them is stationary. It was first suggested by Granger (1981) that a vector of time series that become a stationary process when differenced, can also have a linear combination that has a stationary process without differencing, it can then be said that such variables are co-integrated, which leads to the question of how much differencing should be carried out on the variables in regards to the combination of the time series considered. It has been identified that when all the variables are differenced from their univariate properties appropriately, then the model no longer has a multi-variate linear time series representation with an invertible moving average. In such a case the model can be said to have been over-differenced. Engle and Granger (1987) pointed out that a co-integrated structure can be represented in an error correction model which includes both the stationary and non-stationary characteristics of macroeconomic time series, that is, a set non-stationary series combinations that have a common economic factor that affects them in the same way, so that they exist a common trend between them and as such will always move linearly together in the long-run even if they drift apart from each other in the short-run. These factors could be inflation, interest rates and/or economic policies. The error correction model provides a methodology that can be used to estimate, forecast and test co-integration. The Engle and Granger method also known as the two-step technique is considered not to be credible enough due to some problems that involved in its procedure. This is evident in the analysis carried out by Xu (2005) which was to check the efficiency of the two-step method used by Lattau and Ludvigson (2001) and the Vector Error Correction Model (VECM) method to check for the co-movement in both German and US data. It was concluded that the VECM is more appropriate method to study the effect of consumption-wealth ratio (cay) on stock return and the excess returns in both data set significantly.

The aim of this paper is to use the VECM to analyse for co-integration between three European stock markets, namely; UK, Germany and French stock markets, in an attempt to replicate to an extent the analysis carried out by Pascual (2003) in his paper "Assessing European stock markets (co)integration" using the Johansen test. However, although Pascual (2003) uses the quarterly data of the European stock market indices from 1960 to 1999, this paper will use a sample size of 192 observations from 1963 to 2010 of the same stock market indices, this is due to data availability issues. Also, this paper will concentrate mainly on using the Johansen test to measure the co-movement of the markets, comparing co-integration results at different point in time to find out if there exists evidence of an increasing convergence of the European stock markets as the observations increases. The following section is the review of literatures on various analysis undertaken to investigate for co-integration using VECM, next is the description of the methodology that will be used in this papers' analysis, followed by presentation and interpretation of results.

Literature Reviews

Financial market integration has been a subject of extensive research in economic literatures for a long time, with the aim of investigating the evidence of the co-integration relationship between national stock indices by studying the long-run co-movements of these markets. According to Corhay et al (1993), this interest is spurred from the "increase in the flow of capital across national boundaries, possible gains from international diversification and the existence of lead-lag interrelationships among stock exchanges". However, different methods have been used and improved upon with time. Pascual (2003) attempts to prove that an increase of convergence between the stock indices of the selected European stock markets should not be considered as an accurate inference from the recursive approach proposed by Rangvid (2001). In his opinion the results from the Rangvid (2001) analysis could be misleading because an increase in the convergence of the European markets could be interpreted to be as a result of the increase in the power of the Johansen test as the sample size increases from 20 to 156 observations. So therefore, it can be said that exist no evidence of an increasing co-integration. He then suggested an alternative method to check for increasing stock market integration. He proposes that the error correction term (ECT) should be estimated as it can reflect the speed of adjustment to deviations from the long-run co-integration relationship. A higher value of the coefficient of the ECT, could be interpreted as a higher level of the stock market integration, as the sample increases.

Corhay et al (1993), in their analysis recognises that the best approach to analyse stock prices when the variables involved are non-stationary is the use of the co-integration concept or the common stochastic trends, which suggest that various non-stationary variables do linearly move together in the long-run. It is in their opinion that since it is expected that the stock markets of two or more European countries are subject to a common market trend, then it can be said that the markets are co-integrated. Their analysis involved 389 biweekly observations, that is, from the 1 March 1975 to 30 September 1991, of stock price indices of five major European stock markets (Germany, France, Italy, UK and the Netherlands). Using the VECM approach that would be used later on in this paper, which was proposed by Johansen (1988), and Johansen and Juselius (1990) which is a maximum likelihood approach to estimate and test the number of con-integration in VAR model. In their conclusion they found evidence that reveals that they exists some long-run stochastic trends between several European stock market indices, although it was also discovered that the Italian stock prices seem not to influence this long-run trend.

Pukthuanthong and Roll (2009) in their study proposes an alternative measure of the integration of global markets. They suggest using empirically the explanatory power of multi-factor model to investigate the increasing integration of global markets as the correlation of countries market indexes is considered a poor measure. They explain that unless the same global factors affects for instance two countries indexes at the same proportion, their correlation would be imperfect even if the global factors explain the return of the indexes in both countries 100%. They observed that they seem to be an increasing co-integration between the 17 large countries over time, pointing out that simple correlation did not give an efficient result, because it failed to reveal the full extent of integration of the countries indexes over the past 30+ years.

The reason for the interest by economic analyst and economic policy makers in the relationship between stock markets and their convergence could be due to the investigation of whether there is a possibility of gains from international diversification, most especially in the perspective of an investor, for instance, in the case where there exist a long-run linear common trend between national stock markets, then the possibility of gaining from international diversification in the long run is less likely. Fraser and Oyefeso (2005) in their study investigate the long-run convergence between U.S., UK and seven European stock markets. From the Johansen multivariate co-integration tests conducted which was used on a sample of monthly data over the period from 1974 to 2001of the stock market price indices of a selected set of European countries including the UK and U.S.; France, Denmark, Belgium, Germany, Italy, Sweden and Spain, shows that they exists a long-run relationship between the stock markets due to the presence of a single common stochastic trend. The suggested inference from their analysis confirms that stock markets examined are completely correlated in the long-run or the future. It was also noted that the results obtained from their investigation shows a much more degree of integration than those obtained by Corhay et al. (1993) carried out on a specified set of European markets, in their opinion, this might be as a result of the extended time period. Other paper that have supported the view that the main stock markets of the world have converged over the long-run includes that of Kasa (1992), where the observation sample are from the monthly and quarterly data of equity markets of U.S., Japan, Germany, England and Canada from 1974 to mid-1990. In Taylor and Tonks (1989) they investigated the effect of the abolition of the UK exchange control on the degree of integration of the UK and overseas stock markets, using the Engle and Granger (1987) two steps technique to check for co-integration on time series data. Their results show evidence that conforms to that obtained from the previously mentioned co-integration analysis. In this case, with the abolition of the exchange control, the UK stock exchange has become co-integrated with that of Japan, Germany and the Netherlands, in their opinion this might have be due to the fact that since the capital control was now relaxed and as such the unexploited arbitrage opportunities have been utilized.

Syriopoulos (2004) investigates the existence of short and long-run correlation among selected major developed stock markets; Germany and the US and emerging European stock markets; Poland, Hungary, Czech Republic and Slovakia. The VECM technique was used and it was inferred that there exists co-integration relationship between the markets. It was in the authors' opinion that domestic and external forces, which may be referred to as macroeconomic forces, affects the stock markets behaviours, which in turn leads to the long-run equilibrium, it was also observed that there exists more degree of correlation between the individual European markets and the developed markets in comparison with their fellow emerging markets. This implies that the investment strategy of international diversification of risk in order to create an efficient market portfolio return may be limited for investors interested in utilizing this investment strategy.

In Karolyi and Stulz (1996) they investigate the components of cross-country shock return co-movements. U.S and Japan shares returns which are traded in the United States were studied to find out whether macroeconomic announcements and interest rates creates shocks that affects the co-movements between the U.S and Japanese share returns. From the results obtained from the VECM empirical method, it was inferred that these macroeconomic factors do not affect the co-movements and that covariance and correlations in the markets are high when they highly volatile. In their opinion, which is similar to Syriopoulos (2004), this means that international diversification as an investment strategy to spread out risk might not be as effective as expected , as their analysis shows that diversification in this case does not provide enough cover against large shocks to national indices as one might have expected. It was also suggested by Karolyi and Stulz (1996) the covariances between countries are not constant, because they change over time and can be forecasted.

The question of what could be the reason for the increase in the co-integration in the stock markets arises. What are the macroeconomic or global factors that have led to the co-movement of the stock market indices of emerging and developed countries? Yang et al (2003) study of the effect of the establishment of the Economic and Monetary Union (EMU) on the short and long-run integration among eleven European stock markets and US stock market. Their results were similar to that obtained by Taylor and Tonks (1989) and Corhay et al (1993). It was in that opinion that modern information technology and merger 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 using the methodology proposed by Lettau and Ludvigson (2001), which is the two-step method, examines three countries; Australia, UK and Canada. They confirmed the results from the Lettau and Ludvigson (2001) analysis that suggest that the lagged co-integration variable (cay) is a significant predictor of the expected return or excess return of the stock markets of the specified countries, just as the case in U.S. Although, Xu (2005) uses the VECM to investigate the relationship between the consumption-wealth ratio (cay) on German stock returns. The purpose of Xu (2005) analysis was to compare the efficiency of the methodology proposed by Lettau and Ludvigson (2001) and the VECM using German and U.S. data, and it was concluded that the VECM is a more appropriate method to study the effect of cay on stock returns and excess returns in both data set significantly.

It may then be said that cay might be regarded a macroeconomic factor that determines the linear trend of stock market returns in the long-run, since there are evidence that they exist a correlation between these variables and the financial markets returns. With this evidence, the stock market returns could be predictable by business cycle at rotational frequencies in the long-run.

Methodology

The methodology that would be used is the Vector Error Correction Model (VECM) which has been used most frequently in the analysis of economic time series data. Engle and Granger (1987) elaborate on the fundamental of the co-integration aspect. In this paper, the co-integration analysis in the framework of vector autoregressive model (VAR) as proposed by Johansen (1988), and Johansen and Juselius (1990) would be used.

The following is a statistical explanation of the VECM analysis using the Johansen technique as denoted by Brooks (2008). In order to use the Johansen approach, a VAR with k lags containing a set of g variables (g ≥ 2) which are assumed to be I(1) and cointegrated, would have to be converted into a vector error correction model (VECM), such that the set up:

yt = β1 yt−1 + β2 yt−2 + · · · + βk yt−k + ut

g Ã- 1 g Ã- g g Ã- 1 g Ã- g g Ã- 1 g Ã- g g Ã- 1 g Ã- 1 (3)

is transformed to a vector error correction model (VECM) as below:

∆yt = ∏yt−k + Г1∆yt−1 + Г2∆yt−2 +· · ·+Гk−1∆yt−(k−1) + ut (4)

where ∏= () - Ig and Гi = ( - Ig . From the above VAR equation the g variables are in the first differenced form on the left hand side and on the right hand side the k-1 are the lags of the dependent variables in their differenced form, each contains a Г coefficient matrix that accompanies them.

The matrix П in the Johansen test can represent the long-run coefficient matrix, since all the ∆yt−i will be zero and the error term. ut will be set to their expected value of zero will leave ∏yt−k = 0, in equilibrium. The rank of the П matrix from its eigenvalue is used to calculate the number of co-integration between the ys. The eigenvalues, which are the number of its characteristic roots that are different from zero equals to the rank of a matrix. The symbol λi denotes the eigenvalues, which are set in ascending order as thus; λ1 ≥ λ2 ≥ . . . ≥ λg. In the case where the eigenvalues ( λs) are roots they have to be less than 1in absolute value and positive, and λ1 will be closest to 1 which is the largest, while λg will be closest to 0 which is the smallest. When the analysed variables are not co-integrated, the rank of the matrix П will not be different from zero considerably, such that λi ≈ 0 ∀ i.

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

λtrace(r ) = − Ti) (4)

(λtrace = 0 when all the λi = 0, for i = 1, . . . , g.)

and

λmax(r, r + 1) = − T ln(1 −r+1) (5)

where r is the represents the number of co-integrating vectors under the null hypothesis and i represents the estimated value for the i-th eigenvalue from the matrix П. In the λtrace, which is a joint test has a null hypothesis where the number of co-integrating vectors is less than or equal to r against an alternative hypothesis that there are more than r. In the λmax tests a separate test is conducted on each eigenvalue with a null hypothesis that is the number of co-integrating vectors is r and an alternative hypothesis of r + 1. The trace test starts with p eigenvalues, and then in succession the largest is removed. Every eigenvalue has with it an attached different co-integration vector, which is known as the eigenvectors. A significantly non-zero eigenvalue shows a significant co-integration vector.

The critical values used for the two test statistics depends on the value of the g - r, the number of non-stationary elements and how constants are included in each of the equations. When the critical value is less than the test statistics, reject the null hypothesis that there are r co-integrating vectors in support of the alternative hypothesis (r +1 for the λtrace test or more than r for the λmax test). The test is conducted in a sequence and under the null, r = 0, 1, . . . , g - 1 so that the hypotheses for λmax can be represented as below as:

H0 : r = 0 versus H1 : 0 < r ≤ g

H0 : r = 1 versus H1 : 1 < r ≤ g

H0 : r = 2 versus H1 : 2 < r ≤ g

. . .

. . .

H0 : r = g − 1 versus H1 : r = g

From the above, the first test means the null hypothesis of no presence of co-integrating vectors, therefore the corresponding П matrix have a 0 rank. In the case where the null hypothesis (H0: r = 0) is rejected, then the null that there is one co-integrating vector (H0: r = 1) is tested and the process continues, and as such the value of r is continually increased until the null hypothesis is not rejected. The matrix П can never be at full rank (g) as this would mean that yt is stationary. In the case where the matrix П has 0 rank, then by correspondence to the univariate case, ∆yt depends only in ∆yt − j and not on yt - 1, which will result to no long-run relationship between the elements of yt - 1, which in turn means no co-integration. For instance, in 1< rank (П) < g, there are r co-integrating vectors. The matrix П is then characterised as the product of two matrices, α and β', of the dimension (g Ã- r ) and (r Ã- g), respectively, that is,

П = αβ' (6)

where matrix β denotes the co-integrating vectors, while α , which is known as the adjustment parameter, gives the amount of each co-integrating vector associated with each equation of the vector error correction model.

In the following section the VECM approach using the Johansen technique as explained, will be carried out on the selected three European stock markets; UK, France and Germany to investigate the possibility of an increasing market co-integration, using to an extent the recursive approach done by Pascual (2003) which is similar to that done by Rangvid (2001). The Johansen approach is then applied to the vector error correction model;

∆xt = A + ∏0xt−1 + i ∆xt−1 + ut (7)

here x represents the vector containing the logarithm value of the stock market indices for the selected European countries. A larger number of the significant co-integrating vectors will be observed as time goes on if the markets are converging.

Data

The data used were used to investigate for co-integration are the quarterly data of the European (UK, Germany and France) stock market indices from 1963 to 2010 which results to a total sample size of 192 observations obtained from DATASTREAM. The reason for starting this analysis from 1963 instead of 1960 as carried out by Pascual (2003) is due to data availability problems. Starting with a sample of 20 quarters from 1960:Q1 to 1964:Q4 for three European stock indices is estimated recursively by adding one extra observation at a time up to 2010:Q4. In Appendix 1, it can be observed by eye-balling the data, that as more observations are added the lines representing each variable seem to draw closer to each other and have an upward trend. According to Pascual (2003), the upward trend can be attributed to two reasons. Firstly, is the number of existing stochastic trends conducting the three dimensional systems are decreasing with time as markets become increasingly integrated. Secondly, as the observations increase from 20 to 156 the trace statistics merge to the long run values. This may be interpreted as the existence of cointegration between the variables, although the necessary analysis must be undertaken to justify this assumption. In the result representation section, four different lag windows, corresponding to 20, 60, 100, 140 and 192 observations, are analyzed.

Results Presentation

The first step in the VECM analysis is to check for stationarity in the variables. Unit Root test was carried out on the log of the variables using Augmented Dickey-Fuller (ADF), Philips-Perron (PP) and Kwiatkowski-Philips-Schmidt-Shin (KPSS) test. The results are presented below:

Unit Root/Stationary Test

Test Type

UK

G

F

Critical Value

-0.804

-0.420

-0.568

ADF

1% Level

(-3.464)

Fail

Fail

Fail

5% Level

(-2.876)

Fail

Fail

Fail

10% Level

(-2.574)

Fail

Fail

Fail

Critical Value

-0.823

-0.473

-0.434

PP

1% Level

(-3.464)

Fail

Fail

Fail

5% Level

(-2.876)

Fail

Fail

Fail

10% Level

(-2.574)

Fail

Fail

Fail

Critical Value

1.650

1.604

1.615

KPSS

1% Level

 0.739

Reject

Reject

Reject

5% Level

 0.463

Reject

Reject

Reject

10% Level 

0.347

Reject

Reject

Reject

Conclusion

I(I)

I(I)

I(I)

Table 1: Unit root results

From the above table, UK, G and F represents United Kingdom, Germany and France respectively, they denote the log of the European stock market. From the above results one can infer that the variables are I(1), meaning there exist unit roots and therefore the variables are non-stationary. These results can be illustrated in a unit root graph as below:

Figure 1: Unit Root graph

Since, one of the blue points touch the circle, we can conclude that the variables are non-stationary. The next step will be to specify the optimal lag. The below table contains the lag structure of 20, 60, 100 and 140 observations. The optimal lag is obtained when the Akaike criterion has minimum value. The Akaike Information Criterion is appropriate for this analysis since the ample size is quite small.

Akaike Information Criterion for VECM with lag 2 to lag 10

Lag

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 table, comparing the information criteria shows that VAR (1, 2) gives the smallest information criteria for all the different categories of observations and so it is the best linear unbiased estimation. For 20 observations only VAR (1, 2) was obtainable because it is a very small sample size. Following is the cointegration analysis of the variables. Using the Johansen FIML approach for testing the cointegration, there are two basic tests results. The max-eigenvalue and the trace test as explained earlier in this paper. The results of this test are presented below using the given hypothesis decision rule:

H0: R=0 H1: R>0→R>0

H0: 0˂R≤1 H1: R>1

H0: 0˂R≤2 H1: R>2→R>2. where R represents rank and is less than 3.

Co-integration test - 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 *

 0.740448

 40.38981

 29.79707

 0.0021

At most 1 *

 0.544995

 17.46023

 15.49471

 0.0250

At most 2 *

 0.213077

 4.073633

 3.841466

 0.0435

Trace test results shows that there exist 3 co-integrating equations at the 5% level

Table 4: Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized No. of CE(s)

Eigenvalue

Trace Statistic

5%

Critical Value

Prob.**

None *

 0.740448

 22.92958

 21.13162

 0.0276

At most 1 *

 0.544995

 13.38660

 14.26460

 0.0685

At most 2 *

 0.213077

 4.073633

 3.841466

 0.0435

Max-eigenvalue test results shows that there exist 1 co-integrating equation at the 5% level.

Co-integration test - 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 *

 0.192783

 24.73114

 29.79707

 0.1713

At most 1 *

 0.123292

 12.52387

 15.49471

 0.1335

At most 2 *

 0.084363

 5.023705

 3.841466

 0.0250

Trace test results shows that there exist no co-integrating equations at the 5% level

Table 6: Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized No. of CE(s)

Eigenvalue

Trace Statistic

5%

Critical Value

Prob.**

None *

 0.192783

 12.20727

 21.13162

 0.5273

At most 1 *

 0.123292

 7.500164

 14.26460

 0.4318

At most 2 *

 0.084363

 5.023705

 3.841466

 0.0250

Max-eigenvalue test results shows that there exist no co-integrating equation at the 5% level.

Co-integration test - 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 *

 0.188373

 27.20639

 29.79707

 0.0967

At most 1 *

 0.069223

 6.961074

 15.49471

 0.5822

At most 2 *

 2.85E-05

 0.002769

 3.841466

 0.9555

Trace test results shows that there exist no co-integrating equations at the 5% level

Table 8: Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized No. of CE(s)

Eigenvalue

Trace Statistic

5%

Critical Value

Prob.**

None *

 0.188373

 20.24532

 21.13162

 0.0662

At most 1 *

 0.069223

 6.958305

 14.26460

 0.4941

At most 2 *

 2.85E-05

 0.002769

 3.841466

 0.9555

Max-eigenvalue test results shows that there exist no