Nikolaos Vlastakis

George Dotsis

Raphael N Markellos

Nonlinear Modeling of European Football Scores Using Support Vector Machines

Keywords: C45 -Neural Networks and Related Topics &lt, C4 -Econometric and Statistical Methods: Special Topics &lt, C -Mathematical and Quantitative Methods, C53 -Forecasting and Other Model Applications &lt, C5 -Econometric Modeling &lt, C -Mathematical Betting, Football, Forecasting, Nonlinear Models, Support Vector Machines JEL Classification : L83, C53, C45, G14

published or not. The documents may come

Introduction

The usefulness of nonlinear models in forecasting financial variables and challenging market efficiency has been extensively investigated over the past twenty years (see Mills and Markellos, 1997, inter alia). Despite the fact that the efficiency of wagering markets has been also widely studied, most of this research has employed linear parametric models (see [START_REF] Sauer | The Economics of Wagering Markets[END_REF]Vaughan Williams, 2005;[START_REF] Vlastakis | How efficient is the European Football Betting Market? Evidence from arbitrage and trading strategies[END_REF]. As was the case with financial markets, one could reasonably argue that the inconclusiveness of results with respect to betting market efficiency may be due to a misspecification of the models used.

The present paper extends the literature on market efficiency by evaluating the statistical and economic performance of a new class of nonparametric regression models, namely Support Vector Machines (SVMs), in forecasting the outcome of European football matches using odds information from 5 UK bookmakers. The results are compared to those obtained by a standard parametric approach based on Poisson count regression. SVMs belong to the family of neural networks which have been widely applied within the financial literature (see, for example, [START_REF] Mcnelis | Neural Networks in Finance: Gaining Predictive Edge in theMarket[END_REF]. [START_REF] Chen | Expert Prediction, Symbolic Learning and Neural Networks: an Experiment on Greyhound Racing[END_REF] were among the first to employ such techniques for predicting the results of greyhound races. They adopted a decision tree building algorithm along with a backpropagation Artificial Neural Network (ANN) and tested their performance against predictions from human specialists. They found that their techniques were able to outperform their human competitors, with the ANN achieving the best performance. Johansson and Sonstrod (2003) also used a similar ANN approach in modeling greyhound race outcomes and were able to "beat the 3 market" and achieve positive returns. [START_REF] Rotshtein | Football Predictions Based on a Fuzzy Model with Genetic and Neural Tuning[END_REF] employed a fuzzy knowledge base with genetic and neural tuning in order to predict the outcome of European football matches from a dataset of Finnish championship matches for the period 1994-2001. Their somewhat complex model displayed superior predictive ability on the basis of statistical criteria. Recently, [START_REF] Edelman | Adapting Support Vector Machine Methods for Horserace Odds Prediction[END_REF] adapted the methodology of SVMs for predicting the outcome of horse races. His model employed past performance data and bookmaker's odds in a two-stage approach. A "Winningness Index" forecast was first obtained from the SVM and then this was used along with bookmaker odds in a multinomial logit model in order to obtain probability forecasts. The author tested his methodology on a small sample of Australian horseracing data are reported promising results.

The rest of the paper is organized as following. The next section discusses the methodology used. Section 3 presents the empirical results, whereas the final section concludes the paper.

Methodology

Poisson Count Regression

As has been widely discussed in the literature, the Poisson process is a natural assumption when dealing with count data, such as the number of goals scored by a team in a football match. The density of the distribution of the number of occurrences of the event is given by ( ) where λ is the mean and variance parameter. It has been shown in sports betting literature (eg., see [START_REF] Dixon | Modelling Association Football Scores and Inefficiencies in the Football Betting Market[END_REF][START_REF] Cain | The Favourite -Longshot Bias and Market Efficiency in UK Football Betting[END_REF] that the goal scoring processes of the home and away teams can be approximated by independent

Pr ! y e Y y y λ λ - = = , y = 0,1,…, (1)
Poisson processes.

The Poisson regression model is derived from the Poisson distribution through the parameterization of the relation between the mean parameter λ and the regressors x (for a comprehensive description of count regression see [START_REF] Winkelmann | Econometric analysis of count data[END_REF]. The standard method involves an exponential mean parameterization:

i x i e β λ ′ = , i = 1,…,n (2)
The natural estimator for the Poisson regression model is maximum likelihood (ML) using the following cost function

( ) { } 1 ln ln ! i n x i i i i e L yx y β β β ′ = - ′ = - ∑ (3)
The Poisson ML is the solution to k -the number of covariates -nonlinear equations corresponding to the first-order condition for maximum likelihood ( )

1 0 i n x i i i y e x β ′ = - = ∑ (4)

Support Vector Machine Regression

The SVM methodology is similar to that used for building ANNs. Given a sample of training data ( ) ( )

{ } 1 1 , ,..., , m n n
x y x y ⊂ × , where m is the number of regressors, the goal in the so-called ε-SV regression is to find a function ( ) f x that deviates from the targets y i by a maximum of ε, and, at the same time, is as flat as possible (see [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. In the case of linear functions, f takes the form ( )

, f x w x b = + , with , m w b ∈ ∈ (5)
Where , ⋅ ⋅ denotes the dot product in m . Flatness in this context means that the goal is to find a small w. In order to ensure this, one needs to minimize the norm,

i i i i i i i i y w x b w x b y ε ξ ε ξ ξ ξ ⎧ - -≤ + ⎪ ⎪ + -≤ + ⎨ ⎪ ≥ ⎪ ⎩ (7)
The constant C > 0 determines the trade-off between flatness and the amount up to which deviations larger than ε are tolerated. In order to reach the nonlinear form of f , the dual formulation of the optimization problem in (6) is needed. To this end, some kind of dualization method is required, the most common utilizing Lagrange multipliers. This method constructs a Lagrange function from the objective function and the corresponding constraints, by introducing a dual set of variables. The dual optimization problem takes the following form maximize

( )( ) ( ) ( ) * * , 1 * * 1 1 1 , 2 n i i j j i j i j n n i i i i i i i a a a a x x a a y a a ε = = = ⎧ - - - ⎪ ⎪ ⎨ ⎪ - + + - ⎪ ⎩ ∑ ∑ ∑ F o r P e e r R e v i e w 7 subject to ( ) * 1 0 n i i i a a = - = ∑ and [ ] * , 0, i i a a C ∈ (8)
In this formulation, ( )

* 1 n i i i i w a a x = = - ∑ , thus ( ) ( ) * 1 , n i i i i f x a a x x b = = - + ∑ (9)
This is the so-called Support Vector Expansion. Now w can be completely described as a linear combination of the training patterns x i . For the transition to the nonlinear form of the algorithm, the best method is the so-called implicit mapping with the use of kernels. Since the SV algorithm depends only on dot products between patterns x i , it is enough to know ( ) ( ) ( ) , : ,

i i k x x x x = Φ Φ
, rather than Φ explicitly, so that w is a nonlinear combination of the training patterns x i and the optimization problem becomes maximize

( )( ) ( ) ( ) ( ) * * , 1 * * 1 1 1 , 2 n i i j j i j i j n n i i i i i i i a a a a k x x a a y a a ε = = = ⎧ - - - ⎪ ⎪ ⎨ ⎪ - + + - ⎪ ⎩ ∑ ∑ ∑ subject to ( ) * 1 0 n i i i a a = - = ∑ and [ ] * , 0, i i a a C ∈ (10)
where

( ) ( ) * 1 n i i i i w a a x = = -Φ ∑ and ( ) ( ) ( ) * 1 , n i i i i f x a a k x x b = = - + ∑ (11)
In our case a Gaussian kernel is employed, hence ( ) for the score of each team.

2 2 2 , i x x i k x x e σ - - = (12)

Data and Empirical Results

Our dataset contains match results from the English Premier League for the period 13/8/2005 to 10/5/2007, a total of 750 observations. We also have the corresponding 1X2 odds from 5 online bookmakers: Gamebookers, Interwetten, Ladbrokes, Sportingbet, and William Hill, coded hereafter as A, B, C, D, and E, respectively.

Data on Asian Handicap odds along with the handicap size were also available for the matches under study.

In order to gain a clearer understanding of the relationship between odds, Finally, the third segment contains the matches for which the away team is the favourite and is the part of the distribution that contains high values for the odds on the home team. If we look at the first and the third segment we realize that, in each of the two, one color is dominant and it is the color that corresponds to the favourite, whereas the second segment is a more of a random mix in term of coloring or outcomes.

[Insert Figure 1 about here] This could either be an indication that the implied margin is not a good proxy for the actual margin or a sign of market distortion.

[Insert Table 2 about here]

The data was divided in two samples. Roughly 78% of the data (588 observations)

was used for the estimation of the models, whereas the remaining data (162 observations) were used for out-of-sample evaluation purposes. Table 3 presents [Insert Table 3 χ -, where k is the order of the power of the fitted values used in the regression. In this paper, we will consider powers up to four, i.e., k = 2,3,4.

Table 4 summarizes the results of the misspesification tests. None of the reported statistics are significant in the 5% level, which indicates that the model is well-specified. Although the Generalised RESET (GRESET) should be sensitive also to departures from linearity, it may not be able to capture the complex nonlinear structure of the data suggested by the graphical analysis.

[Insert Table 4 about here] For the estimation of the SVM regression model, the independent covariates that were used in each of the two regressions were the ones found significant for the respective Poisson models. This choice of regressors was made so that the information available to each model was the same and direct comparisons can be drawn. The value for 2 σ in equation ( 12) was set via trial-and-error to 20 using a crossvalidation in the fist sub-sample. The magnitude of this parameter is very important since it determines the flexibility of the model. A very low value may lead to overfitting the estimation sample, a common problem for nonparametric and nonlinear models.

Table 5 presents the evaluation of model performance using statistical error functions for the in-sample and out-of-sample data, respectively. Although the two [Insert Table 5 about here] Although the results reported in Table 5 suggest that the two models have similar forecasting performance this may not be true if other non-statistical cost functions are used. As argued by [START_REF] Satchell | An Assessment of the Economic Value of Non-linear Foreign Exchange Rate Forecasts[END_REF], standard statistical error functions may not be suitable for assessment of the economic value of predictions of non-linear models. Thus, the final test of the forecasting performance of the two models will be the economic evaluation of the forecasts provided by the models, through the implementation of a betting strategy.

For the formulation of betting strategies, the data are divided again in two samples. One is used for the estimation of the models and the other for out-ofsample evaluation. Once the models are fitted the out-of-sample data are used to provide forecasts of the goals scored by each team. Then, a forecasted goal difference is calculated for every match. The goal difference is used as the variable under consideration for the formulation of the betting strategy. The strategy consists of a straightforward rule: "If the forecasted goal difference is positive and greater than some threshold T 1 , bet on the home team; if it is negative and less than some threshold T 2 , bet on the away team; and, if its absolute value is less than some threshold T x , bet on x". Thresholds are estimated numerically so as to maximize total profits using in-sample data. The same thresholds are used on the out-of-sample data and the two models are evaluated with respect to economic performance. 6 summarizes the results of the betting strategies for both models. It can be easily seen that Table 6 reveals a different picture than before with respect to the comparative performance of the models. Although the Poisson model is able to outperform the SVM model on in-sample data and return higher total profits with higher expected return per bet, the SVM model greatly outperforms the Poisson model on out-of-sample data. In fact, the Poisson model is unable to produce positive profits in the out-of-sample data, whereas the SVM model returns significantly high profits in both samples. Consequently, the performance of the SVM model is more stable and the model itself more robust than the Poisson.

[Insert Table 6 about here]

Conclusions

This paper examined the weak form efficiency of the UK football betting market using a sample of odds from 5 bookmakers. Motivated by the financial literature and in order to asses the nonlinear forecastability of match outcomes we employed a Support Vector Machine modeling approach. This relatively new class of neural network models have been found in the literature able to capture a wide variety of nonlinear relationships. The performance of the SVMs was compared to that of a standard Poisson count regression. Our preliminary descriptive analysis demonstrates that regressors based on odds variables are likely to be strongly dependent in a nonlinear fashion not only between them but also against match outcomes. Although econometric tests suggest that the Poisson model regressions estimated via a stepwise procedure are well specified, these models have an inferior 0.99 0.99 -0.03 0.99 -0.07 -0.04 -0.07 -0.03 -0.06 -0.53 -0.57 -0.56 0.00 -0.55 1(B) 0.86 -0.11 0.11 0.99 0.98 -0.03 0.98 -0.12 -0.10 -0.13 -0.03 -0.11 -0.57 -0.62 -0.61 0.00 -0.60 1(C) 0.84 -0.11 0.10 0.99 0.98 -0.03 0.99 -0.09 -0.06 -0.09 -0.03 -0.08 -0.54 -0.58 -0.58 0.00 -0.57 1(D) -0.03 0.00 -0.01 -0.03 -0.03 -0.03 -0.03 0.01 -0.01 0.01 0.31 0.00 0.02 0.01 0.03 0.63 0.02 1(E) 0.83 -0.11 0.11 0.99 0.98 0.99 -0.03 -0.08 -0.05 -0.08 -0.03 -0.07 -0.54 -0.57 -0.57 0.00 -0.56 X(A) -0.54 -0.04 0.05 -0.07 -0.12 -0.09 0.01 -0.08 0.96 0.95 0.04 0.96 0.85 0.83 0.82 -0.01 0.83 X(B) -0.53 -0.03 0.05 -0.04 -0.10 -0.06 -0.01 -0.05 0.96 0.92 0.04 0.94 0.84 0.83 0.81 -0.02 0.82 X(C) -0.53 -0.03 0.04 -0.07 -0.13 -0.09 0.01 -0.08 0.95 0.92 0.04 0.94 0.84 0.80 0.81 -0.01 0.82 X(D) -0.05 0.00 0.01 -0.03 -0.03 -0.03 0.31 -0.03 0.04 0.04 0.04 0.04 0.06 0.04 0.06 0.00 0.06 X(E) -0.52 -0.03 0.04 -0.06 -0.11 -0.08 0.00 -0.07 0.96 0.94 0.94 0.04 0.83 0.81 0.80 0.00 0.81 2(A) -0.87 0.02 0.00 -0.53 -0.57 -0.54 0.02 -0.54 0.85 0.84 0.84 0.06 0.83 0.98 0.98 -0.01 0.98 2(B) -0.89 0.03 -0.01 -0.57 -0.62 -0.58 0.01 -0.57 0.83 0.83 0.80 0.04 0.81 0.98 0.98 -0.01 0.97 2(C) -0.89 0.03 -0.01 -0.56 -0.61 -0.58 0.03 -0.57 0.82 0.81 0.81 0.06 0.80 0.98 0.98 0.00 0.98 2(D) 0.01 0.03 -0.03 0.00 0.00 0.00 0.63 0.00 -0.01 -0.02 -0.01 0.00 0.00 -0.01 -0.01 0.00 -0.01 2(E) -0.88 0.03 0.00 -0.55 -0.60 -0.57 0.02 -0.56 0.83 0.82 0.82 0.06 0.81 0.98 0.97 0.98 -0.01 Mean -0.14 -0.08 -0.03 0.08 0.05 0.07 0.05 0.07 0.32 0.32 0.31 0.03 0.31 0.20 0.18 0.18 0.03 0.19 Odds for Home Victory, Draw and Away Victory for bookmaker A, B, C, D, E, and Asian Handicap (AH) are denoted by 1, X and 2, respectively. There are no odds for Draw in Asian Handicap. AHS is the Asian Handicap Size. The 5% and 1% two-sided critical values for the correlation coefficient are 0.062 and 0.081, respectively.

  are estimated, one for the home team score and the other for the away team score.

  case for the Poisson model, two separate SVM models are estimated, one

Figure 1

 1 Figure 1 presents 3-dimensional (3D) scatter plots of the odds quoted by each

  the estimation results of the two Poisson models. The method used for the estimation was stepwise regression using a 5% level of significance. As can be seen, the statistically significant regressors for the home score model are the constant, the size of the Asian Handicap, the odds for away team victory from bookmakers A and E and the margin of bookmaker C,. For the away team score, only the Asian Handicap size and the odds for away team victory from bookmaker A enter the regression as statistically significant variables. The signs of the coefficients are as expected, with the odd on away team victory from bookmaker A in the home score regression. The adjusted R 2 and log likelihood statistics reveal a reasonably good fit of the models.

θ

  is the maximized loglikelihood function for the original model, and ( ) a l Ψ that for the extended model,

  comparable performance, the SVM model outperforms the Poisson regression in almost all instances.

  Figure 1. 3D Scatter Diagrams of Odds and Outcomes

Table 1

 1 As the variables in Table1are to be used as regressors, problems from multicollinearity are very likely to arise, certainly in the Poisson count regression. Indeed, as suggested by the graphical analysis the multicolinear relationship between the regressors is likely to be nonlinear.When examining the predictive power of odds, the bookmaker margins are a very important metric, since the size of the margin directly affects the odds. However, it is not possible to know the actual margin without knowing the actual distribution of bets across outcomes. The standard practice in the literature is to calculate an implied margin by assuming an equal distribution of bets. Table 2 presents the descriptive statistics of implied margins for all bookmakers in the sample. An examination of the table reveals that bookmakers B, C, and E operate at comparable levels, whereas bookmakers A and D operate at considerably lower profit levels.

	[Insert Table 1 about here]

presents the linear correlation matrix of odds for all bookmakers in the sample. As expected, odds on the same outcome between bookmakers are generally highly correlated. The same holds for the odds on different outcomes for each bookie. Asian Handicap odds Exhibit small correlation with 1X2 odds, in contrast to the handicap size, which is significantly correlated to the odds on all outcomes for most bookmakers. In terms of the correlation structure, bookmaker D variables appear to have a somewhat different behavior.

  about here] In order to investigate model misspecification two sets of tests were undertaken. As mentioned, one of the most important assumptions of the Poisson regression model is the equality of mean and variance. If this assumption is violated, then the Poisson model is deemed unsuitable for the specified application and other, less restrictive

		si e -on ˆi y ,
	where si e are the standardized residuals and ˆi y the fitted values for the dependent
	variable, from the original model. If the resulting coefficient is found statistically
	significant, then the model is over-or under-dispersed, depending on the sign of the
	coefficient.
		The other misspesification test is a generalized version of Ramsey's RESET
	test for ML models, originally proposed by Peters (2000). This is a Likelihood Ratio
	(LR) test that is conducted by adding RESET variables, i.e., powers of fitted values
	ˆ,	1..

models, such as the negative binomial, should be used. To test the equality of mean and variance, we follow the specification by

[START_REF] Wooldridge | A Note on the Lagrange Multiplier and F-statistics for Two Stage Least Squares Regressions[END_REF]

. This is a regression-based overdispersion test that is carried out by regressing 1

Table 2 .

 2 Descriptive statistics of margins

		Bookmaker
		A	B	C	D	E
	Mean	0.0807	0.1184	0.1230	0.0838	0.1249
	St. Dev.	0.0041	0.0101	0.0038	0.0725	0.0025
	Min.	0.0563	0.0998	0.1113	-0.8814	0.1206
	Max.	0.1012	0.1413	0.1319	0.1067	0.1313

Table 3 .

 3 Poisson Count ML estimation results

	Dependent Variable

Absolute values of z-statistics appear in brackets below the estimated coefficients.

Table 4 .

 4 Poisson Count regression Misspecification Tests

		Overdispersion Test	Generalized RESET Test
	Dependent Variable Coefficient t-statistic	Test Variables	LR statistic
	Home Team Score	-0.0608	-1.6447	ˆi y	2		1.1827
				2 ˆ, i i y y	3	3.7041
				2 ˆˆ, , 3 i i y y y i	4	3.7042
	Away Team Score	-0.0595	0.0489	ˆi y	2		3.9189
				2 ˆ, i i y y	3	2.3012
				2 ˆˆ, , 3 i i y y y i	4	1.3567

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Table 5 .

 5 Statistical error functions for both models

	POISSON	SVM
	In-

sample Out-of-Sample In-sample Out-of-Sample

  MSE, MAE, and RMSE stand for Mean Squared Error, Mean Absolute Error, and Root Mean Squared Error, respectively.

	Home Team Score
	MSE	1.2634	1.4999	1.2176	1.4687
	MAE	0.9020	0.9690	0.8741	0.9447
	RMSE	1.1240	1.2247	1.1035	1.2119
	Away Team Score
	MSE	0.9416	0.9627	0.9662	0.9509
	MAE	0.7667	0.7572	0.7712	0.7662
	RMSE	0.9704	0.9812	0.9662	0.9751

Table 6 .

 6 Economic evaluation of models

	POISSON	SVM
	In-

sample Out-of-Sample In-sample Out-of-Sample Number of Bets

	Bets 1	363	93	204	57
	Bets X	30	9	48	15
	Bets 2	99	25	67	16
	All Bets	492	127	319	88
	Expected Return
	Bets 1	9.87%	0.31%	13.32%	4.98%
	Bets X	8.83%	7.22%	-12.29%	8.00%
	Bets 2	3.65%	-7.64%	0.16%	7.56%
	All Bets	8.56%	-0.76%	6.70%	5.97%
	Total Profit
	Bets 1	35.84	0.29	27.17	2.84
	Bets X	2.65	0.65	-5.9	1.2
	Bets 2	3.61	-1.91	0.11	1.21
	All Bets	42.10	-0.97	21.38	5.25

Page 22 of 21 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Christian Klein
email: cklein@uni-hohenheim.de

Bernhard Zwergel
email: bernhard.zwergel@wiwi.uni-augsburg.de

Henning John

J Henning Fock
email: john.henning.fock@wiwi.uni-augsburg.de

G10 -General < G1 -General Financial Markets < G -Financial Economics, G12 -Asset Pricing < G1 -General Financial Markets < G -Financial Economics, G14 -Information and Market Efficiency|Event

Keywords: Sentiment, Replication Studies, Event Studies, Data Mining, Market Efficiency

  L'archive ouverte pluridisciplinaire

In past decades, many empirical studies revealed return anomalies in many different asset classes and markets. Very recent publications have, however, even found evidence that stock markets react to the results of soccer matches.

In this paper, we argue that such empirical studies should be analyzed carefully; we thus endorse the use of replication studies to verify results. Consequently, by rebuilding the study of [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF], we are able to detect mistakes in the empirical set-up. Based on these findings, we demonstrate how even minor flaws can have a crucial influence on the results of such studies and point out pitfalls that are frequently encountered. We furthermore emphasize the importance of robustness checks to validate the results of empirical studies.

Introduction

In recent years, empirical researchers have put much effort into uncovering stock market anomalies. The results often violate the traditional efficient market hypothesis and sometimes result in the mistrust of traditional asset-pricing models. Today, we believe that many irrational aspects do indeed play a role in the formation of returns: daily stock returns are, for example, correlated with sunshine (see [START_REF] Hirshleifer | Good day sunshine: Stock returns and the weather[END_REF], daylight saving changes (see [START_REF] Kamstra | Losing sleep at the market: The daylight saving anomaly[END_REF], and sporting events. [START_REF] Edmans | Sports sentiment and stock returns[END_REF] demonstrate a significant relation between soccer match results and stock returns. They also document a loss effect after other international sports events like cricket, rugby, and basketball games. The authors argue that the literature on psychology indicates that sport results have a significant effect on investors' mood in general.

In their study on the influence that the performance of England's national soccer team has on the British FTSE 100 index, [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] find a statistically significant relationship. For these authors, the economic importance of sport events is the rationale for stock markets reacting to a soccer team's performance.

The "soccer industry" indubitably plays an important economic role. The World Cup 2006 in Germany exemplified this when approximately 4 million fans came from abroad to support their teams. FIFA (Fédération Internationale de Football Association), the international governing body of the soccer association, generated total revenues of CHF 2.858 billion ($ 2.594 billion) with this World Cup. Could this have had an impact on the stock market? Since FIFA is neither capital market oriented, nor listed, we assume that in terms of GDP, the role of the "soccer industry" is not a significant one at all. In terms of psychological factors, the assumption that the outcome of a soccer game does have an influence on the stock market the day following the match is very unlikely as well -however compelling the argument might look at first glance. This is exemplified, for example, by Germany's defeat by Italy in the 2006 World Cup semifinal. This was a shock for Germans (and, of course, for German shareholders); nevertheless, the World Cup event still had an overall positive effect on the German economy. Furthermore, a large portion of the German DAX free float is traded by foreign investors who were arguably not interested in the outcome of that game (they might even have been Italians!). Was there therefore a rationale for the DAX index to reflect the outcome of this game?

We replicated the study by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] and were able to detect several inconsistencies in their event study set-up. Based on these findings, we present typical mistakes found in such empirical studies, especially in event studies, which are extremely susceptible to errors and assumptions. These errors should be avoided, as even small variances in the measuring of an event can influence the results strongly.

Our aim with this paper is to encourage the utilization of replication studies. 1 We want to stress the importance of publishing empirical papers that do not find statistically significant results (especially when studying topics related to market efficiency, see, e.g., [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF][START_REF] Fama | Efficient capital markets: II[END_REF].

The outline of this paper is as follows: in the next section we demonstrate, by means of a replication study, that the original study by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] contains several basic mistakes and could lead to quite different results. Section 3 applies robustness checks to the corrected results and the last section summarizes our findings and concludes the paper.

1 On the importance of replication studies see [START_REF] Hubbard | An empirical comparison of published replication research in accounting, economics, finance, management, and marketing[END_REF] In their empirical study, which covers the period January 6, 1984to July 3, 2002[START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] find a strong link between the performance of England's national soccer team and daily changes on the London Stock Exchange's FTSE 100 index. The authors arrive at this finding after performing an event study analysis of the London Stock Exchange's stock index movements following international matches played by the national soccer team. In conclusion, they use a binomial test to check whether the return of the trading day following a soccer game differs from the unconditional mean return of all other trading days.

We rebuilt the above-mentioned study in respect of the same time period, with daily prices sourced from Datastream and soccer game results from various media. We subsequently arrived at results that differed from the ones reported by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]. In order to explain this divergence, we studied possible distortions in event studies in greater depth. By not correcting them, we were able to reproduce some of the results produced by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]. The differences between our (corrected) results and those of Asthon et al. (2003) are noteworthy. [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] used Datastream as the source of their capital market data.

Holiday Return Effect

Generally, the data quality is very good, but there is one problem: Datastream always reports the last price if there has been no trading on a holiday. This effect produces a zero return for every holiday within the data series, which can seriously distort an empirical investigation if this effect is not corrected. A possible solution is to use Datastream's holiday function, which shows a price of "#NA" for holidays when the data file is downloaded. Another solution is to select all returns with a zero size and 6 check them against a calendar, removing them if they fell on holidays. Checking against a calendar is crucial because it is also possible, although not very likely, that two real closing prices could be identical. The latter could occur since, for example, genuine zero returns were not all that uncommon in the early days of the FTSE 100 index when its level was not particularly high.

It is definitely a grave mistake not to correct zero returns in the data set. When using the data of final match days, we were closer to the results provided by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] than when we did correct zero returns on holidays. This affected, for example, the data regarding the game played on Sunday, June 2, 2002 (World Cup, England vs. Sweden). The following two weekdays were public holidays: on June 3, the Queen's Golden Jubilee was celebrated and June 4 was the spring bank holiday. [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] apparently used a zero return to compute the abnormal return for this final game, as our results converged with theirs when we did the same. We thereafter checked other pitfalls and were able to replicate more of their results. [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] report that within the study period, England's national soccer team participated in 35 "tournament final games" of which they won 12 games, lost 12, and drew 11. This information is not actually correct, but by accessing the national soccer team's official homepage and doing a "copy and paste" of the "fixtures and results," it was possible to replicate the above results. The correct distribution of the 35 "tournament finals games" was, however, 13 wins, 14 losses, and 8 draws. There is, however, a further problem: the list of games within the study period ends with a semi-final at Wembley Stadium on June 26, 1996, which resulted in a draw (1:1). This is, of course, not possible in a semi-final. The game was won 6:5 by Germany after a penalty shootout and indicated as such in the list by an 7 asterisk, which could be overlooked when "copying and pasting." The impact on the results is, however, enormous. Since Ashton and his colleagues argue that there is a "feel-good factor," it should, in this case, have been quite low as the national team had lost a semi-final. However, it is quite obvious that all the errors made had a strong influence on the empirical analysis results.

The "Copy and Paste" Effect

The Globalization Effect

In every event study, the point in time is of extraordinary importance when measuring the event's impact on stock returns. Efficient markets will immediately reflect the effect of the event in the stock prices [START_REF] Mackinlay | Event studies in economics and finance[END_REF]. The London Stock Exchange does not trade around the clock; consequently, if the result of a soccer game is known before 4.30 pm BST on the day, this result will be reflected in that day's closing price, otherwise in the next trading day's closing price. It is therefore important to identify the exact time of the event to group it accordingly. In their article, [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] mention that they always used the return of the trading day following the soccer game. In general, this procedure cannot be criticized, because the majority of soccer games in Europe (with only a few exceptions) are decided after stock exchange trading hours. An exception would be the World Cup 2002 hosted by Japan and South Korea. The time difference between the UK and Japan is +8 hours. During this World Cup, some kick-offs took place during the evening, which was around midday British time. Consequently, the results were known on the day of the soccer game and were already digested in the day's closing price. Any measurement of the effect of the match result on the next trading day would therefore distort the results. It is, however, possible to argue against our assertion. Without using market microstructure data, it is indeed difficult to assess the effects of these games in an entirely undistorted way.

The Correct(ed) Outcomes

By including the aforementioned errors, we were able to replicate the results of [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] in respect of tournament final matches' draws and losses. These are presented in tables 1 and 2, while tables 3 and 4 reflect the new results after the distortions had been corrected.

As illustrated in table 1, [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] indicate that all games have a positive mean return after a win and a negative mean return after a defeat. When comparing these results with ours (table 3), the difference becomes evident. In our study, the mean returns after losses are considerably less negative or even positive. This effect is also evident with regard to wins, which become less positive. The results are even positive in games that end in a draw, with the exception of tournament finals.

We also tested the significance of our results and found that compared to the [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] study, the statistical significance had changed tremendously. In duplicating these authors' study, we used binomial statistics to test whether the expected return on the next day's 2 trading was greater (or less) than the unconditional mean return after a win (or loss) by considering what proportion of returns after a particular game outcome exceeded the unconditional mean return. In the original study (table 2), all mean returns after a loss are significant on a 5% or better level. In our study (table 4), only one result is significant on a 5% significance level. We also find no significant results in respect of a win, as demonstrated in table 2.

It is important to note that in both studies significant results are not independent of one another: "tournament final games" are, of course, a subgroup of "tournament finals and qualifying games." Further, it is evident that the significant results of "tournament finals games" are the only reason why the larger sample is significant.

2 Or the same trading day, depending on the time of the event. Ultimately, we could therefore not support Ashton and colleagues' conclusion that there is a "statistically significant relationship between the performance of the English national soccer team and the change in the price of shares traded on the London stock exchange" (Ashton et al., 2003, p. 785). The only appropriate significant results seem to be the stock market reactions after the English soccer team suffered losses in tournament finals. This result is astonishing insofar as we doubt that soccer fans are equipped with large short-selling capacities, since this instrument is not much used by private investors and was certainly not available some years ago.

[Insert Another interesting finding is that even small variations or mistakes in the data set can have a huge impact on the results of an event study. After notifying Ashton and colleagues of our findings, they obtained almost the same results using the suggested corrections. However, there was still an asymmetry between the two studies concerning the time at which the return should be measured. In their study, the authors always use the next available trading day, whereas in our study, if the game results were known during stock exchange trading hours, that day was used rather than the following day. We therefore computed a mean return of 0.0186% in respect of the thirteen wins in tournament finals (see table 3), whereas [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] computed 0.1795%3 . We could then replicate their results by employing different returns in respect of two of the 13 matches -those held on June 7, 2002 and June 15, 1998. The first game (June 7, 2002) was the world championship in Sapporo, Japan where the English team defeated Argentina (1:0). Since, the kickoff was at 8:30 pm Japan time, the result was known during the London Stock Exchange trading hours on the day of the game (Friday at about 2.15 pm GMT). We therefore took a return of -0.750964% to compute the average. Ashton et al. used the return of the following trading day (0.158727%), which was a Monday, to compute their average. On the second date (June 15, 1998), the game in Marseille kicked-off at 2:30 pm local time on a Monday. Hence, the result was known at the London Stock Exchange that same day (approximately 3.15 pm). Consequently, we used that return (-0.937641%). [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF], however, used the return (0.244939%) of the next day, Tuesday, to compute their average. Which of the two standpoints is the correct one is not relevant in this case, but this example does indicate how sensitive, and therefore easy to influence, such results are, especially when sample sizes are very small.

Robustness Checks

In the following section, we address the problem of data mining that make robustness checks essential. We discuss a variety of possible robustness checks and, by means of the study by [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF], demonstrate how formerly significant results disappear when robustness checks are applied.

Using large financial databanks makes research much easier for investigators. However, they also run the risk of finding significant correlations between variables that are no more than a statistical fluke. Moreover, it is known that papers with To avoid reporting significant but wholly spurious correlations, sophisticated researchers apply robustness checks to their results to counter the data-mining argument.

Sub-Samples

An easy way to check the robustness of empirical results is to divide the entire sample into sub-periods, Sub-period make it possible to check if the discovered correlations are stable over time. If they are not, there are two possible explanations:

1. For some reason, the correlation regime has changed over time. This may be due to structural changes, behavioral changes, or something else. Whatever the reason, the authors have to find a logical economic explanation for these changes.

2. The correlations are pure coincidence.

We divided the entire [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] sample into two sub-periods to check the robustness of the empirical results. The first sub-sample includes the period 1984 to 1993, the second the one 1994 to 2002. In the corrected results (table 4), a significant relationship is found between games lost and the FTSE returns on the following trading day.

Table 5 presents the results of this robustness check. There are (still) significant results in the first sub-sample: an English soccer team defeat produces a FTSE stock market return that is below the average FTSE return. Please note that "tournament finals games" is a sub-sample of "tournament finals and qualifying games" -the significant results of "tournament finals games" are the only reason why the larger sample is significant. In the second sub-sample (1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002), the significance does 12 not only disappear, the effect is also reversed: there are more days with a higher than average return after a game ending in defeat than days with a lower return. Since there is no logical explanation for this finding, we attribute the first sample's significant results to pure coincidence.

[Insert Table 5 about here]

Outliers

A common characteristic associated with financial time series like stock market series is the presence of extreme observations. Consequently, statistics derived from financial data sets include outliers and is often misleading. This is a real problem when computing OLS regressions or -like [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] did -calculating means. A common approach to check for the influence of outliers is to eliminate the extreme quantiles (e.g., the 1% quantiles) of the observations. Table 6 is thus similar in structure to table 3, but the largest and smallest returns were eliminated when the average mean returns were computed. It becomes clear that the mean return after a victory in a final turns negative, which is contrary to [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] perception.

[Insert Table 6 about here]

Confounding Events, Seasonal Effects, and Economic Relevance

Additional robustness checks would be possible: We could check for confounding events, like ad-hoc announcements or changes in interest rates on the day of a soccer game. Furthermore, it might be logical to estimate and remove seasonal effects from the time series: discovered correlations may, after all, be attributable to business Finally, even if there were a relationship between soccer results and stock market outcomes, there is still one unanswered question: so what?

From a theoretical point of view, such a finding would challenge the efficient market hypothesis. We call financial markets "efficient" if they don't allow investors to earn above-average, risk-adjusted returns. From a practical point of view, investors could be interested in such a finding if this knowledge were to allow them to earn money.

Whatever the case, empirical researchers would need to find a trading strategy in order to generate abnormal returns on the basis of a discovered anomaly. We doubt that this is possible, even if trading costs were not taken into account.

We do not see the necessity of doing additional checks on the [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] study. In our opinion, we have already supplied sufficient evidence that there is no link between England's national soccer team's achievements and the British FTSE 100 index.

Conclusions

The aim of this study was twofold. Primarily, we wanted to point out some errors in previous empirical studies. By means of a replication study, it was demonstrated that even small mistakes could have a strong influence on empirical studies results.

Consequently, the importance of replication studies was illustrated.

In our opinion, replication studies have two predominant effects: first, they evaluate the set-up of the original empirical study and find possible mistakes -as we did in respect of the [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF] paper. Second, by using new (independent) data, it [START_REF] Marquering | Disappearing anomalies: a dynamic analysis of the persistence of anomalies[END_REF], for example, examined several wellknown stock market anomalies before and after publication. They demonstrate that most of the effects disappeared after publication, or may never have existed at all. The problem with these kinds of papers is that they culminate in non-significant results, which are rather difficult to have published.

In general, studies on market anomalies are always prone to the accusation of "data mining," i.e. searching for coincidently significant results in large data sets. If data sets are large enough and contain many variables, it is obvious that significant (although useless) interrelations can sometimes be found by sheer coincidence. This can be a problem, as "publication bias" (or the "file-drawer problem"), known from medical publications, also plays an important role in economics. Studies with significant results are therefore more likely to be published; conversely, studies without such results are not publishable and are ignored. However, the efficiency of capital markets can never be proven significantly, only their failure can. One can only hope that the reputation of replication studies, as well as studies without significant results will improve in future. 5 See, e.g., [START_REF] Lindgren | Statistical theory[END_REF] The table shows the results of [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]. "Tournament finals games" are all games of international tournaments of the English national soccer team. For "Tournament finals and qualifying games" qualifying competitions for major international tournament are added. "All internationals" include all types of international soccer games of the English national soccer team, including non-tournament 'friendly' matches. This table presents the binomial statistics of [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]. The intuition in column "win" is that the average return for the index after a win is greater than the unconditional mean return. The number in brackets is the number of won games of the English team, the number without brackets is the amount of cases where the average return for the index after a win is greater than the unconditional mean return.

In column "loss" they test whether the average return for the FTSE 100 index after a loss of the national soccer team is significantly less than the unconditional mean return for all types of matches.

For instance, the English soccer team has lost 38 games, the FTSE 100 index has fallen by an amount greater than the unconditional mean return on 27 of these occasions. The table shows our corrected the results and is organized like table 1. "Tournament finals games" are all games of international tournaments of the English national soccer team. For "Tournament finals and qualifying games" qualifying competitions for major international tournament are added. "All internationals" include all types of international soccer games of the English national soccer team, including non-tournament 'friendly' matches.

Compared to Ashton et al. the mean returns after losses are considerably less negative or even positive. This effect is also visible for the wins, which become less positive. The results are even positive in games that end in a draw, with the exception of tournament finals. This table presents our corrected binomial statistics of [START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]. Like in table 2 the intuition in column "win" is that the average return for the index after a win is greater than the unconditional mean return. The number in brackets is the number of won games of the English team, the number without brackets is the amount of cases where the average return for the index after a win is greater than the unconditional mean return.

In column "loss" they test whether the average return for the FTSE 100 index after a loss by the national soccer team is significantly less than the unconditional mean return for all types of matches.

In truth the English soccer team has lost 42 games, the FTSE 100 index has fallen by an amount greater than the unconditional mean return on 24 of these occasions.

  much more likely to be published; an effect also known as the "file-drawer bias." 4

4

  See McQueen et al. (1997), p. 69.

  anomalies. In addition, criticism in respect of data snooping could be avoided by studying out-of-sample data.

  rule out the occurrence of a statistical Type I 5 error in the original empirical study.

  Table 1 about here] [Insert Table 2 about here] [Insert Table 3 about here] [Insert Table 4 about here]

14 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  , p. 308 for a description.

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Table 1 Descriptive

 1 Statistics (Ashton et al., 2003)

		Number of games	Mean return after win	Mean return after	Mean return after loss
			(No. of wins)	draw (No. of draws)	(No. of losses)
	The FTSE 100 index: Period of Investigation 6 January 1984-3 July 2002
	Unconditional Daily Return on the index 0.03511%
	All internationals	210	0.1705% (102)	-0.000481% (70)	-0.1289% (38)
	Tournament finals and	99	0.3085% (49)	-0.1843% (33)	-0.1863% (17)
	qualifying games
	Tournament finals games	35	0.2698% (12)	-0.2129% (11)	-0.4150% (12)

  Ashton et al compute positive mean returns of the FTSE 100 index after wins and negative mean returns after losses.

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Table 2

 2 Binomial Statistics[START_REF] Ashton | Economic impact of national sporting success: evidence from the London stock exchange[END_REF]

				No. of games played	No. of returns after a win/loss which are >/≤ the
					unconditional mean. (Total no. of wins/losses)
					Win	Loss
	All internationals		210	50 (102)	27*** (38)
	Tournament	finals	and	99	30* (49)	13** (17)
	qualifying games
	Tournament finals games		35	8 (12)	9** (11)

Table 5

 5 *, **, *** indicates significance at the 10%, 5% and 1% level, resp.Sub Samples. Binomial StatisticsNo. of returns after a win/loss which are >/≤ the unconditional mean. (Total no. of wins/losses) To check the robustness of the (corrected) results we split the sample in two sub samples. When comparing these results to the results of table 4 it can be seen that the significance can be explained by the first sub sample -the effect reverses in the second subsample.

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21 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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Table 6 Descriptive

 6 Statistics without extreme values (i.e. highest and lowest return) We check the robustness of the (corrected) results of table 3 with removing the largest and the smallest value of the returns. In one out of six cases the algebraic sign changes.

					Page
		Number of games	Mean return after win	Mean return after	Mean return after loss
			(No. of wins)	draw (No. of draws)	(No. of losses)
	The FTSE 100 index: Period of Investigation 6 January 1984-3 July 2002
	Unconditional Daily Return on the index 0.03655%
	Tournament finals and	91	0.1105 (46)	0.0809 (27)	-0.1303% (18)
	qualifying games
	Tournament finals games 29	-0.1258% (11)	-0.5402 % (6)	-0.2286% (12)

22 of 22 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Ashton et al. provided us with the result of one game corrected in respect of the "copy and paste" effect.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Martin Chalkley
email: m.j.chalkley@dundee.ac.uk

L13 -Oligopoly and Other Imperfect Markets < L1 -Market Structure, Firm Strategy, and Market Performance < L -Industrial Organization, F12 -Models of Trade with Imperfect Competition and Scale Economies < F1

Keywords: Merger, International trade, Oligopoly JEL classi…cation: L4, F2 merger, international trade, oligopoly

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In an important contribution to the literature on incentives for …rms to merge, Salant, Switzer and Reynolds (1983) establish that a bilateral merger from an initial Cournot equilibrium with linear demand is unpro…table, except in the case of duopoly. One strand of subsequent research has explored conditions that might augment the pro…tability of merger, such as the existence of cost savings (Perry and Porter, 1985), product di¤erentiation advantages [START_REF] Deneckere | Incentives to form coalitions with Bertrand Competition[END_REF] or more complex (non-linear) demand functions [START_REF] Cheung | Two remarks on the equilibrium analysis of horizontal merger[END_REF][START_REF] Faulí-Oller | On merger pro…tability in a Cournot setting[END_REF]. In this paper we revert to Salant, Switzer and Reynolds's constant cost, homogeneous product, linear demand framework to examine the pro…tability of merger in an open economy. More speci…cally, we ask: for what con…gurations of market structure and trade costs, if any, will merger be pro…table? Using a two-country model, we consider both withincountry ("domestic") and cross-border ("international") mergers. We …nd that, under trade conditions, duopoly is a necessary, but no longer su¢ cient, condition for pro…table domestic merger, but international merger can be pro…table from any market structure. 1

The Model

We consider a homogeneous product which can be produced and consumed in either, or both, of two countries, i; j = 1; 2; i 6 = j: We suppose that each country has an identical linear inverse demand function

P i = a Q i :
(1)

Production costs comprise a constant marginal cost which is the same for all …rms and, for simplicity, set to zero. In addition, exports are subject to a trade cost, t, per unit of output. We let n i and n j denote the number of …rms located in countries i and j respectively. We assume that the outcome of competition between …rms is a Cournot equilibrium in quantities and that there is no arbitrage. This implies that the price prevailing in country i can di¤er from that in country j by more than t. Pro…ts for a representative …rm based in country i are given by i = P i y i + (P j t)x i ;

(2) where y i denotes the sales of a …rm in country i to its home market and x i its exports to country j.

The problem of a representative …rm in country i is to maximise (2), holding outputs of rival …rms …xed and subject to non-negativity constraints on x i and y i . We assume, henceforth, that domestic output is strictly positive in equilibrium. The …rst order conditions, using an asterisk to denote Cournot equilibrium values, for this problem are

P i y i = 0 (3) and P j t x i 0 x i 0 ; (4)
where a right hand brace indicates a pair of complementary inequalities, one of which must hold with equality.

Noting that P i is a function of y i and x j alone, whilst P j is a function of y j and x i alone, conditions (3) and ( 4) can be solved simultaneously to yield x i (n i ; n j ) = max 0; a (n j + 1)t n i + n j + 1 (5) and 1 A related, but di¤erent, issue is the change in the pro…tability of merger that would result from a change in trade cost or unilateral tari¤. This question has been examined by Long and Vousden (1995), [START_REF] Falvey | Mergers in open economies[END_REF] and Gaudet and Kanouni (2004). y i (n i ; n j ) = (a + n j t)=(n i + n j + 1), x j (n i; n j ) > 0; y i (n i ; n j ) = a=(n i + 1);

x j (n i; n j ) = 0:

In the case where exports from both countries are zero, (5) -( 6) reduce to the standard conditions for Cournot equilibrium in each country in which …rms make pro…ts of

A i (n i ) = a n i + 1 2 : (7)
By inspection of (5), the maximum trade cost compatible with international trade taking place (i.e. x i ; x j > 0) is given by

t < max a n i + 1 ; a n j + 1 t (8)
and for the symmetric case where n i = n j = n this threshold level of trade cost is

t = a n + 1 : (9)
For an equilibrium involving trade, substitution from ( 5) and ( 6) into (1) and (2), after some rearrangement, yields:

i (n i ; n j ; t) = 2a 2 + 2n 2 j t 2 2an j t 2 + t 2 n i + n j + 1 : (10)
It is convenient to decompose this pro…t into the elements deriving from the home and overseas markets. For a …rm located in country i, we denote these pro…ts as i i (n i ; n j ; t) and j i (n i ; n j ; t) respectively and note that i (n i ; n j ; t) i i (n i ; n j ; t) + j i (n i ; n j ; t): Substitution from ( 5) and ( 6) into (1) yields

i i (n i ; n j ; t) = a + n j t n i + n j + 1 2 (11) and j i (n i ; n j ; t) = a (n j + 1)t n i + n j + 1 2 : ( 12
)
Similarly for a …rm in country j; j (n i ; n j ; t) j j (n i ; n j ; t)+ i j (n i ; n j ; t); where

j j (n i ; n j ; t) = a + n i t n i + n j + 1 2 (13) and i j (n i ; n j ; t) = a (n i + 1)t n i + n j + 1 2 : ( 14

Merger under Trade

We consider …rst an initial equilibrium with trade taking place and consider two possible forms of merger. In what we term a domestic merger two …rms in country i form a single …rm and in what we term an international merger one …rm in country i joins with one …rm in country j: With a domestic merger two …rms forgo their individual pro…ts i (n i ; n j ; t) but take a share of pro…ts that are enhanced through the reduction in domestic competition. Thus, the gain to a domestic merger in a trade equilibrium in which n i = n j = n, can be written as

G D (n; t) = i i (n i 1; n j ; t)+ j i (n i 1; n j ; t) 2 i i (n i ; n j ; t) 2 j i (n i ; n j ; t): (15)
With an international merger, we need to consider three types of …rms. Following such a merger there will be n 1 …rms located wholly in country i, n 1 …rms located in country j and one newly merged …rm which has a production base in each country. The newly merged …rm is not the same as existing …rms in that it can supply either market from a domestic production unit. Therefore, in setting its output in each country it can act like a domestic …rm in that country; there is no incentive for it to produce for export because by producing domestically it can save trade costs. The multinational …rm's impact on total production can therefore be deduced by analogy with a Cournot equilibrium in which domestic output in country i is determined as if there are n domestic Cournot competitors and n 1 overseas competitors, whilst output for export from country i is determined as if there are n 1 …rms competing over exports facing n overseas competitors. The position in country j is symmetric to this. International merger, therefore, has the e¤ect of reducing by one the number of exporters serving each market but leaving the number of domestic producers unchanged. Thus, the gain to an international merger in a trade equilibrium in which n i = n j = n, can be written as

G I (n; t) = i i (n i ; n j 1; t) + j j (n i 1; n j ; t) i (n i ; n j ; t) j (n i ; n j ; t): (16)
In their autarky setting, Salant, Switzer and Reynolds (1983) demonstrate that a merger is pro…table only if there are two …rms in the initial Cournot equilibrium. Does this result hold under trade? A …rst thought might be that merger would clearly be unpro…table since the number of competitors is doubled relative to the Salant, Switzer and Reynolds setting. However, the situation under trade is complicated by the fact that merger generates an asymmetry between the merged and unmerged entities. Proposition 1 addresses the question in regard to domestic and international merger.

Proposition 1 For domestic merger, G D (n; t) < 0 for all n and 0 t < t whilst for international merger there exists a threshold level of trade costs, b t that G D (n; t) is maximised at a trade cost of t(n) = a 2n 2 +2n+1 . Substitution of this value of t into G D (n; t) shows that the latter is negative for n 2 thus establishing the result for domestic merger. After setting n i = n j = n, substituting from ( 11) and ( 12) into G I (n; t) and equating to zero, the critical (zero-gain) value of trade cost can be solved for as b t I (n) = 8an 3 10an 2a 2(4n 4 +12n 3 +7n 2 2n 1) : For all n 2, b t I (n) lies in the range (0; t ): Since, as can readily be con…rmed, G I (n; t) is negative at t = 0, concave and approaches zero from above as t approaches t it follows that b t I (n) constitutes a threshold such that G I (n; t) < 0 when t < b t I (n), but G 1 (n; t) > 0 when b t I (n) < t < t , thus con…rming the result for international merger.

I (n) such that G I (n; t) < 0 when t < b t I (n), but G I (n; t) > 0 when b t I (n) < t < t Proof. Setting n i = n j = n,
The Proposition establishes that from an initial equilibrium in which there is international trade then, whatever the market structure, there is no incentive for domestic merger. However, with an international merger the situation is somewhat di¤erent. Speci…cally, Proposition 1 admits the possibility that mergers from even relatively competitive markets may be pro…table provided that trade costs are of an appropriate magnitude. The intuition for this di¤erence can be explained in terms of the impact of merger on cost and revenue. For t = 0, domestic and international mergers are equivalent; neither generates a saving in trade cost whilst both lead to a fall in market share and revenue (the standard Salant, Switzer and Reynolds result). Now consider the e¤ect of raising t. Both domestic and international mergers will generate a saving in trade costs, but the reduction will be greater in the latter case since both markets will be served from a domestic plant. For both types of merger the relationship between t and the trade cost saving is non-monotonic, falling to zero as t approaches t , the point at which trade ceases. The impact of t on the consequences of merger for revenue is markedly di¤erent in the two cases: for a domestic merger the loss of revenue increases with t whilst for an international merger there is an inverse relationship. This di¤erence can be understood by considering the situation as t approaches t . In the case of domestic merger the e¤ect becomes concentrated on the home market to the point where, when t = t , the impact of the merger is simply to reduce the number of domestic competitors from n to n 1, with the associated loss of market share for the merging entities. With an international merger, by contrast, there is no loss of market share at t = t since the number of competitors remains at n in both markets. The net result is that for an international merger the cost saving will outweigh the loss of revenue if t is su¢ ciently high, whilst a domestic merger is always unpro…table.

The following …gure (in which a = 1) shows how the pro…tability of an international merger is related to trade cost and initial market concentration; an international merger is pro…table between the locus t and the locus b t I .

Merger Initiating Trade

When a merger takes place from autarky (i.e. t > t ) the resulting change in market structure may initiate trade. To understand this e¤ect, let X i and X j denote the total volume of exports from i to j and j to i respectively. Using ( 5 for the representative …rm in i; and an equivalent expression for that in j, these are given by

X i (n i ; n j ; t) = n i (a t(n j + 1)) n i + n j + 1 (17)
and

X j (n i ; n j ; t) = n j (a t(n i + 1)) n i + n j + 1 : (18)
With a symmetric initial structure with n …rms in each country, the ‡ows of exports from i to j and j to i following a domestic merger in country i are, respectively,

X D i (n; t) = max 0; (a t(n + 1))(n 1) 2n (19)
and

X D j (n; t) = max 0; a nt 2 : (20)
Inspection of ( 19) and ( 20) reveals that the former is positive for t < t whilst the latter is positive for t < t A where

t A a n : (21)
Thus, if t lies in the range t < t < t A a domestic merger in country i will initiate a one-way ‡ow of trade from j to i.

Consider now an international merger. As explained above, this has the e¤ect of reducing by one the number of exporters serving each market but leaving the number of domestic producers unchanged. Assuming a symmetric initial structure with n …rms in each country and using, ( 17) and ( 18), we can write the post-merger trade ‡ows as

X I i (n; t) = max 0; (a t(n + 1))(n 1) 2n (22)
and

X I j (n; t) = max 0; (a t(n + 1))(n 1) 2n : (23)
For either of these ‡ows to be positive requires t < t , which is the same condition that pertained pre-merger. An international merger will thus neither initiate, nor cause the cessation of, trading.

We may thus consider whether from a position of autarky, but where t < t A , a domestic merger is pro…table. The gain from such a merger is and, in rather special circumstances, this may be positive as the following Proposition establishes.

G DA (n; t) = i i (n i 1; n j ; t) 2 A (n i ) (24
Proposition 2 For n = 2 there exists a threshold level of trade costs, b t D , such that G DA (n; t) > 0 when b t D < t < t A (n): In all other circumstances, G DA (n; t) < 0.

Proof.

After setting n i = n j = n, substituting from ( 11) and ( 7) into G DA (n; t) reveals directly that this is negative for n > 2: Setting n = 2 and solving G DA (n; t) = 0 for t yields a critical (zero-gain) value for trade cost of b t D = a(4 p 2 3) 6

: Since G DA (n; t) is increasing in t it follows that for b t D < t < t A (n) it is positive, thus establishing the claim.

The Proposition demonstrates that for trade costs in the range where a domestic merger would initiate trade, duopoly in the home market is a necessary but not su¢ cient condition for the merger to be pro…table. This contrasts to the position under autarky where, as established by Salant Switzer and Reynolds, a bilateral merger from duopoly is always pro…table.

Conclusions

In an autarky setting, Salant, Switzer and Reynolds (1983) demonstrated that a merger would be pro…table if there were two …rms in the initial Cournot equilibrium, but not if there were three or more …rms. Using a two-country model, we show that in an open economy with international trade, the condition for a merger between two …rms in the same country to be pro…table is more restrictive. Speci…cally, duopoly is no longer su¢ cient for a merger to be pro…table; an additional requirement is that trade costs exceed a threshold level. An international merger, by contrast, can be pro…table from any market structure, provided that trade costs lie in a certain range. For simplicity, it has been assumed that both market size and initial market structure are the same in both countries, and one direction for further research is to consider an asymmetric setting.

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  substituting from (11) and (12) into G D (n; t) and di¤erentiating with resect to t the resulting …rst order condition implies 4

  Gaudet, G.,Kanouni, R., 2004. Trade liberalization and the pro…tability of domestic mergers. Review of International Economics 12, 353-358. Long, N.V., Vousden, N., 1995. The e¤ects of trade liberalization on costreducing horizontal mergers. Review of International Economics 3, 141-155. Perry, M.K., Porter, R.H., 1985. Oligopoly and the incentive for horizontal merger. American Economic Review, 75, 219-227. Salant, S., Switzer, S., Reynolds, R., 1983. Losses from horizontal merger: the e¤ects of an exogenous change in industry structure on Cournot-Nash equilibrium. Quarterly Journal of Economics 98, 185-199.

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