Does distance still matter?
An empirical analysis of 24 stock
markets.
Robin van Amsterdam
Supervisor: Dr. Diego Ronchetti
Combined master thesis for the Msc Finance and Msc Economics Faculty of Economics and Business, University of Groningen
July 2015
JEL: C01, C21, C23, G12
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1. Introduction
Volatile – That is the word that best describes the current state of the financial world. A global economy that is constantly evolving, with new challenges on the horizon. Economic-, credit- and banking- crises come and go. To challenge, understand, survive and succeed the current financial challenges one must incorporate and adapt to the newest techniques. Consider an equity manager that must predict and understand stock market returns and volatility to protect his portfolio against risks. Multiple stock prediction techniques have been used in the past. One group of models incorporate macro-economic factors, such as FDI, exchange rates and inflation rates to predict stock returns. Most of these models do not incorporate the spillover effects amongst stock markets however. Ignoring spillover effects can lead to the wrong predictions. When the interest rate in Germany rises for example, this will not only impact the German stock market index, but will also affect stock markets in other countries. These effects should be incorporated when analysing stock returns. The effect and magnitude of these spillovers effects are best determined through spatial econometric techniques, an econometric methodology which incorporates spatial dependencies between subjects. This technique is young, but expanding rapidly.
Spatial econometricians make use of a measurement of closeness between subjects (e.g., geographical distance or bilateral trade) to determine spatial interaction and spatial spillover effects between countries. A spatial weights matrix is used to measures these interdependencies. Take for example a spatial weights matrix that measures how economically dependent countries are on other countries. By using this weight matrix the interdependences of European stock markets can be analysed. Because this type of modelling includes both direct effects and indirect (spatial spillover effects) this will enable the equity manager to better understand the market and hedge risk more appropriately.
3 This paper finds evidence that geographical distance is still an important measure of closeness between countries today that should be taken into account when examining spillover effects. Even though no spillover effects are obtained when using an unrestricted inverse distance matrix, these results are shown to be biased. After restricting the inverse distance matrix such that only geographical relations between strongly tied countries are examined, clear evidence is found in favour of the hypothesis that geographical distance still matters.
1.1.
Stock market predictions
Extensive literature exists on the prediction of stock returns. One of the most important models used is the Capital Asset Pricing Model (CAPM) developed by Lintner (1965) and Sharpe (1964). The CAPM can be used to price individual assets based on the relation between risk and expected returns of assets. Another important asset pricing model is the Arbitrage Pricing Theory (APT) derived by Ross (1976). The APT tries to determine mispricing of assets by using macroeconomic factors or other financial market indices to derive the correct asset prices. If the current price of an asset is different from the predicted price, there is an arbitrage opportunity which can be used to derive a risk free profit.
While these two building blocks of modern day stock pricing are still commonly used, many other strategies have been developed to evaluate stock prices. Fundamental analysis is one example. Fundamental analysts focus on the companies underlying the stocks. By analysing valuation ratios such as P/E ratios forecasters try to predict future stock prices. Another important group of forecasters are called technical analysts. These researchers do not look at underlying company statistics, but rather make their predictions based on trends in historical stock market data, for example by looking at exponential moving averages.
While some studies using these methods have found some proof for stock market predictability to some degree; stock markets stay hard to predict and cannot be captured by a single model. While fundamental and technological analysis are useful when predicting stock returns for a few set of firms or markets, when trying to predict aggregate market indices of several countries these methods are harder to apply and would require an extremely large amount of data. Consider for example an fundamental analysis of the returns of the S&P500. Fundamental analysts would have to examine a large amount of underlying companies to properly predict stock market returns. While this is possible, there are easier ways to predict index returns.
4 However, one of the problems surrounding these indicators is the availability of data. While an increasing amount of economic indicators is becoming available, it is still a common issue that data is not available for certain countries or only for a limited time horizon. Another problem relates to the significance of some indicators. While for a lot of indicators a significant effects can be found while predicting market indices, the sign and magnitude of the estimates can differ a lot.
1.2.
Spatial spillovers amongst stock markets
This study uses macroeconomic indicators to determine stock prices in 24 countries. The focus lies on the examination of spatial dependencies between countries, which can be measured through spatial spillover effects. The existence of spillover effects is not new. Take for example the economic crisis in 2007. The economic crisis showed that now, more than ever, countries are dependent on each other and influence each other’s economies. A vast amount of research has been done on the contagion effects between economies that spread the economic crisis. As globalization continues and economies become more integrated, spillover effects amongst countries will become a more important factor that should be accounted for when doing research.
The spillover effects amongst stock markets have been examined in the past in many different studies (e.g. Baele, 2005; Tai, 2007). The focus of most of these studies lies on the degree of dependence amongst stock markets. Some studies focused on time-varying spillovers between international stock markets (e.g. Égert and Kočenda, 2011). Another example is given by Hamao et al. (1990), whom examined short-run interdependences of prices across three important stock markets using an autoregressive conditionally heteroskedastic estimation technique. Masih and Masih (1999) use a vector error correction model and level vector autoregressive model to examine the short- and long-run interaction effects amongst Asian emerging stock markets and several OECD countries. These models do not incorporate spatial spillover effects however.
One popular model used that incorporates spatial dependencies is the gravity model. Correlation between stock markets are regressed on economic size measures such as GDP and on bilateral distances. These bilateral distances are estimated by cross-country distances such as geographical distance or net exports (Flavin et al., 2002).
Spatial econometrics are tools to investigate spatial spillover effects more formally. One of the first contributions to spatial econometrics was from Anselin (1988) in his seminal book on spatial econometrics. Spatial econometrics make use of a spatial weights matrix, which measures the relative closeness between subjects. With the use of this spatial weights matrix spatial spillover effects can be examined. The key benefit of using a spatial weights matrix over the gravity model is that the relative distance factor does not have to be bilateral: The effects of linkages of the relationship between one country and many other countries at once are now captured.
5 accounts for relative distance between observations (as a proxy for economic or financial integration). They find evidence that bilateral trade relations is the best measure of closeness amongst stock markets. Other closeness measures can also be used to analyse spatial spillovers. Examples of these measures are geographical distance, exchange rate volatility and inflation expectation divergence.
This research uses spatial econometric techniques to examine the impact of geographical distance on spatial spillovers amongst stock markets in more detail. To be able to answer the research question the study is structured as following. Section 2 describes the concept of spatial spillovers in more detail and explains how spatial dependencies can be modelled using spatial econometric techniques. Section 3 describes the methodology and estimation strategy used to determine the impact of geographical distance. Section 4 describes the data used. Section 5 presents and discusses the results of the analysis. Section 6 concludes.
2. The spatial econometric model
Section 2.1 starts with a definition of spatial dependence and describes how this concept is related to spatial interaction effects and spillover effects. Furthermore, Section 2.1 summarizes all spatial econometric specifications that can be computed. Section 2.2 discusses the limitations of several spatial econometric models. Lastly, Section 2.3 concludes by describing some determination and estimation strategies of the spatial econometric models used in previous research.
2.1.
Spatial econometric specifications
The most commonly estimated model in empirical research is the linear regression model. The spatial econometric model extends this expression by adding spatial interaction effects. These interaction effects are obtained with the use of a spatial weights matrix, 𝑊 (NxN matrix), capturing the spatial interdependencies between countries, regions or subjects. This way for example the level of economic integration between countries is determined, i.e. a spatial weights matrix can be seen as a measure that describes the ‘closeness’ between countries.
6 Therefore, any combination of these spatial effects measurement is possible. Adding all spatial interaction effects to the linear regression model and writing the equation in matrix notation yields:
𝑌𝑡 = 𝜌𝑊𝑌𝑡 + 𝛼𝐼𝑛+ 𝑋𝑡𝛽 + 𝑊𝑋𝑡𝜃 + 𝑢𝑡, (1)
with 𝑢𝑡 = 𝜆𝑊𝑢𝑡+ 𝜀𝑡,
where 𝑌𝑡 is a 1xN vector of endogenous variables, 𝐼𝑛 is the identity matrix (NxN matrix with ones on
the diagonal and zero on all the non-diagonal elements), α is an 1xN vector containing the constants or
fixed effects, 𝑋𝑡 is an NxK matrix of exogenous variables, 𝛽 is an Kx1 matrix of coefficient estimates
and 𝜀𝑡 is a Nx1 vector of disturbance terms that are identically and independently distributed.1
Furthermore, the parameter ρ determines the endogenous interaction effects (W𝑌𝑡), θ the exogenous
interaction effects (W𝑋𝑡) and λ the spatial interaction effects amongst the residuals (W𝑢𝑡).
The model described by Eq. (1) is called the general nesting spatial (GNS) model. This is the spatial econometric model that includes all types of spatial interaction effects. By restricting ρ, θ and λ all other spatial models can be obtained which include one or two types of spatial interaction effects. For example, if θ = 0 the model examines endogenous interaction effects and interaction effects amongst the error terms, but does not include exogenous interaction effects. This model is called the spatial autoregressive combined model (SAC, also known as SARAR). Note that when ρ, β and λ are put to zero, the OLS model is achieved. In total there exist seven spatial econometric models (excluding OLS), the models are summarized in table 1.
Table 1. Description and type of interaction effects for each spatial econometric model.
Model Description Endogenous interaction effects (WY) Exogenous interaction effects (WX) Interaction effects amongst the error terms
(Wu)
OLS Ordinary leased squares - - -
SAR Spatial autoregressive model X - -
SLX Spatial lag of X model - X -
SEM Spatial error model - - X
SAC (SARAR)
Spatial autoregressive
combined model X - X
SDM Spatial Durbin model X X -
SDEM Spatial Durbin error model - X X
GNS General nesting model X X X
1 The disturbance terms are assumed to have zero means and variances equal to σ². This assumption is relaxed
7 While spatial interaction effects are used to measure spatial spillover effects, a clear distinction has to be made between the concepts of interaction effects and spillover effects. Elhorst (2015) notes that spillover effects are measured by either exogenous interaction effects or endogenous interaction effects or both. This means that spillover effects, also referred to as indirect effects, do not exist in models that only contain interaction effects amongst the error terms (SEM).
Furthermore, spillover effects can either be local or global in nature. Global spillovers are interaction effects of a change in 𝑋𝑡 in one country that affect all other countries, even if they are
unrelated according to the spatial weights matrix. In stark contrast, local spillovers only occur when countries are closely related to each other according to the spatial weights matrix. In the case of an inverse distance matrix, this means that spillover effects only occur amongst neighbouring countries. Global spillover effects exist when ρ ≠ 0 and for local spillovers to exist it is required that θ ≠ 0. For a thorough discussion of local and global spillover effects see Anselin (2003) and Elhorst and Vega (2013).
Interpretation of the outputs of the spatial econometric models can be difficult and differs per type of model. Following Elhorst (2012), rewriting Eq. (1) yields:
𝑌 = (𝐼 − 𝜌𝑊)−1(𝑋𝛽 + 𝑊𝑋𝜃) + 𝑅, (2)
with 𝑅 being a rest term including the disturbance terms, fixed effects and constants.2 The marginal
effects of a change in one of the exogenous variables on one of the endogenous variables is therefore equal to: [ 𝜕𝐸(𝑦1) 𝜕𝑥1𝑘 ⋯ 𝜕𝐸(𝑦1) 𝜕𝑥𝑁𝑘 ⋮ ⋱ ⋮ 𝜕𝐸(𝑦𝑁) 𝜕𝑥1𝑘 ⋯ 𝜕𝐸(𝑦𝑁) 𝜕𝑥𝑁𝑘 = ] (𝐼 − 𝜌𝑊)−1[ 𝛽𝑘 𝑤11𝜃𝑘 … 𝑤1𝑁𝜃𝑘 𝑤21𝜃𝑘 𝛽𝑘 … 𝑤2𝑁𝜃𝑘 ⋮ ⋮ ⋱ ⋮ 𝑤𝑁1𝜃𝑘 𝑤𝑁2𝜃𝑘 ⋯ 𝛽𝑘 ], (3)
where the diagonal elements of the last matrix on the right hand side indicate the direct effects of the exogenous variable 𝑥 on the dependent variable 𝑦 and the off-diagonal elements indicate the spatial spillover effects, of variable 𝑥 on the dependent variable 𝑦 in all other countries. One should note that the error term and possible fixed effects in time and or space drop out when calculating the marginal effects due to taking the expectations of 𝑌𝑡. The direct and indirect effects for each model are
summarized in table 2.
8 Table 2. Direct and indirect effects comparison per spatial model.
Model Direct effects Indirect effects
OLS / SEM 𝛽𝑘 0
SAR / SAC Diagonal elements of
(𝐼 − 𝜌𝑊)−1𝛽𝑘 Off-diagonal elements of (𝐼 − 𝜌𝑊)−1𝛽𝑘 SLX / SDEM 𝛽𝑘 𝜃𝑘 SDM / GNS Diagonal elements of (𝐼 − 𝜌𝑊)−1[𝛽 𝑘+ 𝑊𝜃𝑘] Off-diagonal elements of (𝐼 − 𝜌𝑊)−1[𝛽 𝑘+ 𝑊𝜃𝑘] Th e d i r ec t a n d i nd i rec t ef f ec t s of ea c h sp a t i a l mod e l. Sou rc e : Elh or st a n d Vega , 2 0 1 3 .
In recent literature most focus has been on models that include endogenous interaction effects or interaction effects amongst the error terms; these are the SAC, SAR and SEM models. However each of these last mentioned models have their limitations. While most of the spatial models are used frequently in spatial econometrics, the SLX model and SDEM models are overlooked according to Elhorst et al. (2012).
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2.2.
Spatial weights matrices
All of the spatial econometric specifications discussed so far rely on the usage of a spatial weights matrix, 𝑊, which measures the closeness between countries. Several strategies exist to create a spatial weights matrix. In his book about spatial econometrics, Elhorst (2014, p.10) describes some commonly used specifications spatial weights matrices. Examples of weight matrices are inverse distance matrices and block diagonal matrices. This study examines the impact of geographical distance using an inverse distance matrix.
Most spatial weights matrices are hard to interpret without further adjustments however. Consequently, it has become common practise to row (or column) normalize the weights matrix to unity. While row-normalising weights to unity is valid for most types of spatial weights matrices, this normalization procedure is not valid for inverse distance matrices. Elhorst, (2014, pp.11-12) gives two reasons why row-normalising an inverse distance matrix can lead to misspecification issues. Firstly, row-normalising the matrix results in an asymmetric matrix. As a result of row-normalization the distance from country 1 to country 2 is no longer the same as the distance from country 2 to country 1. The economic interpretation of distance decay is thereby lost. Secondly, centrally located countries and remotely located countries will be having the same impact. To avoid these issues, two different normalisation methods have been developed. Elhorst (2001) and Kelejian and Prucha (2010) propose to divide every element of the matrix by its largest eigenvalue. As a results largest eigenvalue will be equal to one, which is equal to the largest eigenvalue of a row-normalized matrix. A second method was
developed by Ord (1975), who normalizes 𝑊𝑟 by 𝑊𝑛= 𝐷−1/2𝑊𝑟𝐷−1/2, where 𝑊𝑟 is the original inverse
distance matrix and 𝐷 is a diagonal matrix containing the row sums of the 𝑊𝑟. The matrix created by
this method will have the same eigenvalues as the row-normalized inverse distance matrix. Both methods share the common characteristic that the mutual proportions between countries stay intact and are therefore preferred over row-normalization to unity. In this paper the normalization procedure by
Ord is adopted.3
2.3.
Estimation and testing procedures for spatial econometric models proposed in literature
Several estimation techniques can be used in spatial econometrics. Amongst them are ordinary least squares (OLS) maximum likelihood estimation (ML), instrumental variables estimations (IV), generalized method of moments (GMM) and Bayesian estimation techniques. In this paper OLS and Quasi Maximum Likelihood (QML) estimations are used to determine the direct and spillover effects
3 Note that the procedure described by Elhorst (2001) and Kelejian and Prucha (2010) can lead to different
10 for each model. Bayesian Markov Chain Monte Carlo (Bayesian MCMC) simulation techniques are used to compare non-nested models and to compare the strength of difference weight matrices.
Choosing the correct spatial econometric specification is a challenge. Several statistical tests, procedures and estimation techniques can be used. In one of the first papers about spatial econometrics, Anselin et al. (1996) propose to determine the correct specification by applying robust Lagrange Multiplier tests. Other testing procedures have since been developed, such as likelihood ratio tests and Wald-tests. The previous tests are only appropriate however when one model is nested in the other. Non-nested models can be compared using the Akaike information criterion (AIC) or Bayesian information criterion (BIC). The model which has the lowest criteria should be preferred. Another procedure to compare non-nested models is proposed by LeSage (2014) and is based on Bayesian comparison methods.
Many estimation strategies exist. Most of them are based on testing restrictions. Figure 1 in Appendix A shows how models are linked to each other and which restrictions can be tested. One approach would be to estimate the GNS model, which includes all types of spatial interaction effects. Then one can test restrictions to determine whether the GNS model can be simplified to a model with two spatial interaction effects. If this is the case, the simplified model(s) can be compared with models with only one spatial interaction effect. Extending this testing procedure, ultimately one could end up with the OLS model. This is also referred to as the general-to-specific approach. One could also impose a specific-to-general approach: start with the OLS model, then estimate the SAR, SEM and SLX model and test restrictions, etc. Due to the weak identification problem of the GNS model the top down approach is seldom used.
While the specific-to-general approach is a commonly used strategy, many other strategies can be used. For example, some strategies limit their estimation strategy to consider only a few models. As expressed above, the SAR, SAC, SEM and GNS model have their limitations. The SDM, SDEM and SLX models are more flexible however. LeSage (2014) proposes a strategy that only compares two models: The spatial Durbin model (SDM) and the spatial Durbin error model (SDEM). One should determine the Bayesian posterior model probabilities of each model and choose the model with the highest resulting probability. Another strategy proposed is given by Elhorst and Vega (2015), who take the SLX model as a departure point. One of the benefits of the SLX model is that this model can be used to test for possible endogeneity of the independent variables.
3. Model estimation
11 weights matrix. Section 3.2 shows how the unrestricted matrix will be restricted based on bilateral trade data. Section 3.3 explains the strategy used to determine the correct spatial econometric specification for each spatial weights matrix. Section 3.4 concludes with a strategy to compare the results and strengths of each spatial weights matrix used.
3.1.
Unrestricted spatial weights matrix
The inverse distance spatial weights matrix used to determine spatial spillover effects amongst stock sections is derived in this section. As explained in the literature review, spatial weights matrices are normalized for reasons of interpretability. Which normalization technique should be used depends on the data at hand.
To obtain the inverse distance matrix, 𝑊𝐷, the matrix containing the geographical distances between
capital cities, 𝑊𝐺 must be inversed. For each element the following transformation takes place: 𝜔𝑖𝑗,𝐷=
1/𝜔𝑖𝑗,𝐺, where 𝜔𝑖𝑗,𝐷 describes the individual elements of the inversed matrix and 𝜔𝑖𝑗,𝐺 describes the
elements of the geographical distance matrix. Following, the matrix is normalized using the normalization procedure created by Ord (1975). See Section 2.2 for a description of this procedure.
3.2.
Restricted spatial weights matrices
The restricted spatial weights matrices are created based on one of the results found by Asgharian et al. (2013), who state that a bilateral trade weight matrix outperforms an inverse distance weight matrix. Therefore restrictions will be put on the spatial weights matrix by looking at bilateral trade relations amongst countries.
A weights matrix based on bilateral trade relations is created for this purpose. Yearly import and yearly export data are averaged over the 15 years of the data used. These averages of import and export data between countries are used to compute spatial weights between countries, describing the bilateral trade relations. Section 4.2 describes the estimation technique of the bilateral trade matrix in more detail.
Because each row of the bilateral trade matrix can be adjusted separately without loss of interpretation, a row-normalization to unity is applied. The resulting matrix is referred to in this paper
as 𝑊𝐵𝑇. To apply restrictions on the inverse distance matrix, the bilateral trade matrix is analysed. Two
strategies are adopted.
1. For each element 𝜔𝑖𝑗,𝐵𝑇 of the bilateral trade weights matrix 𝑊𝐵𝑇 that is smaller than 0.05, the
corresponding element in 𝜔𝑖𝑗,𝐷 is put to zero. After normalizing the matrix using the same
procedure as in Section 3.1, the resulting matrix is referred to as 𝑊𝑅𝛼. Where R stands for
12 2. While some countries depend a lot on imports and exports with other countries; other countries are less dependent on trade. To account for this a relative measure of trade dependencies is created by determining the ratio of total trade over GDP for each country.
Similar to strategy one, restrictions are placed on 𝑊𝐷 based on the bilateral trade matrix
𝑊𝐵𝑇. However, instead of using 0.05 as a reference weight for each element a new vector, 𝑅𝑅,
containing country specific reference weights is used. These country based reference weights are created by multiplying the reference weight by a vector, 𝑅𝑁, which measures how trade
dependent each country is.
The steps taken to create vector 𝑅𝑁 are described below:
1. Firstly, the trade dependency of country 𝑖 is calculated as the sum of all imports and
exports of country 𝑖 with all other countries 𝑗𝑗≠𝑖 divided by the average yearly GDP of
country 𝑖. The resulting weights are a raw measure for trade dependencies.
2. Secondly, each individual weight is divided by the average trade dependence weight of
all countries. This way the median country will have a weight of one, whereas countries that hardly trade at all with have a weight close to zero.
3. Thirdly, all weights are clustered into groups of 0.25, such that a country with an
individual weight of 0.10 is assigned a weight factor of 0.25; whereas a country with an individual weight of 1.20 is assigned a weight factor of 1.25.
4. Lastly, each weight is inversed. The inversion of the weights enables the correct
interpretability of the spatial weights matrix. This way countries that are very trade dependent will obtain a low reference weight, while countries that hardly trade obtain a large reference weight.
Table 3 summarizes the reference weights 𝑅𝑅 and 𝑅𝑁, their frequency and the resulting
reference weights used to restrict the inverse distance matrix. The resulting matrix is referred to
as 𝑊𝐺𝛼. Where G stands restrictions based on GDP adjusted bilateral trade relations and α is the
cut-off point used to place restrictions.
Table 3. Summary statistics of reference weight vectors 𝑅𝑅 and 𝑅𝑁.
Weights 𝑹𝑹
Frequency Reference weights Inverse weights (𝑹𝑵) For α = 0.05 For α = 0.15
3 0.25 4 0.200 0.600 3 0.50 2 0.100 0.300 4 0.75 1.33 0.067 0.200 4 1 1 0.050 0.150 3 1.25 0.80 0.040 0.120 4 1.50 0.67 0.033 0.100 2 2 0.50 0.025 0.075 1 3.75 0.27 0.013 0.040
13 Table 3 shows that one country in particular is very trade dependent, namely Belgium. This country has a very high level of trade relative to its GDP. Belgium is surrounded by many large economies, such as Germany, France and the UK. Because the population of Belgium is relatively small and a lot of trade opportunities exist in neighboring countries the relatively high trade dependency is not that strange. As a result of the high trade dependency, Belgium will only restrict inverse distance relations with coefficients smaller than 0.013 (when the cut-off point is 5 percent). As a comparison, the USA is not very trade dependent. For the USA only inverse distance relations that are larger than 0.20 are not set to zero.
3.3.
Determination of spatial econometric specification
Before the effects of the unrestricted model can be compared with the other models a strategy is used to determine the correct specification of each model. Firstly the correct spatial econometric specification will be deduced by using maximum likelihood and OLS estimations in Stata. This is done by means of testing restrictions on ρ, θ and λ. For each spatial weights matrix the types of spatial interaction effects are determined and used to compute the direct and indirect and it is determined
whether the spillover effects are global or local spillovers.4 Secondly, by means of information criteria
and Markov Chain Monte Carlo simulations the strengths of the weight matrices are compared. The estimation strategy is explained in more detail below in Section 3.4.
To compute the direct and spillover effects amongst stock returns it is necessary to determine the correct spatial econometric specification. To derive this specification a specific-to-general approach is used. By testing restrictions for ρ, θ and λ the optimal model will be derived. Figure 1 in Appendix A describes how models are related to each other and which tests can be performed to compare a model with a model that is nested in the former model.
Testing can be done in three steps. Step 1 compares models without spatial interaction effects (OLS), with models containing one type of spatial interaction effect (SAR, SLX and SEM). Step 2 compares models with one interaction effect with models with two interaction effects (SAC, SDM and SDEM). The analysis used in this paper always starts with step 1 and possibly extends to step 2. Only when step 2 leads to inconclusive evidence for one of the specifications, step 3 is used. The procedure is described in further detail below. Note that models with two interaction effects will not be compared with the model that contains all types of interaction effects (GNS), because this model suffers from over parameterization problems.
14
Step 1
Step 1 compares models with one type of interaction effects with the OLS model. Firstly, the SAR, SLX and SEM models are estimated. For each of these models model, the corresponding restriction is tested according to figure 1 in Appendix A. For example, after estimating the SAR model, restriction ρ = 0 is tested. If the null hypothesis of ρ = 0 can be rejected, the SAR model should be preferred over the OLS model. Similar analysis is done for the SLX and the SEM model. If the OLS model is preferred over all models that include one type of spatial interaction effect, the OLS model is the correct specification and step 2 will not be executed. If one or more models with one type of spatial interaction effects are preferred over OLS, step 2 follows.
Step 2
Step 2 compares the models with two types of interaction effects that nest one or multiple models with one type of interaction effects that have been found significantly preferred over OLS in step 1. For example, if the SAR model is found significant in step one, the SAC and SDM model will be estimated. For both the SAC and SDM model all possible restrictions will be tested. Note that even if in step 1 the SLX model is found to be insignificant, the combination of exogenous interaction effects and endogenous interaction effects in the SDM model can still be significant. By testing restrictions on the SAC model it is determined whether the model can be simplified to the SAR and SEM model; similarly it will be tested whether the SDM model should be simplified to the SAR, SLX or SEM model. The results of these tests can lead to two conclusions. Firstly, it is possible that the tests in step 1 and step 2 confirm that one model cannot be rejected in favor of another model, while all other models can be rejected. In this case there is significant evidence for the not-rejected model, which is preferred over all other models. It is also possible that more than one model cannot be rejected. In this case there is inconclusive evidence and step 3 follows.
Step 3
To compare the models that have not been rejected in step 1 and 2 AIC and BIC criterions are used. The model with the lower AIC or BIC criterion is preferred. If these criterions do not provide enough evidence for one or the other model Bayesian model comparison techniques will be used. By estimating both models using Markov Chain Monte Carlo simulations, Bayesian posterior model probabilities can be calculated (also referred to as Bayesian MCMC). The model with the highest posterior model probability is the preferred model.
3.4.
Comparison of restricted and unrestricted spatial weights matrices
15 previous sections is preferred. By comparing the strengths of the resulting models for each spatial weights matrix based on a combination of statistical tests and economic reasoning one spatial weights matrix is found to be superior to all others. The results of this model are used to determine whether geographical distance still matters.
Whenever possible, statistical techniques are used to compare the models derived from each spatial weights matrix. Following Elhorst (2015), Bayesian MCMC techniques can be used to compare spatial weights matrices, given that the preferred spatial econometric specification is either an SDEM, SDM or SLX model. Following the results of the Bayesian MCMC analysis used to compare non-nested models described in Section 3.3 step 3, the marginal likelihoods can also be used to compute Bayesian posterior model probabilities for the spatial models of several spatial weights matrices. The weighting matrix that is used to develop the spatial econometric model with the highest posterior probability is the preferred weighting matrix.
If the results of the statistical tests are inconclusive or the tests cannot be used due to lack of statistical routines, a combination of careful examination of the estimation results produced by the spatial weights matrices and inspection of the weaknesses and limitations for each weight model specification will be used to argue which of the models is preferred.
4. Data
This section presents the data that is used to examine the impact of geographical distance on spatial spillovers amongst stock markets. Firstly, Section 4.1 describes the computation and choice of the dependent and independent variables. Secondly, in Section 4.2 the data used for estimating the spatial weights matrices is put forward. Lastly, Section 4.3 describes the data used for the empirical analysis.
4.1.
Data
This section describes the dependent and independent variables used in this research. The data used to compute the variables and weight matrices in this research are summarized in table 1.
The dependent variable measures monthly returns of the main equity indices in each country. Lognormal returns are calculated from monthly index values as described in table 1.
In previous literature several economic variables have been found to have a significant effect on stock market returns. The control variables used in this study are changes in the index of industrial production, unexpected inflation, changes in a relative competitiveness measure based on prices and exchange rates, changes in interest rates and lagged stock returns.
16 market returns can be predicted by looking at changes in current levels of industrial production, this is why lagged changes in the index of industrial production are used. Benaković and Posedel (2010) find a significant positive relation between industrial production and stock returns. It is expected that an increase in the index of industrial production leads to an increase in the returns of equity indices in the next period.
Table 1. Data sources and transformations.
Basic series
Symbol Variable Source
𝐌𝐈𝐭 Market index values of main stock
indices
Morgan Stanley Capital International (MSCI)
𝐈𝐈𝐏𝐭 Index of Industrial Production OECD Main Economic Indicators (MEI)
𝐂𝐏𝐈𝐭 Consumer Price Index OECD Main Economic Indicators (MEI)
𝐈𝐑𝐭 Short term interest rates OECD Main Economic Indicators (MEI)
𝐑𝐄𝐋𝐀𝐭 Relative competitiveness measure OECD Main Economic Indicators (MEI)
𝐏𝐃𝐢𝐣 Geographical distance between capital
cities
Centre d’Etudes Prospectives et
d’Informations Internationales (CEPII)
𝐈𝐌𝐏𝐢𝐣 Average import STAN Bilateral Trade Database (OECD)
𝐄𝐗𝐏𝐢𝐣 Average export STAN Bilateral Trade Database (OECD)
𝐆𝐃𝐏𝐭 Yearly GDP OECD Main Economic Indicators (MEI)
Transformed series
Symbol Variable Data transformation
Dependent
𝐒𝐑𝐭 Monthly return of the market index SRt = ln(MIt/MIt−12)
Independent
𝐈𝐍𝐅𝐭 Annual change in CPI INFt = ln(CPIt/CPIt−12)
𝐔𝐈𝐍𝐅𝐭 Unexpected inflation at time t UINFt = INFt-INFt−12
𝐂𝐈𝐈𝐏𝐭 Annual change in IIP CIIPt = ln(IIPt/IIPt−12)
𝐈𝐍𝐓𝐭 Monthly change in IR IRt = ln(IRt/IRt−1)
𝐂𝐎𝐌𝐏𝐭 Monthly change in RELA COMPt = ln(RELAt/RELAt−1)
Spatial weights
𝐈𝐃𝐢𝐣 Inverse distance weights IDij=
1 PDij
𝐁𝐓𝐢𝐣 Bilateral trade weights BTij =
IMPij+ EXPij ∑k=j,j≠ik=1 (IMPij+EXPij)
17 Inflation is calculated as the change in monthly CPI levels in one year compared to the change in monthly CPI levels in the previous year. Furthermore, expected inflation of month 𝑡 is set equal to the inflation level of twelve months earlier. See table 1 for the computation of unexpected inflation.
A third control variable is a relative competitiveness index, COMP, which weighs relative consumer prices and unit labour cost for the overall economy of one country based on the structure of competition
in both export and import markets of the goods sector with other countries.5 An increase in this index
corresponds to a real appreciation of the countries’ competitive position with respect to other countries. As a result, it is expected that an increase in the relative competitiveness index will be accompanied by increasing stock returns.
A fourth control variable is measured by a change in short-term interest rates (INT). An increase in interest rates impact a country’s economic performance negatively in three ways. Firstly, economic theory suggests that when interests increase the present values of future cash flows decrease (see for example Chen et al., 1986); thereby making it less attractive for investors to invest in the stock markets. Another negative effect of an increase in interest rates is the ability for companies to attract capital. When interest rates are high, the cost of capital increases. Thirdly, consumers will save more when interest rates increase and consume less. Because of the increased cost of capital and the decrease in consumption the stock market will be negatively affected. The empirical literature however is mixed on the sign and magnitude of the effect of changes in interest rates on stock returns. For example, while Binti (2011) find a significant negative effect of changes in interest rates on stock returns; Benaković and Posedel (2010) find a positive effect of changes in 3-month interbank interest rates on stock returns. The interest rates used in this study are three month interbank offer rates, but can also be the rate of treasury bills or certificates of deposits with a maturity of three months, depending on the data obtained from the central bank of a country. The 3-month European Interbank Offered Rate is used for euro area countries. This last property could become a potential problem, as these interest rates will be spatially dependent.
Lastly, due to the large amount of time series data in the panel data set used in this study one must also correct for possible auto correlation in returns and spatio-temporal interdependencies amongst stock returns Asgharian et al. (2013). This problem is illustrated by Rapach et al. (2013), who show that lagged US stock returns contains strong predictive power over US stock returns and stock returns of other industrialised countries. To take care of this problem stock returns lagged by one month are used.
18
4.2.
The spatial weights matrix
To compute the spatial spillover effects a spatial weights matrix is used. This matrix is an NxN matrix describing the relationship between individual countries. The rows of the matrix can be
interpreted as the way country 𝑖 is affected by all other countries 𝑗𝑗≠𝑖, whereas the columns describe
how country 𝑖 affects all other countries 𝑗𝑗≠𝑖. By definition all diagonal elements are zero.
In this paper two spatial weights matrices are computed. Firstly a matrix based on geographical distance between capital cities of countries is used. Countries that are geographically close are assumed to have a larger impact on each other than countries that are situated far away from each other. To translate these relations into relative weights between countries inverse distances are taken. This way neighbouring countries will have a value close to one; while countries that are located far from each other will have a value close to zero.
The second spatial weights matrix is based on bilateral trade data between countries. Bilateral trade data captures the relations between two countries through their levels of import and export. It is expected that countries that trade a lot with each other have more spatial spillover effects than countries that hardly trade with each other. The average level of import (IMP) and the average level of export (EXP) between the countries analysed over 1999 up to 2014 are used to create the spatial weights matrix.
Bilateral trade (BT) between country 𝑖 and country 𝑗 is then calculated as: BTij =
IMPij+ EXPij
∑k=j,j≠ik=1 (IMPij+EXPij).
Due to this definition all weights of each row add up to one and the relations between countries can be easily interpreted. Restrictions can be easily imposed due to this useful property.
4.3.
Data description
Panel data is used for the empirical analysis. It consists of monthly data for 24 countries from
February 1999 up to December 2014.6 The econometric software package used requires a balanced panel
data. To obtain a strongly balanced dataset some 78 missing data points were imputed. Imputing data can be time-consuming as it hard to justify and determine the proper imputation technology. Because the number of missing observations is small relative to the total number of observations of 4582, missing data is imputed using simple forms of inter- and extrapolation. Imputations for missing data between observations is done through interpolation of its closest values (mostly the case for interest rate data). Imputations for missing values at the start or the end of the time frame are done based on the weights found by the bilateral trade weight matrix using a weighted extrapolation techniques. For each missing
6 Data from February 1997 up to December 2014 is used to compute all variables. However, due to the data
19 variable two returns based on the two countries with the highest weights are combined into one return rate. This custom rate is used to extrapolate these missing values. Table A.2 in Appendix A summarizes the missing observations and describes the data imputations techniques.
Table 4. Descriptive statistics for each country. Mean and standard deviations for each variable.
Country SR COMP INT UINF IIP Mean Sd Mean Sd Mean Sd Mean Sd Mean Sd
Belgium -0.001 0.072 0.000 0.007 -0.019 0.096 0.000 0.019 0.026 0.054 Canada 0.005 0.056 0.001 0.016 -0.008 0.077 0.001 0.015 0.009 0.049 Chile 0.007 0.053 -0.001 0.022 -0.006 0.130 0.000 0.032 0.022 0.053 China 0.005 0.092 0.001 0.014 0.001 0.151 0.002 0.030 -0.004 0.047 Denmark 0.007 0.064 0.000 0.007 -0.013 0.075 -0.001 0.011 0.000 0.059 Finland 0.000 0.099 -0.001 0.008 -0.019 0.096 0.000 0.017 0.011 0.073 France 0.001 0.064 0.000 0.007 -0.019 0.096 0.000 0.011 -0.005 0.048 Germany 0.001 0.074 -0.001 0.009 -0.019 0.096 0.000 0.010 0.017 0.071 Hungary 0.002 0.088 0.001 0.020 -0.011 0.076 -0.008 0.025 0.046 0.087 Indonesia 0.012 0.091 0.002 0.034 -0.009 0.048 -0.026 0.124 0.033 0.056 Ireland -0.005 0.072 0.000 0.010 -0.019 0.096 -0.001 0.031 0.046 0.087 Israel 0.004 0.063 0.000 0.015 -0.021 0.121 -0.003 0.032 0.027 0.059 Italy -0.003 0.072 0.000 0.007 -0.019 0.096 -0.001 0.011 -0.013 0.065 Mexico 0.012 0.066 0.001 0.022 -0.012 0.059 -0.007 0.020 0.014 0.030 Netherlands 0.001 0.069 0.000 0.008 -0.019 0.096 -0.001 0.010 0.010 0.045 Norway 0.005 0.075 0.000 0.014 -0.008 0.055 0.000 0.018 -0.011 0.047 Poland 0.003 0.075 0.001 0.022 -0.009 0.046 -0.007 0.024 0.051 0.057 Portugal -0.006 0.059 0.000 0.006 -0.019 0.096 -0.002 0.017 -0.008 0.044 Russia 0.013 0.122 0.004 0.021 -0.005 0.169 -0.012 0.184 0.036 0.060 South Africa 0.011 0.057 -0.001 0.033 -0.005 0.034 -0.001 0.041 0.012 0.056 Spain 0.001 0.070 0.000 0.008 -0.019 0.096 -0.001 0.018 -0.011 0.060 Sweden 0.005 0.075 -0.001 0.014 -0.021 0.163 0.000 0.017 0.002 0.070 United Kingdom 0.001 0.053 -0.001 0.014 -0.013 0.066 0.000 0.011 -0.007 0.032 United States 0.003 0.055 -0.001 0.012 -0.018 0.124 0.000 0.019 0.012 0.046 Average 0.003 0.072 0.000 0.015 -0.014 0.094 -0.003 0.031 0.013 0.057
20 standard deviations. While interest rates fluctuate a lot, as can be seen from the high average standard deviation of 9.4 percent per month, interest rates have shown a clear decreasing trend during most of the months of the 15 years analyzed. Even though the mean of unexpected inflation is close to zero, Indonesia is a distinct outlier, which dealt with on average 0.026 points of unexpected inflation. Another country that also has a large value for unexpected inflation is Russia. These high levels of unexpected inflation go along with high standard deviations; whereas most countries have a standard deviation of about 3.1 percent; Russia and Indonesia have a significantly larger standard deviation (18.4 percent and 12.4 percent respectively). The change in industrial production is on average 1.3 percent per month. Poland, Ireland and Hungary have relatively high growth rates in industrial production, whereas Italy, Norway and Spain have decreasing rates of industrial production. While these high growth rates are accompanied by high volatility in industrial production values for Ireland and Hungary, this is not the case for Poland.
Fixed versus random effects
Table 5. Determination of fixed or random effects.
Random versus Fixed effects
Robust hausman test comparing fixed with random effects 𝐻0: difference in coefficients not systematic
Chi²(10) 22.08
P-value 0.0147
Fixed effects specification
Time fixed effects
𝐻0: fixed effects are jointly insignificant
F(5, 190) 3295.85
P-value 0.0000
Spatial fixed effects
𝐻0: fixed effects are jointly insignificant
F(23, 190) 2.45
P-value 0.0005
Time and spatial fixed effects
𝐻0: fixed effects are jointly insignificant
F(28, 190) 1033.61
P-value 0.0000
21 preferred. Table 5 also shows that both time and spatial fixed effects should be adopted. The F-test testing the hypothesis that all time (spatial) dummies are jointly insignificant is strongly rejected.
Besides from statistical reasons, the addition of both spatial and time fixed effects is also intuitively appealing. Countries are different from each other in many ways. Compare for example Russia and the Netherlands. The differences in culture, politics and levels of wealth are large. Because countries are so different in many ways, it is also to be expected that a change in interest rates for the Netherlands will not have the same effect on stock returns as a similar change in interest rates on stock returns in Russia. Spatial fixed effect dummies account for these differences between countries. The inclusion of time dummies can be explained by the fact that the world is constantly changing. Stock markets in January 2000 reacted differently than stock markets in March 2008. To account for these differences over time, dummies for each month (except for one) are used.
Heteroskedasticity
Heteroskedasticity can be a serious problem in panel data analysis. When a group of variables have differing variances for each subject the variables are referred to as heteroskedastic. To determine the existence of heteroskedasticity a group wise heteroskedasticity test is performed (table 6). As becomes clear from the table, the null hypothesis of homoskedasticity is soundly rejected. Consequently, heteroskedasticity is a serious problem.
Table 6. Group wise heteroskedasticity test.
Group wise heteroskedasticity
𝐻0: homoskedasticity
Chi² 1813.47
P-value 0.0000
Spatial autocorrelation
22 Table 7. Moran’s I tests for spatial autocorrelation.
Moran's I
𝐻0: Residuals are spatially autocorrelated
Rejected 37 Average p-value 0.2151
Not rejected 154
Total 191
Non-normality of data
Most of the spatial models used in this paper (SAC, SAR, SDM, SEM and SDEM) are estimated using maximum likelihood. One of the assumptions underlying ML estimations is asymptotic normality. To verify whether the residuals are normally distributed a kernel density plot (figure 2) is used. This graph compares the distribution of the residuals found in the estimations with the normal distribution. As becomes clear from figure 2, the residuals are definitely not normally distributed.
Figure 2. Kernel density plot.
To deal with the econometric problems of non-normality, spatial autocorrelation and heteroskedasticity the standard errors are adjusted. Estimates based on robust standard errors, also referred to as clustered sandwich estimates, can be used for this purpose. For example, these robust standard errors be used to perform Quasi Maximum Likelihood (QML) estimations. The QML estimation is similar to the ML estimation, but differs in the computation of the covariance matrices that are used for the estimates. Whereas the covariance matrices of the estimates under ML are based on
0 2 4 6 8 D e n sit y -1 -.5 0 .5 e[CountryID,t]
Kernel density estimate Normal density
kernel = epanechnikov, bandwidth = 0.0095
23 asymptotic normality, the covariance matrices for QML can be computed using other distributions. As a result, QML estimation is used to produce consistent estimates.
Robust standard errors are also used to deal with problems such as the lack of independence of observations and heteroskedasticity of the data. One of the most commonly used robust standard error estimates are the Huber-White estimates (Huber, 1967; White, 1982, 1980). These standard errors resolve the issue of heteroskedasticity and are most appropriate when the residuals are spatially and temporally independent (Hoechle, 2007). To deal with the problem of spatially dependent residuals, Driscoll and Kraay (1998) adjust the standard errors in a different way. In a study comparing Huber-White estimates with Driscoll and Kraay estimates for panel regressions, Hoechle (2007) shows that when residuals are cross-sectionally dependent Driscoll and Kraay estimators should be used. Because of the existence of both spatial and temporal dependence, Driscoll and Kraay standard errors will be used in this study.
5. Analysis
To be able to answer the research question the results are structured into four subsections. In Section 5.1 the unrestricted inverse distance model is created and analysed. This model examines whether spillover effects amongst stock markets can be explained by geographical distance relations between countries by using an unrestricted inverse distance matrix. The unrestricted model is used to showcase the differences between the various spatial econometric specifications and the process of determining the correct spatial econometric specification. Furthermore, the strength of the explanatory variables are analysed. Section 5.2 determines the impact of geographical distance more carefully, by restricting some of the weights of the spatial weights matrix to zero. The restrictions are placed based on data from bilateral trade relations. For each of the restricted spatial weights matrices the correct spatial econometric specification(s) is determined. Section 5.3 compares the results and the strength of each of the restricted and unrestricted models analysed in Sections 5.1 and 5.2. Lastly, Section 5.4 concludes by formulating an answer to the research question and by providing some suggestions for further research.
5.1.
Results obtained with the unrestricted weighting matrix
24 and magnitude of the interest rate estimate is as expected, an increase in interest rates is followed by a decrease in stock returns. On the contrary, the sign of unexpected inflation is not as expected. An increase in unexpected inflation leads to an increase in stock returns. Lastly, the lagged change in the index of industrial production does not seem to influence stock returns in any of the models and is
therefore excluded from the model.7
Table 8. Estimation results.
OLS SAR SLX SEM SAC SDM SDEM
LSR -0.054* -0.054* -0.052* -0.055* -0.054* -0.053* -0.053* t-stat -1.79 -1.86 -1.80 -1.86 -1.85 -1.88 -1.88 COMP -0.137 -0.135 -0.135* -0.136 -0.136* -0.134 -0.135 t-stat -2.00 -2.01 -1.97 -2.03 -1.96 -1.99 -2.01 LINT -0.055 -0.055 -0.058 -0.055 -0.055 -0.058 -0.058 t-stat -3.11 -3.19 -3.07 -3.20 -3.20 -3.17 -3.15 LUINF 0.069 0.068 0.068 0.068 0.068 0.068 0.067 t-stat 2.57 2.57 2.57 2.59 2.58 2.61 2.6 CIIP 0.015** 0.016** 0.012** 0.016** 0.016** 0.013** 0.013** t-stat 0.67 0.71 0.52 0.72 0.69 0.57 0.58 W * SR 0.300** 0.180** 0.300** t-stat 1.09 0.86 1.09 W * LSR -0.072** -0.064** -0.071** t-stat -0.78 -0.78 -0.78 W * COMP -0.409** -0.396** -0.448** t-stat -0.81 -0.86 -0.90 W * LINT 0.079** 0.079** 0.079** t-stat 1.32 1.45 1.36 W * LUINF -0.072** -0.052** -0.049** t-stat -0.18 -0.14 -0.13 W * CIIP -0.003** -0.008** -0.006** t-stat -0.03 -0.12 -0.07 W * u 0.177 0.220** 0.171 t-stat 2.45 0.83 2.45 R² 0.588 0.697 0.589 0.687 0.692 0.455 0.470 AIC 14.012 14.012 16.011 24.012 24.012 BIC 59.024 59.024 67.454 101.176 10.118 Log-likelihood 7477.8 7479.6 7480.2 7481.6 7483.5
Th e O LS a n d S LX m od e l a r e es t i ma t ed u si n g OLS, wh e rea s a l l ot h er mod e ls a r e e st i ma t ed b y QM L. Ti me a n d sp a c e fi x ed ef f ec t s a r e i n c lu d ed . ** In si gn i fi c a n t at 0 .1 0 . * In si gn i fi c a n t a t 0 . 10 , bu t si gn i fi c an t a t t h e 0 . 0 5 lev e l.
7 The results described by tables 8-10 do not change significantly when excluding the variable CIIP. Table A.3 in
25 The other coefficients in table 8 are estimated results of the spatial interaction effects. Elhorst and Vega (2015) note however that these estimates cannot be compared between certain models due to the difference in nature of the models (global or local) and due to possible feedback effects that exist in models that contain endogenous interaction effects (WY). Feedback effects exist in the SAR, SAC and SDM model because the effects on the dependent variable are passed on to other countries and come back to the original country through the endogenous interaction effect. The direct and indirect estimates account for these feedback effects and were derived by LeSage and Pace (2009, p.39) and given in table 9. The feedback effects can be calculated as the difference between the estimated results in table 8 and the direct and indirect effects in table 9. In most cases the feedback effects are very small. For example, the largest feedback amongst the direct effects effect is for COMP: -0.148 - -0.134 = -0.014. The largest feedback effect for the indirect effects (spillover effects) is a lot larger: This feedback effect is also for a change in the competitiveness measure (COMP): -0.311 + 0.396 = -0.085.
Table 9. Direct and indirect effects.
OLS SAR SLX SEM SAC SDM SDEM
Direct effects LSR -0.054* -0.055* -0.052* -0.055* -0.054* -0.054* -0.053* t-stat -1.79 -1.85 -1.80 -1.86 -1.85 -1.89 -1.88 COMP -0.137 -0.144 -0.135* -0.136 -0.136* -0.148 -0.135 t-stat -2.00 -2.49 -1.97 -2.03 -1.96 -2.42 -2.01 LINT -0.055 -0.055 -0.058 -0.055 -0.055 -0.057 -0.058 t-stat -3.11 -3.06 -3.07 -3.20 -3.20 -3.02 -3.15 LUINF 0.069 0.070 0.068 0.068 0.068 0.070 0.067 t-stat 2.57 2.34 2.57 2.59 2.58 2.29 2.60 CIIP 0.015** 0.012** 0.012** 0.016** 0.016** 0.008** 0.013** t-stat 0.67 0.45 0.52 0.72 0.69 0.32 0.58 Indirect effects LSR -0.013** -0.072** -0.008** -0.056** -0.071** t-stat -0.73 -0.78 -0.12 -0.95 -0.78 COMP -0.038** -0.409** -0.030** -0.311** -0.448** t-stat -0.85 -0.81 -0.16 -0.85 -0.90 LINT -0.013** 0.079** -0.008** 0.034** 0.079** t-stat -0.88 1.32 -0.15 0.95 1.36 LUINF 0.015** -0.072** 0.009** 0.005** -0.049** t-stat 0.81 -0.18 0.13 0.02 -0.13 CIIP 0.004** -0.003** 0.002** -0.001** -0.006** t-stat 0.39 -0.03 0.07 -0.02 -0.07 Se e t h e n ot e o f t a b l e 8 . ** In si gn i fi c a n t a t 0 . 10 . * In si gn i fi c a n t a t 0 .1 0 , b u t si gn i fi ca n t at t h e 0 . 0 5 lev el.
26 they are very different from the estimates found by the SLX, SDM and SDEM models. The SAC and SAR suffer from their inflexibility. As explained in the literature section, the ratio of direct/indirect effects is equal by construction. As a consequence, the indirect effects predicted by the SAC and SAR model are unreliable. The spillover effects amongst the SLX and SDEM are very similar, whereas the magnitudes of the SDM model estimates are further away from the two prior models. This is most likely due to the feedback effects that affect the SDM model, but are not present in the SLX and SDEM model. Table 10. Determination of the model specification.
Model: 𝑾𝑫
Strategy Wald-test Test statistic Test-value p-value Result Step 1 OLS, compared with SAR, SLX and SEM
SAR ρ=0 Chi²(1) 1.19 0.2745 OLS
SLX θ=0 F(5, 190) 0.72 0.6063 OLS
SEM λ=0 Chi²(1) 6.01 0.0143 SEM
SEM
Step 2
SAC, compared with SAR and SEM
SAR λ=0 Chi²(1) 0.05 0.8284 SAR
SEM ρ=0 Chi²(1) 0.03 0.8579 SEM
SEM* SDM, compared with SLX, SEM and SAR
SAR θ=0 Chi²(5) 3.83 0.5745 SAR
SLX ρ=0 Chi²(1) 1.19 0.2752 SLX
SEM θ=-ρβ Chi²(5) 3.66 0.5993 SEM
SEM* SDEM, compared with SLX and SEM
SLX λ=0 Chi²(1) 5.99 0.0143 SDEM
SEM θ=0 Chi²(5) 3.87 0.5685 SEM
SEM
Resulting model: SEM
Se e t h e n ot e o f t a b l e 8 . Fu rt h e r mor e a s i gn i fi c a n c e l e v el o f 0 . 0 5 i s u sed . *Th e S AR mod e l h a s a lr ea d y b e en e xc lu d ed a s a p ossi b i li t y i n st ep 1 .
27 Step 2 estimates the SAC, SDM and SDEM models, models containing two types of spatial interaction effects. The tests confirm that the these last three models can always be simplified to the SEM model and sometimes also be simplified to the SAR or SLX model. But as the SAR and SLX model have been outperformed by OLS in step 1 and the SEM model is preferred over the OLS model, the SEM model is the only spatial model which cannot rejected in favor of any of the other models. As a result, there is strong evidence that the SEM model is the correct model.
Even though the SEM model allows for spatial interaction effect amongst the error terms, it does not allow for the determination of spatial spillover effects. Without placing any restrictions on the spatial weights matrix, no spatial spillover effects amongst stock markets exist. This is contrary to the result found by Asgharian et al. (2013), who found significant spillover effects amongst stock markets when using a geographical distance based weight matrix. They did not specify their usage of standard errors however. The difference in this result is most likely due to the usage of different robust standard errors. As shown in Appendix B, adopting Newey-West standard errors leads to significant different results compared to the results obtained by Driscoll and Kraay standard errors.
5.2.
Results obtained with the restricted weighting matrices
To determine whether geographical distance still matters when deriving the spillover effects amongst stock markets, restrictions will be placed on the spatial weights matrix. Two different methodologies based on bilateral trade data are used to restrict the inverse distance matrix. Strategy one uses raw bilateral trade data between other countries, while strategy two adjusts these bilateral trade relations matrix based on a measure for trade dependency of a country.
Table 11 compares the restrictions used for each of the weights matrices for strategy 1 and 2 as well as for two different cut-off points. As can be seen strategy 1 and 2 impose many similar restrictions on the inverse distance matrix when a cut-off point of 5 percent is taken. This changes a lot however when a cut-off point of 15 percent is taken; almost half of the cells left unrestricted are different for strategy 1 (45 percent) and strategy 2 (43.6 percent). This suggests that the results for strategy 1 and strategy 2 will be more similar when a small cut-off point is taken and differ more when a larger weight based on bilateral trade is required for cells to remain unrestricted.
Table 11. Summary of unrestricted cells per strategy.
α = 0.05 α = 0.15
Statistic Strategy 1 Strategy 2 Strategy 1 Strategy 2
# unrestricted cells 151 145 40 39
# overlap unrestricted cells 124 124 22 22
# different unrestricted cells 27 21 18 17
Percentage difference 17.9 14.5 45 43.6
Percentage of unrestricted cells out of the total
number of observations 28.6 27.4 7.6 7.4
28
A full comparison of the restrictions placed for α = 0.05 is given by table C.1 in the Appendix. It
verifies the trade dependencies found. Take Belgium for example, 6 more countries are left unrestricted in strategy 2 compared to strategy 1, which verifies that Belgium relies a lot on bilateral trade.
To determine the correct model specification and compare the resulting specifications a similar analysis is done as in Section 5.2, however only the relevant models and results will be described. Tables C.2, C.3 and C.4 in Appendix C describe the model determination process, coefficient estimates and direct and indirect effects respectively. Note that the lagged change in industrial production is dropped
from the model due to the lack of explanatory power.8
Table C.2 in the Appendix summarizes the model specification process for both strategies when two different cut-off points are used. The table shows that for each of the four spatial weight matrices analysed two spatial econometric specifications are preferred over all other models. For both strategy 1 models and the strategy 2 model with a cut-off point of 5 percent the SDM and SDEM outperform all other models. For the strategy 2 model based on the 15 percent cut-off point the SEM and SAR outperform all other models. Both the SDM and SDEM models as well as the SEM and SAR models are non-nested and consequently step 3 will be carried out to determine the best model.
The first option when comparing non-nested models is to look at the AIC and BIC criteria. Table C.3 in Appendix C lists the information criteria for each of the non-nested models. For each of the four spatial weights matrices, the two models that are preferred over all other models have the same information criteria. So information criteria do not provide conclusive evidence in favor of any of the non-nested models.
Following LeSage (2014) and Elhorst and Vega (2015), Bayesian techniques are used to compare the non-nested models. This model comparison strategy relies on the examination of Bayesian posterior model probabilities. These probabilities are created by using Markov Chain Monte Carlo simulations of the SDM and SDEM models. A Matlab routine that compares the SDM, SDEM and SLX models used by Elhorst and Vega (2015) is used to compare the Bayesian posterior model probabilities of the SDM
and SDEM models.9 The model with the highest posterior model probability should be preferred. While
it is possible to compare the SDM and SDEM models using this technique, no routine is available to
compare the SAR and SEM model however, so the optimal model for 𝑊𝐺15 cannot be determined by
making use of Bayesian techniques. The results of the comparison between the SDM and SDEM models
8 The analysis in this section has also been done with CIIP included. The results showed that similar to Section
5.2 the coefficient estimates of CIIP are statistically insignificant.
9 The Matlab routine used was originally created by LeSage (2014) and able to compare the SDM and SDEM
29
for the restricted model estimations using spatial weights matrices 𝑊𝑅5, 𝑊𝑅15 and 𝑊𝐺5 are summarized
in table 12.
Table 12. Model determination by means of Bayesian techniques.
Model Statistic SDM SDEM
𝑾𝑹𝟓 log marginal likelihood 5503.20 5504.45
model probabilities 0.2234 0.7766
𝑾𝑹𝟏𝟓 log marginal likelihood 5469.94 5469.83
model probabilities 0.5271 0.4729
𝑾𝑮𝟓 log marginal likelihood 5503.77 5505.22
model probabilities 0.1910 0.8090
When taking a cut-off point of 5 percent the SDEM model is preferred for both strategy one and two. Furthermore, the SDM specification is only slightly preferred over the SDEM in the case of a cut-off point of 15 percent for strategy 1. Recalling that the SDEM model produces local spillovers and the SDM produces global spillover effects, the result for strategy one is quite interesting. As more restrictions are applied based on pure bilateral trade relations, a switch is made from local to global spillovers. This result is not unexpected however, as evidence for global spillovers is more easily obtained when a spatial weights matrix contains only a few number of unrestricted cells, which is clearly the case when using a cut-off point of 15 percent (see table 11).
It is unclear whether such a relation also exists for strategy 2. While for 𝑊𝐺5 significant evidence
is found for local spillover effects through the SDEM model, the case is not so clear for the 𝑊𝐺15 model.
If the SAR model is to be preferred over the SEM model, the spatial interaction effects occur through the endogenous interaction effects; in this case the spatial spillover effects turn from local to global, similar to the relation found in strategy 1. However, when the SEM model outperforms the SAR model no spillover effects can be measured.
The local spillovers found in 𝑊𝑅5 and 𝑊𝐺5 are more appealing for a spatially dependent model
based on geographical distance than a global spillover model found by 𝑊𝑅15. This is because the local
nature of the SDEM model can be seen as evidence for the fact that geographical distance still matters, as local spillovers imply that spillovers are most common amongst closely situated countries. The global spillover model on the contrary predicts that a spillover effect in one country impacts all other countries, suggesting that spillover effects impact neighboring countries in a similar way as the spillovers affect
countries that are located far away. For 𝑊𝐺15 no conclusive evidence could be found. If the SAR model
