Stock Returns’ Co-Movement: A Spatial Model with Convex Combination of Connectivity Matrices

Abstract

This paper examines the extent of stock-returns’ co-movements among firms in different countries and explores how various measures of closeness affect those co-movements by estimating a spatial autoregressive (SAR) convex combination model that merges four weight matrices—geographical distance, bilateral trade, sector similarity, and company size—into one global matrix. Our results reveal strong spatial stock-market dependence, show that spatial proximity is better captured by financial-distance measures than by pure geographical distance, and indicate that the weight matrix based on sector similarities outperforms the other linkages in explaining firms’ co-movements. Extending the traditional SAR model, the study simultaneously evaluated cross-country and within-country dependencies, providing insights valuable to investors building optimal portfolios and to policymakers monitoring contagion and systemic risk.

1. Introduction

Globalization has important implications for various economic entities. It allows investors to diversify their portfolios by buying and selling financial assets in different markets. However, its impact is not so straightforward (Endri et al. 2024). On the one hand, as markets become increasingly integrated, they become more vulnerable to global shocks. As a consequence, co-movements tend to rise reducing the benefits of diversification. On the other hand, financial liberalization is often accompanied by a wave of crises that has transmitted to other countries causing simultaneous collapses. These crises are characterized by their quick and violent spread across countries, creating a contagion risk (Ben Abdallah et al. 2023). Endri et al. (2024) reveal that high market synchronization makes the benefits of integration meaningless by spreading crises across global markets.

The rapid growth of globalization causes increased correlations among financial markets, especially stock markets (Bekaert et al. 2009; Chuluun 2017). This has encouraged academics to study the nature and structure of linkages and explain stock market co-movements. Recent studies argue that corporate similarities—such as being in the same sector or sharing strong trade relationships—can be even more influential in explaining co-movements than geographic proximity (Asgharian et al. 2013; Fernandez 2011; Fernández-Avilés et al. 2012; Abate 2015; Bera et al. 2016; Ben Abdallah et al. 2022; Gu 2020). For example, two geographically distant firms operating in the same industry may exhibit higher return correlations than adjacent firms in unrelated sectors.

Motivated by these issues, this paper examines the extent to which stock returns’ co-movements exist within firms in different countries, using a novel econometric approach, i.e., spatial econometrics.

The recent surge of interest in spatial econometrics has resulted in the application of these tools in a wide range of research fields, including, among others, economic geography, environmental sciences, environmental criminology, and urban economics. Even though spatial phenomena have become very popular in recent years, they have received not much attention in financial applications (Fernandez 2011; Abate 2015). In the wake of the global financial crisis, studying stock market co-movements through the lens of spatial econometrics has been an increasing area of interest (Tam 2014).

Spatial econometrics is used to analyze the co-movement of stock returns by constructing spatial weight matrices that capture various types of proximity. Following the convex combination approach of Debarsy and LeSage (2022), we integrate simultaneously many weight matrices (geographic, trade, sector, and size) in our model to identify their relative importance. In doing so, our paper not only extends the conventional SAR framework but also bridges the gap between firm-level and country-level analyses, thus providing a more comprehensive view of stock return co-movements.

Previous studies used different methodologies to analyze interdependencies among stock markets such as correlation methods, Granger causality test and volatility, network analysis, wavelet techniques, VAR and GARCH models …. However, these traditional econometrics treat economic units as independent entities and typically do not consider spatial dependencies (Breitenecker and Weyh 2013; Ben Abdallah et al. 2022). Their scope is limited to temporal correlations and does not extend to spatial interactions. These models often fail to capture the spatial dimensions of economic phenomena, which are crucial for understanding stock market dynamics (Breitenecker and Weyh 2013; Gu 2020; Zhang et al. 2022). In contrast, spatial econometrics offers significant advantages in analyzing spatial dependencies and interactions. Spatial models explicitly account for spatial dependencies and interactions between economic units (Ben Abdallah et al. 2023). This means that the economic activities in one region can influence those in neighboring regions, leading to more accurate and comprehensive models, which are particularly useful in studying stock market transmission and interdependencies. More specifically, to model the relationships between non-adjacent economic entities, spatial econometrics uses spatial weight matrices which allow analysis of global spatial interactions and agglomeration phenomena in stock markets.

This paper innovates by using the convex combination approach in spatial econometrics to take into account the interaction structure of the financial system. This methodological contribution avoids potential bias that could arise from spatial weight matrix comparison and the selection of a single type of interaction matrix (Fernández-Avilés et al. 2012; Asgharian et al. 2013; Tam 2014; Chulia et al. 2017).

The convex combination approach involves constructing a single weight matrix combining multiple types of connectivity through a linear combination of different weight matrices. It allows the integration of various spatial relationships into one model, which can be used to infer the relative importance of each type of connectivity (Debarsy and LeSage 2021). The coefficients derived from the convex combination provide insights into the analysis of global cross-sectional dependence (Hazır et al. 2017; Debarsy and LeSage 2018, 2021, 2022). The traditional spatial autoregressive model (SAR model) is extended to include various spatial weight matrices. More specifically, 4 W-matrices based on geographic proximity, bilateral trade, sector similarity, and company size are employed in our model. Through an examination of the associated scalar dependence parameters, the convex combination approach allows us to draw informative inferences about the design of spatial dependence and the role played by each weight matrix in stock returns’ co-movements of companies across different countries.

Although previous studies analyze stock returns’ dependence between countries or companies in a given country, our study explores both dependencies simultaneously. As far as we know, this study is the first attempt to use the convex combination approach in the field of finance to give an overall assessment of the role played by spatial dependencies in analyzing stock returns’ co-movements. Also, this analysis combines firm-specific and country-specific variables and assesses which drivers affect its magnitude.

The remainder of this paper is organized as follows. Section 1 reviews the relevant literature on stock returns’ co-movements. Section 2 explains the spatial methodology and describes weights matrices and data. Empirical results and robustness analysis are presented and discussed in Section 3. Finally, concluding remarks and contributions are developed.

2. Literature Review

2.1. Theoretical Underpinning

The stock market plays a crucial role in the financial system, while the assessment of stock returns’ co-movements is a fundamental topic in finance (Ozdemir 2009; Rua and Nunes 2009; Chuluun 2017; Bai et al. 2018). Stock returns’ co-movement describes the tendency of asset prices to move together in synchronized patterns. This phenomenon is crucial to modern finance theory and has significant implications for portfolio construction, risk management, and international investing. In other words, to conduct risk hedging (through diversification), policymakers and financial investors need to understand the linkages and co-movements of financial markets (Rua and Nunes 2009; Asgharian et al. 2013; Chuluun 2017). In this regard, Weng and Gong (2016) assert that “an intensive analysis of the dependence structure among stock markets is invaluable to financial experts, policy makers, and academic researchers, providing them with important implications for portfolio management, policy-making, and risk assessment”.

Several theoretical underpinnings have been proposed to explain this co-movement ranging from classical asset pricing models to behavioral finance perspectives. To begin with, The portfolio theory plays a significant role in explaining the co-movement of stock returns. Particularly, and according to the capital asset pricing model (CAPM), the co-movement between different stocks is largely explained by their relationship with the market index, and the market itself is a primary driver of stock return correlations, even without considering fundamental factors of the stocks (Fama and French 1992; López-García et al. 2020). Subsequently, modern portfolio theory emphasizes the benefits of international diversification in reducing risk. It is another critical aspect of understanding stock return co-movement. However, increasing globalization has led to higher degrees of integration, which can reduce the benefits of international diversification (Das et al. 2018; Kaffel and Abid 2020). Afterward, the international asset pricing model (IAPM) extends traditional CAPM to consider the global context and the integration of international financial markets. This theory allows an understanding of the dynamics of stock returns and their risk characteristics by underscoring the impact of global economic forces on the co-movement of stock returns (Chen et al. 2016). Finally, behavioral finance theory assumes that stock return co-movement can be influenced by non-fundamental factors such as investor sentiment (Frijns et al. 2017).

2.2. Empirical Literature

The study of co-movements among financial markets has been a widely debated issue since the famous work of Grubel (1968) highlighted the benefits of international portfolio diversification. It has been recognized that diversification benefits can be accomplished when the co-movements among financial markets are taken into account.

Scientists use different denominations to designate co-movement, such as interdependence, correlation, interaction, integration, co-exceedance, synchronization, jump, and spillover. Overall, studies on stock market co-movements are split into two broad strands of literature focusing on country-level and company-level analysis. There are a variety of methodologies used to analyze interdependencies among stock markets. The most important are correlation methods (Bekaert et al. 2009; Das et al. 2018), Granger causality test and volatility analysis (Letteri 2024), network analysis, Markov-switching copula model (Czapkiewicz et al. 2018; Just and Echaust 2020), wavelet techniques (Dajčman and Festić 2012), VAR and GARCH models (Abuzayed et al. 2021).

The literature is too broad to survey here; therefore, we review only the existing literature related to spatial econometrics. It is a subfield of statistics that have been overlooked in finance, but in the wake of the global financial crisis, it has received more attention (Selan and Kalatzis 2017; Kutzker and Wied 2024).

Spatial econometrics has been increasingly utilized to analyze the complex interdependencies among stock markets, using spatial relationships and economic linkages to understand market dynamics. This section presents a comprehensive overview of existing stock returns’ co-movement literature on both the country level and the company level through the lens of spatial econometrics.

2.2.1. Country-Level Stock Returns Co-Movements

Since the introduction of spatial econometric techniques, spatial modeling of dependence structures has become well received. Nevertheless, this technique is fairly new to financial data and especially to the analysis of stock markets’ co-movements. For instance, Fernández-Avilés et al. (2012) investigated the complex global dependencies in international financial markets using spatial techniques. By analyzing daily returns on 17 stock market indexes for the period ranging from January 2002 to March 2010, they compared two different measures of distance based on the kriging predictions achieved from the respective semivariograms. They found that financial proximity obtained from the foreign direct investment (FDI) linkages between countries are more relevant in accounting for market co-movements rather than physical distance.

Similar evidence was established by Asgharian et al. (2013), who shed further light on financial integration and stock market co-movements. They investigated how different stock markets affect each other. More specifically, Asgharian et al. (2013) analyzed how various geographical, economic, and financial linkages between 41 equity markets from January 1995 to December 2011 influenced co-movements among these markets. Applying the spatial Durbin model (SDM), they concluded that most linkages, which are bilateral trade, bilateral FDI, interest rate, exchange rate, expected inflation, purchasing power parity, and geographical distance between the capital cities for every pair of countries, could capture co-variations in returns, and that bilateral trade is the most significant factor.

Tam (2014) explored dynamic linkages among East Asian equity markets using an advanced spatial–temporal model with an error correction specification. By analyzing 12 East Asian equity markets, he found that they are characterized by linkages through significant spatial effects, and Japan is a dominant driver of market ties in this region.

More recently, Chulia et al. (2017) investigated international risk synchronization in 23 global stock markets from January 1995 to July 2015. Using spatial correlations, they constructed a global index describing the risk synchronization based on economic and geographical considerations. They discovered that mature and emerging markets present different risk-profile dynamics. Also, they documented that international financial crises have different effects on these two markets.

Finally, Zhu and Milcheva (2020) estimated a spatial multi-factor model (SMFM) that mixes asset pricing techniques with spatial econometrics to evaluate systemic implications for REIT index returns in 14 countries from 1993 to 2015. They distinguished between systematic risk, which describes co-movements due to market risk exposure, and spillover risk, which means co-movements due to linkages between markets. They employed a variety of bilateral linkages and found that economic integration plays a more significant role in explaining the interconnectedness among markets than other linkages, such as geographic distance. Also, they outlined that idiosyncratic risk was the most important type of risk in real estate stocks during the tranquil period, while during the global financial crisis, the spillover risk increased dramatically.

2.2.2. Company-Level Stock Returns Co-Movements

Stock returns’ co-movements among companies are analyzed by several studies using traditional econometrics such as the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT). Recently, some approaches explored the relationship between corporate finance and stock returns analysis using spatial econometrics.

Some important contributions include that of Fernandez (2011), which is considered one of the first examples of the use of spatial econometric models to investigate how a specific firm’s stock returns are affected by the risk of its peers. To do so, she formulates a spatial version of the capital asset pricing model (S-CAPM). The proposed spatial model makes it possible to consider alternative measures of spatial distances between firms based on firm characteristics variables, such as market capitalization, the market-to-book value, debt maturity, and dividend yield. She analyzed a panel of 126 Latin American firms from 1997 to 2006. She noticed evidence of spatial effects, and that beta is significant even after considering spatial dependence.

Moreover, Eckel et al. (2011) conducted an analysis based on firms listed on the S&P 500 on August 2005 for the period ranging from 2000 to 2008. They measured the effects of geographical distance on stock market correlation. They concluded that the beyond 50 miles’ geographical proximity does not influence stock return correlations.

Another contribution is Arnold et al. (2013), who modeled spatial dependencies in stock return of companies listed on Euro Stoxx 50 from 2003 to 2009. They developed a spatial autoregressive (SAR) panel model that allows for distinguishing between three types of spatial dependence: general dependencies, dependencies inside branches, and local dependencies. This model was extended on three occasions. Wied (2013) takes into consideration structural breaks. Schmitt et al. (2016) combined this model with local normalization techniques. While Gong and Weng (2016) utilized it for value at risk forecasts in the Chinese stock market. One common conclusion drawn from these studies is that combining spatial modeling and finance can lead to superior risk forecasts in portfolio management.

Bera et al. (2016) analyzed spatial linkages of the Istanbul Chamber of Industry (ICI) firms’ sales and net asset profitability measures for the period spanning from 1997 to 2011. They constructed the weight matrix using geographical contiguity and firm-specific measures, such as industrial branches, equity, and productivity levels. They analyzed how various weight matrices lead to very different results of spatial dependence. The results show significant partial dependence when distance is measured by financial proximity, such as the sector or productivity similarities of the firms, rather than geographic closeness.

Finally, Selan and Kalatzis (2017) investigated the peer effects in 166 Brazilian companies listed on the Sao Paulo Stock Exchange. They estimated a spatial autoregressive model (SAR) and a spatial Durbin model (SDM) with static spatial weight matrices and used quarterly data from 2007 to 2014. They detected a positive spatial dependence among stock returns from peer companies but a negative feedback effect from fundamental characteristics (book-to-market and dividend-price ratio).

3. Methodology

The spatial econometrics approach, first put up by Paelinck and Klaassen (1979), is a subfield of statistics that deals with spatial dependence between observations collected from points or regions located in space (LeSage 2008). Its essence is that space matters. It tries to understand how the values of the dependent variable at a specific location depend on the values of observation at other locations. The spatial econometrics approach has provided a powerful tool to capture the neighborhood effects by setting up an interaction matrix, known as the spatial weight matrix, and noted W. This matrix describes the spatial interaction effects among neighboring units and captures the spillover effects among them.

Spatial econometrics has been continuously enriched and developed (Arbia 2016). Even though it is a popular topic in a variety of areas of economics, its use in financial applications is relatively innovative regarding the difficulty of defining the spatial weight matrix in the context of financial markets (Fernandez 2011; Arnold et al. 2013; Asgharian et al. 2013; Bera et al. 2016; Weng and Gong 2016; Selan and Kalatzis 2017). The key point of spatial econometrics is the choice of an adequate measure of closeness. The weighting matrix W plays a crucial role in modeling the spatial correlation nerveless it is not evident how distance should be gauged. There is little guidance on how to select the appropriate spatial weight in an empirical application (Abate 2015; Weng and Gong 2016; Zhu and Milcheva 2020).

The usual tradition is constructing the spatial weight matrix is using the geographic proximity or contiguity between spatial units (Fernandez 2011). This spatial proximity has advantages. It is objective, easy to determine, fixed over time, and in most cases exogenous (Debarsy and LeSage 2018; LeSage 2021). However, there is nothing in the basic framework for why spatial distance should need to be limited to geographic distance.

The spatial weight matrix plays a fundamental role in spatial models and might not be unique. In the context of financial markets, W can be defined using non-geographically based connectivity matrices like economic and/or financial distance; see, for instance, Fernandez (2011), Arnold et al. (2013), Asgharian et al. (2013), Bera et al. (2016), Zhu and Milcheva (2020). Thus, it would be interesting to investigate how various weight matrices lead to very different results in terms of the strength of spatial dependence.

Spatial regression models generally rely on a single weight matrix which leads to a great deal of criticism. In their article “The Biggest Myth in Spatial Econometrics”, LeSage and Pace (2014) present the criticism of spatial regression models as being sensitive to weight matrix specifications. They highlight that different choices for W could lead to very different estimates and inferences.

One response to dissatisfaction regarding using a single spatial weight matrix has been the introduction of multiple weight matrices in one spatial model. One such example is the spatial autoregressive model (SAR model), which relies on more than one W, and has been specified and estimated in numerous studies (see Arnold et al. 2013; Schmitt et al. 2016; Gong and Weng 2016). Different weight matrices may capture different types of cross-sectional dependence within the same model (Debarsy and LeSage 2018). To select the most appropriate spatial weight matrix, different approaches, such as the J-tests approach (Kelejian and Piras 2011), the Bayesian approach (LeSage and Pace 2009), log-marginal likelihoods (LeSage 2014), Mallow’s type criterion (Zhang and Yu 2018), Akaike information criterion, were developed (for more details, see Zhang and Yu 2018; Hazır et al. 2017; Debarsy and LeSage 2018).

Spatial weight matrix comparison techniques were criticized by Hazır et al. (2017). On the one hand, they focus on selecting a single best interaction matrix and neglect the simultaneous effects of the different interaction structures. On the other hand, when using multiple weight matrices in a spatial model, one cannot correctly explain partial derivatives. Direct and spillover effects cannot be measured, and the relative strength of the different dependencies cannot be assessed (LeSage and Pace 2014; Hazır et al. 2017).

In this work, we use an innovative approach to avoid potential bias that could arise from selecting a single spatial weight matrix. More precisely, the convex combination approach developed by Debarsy and LeSage (2018) is used.

Contrary to spatial weight matrix comparison techniques, the convex combination approach authorizes for simultaneous effects arising from various interaction structures (Hazır et al. 2017). The coefficients associated with each matrix quantify the relative importance assigned to each type of connectivity in the global cross-sectional dependence scheme (Debarsy and LeSage 2021). This innovative approach uses convex combinations of various connectivity matrices to form a single weight matrix. This matrix can be used in conventional spatial regression estimation and thus, interpretations of direct and indirect effects based on conventional matrix derivatives are preserved (Debarsy and LeSage 2018).

3.1. The SAR Convex Combination Model

In this paper, we use an innovative approach to avoid potential bias that could arise from selecting a single spatial weight matrix (Hazır et al. 2017). More precisely, the convex combination approach developed by Debarsy and LeSage (2018) is used. The convex combination approach authorizes for simultaneous effects arising from various interaction structures. The coefficients associated with each matrix quantify the relative importance assigned to each type of connectivity in the global cross-sectional dependence scheme (Debarsy and LeSage 2021).

Depending on the source of spatial dependence among cross-sectional units, various spatial econometric models were developed in the spatial econometrics literature (Anselin 1988; Elhorst 2010). Among them, we select the spatial autoregressive (SAR) model, which incorporates a spatially lagged dependent variable as shown in (1):

$$Y=\rho W y+X\beta +\epsilon ; \epsilon \sim (0, {\sigma }^2{I}_n)$$

(1)

where

• Y is an NT × 1 vector of the dependent variable.
• ρ is the spatial autoregressive parameter that measures the intensity of the spatial interdependency.
• X is an NT × K matrix of exogenous explanatory variables.
• β is a K × 1 vector of unknown coefficients to be estimated.
• ε is an NT × 1 vector of disturbance terms, where ε_i are independently and identically distributed error terms with zero mean and variance σ².
• W is an NT × NT spatial weights matrix that equals I_T ⊗ W_N, where ⊗ denotes the Kronecker product. I_T is an identity matrix of dimension T, and W_N is an N × N spatial weight matrix describing the spatial arrangement of the cross-section units.

To this model, we introduce an extension that relies on the convex combination approach, and the SAR convex combination model can be written as follows:

$$Y=\rho W_C\left(\Gamma \right)Y+X\beta +\epsilon ; \epsilon \sim (0,{\sigma }^2{I}_n)$$

(2)

$$W_C\left(\Gamma \right)=\sum _{m=1}^M{\gamma }_m{W}_m {0\le \gamma }_m\le 1, \sum _{m=1}^M{\gamma }_m=1$$

(3)

The matrix W_c is a combination of M different types of connectivity describing cross-section dependencies. Each W_m is an NT × NT matrix measuring the spatial distance between observations. It takes a block diagonal form (I_T ⊗ w_m): w_m represents an N × N connectivity matrix with the main diagonal containing zero elements (by convention all the diagonal elements of the weight matrix are equal to zero ($W_{ii}=0$)). m = 1…M and M represents the number of the spatial weight matrix. I_T is a T-vector of ones. When each w_m, is row-normalized, then W_c respects the conventional row-normalization. This allows the use of conventional spatial regression model specifications and estimation methods (Debarsy and LeSage 2018). ${\gamma }_m$: m = 1…M are scalar dependence parameters quantifying the strength of each type of dependence modeled by W_m.

To isolate the parameters $\rho \text{and} {\gamma }_m$ in the (M + 1) × 1 vector W, the SAR model in (2) can be re-expressed as in (4):

$$\begin{array}{l}\tilde{Y}=X\beta +\epsilon \\ \tilde{Y}=(Y, W_{1 }Y, {W_{2 }Y,\dots ,W}_{M }Y)\\ W=\left(\begin{array}{c}1\\ -\rho \Gamma \end{array}\right)\\ \Gamma =\left(\begin{array}{c}{\gamma }_{1 }\\ {\gamma }_{2 }\\ .\\ .\\ .\\ {\gamma }_{M }\end{array}\right); {\gamma }_{M }=1-{\gamma }_{1 }-{\gamma }_{2 }-\dots -{\gamma }_{M-1 }\end{array}$$

(4)

By doing this isolation, we can pre-calculate the NT × (M + 1) matrix $\tilde{Y}$ which contains only sample data prior to the beginning of the MCMC sampling loop (Debarsy and LeSage 2021).

The likelihood of the model in (5) is shown in (6):

$$\begin{array}{l}f(YX, W, \rho , \Gamma , {\sigma }^2, \beta )=\left|R \left(W\right)\right| {(2 \pi {\sigma }^2)}^{NT/2}exp \left({\displaystyle \frac{e\prime e}{2{\sigma }^2 }}\right)\\ e=\tilde{Y}W-X \beta \\ RW I_{NT }-\rho W_{c }( \Gamma )\end{array}$$

(5)

$\left|R\left(W\right)\right|$ is the Jacobian of the transformation from the disturbances to the dependent variable vector. $\rho$ is restricted to (−1, 1) so that ${R\left(W\right)}^{-1}$ = $\sum _{j=0}^{\mathrm{\infty }}{\rho }^j{W_c}_c^j\left(\Gamma \right)$ exhibits an underlying stationary process.

The SAR convex combination model is estimated using Markov chain Monte Carlo (MCMC) estimation via MATLAB R2017 according to the routine developed by LeSage (2021) in his book “A panel data toolbox for MATLAB” (for more details regarding MCMC estimation for this model, see Debarsy and LeSage 2021).

3.2. Weight Matrices and Data

3.2.1. Construction of the Weight Matrices

In our analysis, we use four kinds of connectedness to construct W_c. The first weight matrix, named W_distance, is created based on geographic proximity. This measure has enjoyed much empirical success in explaining market linkages. Although most financial transactions can be performed electronically, geographical proximity may facilitate financial linkages (Fernández-Avilés et al. 2012; Zhu and Milcheva 2020). In this work, geographic proximity between companies is measured according to the Cliff–Ord weight matrix (Anselin 1988) by calculating the inverse distances function as follows:

$$W=\left\{\begin{array}{c} {\left(d_{ij}\right)}^{-1} ; \nabla i\ne j \\ 0 ; \nabla i=j \end{array}\right.$$

(6)

where d_ij is the distance between the capital cities for every pair of countries where they are headquartered.

A second weight matrix, labeled W_trade, is created based on bilateral trade. The motivation for considering the bilateral trade weight matrix is based on the following economic theory: countries which trade more are not independent and are closely linked economically (Abate 2015). Increasing trade among countries indicates that domestic firms’ profitability will be influenced by firms and economic circumstances in other countries (Paramati et al. 2016). In our work, the bilateral trade intensity W_trade between two firms is equal to the bilateral trade intensity between the two countries where they are headquartered. This last is defined as the ratio of the total exports (exp) and imports (imp) of the country i with respect to country j divided by the sum of total exports and imports of the country i during the same year:

$$W_{ij}={\displaystyle \frac{{exp}_{ij}+{imp}_{ij}}{{\sum }_{K=1}^{K=N}{exp}_{iK}+\sum _{K=1}^{K=N}{imp}_{iK}}}$$

(7)

The last two matrices, W_sector and W_size, are constructed based on sector similarities and company size, respectively. Firms operating in the same sector would have stronger co-movements with each other than those operating in different sectors (Bera et al. 2016). Equally, firms of similar size may be more connected than those having different sizes. The similarity in size W_size is calculated as the absolute value of the differences between the size of the two firms, which is approximated by the market capitalization of each firm (Asgharian et al. 2013; Zhu and Milcheva 2020):

$$W_{ij}=\left| {size}_i-{size}_j\right|$$

(8)

While the weight matrix based on sector similarities W_sector is constructed as follows:

$$W_{ij} =\left\{\begin{array}{c}{W}_{ij}=1 \text{if firm} i \text{and firm} j \text{are in the same sector}\\ W_{ij}=0 \text{if firm} i \text{and firm} j \text{aren’t in the same sector}\end{array}\right.$$

(9)

It should be noted that the intention is not to provide an exhaustive analysis of all possible linkages for stock market integration. This study aims to analyze the impact of a selection of possibly important linkages on stock market co-movements. Equally, the focus on these following channels can be explained by two reasons: they have been highlighted in the previous literature, and there are bilateral linkages for which data are available.

Furthermore, it should be mentioned that weight matrices based on financial distance may be endogenous and the statistical properties of the estimator would not hold. To solve this issue, we replicate the methodology proposed by Keiler and Eder (2013). The respective variables used to calculate the weight matrices are lagged by one period. An additional check is required because if the lagged variables still contained information about the contemporary value of stock returns, then the weight matrix would result in being endogenous also after lagging the variables. To address this possibility, the correlation between each lagged variable and the stock returns is calculated. If the resulting values are very small, then the weight matrix is exogenous, and the statistical properties of the estimator are preserved. Once, the connectivity matrices have been computed, every matrix is usually (row) standardized by dividing each element of its row by the sum of its corresponding elements. Thus, each row of W, denoted W_ij, sums to unity.

3.2.2. Similarity of the W_m, m = 1, …, M Weight Matrices

For our analysis, four W-matrices are considered. To produce accurate estimates of the model parameters γ_m, m = 1, …, 4, ρ, β and σ², these four W-matrices must be different. LeSage (2021) notes that there is no sense in using multiple weight matrices that are similar. He demonstrates that two identical matrices will produce an ill-defined problem with no solution.

To resolve this issue, we convert the weight matrices to a vector (W_distance u, W_trade u, W_sector u, W_size u) using a matrix-vector product including the weight matrices and a random normal vector u. Then based on the correlation coefficient, we can judge the similarity of the matrices. The correlation between the different connectivity matrices is calculated and shown in

Table 1. Correlation matrix of the four W-matrices.

	Wdistance	Wsector	Wtrade	Wsize
Wdistance	1.000	−0.008	0.551	0.032
Wsector	−0.008	1.000	0.016	−0.027
Wtrade	0.551	0.016	1.000	0.051
Wsize	0.032	−0.027	0.051	1.000

. Results show that these four W-matrices are not highly correlated. Therefore, our estimations are preserved.

3.3. Data

This research work attempts to investigate to what extent stock returns in one firm affect other firms’ stock returns via various spatial linkages using the SAR convex combination model described above. More precisely, the following model will be estimated:

$$Y=\rho { Y(}_{{\gamma }_1}{W}_{distance}+{\gamma }_2{W}_{sector}+{\gamma }_3{W}_{trade}+{\gamma }_4{W}_{MV})+{\beta }_1\mathrm{S}\mathrm{I}\mathrm{Z}\mathrm{E}+{\beta }_2\mathrm{R}\mathrm{O}\mathrm{A}+{\beta }_3\mathrm{E}\mathrm{X}\mathrm{C}\mathrm{H}+{\beta }_4\mathrm{T}\mathrm{O}+{\beta }_5\mathrm{I}\mathrm{N}\mathrm{F}+{\beta }_6 GDP+\epsilon$$

(10)

Data needed for this analysis are scattered around various databases. The dependent variable is the firm’s stock returns. It is defined as the difference in the logarithm of yearly stock price indices that were collected from Datastream. Firm-level explanatory variables include return on ssset (ROA) and firm size (SIZE). ROA and SIZE are interpolated from Datastream. Country-specific variables comprise four variables. Data on GDP growth and inflation rate (INF) are obtained from the World Development Indicators (WDI) database. Trade openness (TO) and exchange rate (EXCH) are sourced from the Unctad database.

For the data needed to calculate spatial dependence between firms, distances between capital cities are interpolated from the Centre d’Etudes Prospectives et d’Informations Internationales (CEPII). Bilateral trade data are taken from the IMF financial statistics. Firms’ sector data are from the Compustat database. Data on market value are obtained from the Datastream. After removing firms with missing observations and balancing the data, our estimation sample includes 800 firms across eight countries. The estimation period ranges from 2005 to 2020, yielding 12,000 yearly observations. Using annual data is determined by data availability to ensure that fundamentals and macroeconomic variables are measured at the same frequency (Asgharian et al. 2013). Generally, annual return data are required to capture long-term relationships between variables by isolating co-movement driven by genuine economic fundamentals.

4. Empirical Results

4.1. Spatial Correlation Tests

To check whether spatial autocorrelation should be considered, we used two categories of tests. Lagrange multiplier test statistics and their robust counterparts for a spatially lagged dependent variable (hereafter LM-lag and Robust LM-lag) and a spatially autocorrelated error term (hereafter LM-error and Robust LM-error) were conducted (Anselin 1988; Elhorst 2010). All test results are presented in

Table 2. LM test results of spatial econometric model.

	Wdistance	Wsector	Wtrade	Wsize
LM lag	5802.449 ***	9016.101 ***	7303.048 ***	339.561 ***
Robust-LM lag	334.974 ***	411.656 ***	325.483 ***	10.866 ***
LM error	31,422.223 ***	15,632.893 ***	32,613.211 ***	461.984 ***
Robust-LM error	25,954.748 ***	7028.448 ***	25,635.647 ***	133.289 ***

Note: *** indicate statistical significance at 1% significance level.

.

All of the LM test statistics are highly significant at the 1% significance level. Thus, the null hypothesis of no spatial autocorrelation in both the dependent variables and in the error term must be rejected in all estimations regardless of the types of matrix. These results reveal that spatial models are more relevant than non-spatial models.

4.2. Results of the SAR Model with Various Matrices

Table 3. Estimation results of the SAR model with various matrices and with space-fixed effects.

	OLS	SAR-Wdistance	SAR-Wsector	SAR-Wtrade	SAR-Wsize
Variable	Coefficient	Coefficient	Coefficient	Coefficient	Coefficient
ρ	-	0.606 ***	0.655 ***	0.705 ***	0.003 ***
SIZE	536.525 ***	465.994 ***	439.110 ***	465.305 ***	535.930 ***
ROA	−0.090 ***	−0.077 ***	−0.074 ***	−0.077 ***	−0.090 ***
EXCH	−12.407 ***	−5.298 ***	−4.971 ***	−4.030 ***	−12.346 ***
TO	−3.019 ***	−1.649 ***	−1.258 ***	−1.526 ***	−2.998 ***
INF	−2.861 ***	−0.843 ***	−0.739 ***	−0.695 ***	−2.868 ***
GDP	3.617 ***	1.699 ***	1.377 ***	1.533 ***	3.601 ***
R-squared	0.351	0.454	0.490	0.461	0.385
Log-likelihood		−69,258.215	−68,905.466	−69,176.333	−69,957.849

Notes: This table presents the estimation results of the SAR model presented in Equation (1) and the estimation results of the OLS model. *** indicate statistical significance at 1% significance level.

reports the estimation results of the SAR model (Equation (1)) for each connectivity matrix considered separately, along with estimates based on the ordinary least squares (OLS) model. Comparing the R-squared values of the SAR estimations with those of the OLS model indicates that allowing for spatial correlation considerably increases the explanatory power of the model.

Another variable of interest is the spatially lagged dependent variable. Its coefficient rho (ρ) is positive and highly significant for all dependencies measures. Therefore, we can conclude that the spatial dimension is relevant, and that firms with similar return levels cluster together. On average, the changes in stock returns from neighboring firms have a significant positive impact on domestic stock returns. These findings confirm the previous results of the spatial correlation tests presented in Table 2.

The next step is to compare spatial weight matrices. There is a clear divergence in the calculated effects estimates. Comparing the log-likelihood values for the different neighborhood measures shows that the specification based on sector similarities has the largest explanatory power, whereas the specification based on size similarities performs worst in explaining stock returns’ co-movements.

Table 3 also shows the estimated coefficients associated with the explanatory variables. Firm-specific characteristics have a significant impact on stock returns’ co-movements for all models. For instance, firm size has a significant positive effect on stock prices. The size of a firm may reflect its level of development, the degree of market liquidity, information cost, and the transaction cost associated with trading equity in that firm. Furthermore, it may serve as a proxy for portfolio diversification and thus reduce the riskiness of big firms. Investors have confidence that big companies with a large amount of wealth can return funds that have been invested.

In addition to firm-specific factors, the country-industry literature demonstrates that country factors have importance in explaining a company’s stock market returns (Bekaert et al. 2009). Countries differ on a broad range of attributes such as local monetary, fiscal policies, legal regimes, and economic structures. These differences are the main force driving international diversification strategy. The impact of country effects on company stock returns’ co-movements was neglected in previous literature. In our study, we mainly focus on two categories of country-specific variables describing global integration (the degree of development and the degree of trade openness) and macroeconomic indicators (the exchange rate and the inflation rate). These variables constitute the barometers for measuring the performance and success of the securities market. A conducive macroeconomic environment increases a business’s profitability and allows it to access securities for sustained growth. All these variables significantly explain stock returns. For example, the exchange rate and the inflation rate have a significant negative impact on stock returns. These variables affect stock market activity, impinging directly on the state of corporate activities in the country. On the one hand, exchange rates affect resource allocation, prices, production levels, and profitability. Their fluctuation is reflected in share prices. In the case of depreciation, capital flight affects the exchange’s trading volume and the price index. On the other hand, high inflation rates intensify the cost of living and heighten uncertainties concerning future prices and investment. Investors resort to investing in physical assets, leading to a reduction in trading, and thus the demand for shares falls.

4.3. Results of the SAR Convex Combination Model

Spatial weight matrix comparison techniques focus on selecting a single best weight matrix. To allow for simultaneous effects arising from various interaction structures, the convex combination approach is used.

Table 4. Bayesian SAR convex combination model with space-fixed effects.

	SAR-Wsector	SAR Convex Combination Model
Variable	Coefficient	Coefficient
ρ	0.655 ***	0.691 ***
SIZE	439.110 ***	442.518 ***
ROA	−0.074 ***	−0.074 ***
EXCH	−4.971 ***	−4.427 ***
TO	−1.258 ***	−1.264 ***
INF	−0.739 ***	−0.650 ***
GDP	1.377 ***	1.335 ***
$\gamma$
${\gamma }_1$-Wdistance	-	0.0179
${\gamma }_2$-Wsector	-	0.701 ***
${\gamma }_3$-Wtrade	-	0.256 ***
${\gamma }_4$-Wsize	-	0.023 ***
Log-likelihood	−68,905.466	−63,197.917

Notes: This table presents the estimation results of the SAR convex combination model presented in Equation (2) and also the estimation results of the SAR model constructed using the sector similarity matrix. *** indicate statistical significance at 1%significance level.

reports estimation results of the SAR model constructed using the sector similarity matrix and the SAR convex combination model of Equation (10).

At the start, it can be seen that the log-likelihood values show a clear improvement for the SAR convex combination model relative to that based on one spatial weight matrix.

Table 4 reveals that, for the SAR convex combination model, the spatial correlation coefficient is positive and highly significant. We can conclude that there is strong evidence for spatial dependence. Firms’ stock returns commoved together. If we consider the estimated weight assigned to all matrices used simultaneously, three of the weight matrices, which are W_sector, W_trade, and W_size, have a significant impact on stock return’s co-movements, while W_distance does not. Our findings go hand in hand with literature. In the context of financial markets, a geographical weight matrix may not be suitable for characterizing dependencies among different firms. Indeed, through globalization, international financial markets have become more integrated. The evidence suggests that reducing barriers to integrating countries’ economic activities makes firms’ locations look less relevant for variation in stock returns. Therefore, spatial proximity may be better quantified by financial and/or economic distance measures than geographical distance (Fernández-Avilés et al. 2012; Bera et al. 2016).

When analyzing the scalar dependence parameters quantifying the strength of each type of dependence, it is obvious that W_sector outperforms the other linkages and receives the principal weight (0.701), then W_trade (0.256), while W_size has the lowest weight (0.023). In the first place, sector similarity is the most important linkage and has a greater impact on firms’ stock returns. We argue that firms belonging to the same sector are more connected than those operating in different sectors (Arnold et al. 2013; Bera et al. 2016). These firms should display a similar behavior because they are exposed to the same global input factors like commodity prices. In the second place, spatial interaction, defined based on the bilateral trade weight matrix, has a considerable role in explaining firms’ stock returns co-movements. Nowadays, cross-country correlations are significantly higher than they were. Greater trade connectedness increases the correlation of business cycles and, hence, leads to stronger co-movements. Our findings confirm that a large value of trade between two countries signifies higher dependence between them. Consequently, domestic firms in a country are affected by firms and economic circumstances in other countries. The stock returns in these firms are more correlated and display greater co-movements.

Consistent with Asgharian et al. (2013), we find that financial-distance matrices outperform geographic distance. By embedding all four linkages in a single SAR-convex framework, our study demonstrates that sector similarity now explains nearly 70% of spatial dependence, underscoring the rise of global industry factors post-2005.

We conclude that the sector similarity matrix has the dominant effect, followed by bilateral trade, while the weight for geographical distance is negligible. So, in today’s globalized market, sector-specific factors and trade relationships are more critical in driving stock returns’ co-movements than mere physical proximity.

Assume a one-day −10% return for Apple Inc. (GICS 45) on 3 January 2019. With the convex-combination estimate ρ = 0.691 and the weight on the sector matrix ωsector = 0.701, the effective sector spill-over coefficient equals 0.48. A peer technology firm is therefore expected to lose about −10% × 0.48 ≈ −4.8%. For companies outside GICS 45, the combined coefficient attached to the other connectivity matrices is 0.21, so the average contemporaneous impact is only −2.1%. Focusing on this first-round effect (and ignoring higher-order feedback) highlights that a sector shock propagates more than twice as strongly to same-industry peers as to firms in the rest of the market, visually confirming the dominance of sectoral channels documented in Table 4. A real-world precedent is the dot-com bust of 2000–2002: profit warnings at Cisco and Nortel triggered double-digit losses across global telecom and IT stocks, while utilities and health-care shares remained largely insulated (Bekaert et al. 2009; Cavaglia et al. 2000).

Our finding that financial distance matrices dominate pure geography has already been validated by some market-level studies. Using bilateral-trade weights, Asgharian et al. (2013) showed that bilateral trade linkages capture equity co-movements more effectively than distance. Fernández-Avilés et al. (2012) corroborated this with foreign direct investment (FDI) ties, while Zhu and Milcheva (2020) found that asset-level spatial ties matter for real estate assets returns. By embedding all four linkages in a convex-combination SAR at firm level, we confirm and generalize these insights for 2005–2020.

Unlike Forbes and Rigobon (2002), who report country-level contagion during the 1997–98 crises, we find sector channels now dominate. One explanation is the post-2005 expansion of cross-border supply chains and digital platform firms, which blurs national boundaries within each industry.

Synchronized fluctuations in energy and commodity prices (Kirikkaleli and Güngör 2021) and the diffusion of cloud technologies have aligned the valuations of specially listed companies across borders, regardless of geography. The integration of supply chains, highlighted by the International Monetary Fund (2021), further amplifies sectoral shocks internationally.

Large-capitalization firms, frequently included in global indices and ETFs, exhibit stronger co-movement because they are traded by the same global investor base and are subject to an index-arbitrage activity (Barberis et al. 2005). In contrast, small-caps, absent from such vehicles, remain locally anchored, resulting in lower cross-border dependence.

Our finding underscores that investors and policymakers should consider multiple dimensions of connectivity when assessing diversification benefits and systemic risk.

4.4. Robustness Analysis: The SAR Convex Combination BMA Model

For L candidate connectivity matrices, there are M = 2^L − L − 1 possible ways to use two or more of the L matrices in different model specifications. In our case L = 4, so we have M = 11 possible models involving two or more matrices. The convex combination approach raises the issue of which matrices should be employed and which should be ignored. To tackle this problem, Metropolis–Hastings guided Monte Carlo integration procedure, proposed by LeSage (2021) and Debarsy and LeSage (2022), is used. More precisely, Bayesian model averaged estimates are calculated.

Table 5. SAR convex combination BMA model with space-fixed effects.

Models	logm	Prob	rho	Wdistance	Wsector	Wtrade	Wsize
Model 1	−63,219.107	0.000	0.654	0.155	0.845	0.000	0.000
Model 2	−63,376.783	0.000	0.707	0.029	0.000	0.971	0.000
Model 3	−63,437.662	0.000	0.609	0.967	0.000	0.000	0.033
Model 4	−63,203.867	0.000	0.694	0.000	0.705	0.295	0.000
Model 5	−63,209.427	0.000	0.631	0.000	0.972	0.000	0.028
Model 6	−63,357.774	0.000	0.707	0.000	0.000	0.974	0.026
Model 7	−63,208.299	0.000	0.693	0.017	0.706	0.277	0.000
Model 8	−63,207.610	0.000	0.655	0.143	0.832	0.000	0.025
Model 9	−63,362.225	0.000	0.705	0.028	0.000	0.946	0.026
Model 10	−63,193.441	0.988	0.692	0.000	0.700	0.277	0.023
Model 11	−63,197.893	0.012	0.692	0.016	0.701	0.260	0.023
BMA	−63,193.493	1.000	0.692	0.000	0.700	0.277	0.023
Highest	−63,193.441	0.988	0.692	0.000	0.700	0.277	0.023

reports the results of the SAR convex combination BMA model. This table presents ρ estimates for each model, estimated values for the parameters $\gamma$, associated posterior probabilities, and the log-marginal likelihood estimates of all models including two or more matrices in the convex combination.

The highest probability model is shown in the last row of Table 5. Of the 11 different models, model 10 receives the highest posterior model probability of (0.988) and the highest log-marginal likelihood (−63,193.441). Model 11 is the only other model that presents a non-zero probability equal to (0.012). Model 10 produces an estimate $\sigma$ equal to 0.692. W_sector displays the principal weight (0.700) in explaining firms’ stock returns co-movements, then W_trade (0.277), W_size has the lowest weight (0.023), while W_distance has no weight. These findings confirm the presence of spatial dependence, and that it is better quantified by financial distance measures than geographical distance. Additionally, note that the scalar dependence parameters are very close to the weights found in Table 4, which confirms our results.

A linear combination of the parameters is formed using the posterior model probabilities from models 11 and 10. It is shown in the row labeled BMA and represents the model averaged estimates presented in

Table 6. Bayesian model average of the SAR convex combination models with space-fixed effects.

Variable	Mean	Lower 0.01	Lower 0.05	Median	Upper 0.95	Upper 0.99
ρ	0.692	0.650	0.659	0.692	0.725	0.734
SIZE	442.578	420.931	426.173	442.501	458.972	463.113
ROA	−0.074	−0.089	−0.085	−0.074	−0.062	−0.058
EXCH	−4.401	−7.011	−6.366	−4.397	−2.473	−1.923
TO	−1.262	−1.536	−1.479	−1.263	−1.046	−0.978
INF	−0.655	−1.277	−1.123	−0.656	−0.184	−0.061
GDP	1.333	1.043	1.110	1.333	1.562	1.637
${\gamma }_1$-Wdistance	0.000	0.000	0.000	0.000	0.000	0.001
${\gamma }_2$-Wsector	0.700	0.599	0.618	0.698	0.787	0.803
${\gamma }_3$-Wtrade	0.277	0.171	0.186	0.278	0.355	0.379
${\gamma }_4$-Wsize	0.023	0.013	0.015	0.023	0.031	0.034
BMA log-likelihood	−63,193.493

. Since model 10 receives almost all of the posterior model probability, it is not different from the model averaged estimates.

5. Conclusions

In this paper, we used spatial panel econometrics techniques to analyze to what extent different linkages among firms influence the dependence between their stock market co-movements. We employed data on yearly returns for 800 firms across eight countries for the period ranging from 2005 to 2020.

The adopted methodology moves beyond simple models toward complex models that incorporate spatial interaction effects by implementing a novel spatial econometric method, i.e., the convex combination approach. Specifically, we employed the spatial autoregressive (SAR) convex combination model. It allows us to incorporate different weight matrices within one spatial model and combines them to form a single weight matrix. Through an examination of the associated scalar parameter, this approach permits drawing inferences about the design of spatial dependence and gives an overall assessment of the role played by each spatial dependency in analyzing stock returns’ co-movements of companies across different countries.

To quantify their closeness, several linkages among firms were employed: geographic proximity, bilateral trade, sector similarities, and company size. We calculated the correlation between these matrices. They were not highly correlated, guaranteeing accurate estimates of the model parameters. In the first step, we performed a series of spatial correlation tests and found that spatial models are more relevant than non-spatial models. In the second step, the SAR models with various matrices were estimated. Results affirmed the presence of spatial dependence. Its strength varies with the different weight matrices.

In the third step, we estimated the SAR convex combination model. We found strong evidence for spatial dependence indicating that firms’ stock returns commoved together. Our findings highlight that the weights of various matrices depend on the linkages between firms. Also, spatial proximity may be better quantified by financial distance measures than geographical distance.

Moreover, we noticed that the spatial weight matrix constructed based on sector similarities outperforms the other linkages utilized to capture the dependencies between companies. Firms belonging to the same sector are more connected than those operating in different sectors. They display a similar behavior because global input factors like commodity should have a similar effect on firms belonging to the same branch, regardless of where they are headquartered. This is attributed to the growing phenomena of globalization and financial market integration and the dominant role played by information technology.

In the last step, we estimated the SAR convex combination BMA model to test the robustness of our results. We found that our estimations are robust and consistent with the main finding.

Taken together, our estimates and the documented one-hour daylight-saving-time reversal effect reported by Mugerman et al. (2020)—which trims global returns for a single trading day—indicate that equity markets transmit shocks through two complementary channels. Structural, fundamentals-based linkages drive the persistent cross-market co-movement highlighted in this study, whereas transient behavioral disturbances, such as circadian-rhythm shifts, account for the short-lived synchrony observed at the daily frequency.

The results from this work have significant implications for investors and policymakers. Such an approach allows us to recognize the information that matters for international portfolio diversification, as well as financial contagion, international asset pricing, and measurement of systemic risk.

Our results are useful to investors to properly diversify their investment portfolios internationally. They need to know the factors that cause dependencies between stock markets. For international investors, the diversification across countries offers limited protection when portfolios remain concentrated in a single industry. Allocating across sectors, rather than across countries, provides a more effective hedge against common shocks. The results indicate that international diversification strategies should be based primarily on sectoral diversification rather than geographic diversification.

Likewise, our findings can help policy makers. The presence of spatial dependence suggests that any negative shock affecting one firm’s stock returns will rapidly spread to other stock markets. This leads to an increase in global systemic risk and financial contagion. Policymakers can benefit from this information to understand the drivers of the stock market co-movements and develop relevant policies that stabilize the financial system. From a supervisory standpoint, our results suggest that cross-border stress-testing frameworks should incorporate sector-specific scenarios. For example, a distress in the semiconductor supply chain can propagate internationally through sectoral linkages, even when macroeconomic conditions differ.

Finally, we must acknowledge the limitations of our work. Our sample focuses only on eight countries and eight hundred firms. It would be interesting to include other counties and firms to increase the sample size (specifically the N-dimension) to produce accurate estimates regarding the relative importance of various types of connectivity. Future research could extend this approach by incorporating additional dimensions, such as financial flows or real-time supply-chain linkages, to further elucidate the complex dynamics of global stock market co-movements.