Interconnectedness in the Global Financial Market

TL;DR

This paper investigates the interconnectedness of global stock markets by analyzing dependencies among stocks across countries and sectors, revealing core countries, influential sectors, and the volatility of these relationships.

Contribution

It introduces a network-based approach using GARCH and robust regression to measure and visualize global stock market interconnectedness and sectoral influence.

Findings

United States and Germany are central in the global market network.

Energy, materials, and financial sectors are key connectors, with increasing influence over time.

Market dependencies are volatile and exhibit significant heterogeneity among stocks.

Abstract

The global financial system is highly complex, with cross-border interconnections and interdependencies. In this highly interconnected environment, local financial shocks and events can be easily amplified and turned into global events. This paper analyzes the dependencies among nearly 4,000 stocks from 15 countries. The returns are normalized by the estimated volatility using a GARCH model and a robust regression process estimates pairwise statistical relationships between stocks from different markets. The estimation results are used as a measure of statistical interconnectedness, and to derive network representations, both by country and by sector. The results show that countries like the United States and Germany are in the core of the global stock market. The energy, materials, and financial sectors play an important role in connecting markets, and this role has increased over time for the energy and materials sectors. Our results confirm the role of global sectoral factors in stock market dependence. Moreover, our results show that the dependencies are rather volatile and that heterogeneity among stocks is a non-negligible aspect of this volatility.

Tables

Table 1. Table 1: Number of stocks by sector and country. Sectors are: Energy, Materials, Industrials (Indus.), Consumer Discretionary (Cons. D.), Consumer Staples (Cons. S.), Health Care (Health), Financials, Information Technology (IT), Telecommunication Services (Telec.), and Utilities (Util.). Countries are: Austria (AUS), Brazil (BRA), China (CHN), Spain (ESP), France (FRA), United Kingdom (GBR), Hong Kong (HKG), India (IND), Japan (JPN), South Korea (KOR), Holland (NLD), Singapore (SGP), United States (USA), Canada (CAN), and Germany (GER).

Country	Energy	Material.	Indus.	Cons.D.	Cons.S.	Health	Finan.	IT	Telec.	Util.
AUS	12	21	25	17	7	8	29	5	2	3
BRA	2	11	10	13	5	1	5	3	2	14
CHN	10	95	134	90	36	42	36	40	0	17
ESP	1	9	15	6	4	4	11	3	1	6
FRA	6	15	47	53	16	18	42	47	4	9
GBR	17	19	69	54	21	10	68	26	3	8
HKG	1	2	16	8	2	1	37	3	2	6
IND	7	125	105	116	37	28	21	51	4	6
JPN	3	35	125	115	20	17	24	157	3	1
KOR	2	91	101	102	27	35	11	125	4	2
NLD	2	3	17	8	9	0	12	9	1	0
SGP	0	0	11	3	1	1	18	2	2	1
USA	38	26	65	73	43	43	81	55	6	31
CAN	50	43	19	13	11	3	40	6	6	8
GER	6	11	50	29	8	22	15	33	5	4

Table 2. Table 2: Statistics for the Returns Time Series. We calculated the variance, kurtosis and the tail exponent from all returns in each country. In three of the markets, the maximum daily price change is constrained (caps), which permits the analysis of the tail exponent. The far right columns show the average correlation of stocks within a given sector is always larger than the average correlation of all stocks within a given country.

	$N$	$var\left(r\right)$	Kurtosis	Tail	Avg. correlation	Avg. correlation	Correlation
	stocks			exponent	within sector	all stocks	Ratio
Country
AUS	129	0.00064	18.7	3.47	0.35	0.18	1.94
BRA	66	0.00063	14.4	3.77	0.42	0.22	1.91
CHN	500	0.00105	5.0	–	0.27	0.25	1.08
ESP	60	0.00057	12.1	3.61	0.38	0.23	1.65
FRA	257	0.00059	26.4	3.32	0.22	0.14	1.57
GBR	295	0.00064	25.6	3.40	0.22	0.16	1.37
HKG	78	0.00082	18.0	3.35	0.44	0.20	2.20
IND	500	0.00114	8.5	–	0.31	0.25	1.24
JPN	500	0.00120	15.1	2.99	0.37	0.31	1.19
KOR	500	0.00124	8.0	–	0.36	0.27	1.33
NLD	61	0.00059	25.7	3.38	0.40	0.25	1.60
SGP	39	0.00052	38.1	3.37	0.48	0.29	1.65
USA	461	0.00065	24.2	3.25	0.24	0.21	1.14
CAN	199	0.00070	21.1	3.26	0.23	0.15	1.53
GER	183	0.00074	17.1	3.44	0.25	0.15	1.66

Table 3. Table 3: p-values for stock index Correlations versus Median of p-Values from Stock Correlations. Sector denominate values are obtained by first averaging over stocks within one sector, while stock denotes the median p-value of the correlation of all stocks between two countries. For Singapore and the Netherlands, several of the sectors are not populated and therefore do not have sector results.

		AUS	BRA	CHN	ESP	FRA	GBR	HKG	IND	JPN	KOR	NLD	SGP	USA	CAN	GER
Country
	index	0.0000	0.0000	0.0036	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
AUS	sector	0.0823	0.2202	0.6162	0.1357	0.1220	0.1869	0.1285	0.3215	0.3196	0.3571	–	–	0.1190	0.2547	0.1285
	stock	0.0303	0.1856	0.5673	0.0858	0.0993	0.1253	0.0718	0.3125	0.2848	0.3297	0.0481	0.0560	0.0861	0.1615	0.1058
	index	0.0000	0.0000	0.0008	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
BRA	sector	0.2038	0.0095	0.6370	0.0530	0.0852	0.1048	0.0748	0.2759	0.3910	0.2513	0.0616	–	0.0323	0.1516	0.0746
	stock	0.1774	0.0073	0.6129	0.0524	0.0903	0.0977	0.0635	0.2890	0.3657	0.2749	0.0342	0.0595	0.0284	0.1043	0.0812
	index	0.0018	0.0001	0.0000	0.0390	0.0070	0.0040	0.0000	0.0020	0.0097	0.0006	0.0111	0.0002	0.0118	0.0021	0.0026
CHN	sector	0.5655	0.5435	0.0000	0.7299	0.6068	0.6202	0.1665	0.5258	0.6351	0.4136	0.5710	–	0.6674	0.7197	0.6041
	stock	0.5333	0.5367	0.0000	0.7302	0.6150	0.6415	0.1964	0.5441	0.6586	0.4689	0.6305	0.3510	0.6723	0.6686	0.6219
	index	0.0000	0.0000	0.0826	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
ESP	sector	0.1344	0.0570	0.7534	0.0011	0.0163	0.0258	0.1169	0.2566	0.3311	0.2989	0.0097	–	0.0044	0.0968	0.0323
	stock	0.0870	0.0531	0.7557	0.0006	0.0205	0.0243	0.0825	0.2710	0.3163	0.2861	0.0031	0.0554	0.0050	0.0650	0.0275
	index	0.0000	0.0000	0.0094	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
FRA	sector	0.1505	0.1067	0.6852	0.0236	0.0361	0.0697	0.1252	0.2882	0.3002	0.3109	0.0186	–	0.0221	0.1218	0.0552
	stock	0.1173	0.1123	0.6817	0.0287	0.0367	0.0678	0.0983	0.3047	0.2819	0.3147	0.0130	0.0712	0.0335	0.1110	0.0546
	index	0.0000	0.0000	0.0101	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
GBR	sector	0.2002	0.1315	0.6983	0.0345	0.0542	0.0526	0.1603	0.3707	0.3379	0.3612	0.0359	–	0.0249	0.1469	0.0658
	stock	0.1310	0.1169	0.6883	0.0304	0.0611	0.0356	0.1199	0.3596	0.3342	0.3490	0.0120	0.0876	0.0189	0.1113	0.0583
	index	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
HKG	sector	0.1316	0.0844	0.2054	0.0931	0.0936	0.1179	0.0062	0.1318	0.2378	0.1527	–	–	0.0729	0.2223	0.0956
	stock	0.0648	0.0674	0.2215	0.0718	0.0752	0.1001	0.0007	0.1153	0.2022	0.1495	0.0348	0.0046	0.0632	0.1335	0.0741
	index	0.0000	0.0000	0.0063	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
IND	sector	0.2853	0.2375	0.6172	0.2144	0.2346	0.3141	0.1295	0.0018	0.5052	0.3788	0.1466	–	0.2573	0.3515	0.2297
	stock	0.2767	0.2573	0.6367	0.2370	0.2510	0.3057	0.1181	0.0025	0.5175	0.3890	0.1772	0.0986	0.2663	0.3224	0.2395
	index	0.0000	0.0000	0.0448	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
JPN	sector	0.3659	0.3911	0.7258	0.3404	0.2903	0.3597	0.2810	0.5842	0.0431	0.4657	0.1923	–	0.3135	0.4064	0.3113
	stock	0.3023	0.3790	0.7411	0.3186	0.2822	0.3542	0.2487	0.5825	0.0401	0.4655	0.2137	0.1875	0.3062	0.3477	0.2933
	index	0.0000	0.0000	0.0036	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
KOR	sector	0.3535	0.2897	0.5265	0.2886	0.2962	0.3630	0.1710	0.4299	0.4096	0.1015	0.2323	–	0.3335	0.4360	0.3005
	stock	0.3241	0.3063	0.5342	0.2932	0.2947	0.3392	0.1633	0.4333	0.4096	0.1093	0.2310	0.1574	0.3310	0.3962	0.3028
	index	0.0000	0.0000	0.0211	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
NLD	sector	–	0.0835	0.6929	0.0158	0.0205	0.0318	–	0.2056	0.2055	0.2464	0.0156	–	0.0137	0.0994	0.0309
	stock	0.0625	0.0497	0.7091	0.0049	0.0135	0.0146	0.0531	0.2420	0.2191	0.2671	0.0009	0.0248	0.0045	0.0413	0.0157
	index	0.0000	0.0000	0.0007	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
SGP	sector	–	–	–	–	–	–	–	–	–	–	–	–	–	–	–
	stock	0.0609	0.0703	0.4812	0.0586	0.0642	0.0842	0.0067	0.1251	0.1564	0.1603	0.0230	0.0007	0.0585	0.1251	0.0582
	index	0.0000	0.0000	0.0297	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
USA	sector	0.1435	0.0329	0.7274	0.0048	0.0158	0.0207	0.0993	0.3015	0.3114	0.3079	0.0099	–	0.0003	0.0559	0.0151
	stock	0.0803	0.0278	0.7221	0.0044	0.0225	0.0145	0.0683	0.3079	0.2900	0.3152	0.0023	0.0482	0.0001	0.0284	0.0168
	index	0.0000	0.0000	0.0056	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
CAN	sector	0.2674	0.1622	0.7475	0.1035	0.1137	0.1428	0.2351	0.4408	0.3898	0.4429	0.0982	–	0.0653	0.1161	0.1157
	stock	0.1646	0.1177	0.7112	0.0755	0.1044	0.1094	0.1468	0.3770	0.3431	0.3966	0.0389	0.1210	0.0368	0.0487	0.0893
	index	0.0000	0.0000	0.0066	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
GER	sector	0.1544	0.0955	0.7109	0.0376	0.0618	0.0675	0.1276	0.2914	0.3006	0.3407	0.0301	–	0.0238	0.1304	0.0416
	stock	0.1291	0.1105	0.7107	0.0382	0.0564	0.0667	0.1052	0.3056	0.2916	0.3330	0.0151	0.0657	0.0266	0.1010	0.0423

Table 4. Table 4: Summary Statistics for Sector-based Networks Calculated Using Dynamical Analysis. For each window we calculate the number of nodes, n 𝑛 n , average degree, ⟨ k ⟩ delimited-⟨⟩ 𝑘 \langle k\rangle , average clustering coefficient, ⟨ C ⟩ delimited-⟨⟩ 𝐶 \langle C\rangle , average shortest path for the giant component, ⟨ L ⟩ delimited-⟨⟩ 𝐿 \langle L\rangle , and density.

	Nodes	Degree	Clustering	p. Length	Density
	$n$	$⟨k⟩$	$⟨C⟩$	$⟨L⟩$
Window
Jul 06 – Jun 07	26	4.00	0.573	1.76	0.160
Jan 07 – Dec 07	50	7.12	0.730	2.50	0.145
Jul 07 – Jun 08	66	7.80	0.716	2.34	0.120
Jan 08 – Dec 08	80	9.50	0.734	2.81	0.120
Jul 08 – Jun 09	84	9.77	0.732	2.75	0.118
Jan 09 – Dec 09	53	10.32	0.744	2.61	0.198
Jul 09 – Jun 10	63	10.24	0.654	2.50	0.165
Jan 10 – Dec 10	60	10.78	0.683	2.33	0.183
Jul 10 – Jun 11	42	7.35	0.686	1.93	0.179
Jan 11 – Dec 11	71	13.45	0.729	2.57	0.192
Jul 11 – Jun 12	75	13.45	0.794	2.36	0.182
Jan 12 – Dec 12	37	8.03	0.706	2.27	0.223
Jul 12 – Jun 13	36	6.31	0.650	1.92	0.180

Table 5. Table 5: Estimation Results for Determinants of Network Structure on the Level of Stocks. We show the results for six different models. In each case we report the most important coefficients together with their Z-score. Country-based effects and regional segmentation explain large parts of the observed network structure while sectoral effects are also significant, yet on a smaller scale.

model	Base		West vs. East		Development		Sectoral diff.		Sector by sec.		Country by co.
AIC	5,376,507		5,269,102		5,276,926		5,369,822		5,370,925		4,763,994
nvar	25		26		26		26		79		144
	coef.	$Z$	coef.	$Z$	coef.	$Z$	coef.	$Z$	coef.	$Z$	coef.	$Z$
country effects	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$
sector effects	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$
match country	4.6307	(1064.9)	4.0338	(910.0)	3.9155	(857.7)	4.1530	(580.9)	4.6215	(1057.6)
match sector	0.1499	(47.1)	0.1516	(47.0)	0.1491	(46.3)	0.1388	(43.6)			0.1707	(50.0)
match region			0.7971	(328.3)
match devmnt.					0.8799	(319.9)
sectoral diff.							-7.1627	(81.7)
interact. sector									$\bullet$	$\bullet$
interact. country											$\bullet$	$\bullet$

Equations

$$h_t={\alpha }_0+\sum _{i=1}^q{\alpha }_i{\epsilon }_{t-i}^2+\sum _{i=1}^p{\beta }_i{h}_{t-i}.$$

$$r_{i,t}^f=\frac{r_{i,t}}{\sqrt{h_{i,t}}},$$

$$r_i^f={\beta }_{0,ij}+{\beta }_{1,ij}{r}_j^f+ϵ.$$

$$R=(1+{\bar{P}}^w)./(1+{\bar{P}}^d).$$

$$\displaystyle C=min\>\bigl{(}\>1\>,\>R+1-\langle diag^{*}(R)\rangle\bigr{)},$$

$$P=max\left(0,\left((1+P^d)\circ C\right)-1\right).$$

$$A_{ij}\propto (\gamma -p_{ij}).$$

$$s{d}_{i,j}=<|\frac{n_{s,i}}{N_i}-\frac{n_{s,j}}{N_j}|>$$

Full text

1 Introduction

The last decades have in general exhibited an increase in political and economic openness, mostly accompanied by an increase in financial integration. Analyzing and managing macroeconomic financial risks have become increasingly important as global markets have become increasingly connected. The sheer size of these markets and the number of products available require models that can provide a hierarchical or topological simplification of this system.

The financial and economic crisis of the past few decades have shown that contagion can often not be explained by dependencies in the real part of the economy. Distress of financial activity can by itself create contagion which consequences then go beyond the financial sphere. Numerous studies have therefore looked at dependencies of financial markets, especially if they amplify shocks in times of crisis. The transmission mechanisms in financial markets have been claimed to be relatively stable over time (Rigobon, 2003; Bracker et al., 1999), but this claim is debated (Corsetti et al., 2005). Studies that examine determinants of comovement find that structural similarity of the countries’ economies explains only partially the level of comovement of their financial markets. This resulted in a debate about the influence of global sectoral factors (Dutt and Mihov, 2013; Bekaert et al., 2009, 2011). Previous results hint at an increase in the importance of these factors. Forbes and Chinn (2004) find that cross-country factors and global sectoral factors both are important determinants of stock returns. They also note that changes in global linkages over time might make it difficult to disentangle different influences on asset market comovement.

Despite the size and complexity of global financial markets, most studies have used data sets that compare markets on a rather high level of aggregation, mostly in the form of indices. In order to uncover changes in interconnections over time it is however more promising to look at dependencies at the level of assets and their contributions to the comovement of markets and sectors. Networks can describe such large systems and they can also serve as a dimensional reduction.

Interconnections between financial markets play a dual role (see also Martinez-Jaramillo et al., 2019). On the one hand, they can absorb shocks and lead to greater robustness. They can also propagate shocks and create greater fragility. As such, measuring interconnectedness is of value and interest. The methods however, often depend on the availability of data and the kind of underlying financial or economic activity. Thus, measuring interconnections ranges from the use of direct exposures data, usually available only to supervisory and regulatory entities, to publicly available data like market prices. One popular approach, which we also follow, is to investigate the empirical correlation of asset returns and the resulting implied network structure. The resulting networks help to describe the financial system as a whole, its systemic structure, and flesh out possible contagious channels and transmission mechanisms. Thus, we focus on a measure of statistical interconnectedness, based on the econometric analysis of comovement patterns (see also Billio et al., 2012; Diebold and Yilmaz, 2014), which is modified to address a large sample size and cross sectional variations.

The results from this study contribute to the debate about the stability of interdependencies of financial markets and provide evidence of rather volatile relationships (see also Corsetti et al., 2005). Finally, even though the objective of the proposed methodology is not a classical factor model, we also discuss the presence of sectoral effects. We observe that these effects are difficult to model and describe only a limited amount of overall connectivity across stocks, which supports the findings for example presented by Forbes and Chinn (2004).

Apart from analyzing stock comovement, this paper builds on two additional strands of literature. First, it is related to the macroeconomic literature that discusses the effects of market openness on business cycle synchronization (see, e.g., Brooks and Del Negro, 2004; Buch et al., 2005; Claessens and Kose, 2018). These effects depend on similarities in industry structure, although this is often overshadowed by country-specific effects (Imbs, 2004). There are also studies on the transmission of shocks in a crisis situation. In such a situation, the determinants for spillovers can change relative to what is observed in normal times. See, for example, Fratzscher (2012) for an analysis of the 2008 crisis and Kaminsky and Reinhart (2000) for an analysis of crises in the 1990s.

Second, this paper is also related to the literature of network-based analysis of financial markets. In the economics literature the analysis of networks has gained importance since the 2008 financial crisis and the analysis of cascading effects in the financial sector. (see Summer, 2013; Gai et al., 2011; Glasserman and Young, 2016). The comovement of stocks as a network has been analyzed for some time by interdisciplinary researchers. In these approaches, the similarity of stock returns is interpreted as information about linkages between stocks and networks are produced either by applying planar filtering or correlation thresholds (Song et al., 2011; Gopikrishnan et al., 2001; Kenett et al., 2012a). These approaches deal well with the complexity of financial markets and have analyzed cases where the dimensionality is high. A weakness of these studies is that they are often of a more exploratory nature and that the number of links is often chosen ad hoc. We contribute to this literature by proposing an approach where the dimensionality is also high but where each of the connections in this network can be quantified by strength and significance, thus connecting the literature on networks to the literature in financial econometrics.

The remainder of the paper is organized as follows: Sections 2 and 3 describe the stock market data set, the methodology to measure interconnectedness in terms of dependencies between stocks, and how this information can be used to derive networks. Static and dynamic networks and an analysis of their properties are presented in Sections 4 and 5, respectively. In Section 6 we study the structure of the derived networks, and investigate the presence of geographic or sectoral factors.

2 Data

The data used in this study consist of the daily closing prices of stocks listed on exchanges in Australia, Brazil, China, Spain, France, the United Kingdom, Hong Kong, India, Japan, South Korea, the Netherlands, Singapore, United States, Canada, and Germany. We chose stocks that were components of a benchmark index, and were continuously traded with sufficient volume throughout the sample period.111Excluded were stocks that were exempt from trading for more than 10 days in a row as well as stocks without price movement and/or negligible trading volume for more than 8 percent of the total trading days. Also excluded were 5 stocks which at some point showed price behavior or market capitalization similar to penny stocks. Twin stocks were removed. Some countries had a large number of stocks that met our criteria. For this reason, we selected the 500 stocks with the highest market capitalization from exchanges in India, Japan, China, and Korea. We chose U.S. stocks that are in the S&P 500 index. For the UK, we selected stocks in the FTSE 350.222Data were obtained from Standard & Poor’s Compustat and Thomson Reuters Corp. Prices are in local currency. ’China’ refers to mainland China and stocks that are traded within mainland China. For the SAR of Hong Kong stocks are comprised of local stocks as well as other Chinese stocks which legally reside in Hong Kong. A list with the names of all stocks is available upon request. The Global Industry Classification Standard (GICS) sector designation for each stock was used, when available. When GICS was not available, we used the TRBC classification from Thomson Reuters. The number of stocks by country and sector is summarized in Table 1.

Our sample period starts on 1 July 2006 and ends on 30 June 2013. The eight years of data results in $T=1329$ trading days and $N=3828$ stocks. For stocks in all countries, we use data from trading days when the stock markets in London and New York are both open. Because the stocks are traded in different time zones there are limitations in the synchronization of the returns. The international date line in the Pacific necessitates that Asian countries finish trading first in the day, while the Americas finish last. The correlation of daily returns will naturally under-represent the amount of comovement between the most distant countries because trading takes place without an overlap in time. To address this issue, we use two approaches: To analyze the long-run effects, we calculated weekly returns for all time series, which leaves us with $T^{\prime}=365$ observations. To analyze short-run effects, we calculate a correction factor to use with the daily data; details are explained in Section 3.2.

The remainder of this paper uses the returns time series derived from the log price changes of the stocks, $r_{t}=log(p_{t})-log(p_{t-1})$. The number of stocks per country varies from 39 for Singapore to 500 for the larger countries (see summary statistics presented in Table 2). The markets in China, Korea, and India have imposed limits on the maximum daily price changes. In Korea, for example, the limit for the daily price movement was 15 percent. The distributions of the returns time series for these countries are truncated. This does not mean that the volatility is necessarily lower (see sample variances in the table). All time series of asset returns are heavy-tailed, as the values for the kurtosis indicates. The tail exponent was calculated with the Hill estimator, and the values are mostly slightly greater than 3.

To uncover dependencies between stocks in different countries by sector, it is necessary to test if stocks within a specific sector are more correlated than stocks from random sectors. The far right columns in Table 2 show the results from calculating the average of all within-sector correlations, the average of all between-sector correlations, and the ratio between the two. The average of the first is significantly higher for all countries except for China. The dispersion and level vary by country. In Japan, only two sectors show a higher than average between-sector correlation than the within-correlation (see also Figure 1 in the appendix). For the case of China this behavior can mostly be explained with the large influence of the state and significant limitations for foreign investors. Japan, on the other hand is special case among the developed markets. The presence of large horizontally integrated conglomerates and a slightly more inward looking economy are likely the reasons for its unique market structure (see also Chen et al., 2010; Lincoln and Gerlach, 2004).

3 Estimating inter-market and intra-market interconnectedness

For the following, it is important to note that the analysis of comovement across markets differs from the well understood properties of comovement within markets (see, e.g., Barberis et al., 2005; Green and Hwang, 2009). In the latter, one describes the behavior of individual stocks. This is not the case in the analysis of comovement across markets. Most approaches to studying comovement on the global level have focused on the analysis of stock market indices (see, e.g., Baur and Jung, 2006) or other smaller samples including sectoral indices. A wide range of methods has been applied, among these are unit root and cointegration tests, vector autoregression models, correlation-based tests (Forbes and Rigobon, 2002; Fry et al., 2010), causality tests (Billio et al., 2012), multivariate GARCH models (Engle, 2002), and models of variance decomposition (Diebold and Yilmaz, 2009). While advances have been made in applying these models to larger data sets they are still impractical for cases like ours (see also Aielli, 2013; Engle et al., 2019; Greenwood-Nimmo et al., 2015).

3.1 GARCH filtering process

Our approach is based on pairwise comparisons of stock returns. Although an analysis of the correlation of these returns can be informative, changes in volatility in all time series would govern the results, since volatility in financial markets around the world is largely synchronized (see Engle et al., 1990). The long memory in volatility would also complicate the assessment of significance bounds (see also Forbes and Rigobon, 2002).

We choose to investigate dependencies between stocks on a more general level and therefore apply a filtering to the returns. As a filter we use the conditional variance of a univariate GARCH model (Bollerslev, 1986). This process removes volatility changes and autocorrelation from the returns.

This means that we assume that the returns follow a random process with $\varepsilon_{t}=v_{t}\sqrt{h_{t}}$, where $v_{t}$ is white noise and

[TABLE]

We will make use of the conditional variance $h_{t}$ to calculate filtered returns333It should be noted that also the covariances of $h_{t}$ can be used to analyze interconnections, a short comparison of these two measures can be found in C. such that

[TABLE]

for all stocks $i$. We obtain time series with unit volatility.444Except for a few exceptions for stocks from developing markets, the GARCH(1,1) model fits very well and yields the expected coefficients for $\alpha$ and $\beta$. We checked the robustness of our results by omitting the 83 stocks for which the fit of the GARCH model was least satisfactory, but could not find any significant change on terms of the resulting p-values as described in section 3.3 (see also Hansen and Lunde, 1998, on robustness).

3.2 Estimation and correction for non-synchronous trading

By “de-garching” the returns, a time series is obtained that can be treated in an almost standard regression framework. Running a pairwise regression of all the filtered returns generates a measure for the comovement. The only econometric issue of these time series is that the residuals are not normally distributed, which we account for by using a robust regression (Lange et al., 1989) with t-distributed errors.555Note that the sample size would in principle allow to proceed from here with a standard regression analysis or the calculation of correlation coefficients. However, since we need reliable standard errors and significance bounds we rely on a robust regression framework. See also Figure 2 in the appendix for details on the distributions of returns and residuals.

To measure interconnectedness, we estimate pairwise the dependencies for all pairs of stocks $(i,j)$

[TABLE]

Next, we focus on stock-to-stock relationships that are significant with respect to a certain threshold. Therefore, in the following we will use the p-values that can be obtained from this estimation and refer to the elements $p_{ij}$ of the matrix of p-values. Its rows and columns are ordered by countries.666For the time horizons in this study, we do not observe significantly negative correlation.

To test the robustness of these results, which depend on the univariate de-garching, we compare our results with those of a multivariate GARCH model. These models can only be estimated with a limited number of time series. We use the Dynamic Conditional Correlation (DCC) model (Engle, 2002) with pairs of stocks from the sample and compare the average correlation implied by the DCC model with the correlation of our filtered returns. As expected, the results are indistinguishable for long windows. For windows of 190 days, the results on the stock level differ, but the differences are marginal and unsystematic (see B). We have also analyzed the distributions of our p-values and run simulations with permutations of our returns data to check for sensible significance levels and possible problems due to a multiple comparison problem. We found that the distributions of p-values are well-behaved and that an inflation of false positives caused by the large sample size is not to be expected, for details see figure 3 in B.

A challenging issue is the time difference in trading hours across markets. It leads to problems in determining the true dependencies between stocks from different markets. For stocks from Europe, this is a minor issue because UK trading time differs from other European countries by only one hour. The U.S. market opens before EU markets close, for a difference of six hours. Calculating the dependencies between stocks traded in the Americas and Asia is most problematic. The dependencies are biased downwards because the time series are asynchronous (see also Martens and Poon, 2001). In general, this can be dealt with in two ways: by using tick data and calculation of synchronous pseudo closing prices, or by time aggregation. The first method would require massive amounts of data and even then it would be difficult to find a specific time each day when (pseudo) price quotes for all stocks would be available. The second method is easier but limits the time resolution for our analysis.

In the following we will use weekly returns when we analyze the entire time period. An analysis of the dynamics of dependencies (on the time scale of months) however requires the use of daily data. We therefore propose a method that uses the difference between the average estimated daily and the weekly correlations to calculate a correction factor that ties the estimation results for the daily data to the weekly data results. This procedure is related to the works by Christensen et al. (2010) and Hayashi and Yoshida (2008), albeit much simpler and to be understood as an approximation.

We calculate a correction factor for the p-values in the following way: let $\bar{P}^{w}$ and $\bar{P}^{d}$ be $N\times N$ matrices where the elements are the average p-values on a country-to-country level (the matrix contains blocks with identical values resembling all pairs of countries). After adding 1 to each element of these matrices, we can calculate the element-wise (notation:$\>./\>$) ratio of the p-values of the weekly and daily estimates.

[TABLE]

Then, a matrix with correction factors $C$ can be calculated,

[TABLE]

where the last term corrects for differences in the p-values that are not due to non-synchronous trading times (we denote by $diag^{*}$ the diagonal blocks of $R$ capturing the dependencies between stocks within one country). The corrected p-values can then be obtained as

[TABLE]

The $min$ and $max$ ensure that all p-values stay within the $[0,1]$ band.

To summarize, this procedure assumes that overall estimates from the daily and the weekly data are similar and remaining differences are likely to be a result of non-synchronous data. The ratio of the slightly transformed p-values is used to correct for this issue so interdependencies that were otherwise discarded as, say just below the 90 percent confidence bound, will now be accounted for as significant if the correction factor for this country pair is sufficiently smaller than 1.

The calculated correction factors are presented in Figure 1. As expected, the correction increases with geographical distance, even though this does not fully explain the issue. The correction remains relatively small for distant, less-developed markets where the comovement is low anyway (which explains the wide range of the far right boxplots).

3.3 Visualizing and quantifying interconnectedness

In a financial network, entities such as banks, financial institutions, central counterparties, and traders are considered nodes. Their relationships in interbank lending, contractual obligations, and counterparty exposures define the links that connect them. Stocks and their relationships can also be expressed as a network. Each stock is a node and its estimated interdependencies show whether, and how strongly, the nodes are connected to each other. This information is stored by using adjacency matrices, where the entries in row $i$ and column $j$ indicate the strength of the connection between the respective nodes (see Figure 2).

Because the p-values contain this required information and their distributional properties allow a useful weighting, it is relatively easy to obtain adjacency matrices. The matrices of p-values are converted to adjacency matrices $A$ by first removing all entries where the significance level for the stock-to-stock dependence is below a certain threshold $\gamma$. In the following, we use $\gamma<0.1$ if not stated otherwise.777Our qualitative results do not depend on the particular threshold chosen, it is however necessary to allow for a certain range of p-values such that within-market as well as in-between-market interdependencies enter the following analysis. The estimated significance level can be used as a measure of connection strength by defining

[TABLE]

This adjacency matrix has a weighted positive entry if stocks $i$ and $j$ are significantly linked, measured by the estimated conditional correlation. This procedure of creating the adjacency matrix is in fact similar to the method proposed by Billio et al. (2012). Their p-values are however the outcome of a linear regression of raw returns in a Granger-causality analysis.888Note that Granger-causality should not detect any significant relationships in efficient financial markets. Hence, while its use in the above mentioned paper can be justified by market frictions in the time of crisis, it would be less useful in our analysis of comovement, since it would be impractical to dis-entangle the influences that lead to positive results, let alone to compare them over time.

The averages of the p-values of the relationships of the stocks in a specific sector in one country are then used with stocks in a specific sector in another country to map the sector-by sector dependencies.999The averages of the p-values are not p-values anymore. However, in this case it makes sense to use these averages because of low dispersion within the groups of stocks. Also note that in Table 3 median values are used for the comparison because the distribution of p-values for all stocks within one country is slightly skewed. On the sector level, however, the difference between the mean and the median is small, the latter value is mostly slightly higher and would tendencially lead to more significant links. The translation into a corresponding adjacency matrix is done in the same way as before but the dimension is reduced to $150\times 150$ (15 countries, 10 sectors). Sectors in countries that consist of fewer than two stocks are excluded from further analysis.

Finding useful visual representations of adjacency matrices is a complex process and the equivalent of finding a good dimensional reduction of a N-dimensional system, where N is the number of nodes. This process is also related to the problem of community detection in graphs, which is a high-dimensional clustering problem.101010See also the book by Newman (2010) for an in-depth introduction to Network Science. We graph our networks by applying the widely used algorithm developed by Hu (2005). This algorithm arranges nodes in a two-dimensional space in such a way that the total edge length of the graph is minimized, which shows the most pronounced communities of nodes within a graph. This specific algorithm uses the physics of repulsion to generate a visualization.111111We have verified that the qualitative results of the obtained visualizations do not depend on the choice of this specific algorithm.

In order to investigate the determinants that govern the connectivity in these networks we estimate different exponential random graph models (ERGMs, see Strauss and Ikeda, 1990). In these models the links and non-links between stocks are the dependent variable and we estimate the influence of fixed effects on the country and sector-level as well as interaction effects on the likelihood of a link between pairs of stocks. While the estimation results can be interpreted similarly as those of a logistic regression, for numerical reasons these models employ simulated maximum likelihood.121212For an in depth treatment of ERGMs we refer the reader to Lusher et al. (2011).

4 Static analysis of global interconnectedness

We first apply the proposed methodology on the entire sample period, to which we refer as a static analysis. The results obtained from analyzing the weekly returns are depicted in Figure 3. Stocks from Western markets form a hairball in the middle of the network, showing they are highly interconnected. Within this hairball, the mixing is strongest within the European markets, while otherwise regional structures remain visible. The markets of China, Japan, and India are only loosely connected to the central component of the network. These results are in line with earlier studies based on market indices by, for example, Bracker et al. (1999) and also with earlier network-based approaches like Kenett et al. (2012b), where simpler correlation measures were used. The new approach however now allows to observe the interconnectedness across especially the Western countries in more detail. In particular, the proposed methodology provides the means to identify cross-market relationships on a finer grain, namely to identify specific sectors and their centrality in the financial network. We have compared these results based on sectors with those for the stock level interactions (see Figure 1 in the appendix), and have verify that the aggregation procedure does not influence our findings.

It is possible to observe that even the most central sectors in the densely connected middle of the network are still mostly grouped next to sectors from the same market, which is a sign of remaining regional segmentation. When we search for exceptions from this pattern we find examples like the German materials sector or the British and French energy sectors. These nodes are surrounded by related sectors from other markets which hints that in some instances sectoral effects can be as strong as market or regional effects.

The connectivity in this network can be summarized by counting the number of links between sectors and between countries. The results are shown in figure 4. The left panel confirms the visual impression of a very connected U.S. stock market. Most sectors in European markets comove in a significant manner. The right panel shows clear differences on the sector level. Stocks from the financial, industrials, materials, and energy sectors show more interconnections than stocks from other sectors, and this is consistent for all countries.

While we obtain qualitatively similar results from the stock versus the sector level, we can show that this not true if one reverts to stock market indices. We illustrate this by applying the same estimation methodology to stock market indices for the 15 countries. The weekly index returns are de-garched and p-values of the pair-wise estimated dependencies are presented in Table 3. The results are compared with the median of the p-values for the stock-wise analysis and the median when we hierarchically average within sectors and then overall sectors within a country. While the indices are all highly dependent, the stock level shows lower median p-values, indicating pronounced heterogeneity in stock-to-stock dependencies, which are in no way captured by a market index analysis. This means that the findings about market comovement in studies of indices like Hamao et al. (1990) should be interpreted carefully, since they refer to one particular definition of the ‘market’.

[FIGURE:]

5 Dynamic analysis of global interconnectedness

During the eight years covered by the data set, global economic and political events occurred that are likely to lead to significant fluctuations in comovement across markets. The 2008 financial crisis, the euro crisis, and Japan’s 2011 tsunami are a few examples. A dynamic analysis is also necessary to show changes in global financial integration. The data resolution can characterize stock market dependencies in the order of months. We perform a rolling window approach that uses 190 days of data in 13 time steps with a 95-day overlap. We apply the same methodology as before now to daily data with a timing corrections for the p-values as described in Section 3.2. Additionally we will look at the development of some measures for the network structure.

Figures 5 (a–d) display 4 of the 13 resulting networks. Interestingly these representations show pronounced deviations from the long-run representation presented in figure 3. The number of clusters, their relative position, and also the number of sectors (nodes) vary even in tranquil times. The network in the top left panel of Figure 5 is structurally different from the others. US and European stocks form individual groups within the network, although there are multiple links between them. Some Asian markets are loosely connected to the European market. Chinese stocks show strong internal comovement but are not significantly linked to other markets.

The network changes significantly at the end of 2008 with the growing financial crisis. Stocks from all developed markets form one connected component. Some grouping into American, European, and Asian stocks remains, but the borders blur. The markets of South Korea, China, and India appear as weakly connected satellites. In fact, this network looks in many parts similar to the one presented in Figure 3, but the connections are more dense and incorporate more sectors and countries. One can argue that this time period shows strong signs of contagion, since the level of dependencies has clearly shifted upwards, compared to both the previous and the long-run state. One could further refer to this period as that of global contagion, since the stock market network is a bloomed up version of its long-run comparison.

The bottom left panel of Figure 5 shows that in 2011 there is stronger comovement within Asia, these markets are almost separated from the European-American component. Some remaining links now run between American and Asian markets and Australia, while the comovement between European markets becomes more heterogeneous. Its nodes have also lost connections to the Asian markets. This may reflect the Euro crisis, which affected some but not all European countries. This crisis induced shifts in the structure of the network, but does not (at least at this time-scale) lead to clear signs of contagion. By 2012, the network shows comovement almost back to the pre-2008 levels. Some stock markets are very much connected to U.S. stocks, but many sectors are no longer part of this connected core.131313For comparison, networks with links aggregated on the country level are shown in Figure 2 in the appendix.

For a deeper investigation into the origin of this variation, we study the number of links between markets on a country and sector level, similar to Figure 4, but with the addition of the time dimension. The top panels of Figure 6 show that the number of links between stock markets is low at the beginning and at the end of the sample period. It is possible to observe two peaks where many links exist between markets, at the end of 2008 and again around 2010/2011.

There appears to be some synchronization in the dynamics of the number of links, both by country and by sector. It is possible to calculate the fractions of links for each time window, which are presented in the bottom panel of Figure 6. This normalization can help detect shifts in the relative influences of specific countries or sectors. The country-wise view in the left bottom panel shows that the relative number of links is stable for most countries. Only the UK and Hong Kong show fluctuating behavior. The number of links for the United States is steadily increasing towards the end of the sample period. The bottom right panel with a sectoral analysis shows more interesting developments. Only in 2007 and 2008 is the financial sector a driving force for interconnections. Much more dominant are links with the energy and the materials sectors, which gain influence throughout the sample.141414Due to the low overall number of links, the values (fractions) for the first time window are noisy and discarded from our discussion. Stocks related to consumer goods become less important over time and the health industry is not relevant for the entire sample period.

The ranking of the most-connected sectors varies with the data frequency. For example, the analysis based on weekly data showed financial stocks had the most links, followed by industrial stocks. When daily data and shorter time windows are used, stocks from energy and materials sectors dominate the linkages. That indicates fast-moving energy and raw materials markets influence stocks in these sectors more than the factors responsible for comovement with the financial and industrial sectors. The results demonstrate the existence of sectoral influences. While the extent of comovement fluctuates, their relative influence shows some stability. The frequency of the data has a significant influence on the visibility of these sectoral influences. The speed and degree of stock comovements across sectors and across borders varies across time horizons.

Our findings therefore support findings by, for example, Corsetti et al. (2005) who argue that the level of comovement between markets is changing significantly over time, even when taking into account varying levels of market volatility (see also Forbes and Rigobon, 2002).

We can also quantify the changing comovement with network statistics. Table 4 illustrates that the number of nodes with statistically significant relationships fluctuates between 26 and 84. The average number of links of each node, the degree, is related to the number of nodes. This relationship is roughly linear but has significant variation. This indicates that the network is not just undergoing size but also structural changes. This is confirmed by the clustering statistics. Clustering measures how often two nodes B and C, which are both connected to node A, will also be connected to each other. All networks show clustering, the highest levels are observed in 2011. This shows that the networks do change along several dimensions and even in times of crisis not everything is always connected to everything else. This is confirmed by the changes in the average path length. Their changes are not only caused by network size changes but also by the contractions and diversions of single components of the network over time. The density describes the fraction of existing versus possible links between nodes. These values are high because of many links between sectors within countries that are rather stable. We see the highest values during the financial crisis, when the network is large and very connected, and in 2012, when the network is smaller and dense. These findings indicate fundamental changes in the network structure over time which, as we will also see in the next Section, cannot be fully explained by the country-level or sector-level influences described above.

6 Determinants of network structure

The visualizations of the stock dependencies give the impression of a network structure that is governed by country-wise grouping. In the following, we analyze different possible hypotheses that can explain this structure. Furthermore, we investigate the extent to which sector-related effects can help to explain the observed interdependencies.

We estimate different exponential random graph models to compare the hypotheses. The ERGM is employed with the unweighted undirected networks on the stock level. Although the estimation technique is rather involved it helps to realize that the results of this estimation can in fact be interpreted like the outcome of a logistic regression. Hence, our dependent variable are the links in our network and we estimate what influence certain variables have on the probability of having a link between pairs of stocks with certain characteristics.

We start with a baseline model, in which the level of links depends on the country and the sector that the stocks belong to. Further, we estimate the additional effect of pairs of stocks that are part of the same country and the same sector. We find that such a baseline structural model explains a good amount of the network structure. All country and sector effects are highly significant. The largest coefficients on the country level are estimated for the U.S., the Netherlands and Hong Kong. On the sector level the effects are largest for the sectors energy, financials and materials (these are effects on connectivity that are not explained by the number of stocks). The variable for within-country links is highly significant, as table 5 shows. Within-sector links are also significant, yet the effect is one order of magnitude smaller. Next, we analyze which additional determinants might improve the overall fit of the model in terms of the model AIC.151515Due to the large number of observations the AIC and the BIC are in this case almost identical. In all cases the null deviance is 10,154,437 and the AIC/BIC are very close to the residual deviance (minus compensation for the number of variables).

[FIGURE:]

In the first model specification, we assume that regional aspects might help to explain the number of links. The simplest assumption is segregation into countries of the ”west” versus the ”east”.161616BRA, ESP, FRA, GBR, USA, CAN, GER vs. CHN, HKG, IND, KOR, SGP, JPN. Alternatively we have grouped the countries into three groups, ”America”, ”Europe”, and ”Asia”, with almost identical explanatory power. The estimated effect for links within these groups is in fact highly significant and improves the overall fit. Similar results were obtained by Bracker et al. (1999). Alternatively, in our second model specification, we group the countries by economic development, i.e. developed versus developing countries.171717AUS, ESP, FRA, GBR, JPN, NLD, USA, CAN, GER, HKG vs. BRA, CHN, IND, KOR, SGP. The effect and significance of this variable is similar to the regional grouping.

As a third model specification, we focus on an approach to relate differences in linkages between countries to differences in industrial composition (see, e.g., Griffin and Karolyi, 1998). In our case we can measure the difference between countries by the relative number of firms $n$ in each sector $s$. We calculate a proxy for sectoral differences by calculating the average absolute difference of the fractions in each sector, so that for two countries $i$ and $j$181818Using squared differences leads to almost indistinguishable yet slightly less significant results in the estimation.

[TABLE]

When estimating a model that employs these sectoral differences as a covariate for the links, we find that it is significant, yet not on the same level as the previous grouping variables. It should be noted that we only capture effects from composition here, overall differences in connectivity between sectors or size effects are already being accounted for. As a robustness check, we have tested if this changes if we omit the country-fixed effects. Yet, in this case the $Z$-score falls to 60.5 and the AIC worsens to 6,681,810.

In two additional model specifications, we investigate two extreme cases, namely how well we explain the network structure when we estimate effects for all sector-to-sector or all country-to-country interactions. The results for the first case show that sector interaction effects on the global level achieve only a very small improvement in AIC compared to the baseline model. Hence, while the levels for different sectors matter, the interaction effects provide only very little additional information. On the other hand, when we estimate interaction effects for all the countries, the improvement in the AIC is very noticeable. This is of course mostly to be expected, given the number of variables and the structure of the data. Nevertheless, it also shows that effects additional to those that we have just tested are present in the data and that most of those manifest itself on the country-to-country level (a further visualization of the sector-by-sector linkages is provided in appendix E).

Finally, we investigate how the performance of these different models changes when we estimate them for the 13 time windows that were discussed in the previous section. The AICs for the models for each time window are presented in Figure 7. First, we observe that the performance of the models mostly changes together across time windows. All models perform worse during the times of the financial crisis of 2008 and during the European debt crisis than in ”normal” times. Moreover, in these normal times also all of our models perform much better than in the estimation conducted for the entire time period. The only major shift in relative model performance is then in fact observed at the peak of the European debt crisis.191919Plans for a referendum for Greece to leave the EMU we made in October 2011 yet factually abandoned by the end of 2011. At this point the models that propose segregation between the western or developed markets and the rest perform noticeably better than the base model. This also confirms our impression from the network visualizations in the last section that hinted at stronger comovement within the Asian countries in 2011.

These results hint at two things. First, not surprisingly, we can see that the European debt crisis was less of a global crisis than the crisis of 2008. Secondly, the underlying structural properties of market interdependencies are getting cloaked by different temporary effects in the times of crisis. Hence, very long sample periods do not necessary help in the identification of determinants of comovement. For the case of sectoral effects our results tend to support findings by Griffin and Karolyi (1998) and Dutt and Mihov (2013). Effects from sectoral composition do exist, yet their influence is not decisive in explaining the variations in stock interdependencies. We also observed that the network structure shows significant interaction effects on the country-by-country level, while the sector-by-sector level shows only little of such effects. This finding explains why cross-country factors are much easier to identify than sectoral factors and mirrors the findings of Forbes and Chinn (2004) and other factor-model approaches.

7 Conclusions

This paper presents an empirical investigation of comovement, as a statistical measure of interconnectedness, across 15 representative stock markets. We present a framework to measure such statistical interconnectedness on different levels of granularity - the stock, the sector, and the country level. The analysis shows that for asset markets, a global financial market, in terms of cross-country statistical interconnectedness, exists only within certain limits. We observe a mixture of regional and global sectoral effects, with the balance between the two fluctuating over time. For most of the time, regional segregation remains visible, even though in times of stress markets contract to a unisonous behavior. During such contraction periods, some countries still retain a high level of autonomous behavior.

Previous research has focused on factors that determine the level of stock market comovement for various countries. The results, however, were not always as convincing as similar studies on the synchronization of business cycles. Our findings shed some light on this debate: the fine structure of stock market comovement shows significant time variation. Sectoral effects do exist, but they influence only parts of markets. These effects are also overshadowed by bubbles or collapses such as the financial crisis. Tracking the sectoral interconnections over time, we find for example a shift from the financial sector, to the materials and energy sectors. A similar effect has been found during the dot-com bubble of 1997-2000 (see also Imbs, 2004; Raddant and Wagner, 2017). The observation of these shifts help explain why it is difficult to identify determinants for comovement as well es sectoral factors.

Our results also hint that the dimensional reduction on the level of sectors preserves qualitative features of the global stock market, like sectoral shifts, that can be lost in an analysis that is based on market indices. While indices’ comovement is high and persistent, stock and sector-based comovement show more heterogeneity, which is not averaged out in the aggregate.

In general, the methodology presented here can be used to quantify statistical interconnectedness among a large number of assets in many markets. We quantify within and between market interconnectedness, and use network theory to present, quantify, and monitor these relationships and how they change in time. The model presented here is a way to approximate asset-specific risks. First-order risks arise from direct interconnections on the asset level. Second-order risks arise from interconnections on the sector level. Both risk channels must be considered when analyzing the risks for an assets based on regional, global or systemic changes.

Acknowledgments

The authors are grateful for discussions at the Kiel Institute’s staff seminar, the staff seminar at Kiel University, the Center for Polymer Studies at Boston University, and the participants at the NBER summer institute and the conferences of the EEA, CEF, Bank of Finland and WEHIA. MR is grateful for partial funding of this research from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 619255. The article is also part of a research initiative launched by the Leibniz Community. This paper benefited from helpful comments by Mark Flood, Shlomo Havlin, H. Eugene Stanley, Stacey Schreft, Charles Taylor, Julie Vorman, and Peyton Young.