Time-varying correlations and interrelations: Firm-level-based sector evidence

Using firm-level data, we examine stock market correlations and interrelations for the G7 over the period 2000–2013. An examination using aggregate market data supports the view that correlations have risen and particularly so during crisis periods. Using firm-level data, which is tradeable, we establish sector portfolios. We consider three regression approaches. The results support, first, that correlations using firm data are lower than those observed using aggregate market index data. Second, the most important driver for home sector returns is the home market followed by the corresponding US sector. Third, correlations rose during the crisis but have stabilised and even fallen since. This supports the view that markets fall together but rise apart. Fourth, there is evidence that most sector correlations follow a market-wide component, but some sector correlations follow their own component. Subsequently, we examine the key drivers of time-varying correlations. We find that the market-wide component of correlations increases in a US bear market as well as with higher US market volatility and lower US interest rates. However, on a sector basis, there are notable exceptions with some correlations falling in a bear market. Together, these results support the view that diversification benefits remain across market sectors.


INTRODUCTION
This paper examines the nature and determinants of time-varying correlations between the G7 markets over the period beginning from 2000, which captures the large market downturns following the dotcom crash and the financial crisis. In doing so, we pay attention primarily to sector-level correlations, which, in turn, are based on individual firm stock price data, as opposed to aggregate market-level behaviour. We view it important to analyse data based on stocks that can be traded as opposed to index data that cannot. The results presented will, therefore, carry greater information and be of increased relevance not only to academics but also market practitioners. Of note, this paper will show that while correlations have typically increased over the sample period and that movements in correlations are linked to the behaviour of the US market, there remains notable exceptions to this that would allow for diversification opportunities.
The common consensus is that over time correlations between markets have increased. This is largely due to deregulation within financial markets as well as the removal of trade barriers (see, for example, Roll, 1989;King et al, 1994;Longin and Solnik, 1995;Rangvid, 2001;Goetzmann et al, 2001). Further, the literature typically supports the view that correlations rise in periods of market stress (see, for example, King and Wadhwani, 1990;Erb et al, 1994;Longin and Solnik, 1995;Karolyi and Stulz, 1996;Forbes and Rigobon 2002;Opschoor et al, 2014;Karanassos et al, 2016). Given this, an examination of how correlations have varied over the recent past, which includes two notable periods of market stress, is warranted. 1 To date, the majority of the literature uses aggregate market-level index data. While this provides an overview of market behaviour, such index data are not traded and thus may mis-represent the actual strength of correlations. As an exception, Berben and Jansen (2005) and Fasnacht and Loubergé (2007) examine sector-level data, although again using index-based data.
In this paper, we construct sector portfolios using firm-level data for the G7 markets. We then examine the relation between these sector returns with the corresponding USbased series. In addition to sample correlations of the data, we examine regression-based analysis using a CAPM approach, a bi-variate GARCH model and a principal components analysis. Once we establish the nature of the time-varying correlations, we then consider a set of regressions designed to explain such time-variation. In particular, we are interested in considering both the general movement in correlations as captured by the principal components as well as the behaviour of the individual sectors. Notably, we wish to examine whether correlations change with conditions in the US stock market. The paper contributes to the existing literature in three ways. First, as commented, our analysis is based on firm-level data as opposed to index data that are commonly used in the literature. Second, our analysis includes the twin market downturns experienced during the 2000 s as well as the subsequent market recoveries. This offers us a unique opportunity to examine how correlations vary between bull and bear markets. Third, we seek to examine the drivers of time-varying correlations and, in particular, test the view that movements in the US economy and stock market dominate in determining correlations.
The results generally suggest that diversification opportunities remain. Firmlevel-based correlations are lower than those reported for the aggregate market. Moreover, correlations appear to fall during bull markets, but do rise in bear markets. Nonetheless, there is evidence that while most sector correlations appear to follow a market-wide component, some sector correlations follow different components. Furthermore, there is evidence that while the financial crisis has raised correlations across all sectors, some sectors respond differently to movements in the US economy as well as movements in market returns and volatility. Again, such differences will provide diversification opportunities. It is hoped the results here are helpful to both portfolio managers in attempting to obtain diversified portfolios and academics interested in modelling market behaviour.

DATA AND DESCRIPTIVE EVIDENCE
We collect individual stock price data for the G7 markets over the time period 3rd January 2000 to 31st December 2013. The data are obtained from Datastream and in addition to the price data, we obtain information on the market sector. The individual stocks are then sorted into sector portfolios, which are used in the analysis below. The individual stocks are obtained from the main indices in each market, and thus, it allows comparison with previously reported aggregate market index correlations. The aim in the data collection exercise is to obtain a number of firms that is broadly equivalent across the markets but without potentially including very small firms. We take firm data for all available firms in the S&P500 for the US, the FTSE350 for the UK, the DAX, MDAX and SDAX for Germany, the SBF120 for France, the MIBTEL for Italy, the TTOCOMP for Canada and the Nikkei225 for Japan.
To provide a benchmark level of correlation between the G7 markets and an initial view as to how correlations have changed over the recent past, we examine sample correlation coefficients. As such, Table 1 presents the correlations between the returns of national markets (index level data) of the G7 countries over the sample period 1973-2012 on a monthly basis. 2 As can be seen from this table, correlations are high or reasonably high. Notably, the correlation between the two North American markets is high, as it is between the European markets themselves as well as between the European markets (especially the UK) and the US. Relatively lower correlations are found for Japan and, to a lesser extent, Italy.
Analysis of these correlations is further enhanced by considering two sub-samples of the data. First, by examining the correlations over the period 1990-2012 and then 2006-2012, reported in Tables 2 and 3, respectively. The use of samples with later starting dates will allow examination of whether correlations have risen over the more recent period compared to one that captures a longer history. The results in Tables 2 and 3 document the previously noted phenomena of rising correlations between these advanced markets. This is particularly true for the grouping of markets noted above, i.e., between the two North American markets and the European markets both within themselves and with the North American markets (again, most notably with the UK). Furthermore, the correlations are noticeably higher in the latter 2006-2012 period compared to the other sample periods. The 2006-2012 sample covers the financial crisis period, and hence provides a result Notes: Entries are sample correlation coefficients for monthly stock index returns data over the stated sample period.

Time-varying correlations and interrelations
consistent with the established view that correlations increase at times of market stress. These results thus suggest that gains from diversification have diminished over time and more so during a bear market. Set against this prevailing view, we examine the stock return correlations between the firm-level-based market sectors in order to examine whether the correlations at this more disaggregated level follow the same pattern of behaviour. As noted above, these data are not obtained by taking sector-level index data but by obtaining individual firm-level data, with the sector portfolio then constructed. Thus, the correlations here can be obtained by investors. Table 4  The results in Table 4 show the average correlations across the six markets of Canada, France, Germany, Italy, Japan, and the UK compared with the US. Immediately apparent is that the correlations at the sector level are lower than at the market level. This suggests that diversification benefits remain. However, it is equally noticeable that correlations have increased from the first half to the second half of the sample (i.e., from Periods 1 and 2 to Periods 3 and 4). Although it is also noticeable that the rate of increase has declined from Periods 3 to 4 and in five cases the correlation itself has declined. Nonetheless, this result supports the view that correlations rise in crisis periods. It is also noticeable that the majority of correlations decreased (albeit some marginally) from Period 1 to Period 2. The period 2004-2006 is marked by the recovery from the dotcom crash and equally supports the view that bull markets are more likely to be associated with Notes: Entries are sample correlation coefficients for monthly stock index returns data over the stated sample period. Notes: Entries are sample correlation coefficients for monthly stock index returns data over the stated sample period.
falling correlations and a bear market with rising correlations. In terms of the more specific results, we can see that 17 of the 21 sectors have a correlation below 0.5 by the end of the sample. Furthermore, while the correlation of some sectors has increased markedly (e.g., beverages, healthcare and mining) for other sectors, the increase in correlation has been muted (e.g., electricity, technology hardware and telecoms).

REGRESSION-BASED EVIDENCE
To examine the nature of interrelations and correlations in greater detail, we consider three regression-based approaches. First, we consider a market model-type approach and estimate the following equation for each sector and non-US country: where r s,i,t is the return for sector s on market i at time t, r i,t is the return on market i, with the equivalent values for the US denoted USA. Here we are interested in whether the US variables, both sector and market, dominate the home country market in determining the home country sector returns. The results are presented in Table 5. These results suggest an interesting pattern that occurs across all sectors. First, the largest coefficient is attached to the home market. This suggests that the key risk factor for sector stock returns remains the home market return and thus the domestic economy. The corresponding US sector is then more important than movement in the US market as a whole. In considering the individual sectors, we can see that the coefficient magnitude arising from the corresponding US sector is low for many of the sectors (e.g., healthcare, real estate, telecoms), while it is larger for several (e.g., chemicals, industrial metals, technology hardware). Notably, those sectors with a higher degree of conditioning from the corresponding US sector are more open to trade (e.g., automobiles). It is also of interest to note that there is a positive correlation of approximately 0.5 between the home market b and the coefficient on the US sector. This suggests that the riskier sectors are more affected by international markets. Subsequently, we estimate a bi-variate DCC-GARCH model (Engle, 2002) for each market sector with the corresponding US sector. This allows us to obtain the timevarying correlation for each market with the US. The DCC-GARCH model builds upon the constant conditional correlation model of Bollerslev (1990), where the conditional covariance matrix is expressed in terms of the following decomposition: where D t refer to the diagonal matrix of the conditional standard deviations, and C t is the matrix of conditional correlations. Bollerslev (1990) assumes that the correlations were constant, i.e., C t = C. To estimate this model, individual GARCH(1, 1) processes are estimated for each series with the standardised residuals (n t ) computed in the usual way: With the correlations given by, While the assumption of a constant correlation may be useful in certain circumstances, in the analysis here it is not of practical relevance. Hence, we implement Engle's extension whereby the conditional correlation is allowed to exhibit time-variation in a manner similar to the GARCH(1, 1) model. Specifically, conditional correlations are allowed to fluctuate around their constant (unconditional) values as such: where Q is the time-varying correlation matrix. The estimated correlations are standardised, to ensure they lie between -1 and 1. This also ensures both a positive definite matrix as well as readily interpretable correlations. 3 Our interest here lies is whether the estimated correlations have trended upwards over time. To examine this, Tables 6 and 7

Time-varying correlations and interrelations
given its generally lower correlations as noted in Table 1. We consider two types of panel unit root test. Both tests are based on the same principle as the Dickey-Fuller unit root test, with a null hypothesis of non-stationarity (random walk with drift). First, we consider the test of Levin, Lin and Chu (LLU, 2002) that assumes a common unit root process across the different markets, while second, we consider the tests of Im, Pesaran and Shin (IPS, 2003) and Fisher (1932) that allow for individual unit root processes. The results in both tables support the view that correlations are stationary and thus mean reverting. This implies that they are not drifting consistently towards the value of one. A closer examination of the results shows that if we consider the LLU test only, then evidence of non-stationary behaviour does exist. Looking across all sectors and markets, for 11 sectors the null hypothesis is not rejected, while for the tests that exclude Japan, 17 of the sectors exhibit non-stationary behaviour. However, as noted above, this test is more restrictive in assuming a common autoregressive parameter across all markets. The tests that allow the autoregressive parameter to differ across markets all reject non-stationarity. A panel estimate of the trend component in the correlations is positive (and significant) throughout but very small in magnitude.
The above results all examine the relation between the same sector across different markets with respect to the US market. To consider a different approach, we examine whether there is similarity in these sector correlations across all sectors. Hence, we are interested in whether there exits common movement across all sectors that could be more ascribed to a general market movement. To do this, we consider a principal components analysis of the above time-varying correlations between the non-US G7 markets with the US. Principal component analysis allows us to extract common factors (components) from a group of data series. The components are ordered according to how much of the variation across the series they can account for and are orthogonal to each other, thus representing independent information.
The results of the principal components analysis are reported in Table 8. The evidence reported here demonstrates that the first three principal components account for 80 per cent of the variation in correlations across the 21 sectors. Indeed, the first principal component accounts for just over two-thirds of the movement in correlations across all sectors. Furthermore, we can examine the factor loadings to determine how each sector is related to the main principal components. Examining the factor loadings, we can see that for the first, most important, principal component all sectors have an equivalent loading with the exception of technology and telecoms. Instead, these two sectors dominate the third principle component. This supports the view that movements in correlations are typically driven by market-wide events; however, these events do not affect all sectors and thus there remains the potential for portfolio diversification.

EXPLAINING MOVEMENTS IN CORRELATIONS
This section seeks to examine the causes of movements in the firm-based sector-level correlations between the US and the remaining G7 markets. In order to do this we will use the information based on the timevarying correlations obtained from the DCC-GARCH model. In addition, we can also use the results from the principal components exercise to help in understanding why correlations vary over time.
Therefore, we consider two regression approaches to explain the nature of timevariation within the correlations. In the first set of regressions we use the obtained principal components as the dependent variable, while in the second approach we use the correlations obtained from the DCC-GARCH model. In order to explain the movement in correlations, we consider two sets of explanatory variables. First, for the regressions using the obtained principal component variables, we consider the following. A dummy variable designed to highlight the crisis period and thus takes the value of one in the years 2006-2009 inclusive and zero elsewhere. A second dummy variable that represents whether the US market is in a bull or bear state. To obtain this, we take a three-year moving average of the market index and assign a value of one if this is increasing and zero if this is decreasing. We also use a 10-year Treasury bond rate and the standard deviation of the US stock market index, both of these variables enter with a one period lag and will provide a proxy for the state of the US economy and the riskiness of the US market, respectively. Second, for the regressions based on the individual sector correlations, we use the same explanatory variables but expand it to additionally include each individual market's return and standard deviation as well as the specific sectors return and standard deviation.
Examining the results of the first set of regressions, using the principal components as the dependent variable, we can make the following conclusions based on Table 9. For the first principal component, which captures the largest variability across the data, we can see that correlations increase during the financial crisis, when the US market is in a bear state, when US long-term interest rates are failing and when the variability of the US market increases. These results thus accord with the general view from the literature that correlations rise during periods of market stress. Hence, correlations rise during the financial crisis, when the US market is falling and becomes more volatility and when interest rates are lower (low interest rates may signal recessionary conditions, resulting in a capital outflow from the US). Notes: Entries show the first three principal components and the proportion of the variation across all sector correlations that they account for. Entries also report the factor loadings for each sector on the first three components.
The results from the second principal component are largely similar with correlations higher with a declining and more volatile US market and lower US interest rates. The only difference concerns the financial crisis dummy, which is now negative. While the first principal component has a positive factor loading across all sectors (except technology hardware and telecoms), the second principal component has a reasonably sized positive factor loading across only several. One explanation for the negative coefficient on the financial crisis dummy may be that these sectors were either less affected by the crisis (e.g., food and software) or received special government help (e.g., automobiles and banks). For the third principal component, for which only technology hardware and telecoms have a positive factor loading, we see the correlation increase with an increasing US market as well as higher interest rates, in contrast to the previous results. Moreover, US market volatility and the financial crisis are not significant. Thus, for these two sectors, the converse result to that established in the literature is found. Here correlations rise with an expanding economy and a bull market. This supports the earlier view that not all sectors respond in the same direction. Table 10 reports the results for the determinants of time-varying correlations for the individual sectors using a fixed effects panel regression approach. 4 We can see that the dummy variable that relates to the financial crisis is positive and statistically significant for all sectors. Again, this supports the view that correlations rose in the crisis period. For the dummy relating to the US bull/bear market, again we can see some difference across sectors. While this dummy is negative and significant for the majority of the market, indicating that correlations rise during a US bear market. For three sectors, a positive and significant relation is found, such that correlations rise during a bull run and fall in a bear market. For US interest rates, a negative relation is reported for all markets, supporting the view that a weaker US economy is consistent with higher market correlations. For the coefficients relating to market and specific sector returns and standard deviations, the results reveal that no consistent pattern exists, again suggest the potential for markets to move in different directions. In particular, with reference to international market returns, the correlation of only five sectors is significantly affected at the 5 per cent level (with a further four at the 10 per cent level). However, all sectors are affected by the volatility of international market returns. For own sector returns, only three correlations are affected and negatively so, while the own standard deviation significantly affects eight market correlations positively and seven market correlations negatively.

SUMMARY AND CONCLUSIONS
This paper has examined the correlation and interrelations between the G7 markets over the recent past. In particular, the existing Notes: Entries are the regression coefficients (and t-tests) where regressions use the first three principal components obtained from Table 8 as the dependent variables. These are regressed on a dummy variable covering the financial crisis, a dummy variable representing a US bull market, the US 10-year Treasury bond yield and the standard deviation of the US market.

Time-varying correlations and interrelations
literature identifies the view that stock market correlations have risen over time and are also likely to rise during periods of market stress. The key aim of this paper is to reconsider these conclusions and seek to explain movements in time-varying correlations. However, in contrast to the existing literature which typically uses market index data, the results presented here are based on firm-level data. The advantage of using firm-level data is that the correlations obtained are based on data that is tradeable as opposed to non-tradeable index data. Unconditional correlations for both market index data (to provide comparability with the preceding literature) and the firmlevel data appear to show that correlations have increased. Of note, correlations have increased from the pre-crisis period to the crisis and post-crisis period. However, there is some subtly within these results. Notably, there is evidence that correlations have plateaued and even fallen as we move into the post-crisis period. Additionally, correlations fell just prior to the crisis and during a period of economic growth. This supports the view that while markets fall together they rise separately. Also of interest, we note that the correlations based on firmlevel data are lower than those observed at the aggregate market level.
Using three regression-based approaches, we examine both the nature of the interrelations and time-varying correlations between the series. A CAPM style approach suggests that for each market sector the equivalent US sector is more important than the overall movement of the US market in conditioning returns. Although the home market return remains the most important variable. A time-varying correlation model supports the view that correlations exhibit a positive trend over the sample period; however, the slope of the trend is very shallow. Panel unit root tests reject the null of non-stationarity within the sample correlations. A principal components analysis identifies a common component that affects the correlations of all market sectors in the same direction, except two (technology and telecoms), which exhibit their own separate component. Moreover, this principal component accounts for two-thirds of the movement in correlations, while the first three components account for 80 per cent.
Having established the nature of correlations, which indicate the potential for diversification, we seek to explain the nature of the time-variation. Established results argue that correlations rise in times of market stress, which appears to be borne out by those reported here. Notably, correlations across both the market trend identified by the principal components and individual sectors rise during the financial crisis and when the US economy is weak. However, while correlations also generally rise during US bear market periods, this result is not ubiquitous, with some sectors indicting the reverse. Furthermore, the effect of market and sector returns and sector volatility differs across sector correlations.
The analysis of market correlations remains important in the development and understanding of portfolio and risk management. This paper, using firm-level data, examines correlations across market sectors. Notably, we wish to examine the dynamics and determinants of time-varying correlations. Several broad conclusions are reached that suggest diversification benefits may remain. First, firm-level-based correlations are lower than aggregate market index correlations. Second, although timevarying correlations have increased during the crisis period, there is evidence that they may now be falling. Additionally, there is no evidence of a stochastic trend while any deterministic trend is very small. Third, while sector correlations typically move together and follow a general market component, some sectors exhibit a negative relation with that market component with movement governed by a different component. Fourth, evidence exists that while correlations across all sectors rise during periods of weakness in