The Pricing of Spatial Linkages in Companies’ Underlying Assets

As argued above, real estate companies provide a suitable setting to account for spatial linkages as such companies extract a large proportion of their income (80–90%) operating direct real estate assets, mostly through rental income. This enables us to directly relate the locations of the underlying assets of a company to the locations of properties of another company and calculate a spatial weight for each pair of firms. The weight is based on the proportion of the properties held by the two firms that locate within a distance of 25 km:with l = 1, 2, . . ,  L_t and L_t is the total amount of properties held by firm i. \( \sum \limits_{l=1}^{L_t}{q}_{i.l,j,t} \) is the total amount of properties held by firm i in period t that has a distance of less than 25 km with ANY of the properties held by firm j, with:

The same is true for the counterparty firm j:with l = 1, 2, . . ,  K_t and K_t is the total amount of properties held by firm j. q_{j. l, i, t} is the amount of properties held by firm j in period t that has a distance less than 25 km with ANY of the properties held by firm i. In order to make sure that the matrix is symmetric, we use the minimum of these two proportions to calculate the linkage between each pair of firms⁴:

Equation (6) defines the weight between each two firms using their minimum of the individual firm weights, because the minimum distance accounts for the case in which firms have different levels of geographic asset concentration. We assume that the pair of firms would not have a strong relationship as indicated by the average weight if the two individual weights are far apart from each other. In that case, we assume overall weak relationship. In the last step, we combine each element of Eq. (6) in a W_t matrix with the pairwise proportions for each period. We then row-standardize matrix W_t so that for each i we have ∑_jw_{i, j, t} = 1.

Figure 1 shows an example of how the distance between firm A and firm B is calculated based on the individual distances between the properties. Let us assume that company A holds three properties, A1, A2 and A3, and company B holds in two properties, B1 and B2. The dashed lines show the distance between each pair of properties. For firm A, all of the properties locate within a distance of 25 km to any of the properties held by firm B, so the proportion of properties that locate within 25 km to any of the properties held by firm B is 1 (from Eqs. 3 and 4). For B, B2 locates within the 25 km distance and B1 locates beyond the range of 25 km, which leaves us with a weight of 0.5 (from Eq. 5). The spatial weight between firms A and B is then defined as the minimum of 1 and 0.5, and is thus 0.5 (from Eq. 6). If the weight is 0, it implies that none of the properties between the two firms is located within 25 km. If the weight is 1, it implies that all properties of a firm locate within 25 km from the corresponding counterparty.

We first estimate a factor model using the existing factors as described above. We run firm-by-firm regressions using monthly data. We can see that the performance of the model is poor as the R2 is low, at only 3.2% on average, and none of the factors is significant on average (see Model 1 in Table 3). This is compared with R2 of above 60% for stock regressions by Fama and French (2012) using the four Fama and French factors. The reason for the low explanatory power is that the general stock market factors do not include enough real estate companies and a large part of the risk is idiosyncratic. Real estate companies invest predominantly in real estate and can be driven by factors specific to certain buildings such as location, sector, etc. This is reflected in the low and insignificant beta of 0.0019. One of the reasons for the low synchronicity between real estate companies and the market, as argued by Chung et al. (2011), can be due to spatial uniqueness of the underlying assets. In order to capture the comovement in the real estate equity market, we construct factors using the 223 US real estate firms. The results are reported in Model 2 in Table 3. The R2 of Model 2 is now much higher with a value of 49%. However, this value is still not as high as the one observed for stocks. This can point again to remaining idiosyncratic price variations as we account for the REIT market. We can see that the variations in the returns are largely explained by the market index (RM). The average beta across the 74 companies is 0.95 which is very close to the beta of the market of 1. That means that the 74 companies in our sample which we regress individually on a factor constructed using 223 real estate companies commove very closely with the rest of the real estate firm universe.

In a second stage, we use the abnormal returns from the factor model (Model 2) to estimate an unbalanced panel model with 11,031 observations. Model 3 (in Table 4) does not include a spatial term, which Models 4 and 5 regress the residuals on the spatially weighted residuals. The weights capture the spatial linkages between each pair of residuals using the locations of the underlying property holdings of each company. Model 4 includes only one matrix using 25 km as the bandwidth. Spatial models always assume that the co-movement between assets should depend on the strength of their linkages. While we construct the weights based on the strength of the linkages, we also test if that assumption is fair. For this purpose, we apply the distance decay model adding four matrices. The results are presented in Model 5. It shows other bandwidths, such as 25–125 km, 125–1000 km, and distance beyond1000 km. The first matrix is the same as the matrix in Model 4. This means, each weight between a pair of firms reflects the proportion of properties of one firm that locates within 25 km to any of the properties held by the other firm. The second matrix is defined in the same way as the first matrix; the only difference is that the weight is based on the proportion of properties held by one firm that locates between 25 km and 150 km to any of the properties held by the other firm. In the same way, we define matrix three, but the distance is 150–1000 km. Matrix four accounts for the proportion of the properties between two firms that locate further than 1000 km. For each matrix the weight is defined as the proportion of the underlying properties within the respective distance; for the rest of the firms, the weights are set to zero.

In all three models (Models 3, 4 and 5), we control for profitability, liquidity, size, as well as diversification strategies. The degree of market concentration is measured using a Herfindahl-Hirschman Index (HHI). When the index has a value of 1, the firm has a fully concentrated market/sector strategy meaning that all properties are located in the same state or in the same sector. The lower the HHI index, the more diversified the assets the firm holds. We consider two diversification strategies, by location (at the state level) and by sector (at the property type) as similar diversification strategies may yield similar REITs’ returns. Additionally, we control for common economic shocks. For example, Cotter et al. (2015) suggest that performance across geographic locations can be highly correlated because certain MSAs are more exposed to macroeconomic shocks than their others. We follow Ling et al. (2017a) and use the proportion of the properties located in the 25 major MSAs to proxy for the common exposure to top markets of each REIT. The MSAs are Atlanta, Boston, Chicago, Dallas, Denver, Detroit, Houston, Indianapolis, Kansas City, Los Angeles, Miami, Minneapolis, New York, Orlando, Philadelphia, Phoenix, Portland, Sacramento, Saint Louis, San Antonio, San Diego, San Francisco, Seattle, Tampa, and Washington, D.C.

We can see that the spatial coefficient for the 25 km bandwidth is significantly positive and takes the value of 0.2 in Model 4.¹² It means that spatial linkages across the underlying assets of firms significantly drive the comovement across the non-market returns, controlling for systematic factors as well as company-specific characteristics. Model 4 also achieves higher adjusted R2 and lower BIC than the model without spatial consideration (Model 3), confirming significant spatial dependence in the abnormal returns. Given that one standard deviation of Wy is 0.0204, a one standard deviation increase in the abnormal return of ‘neighboring’ REITs is associated with 0.4% increase in the abnormal return of the REIT of interest.

Looking at Model 5, we see that the matrix based on 25 km bandwidth has the highest coefficient of 0.14. The matrix based on the bandwidth between 25 km and 125 km has a smaller but significant coefficient of 0.05. When the distance between the properties exceeds 125 km, we cannot find evidence for spatial dependence anymore. The decrease in the spatial dependence intensity with different bandwidths implies that the comovement in the abnormal returns declines with the distance of the properties held by the two firms. If we compare the goodness of fit between Models 4 and 5, we can see that although the adjusted R2 slightly increases by 0.0005 in Model 5, the BIC is also higher. The additional matrices do not significantly improve the goodness of fit. Therefore, we choose Model 4 as the baseline model.

The control variables have the expected signs but not all of them are significant. We can see that the lagged abnormal returns have a significant effect on current abnormal returns, however the sign is negative. A 10 percentage point increase in the abnormal return would lead to a drop of the abnormal returns in the next period by 0.7 percentage points. Although the effect is small, it is significant and shows that there is some systematic correction over time in the abnormal performance of REITs. This is in line with the assumptions of mean-convergence of the residuals. REITs receives higher abnormal return when they have higher exposure to the top 25 MSAs, consistent with the findings in Ling et al. (2017a).

There is a significantly positive relationship between the size of the company and its abnormal return even after controlling for size as a systematic risk factor. We can see that another important company characteristic, real estate investment growth, has a significantly negative effect on abnormal returns which is also in line with expectations. Small stocks should compensate for the risk exposure. The more the company grows its business, the lower the chances for any additional abnormal returns. The turnover ratio has a significant impact. The abnormal return is positively related to the trading volume of shares of the respective company.

In order to make sure that the physical distance is the best way to capture the relationships in the companies and that the spatial weight matrix does not capture some other linkages or global comovements, we add an alternative weight matrix into the spatial panel model. Those additional weight matrices are based on the similarity in the size across firms, the similarity in their M/B ratios and the similarity in their momentum.¹³ As Bernile et al. (2015) show, the geographic proximity of a firm’s headquarter can also explain the correlation in the abnormal returns. Therefore, we also construct a weight matrix which is defined based on whether the headquarters of each pair of companies are located in the same city.¹⁴ Eq. (2) can be rewritten as:

The results are reported in Table 5. In all cases, the spatial dependence triggered by geographic proximity of the underlying assets remains significant, ranging from 0.1560 to 0.1957. The spatial coefficient of the additional weight matrix ranges from 0.0099 to 0.0751. Except for the weight matrix constructed based on M/B ratio, the remainder are insignificant. This is to show that geographic distance is not capturing other linkages between the companies such as similarities in size, performance or company location. Our measure of company linkages does a good job in capturing comovements as compared to more commonly used spatial measures. Also, it is a call to using spatial distance between underlying assets rather than the spatial linkages between the actual companies.

Concerns may arise as the real estate markets can perform quite differently in each MSA due to variations in local legislations and economies. Consequently, the estimated comovement may not be due to the closeness of the underlying assets but rather to these properties locating in the same region. In the baseline model, we have controlled for the exposure to the top 25 real estate markets. Nevertheless, the comovement might still be caused by a common exposure to regions with similar economic dynamics and industries. We follow Ling et al. (2017a) and account for the exposure to eight geographic regions (Northeast, Mideast, Southeast, East North Central, West North Central, Southwest, Mountain and Pacific) and construct firm-level geographic concentration measures pertaining to each region. The geographic exposure to the eight regions can be proxied by eight variables. Each variable is built based on the proportion of properties located in a given economic region for each firm. In this way, we construct the firm-level exposure to each economic region. However, the sensitivity of abnormal returns to regional common factors can be time-varying. To account for this, we use cluster-fixed time effects. In other words, instead of adding above eight variables, each regional exposure variable is split into 19 years (Model 10), with each variable equaling to the proportion for that year and zero otherwise. This way, we end up with 152 variables (19 × 8). We also perform the analysis with two other groupings as Ling et al. (2017a). One grouping is based on economic activities (Hartzell et al. 1986). We sort the 25 major MSAs into one of the following eight economic activity regions (New England; Mid-Atlantic Corridor; Old South; Industrial Midwest; Farm Belt; Mineral Extraction Area; Southern California; and Northern California). Similarly, we insert the interaction variables of exposure to eight regions and nineteen years. The exposure to eight regions is measured as the share of properties in each region during a given year (Model 11). As an alternative to economic regions, in Model 12, we also sort MSAs into seven industry clusters (Professional and Business; Government; Information and Finance; Leisure and Hospitality; Education and Health Services; Natural Resources, Construction, and Manufacturing; and Trade, Transportation, and Utilities). Additionally, instead of grouping the 25 MSAs into several regions, we insert an interaction variable of the exposure to each of the 25 MSAs and year dummies (Model 14). We also control for the exposure to the 50 most populous American downtowns¹⁵ and add an interaction variable between the exposure to the major 50 CBDs and an year dummy. Finally, we also add a matrix based on the proportion of the properties held by the two firms that are located in the same MSA (Model 15).¹⁶ If the comovement is purely triggered by common factors within one MSA rather than geographic distance, the coefficient for the distance based matrix would be insignificant if the MSA based weight matrix is added.

The results are reported in Table 6. In all cases, the spatial dependence by geographic proximity of the underlying assets remains significant. It ranges from 0.1636 to 0.2013. The spatial coefficient of the weight matrix based on the same MSA is insignificant. This shows that the weight matrix based on the geographic closeness of the underlying assets can capture the comovement between REITs abnormal returns more than the degree that can be explained by common shocks to regions with similar economic and industry bases.

Furthermore, we examine whether any movement still exists after we control for some of the local economic fundamentals such as rent or unemployment rate. In other words, whether the spatial spillover is just because of common exposure to local fundamentals. We proxy MSA level economic shocks using the local unemployment rate and housing returns during the period between 1996 and 2015. We collect unemployment rate data for over 300 MSAs from the Bureau of Labor Statistics and construct the weighted average unemployment rate that each firm’s exposure. The average unemployment rate for each firm is calculated as the proportion of properties in the MSA multiplied with the unemployment rate of that MSA (Model 16). Similarly, in Model 17, we control for an MSA level housing market shock by including the average housing return according to the housing return in each MSA and the exposure of each firm to that MSA. The MSA level housing return comes from Federal Housing Finance Agency.

Besides, we also control for local fundamentals in a more precise way. We calculate the average rent each firm received from their portfolio (Model 18) and the average occupancy of the properties that each firm holds (Model 19). If the returns are no more than just compensation for local exposure to tail risk, the comovement may not exist after controlling for firm level rents and occupancy rate. However, not all firms report the rent or occupancy of their properties, thus the number of observations is sharply reduced, as shown in Models 18 and 19 in Table 7. Even after explicitly controlling for local fundamentals, the results still show significant spatial dependence, ranging from 0.0977 to 0.1944. After the firm level rent is controlled for, the spatial dependence is reduced, which implies that a considerable proportion of spatial dependence could be explained by the similarity in rents. However, the spatial dependence coefficient still remains significant. Overall, the spatial dependence can capture the comovement in firm performance beyond what can be explained by local fundamentals.

Moreover, concerns may arise that REITs’ performance may be affected by the density of the local markets. REITs have higher exposure to markets with higher commercial real estate density may have better performance. We therefore control for the density of properties. We use the average number of properties located with a 25 km radius of each of property (Model 20, Table 7) to measure the density. The result is reported in Table 7. The coefficient for density is significant positive, confirming that investing in more dense locations can yield higher abnormal returns. The spatial dependence coefficient stays robust in all cases demonstrating that the spatial linkages capture different information from density.

Finally, we check for potential endogeneity between REIT’s returns and the weight matrix. One shortcoming of previous papers explaining the comovement across returns with common ownership is that institutional ownership can be endogenous to the model. Investment funds for example can choose to invest in stocks that have common fundamentals. Moreover, different funds can have correlated trading needs and thus naturally commove (Greenwood and Thesmar 2011). Our location-based measure of comovement is less striking because it is based on the location of properties. Although managers can choose the property location taking into account the surrounding properties and their managing companies, making a single spatial linkage endogenous to the model, it is hard to imagine that a company would base the location of its assets in its entire portfolio, which can be more than 100 assets, strategically accounting for spatial linkages of each pair. Due to the nature of the real estate market, most real estate companies albeit following certain investment strategy, take a more ad-hoc approach due to the illiquidity of the market. Therefore, across the entire property portfolio, a pair of firms will end up having an independent asset allocation.

We conduct three robustness checks for the endogeneity issue. First, we regress the abnormal return on lagged weighted abnormal returns (\( {\sum}_{j=1,j\ne i}^N{w}_{i,j,t-1}{\varepsilon}_{i,t-1}\Big) \) and lagged control variables

The second robustness check uses the property holdings in the initial period. In other words, we assume that firms keep a constant property portfolio over the entire estimation period. Under this scenario, as firms cannot change their property portfolio composition, they cannot strategically select the properties by observing the performance for other firms. We estimate thus the following model

The third robustness check adds a variable controlling changes in local supply. If managers can choose the property location taking into account the surrounding properties and their managing companies, the properties held by REITs in top performing markets will gradually increase while the properties purchased by REITs in poor performing markets will gradually decrease. We use the change in the property density to control for the change in local demand and supply. We include the variable of density of 25 km radius interacted with time dummies (Model 23, Table 7).

Table 8 reports the results. The spatial dependence coefficient slightly decreases from 0.19 in Model 4 to 0.17 in Model 21 and 0.14 in Model 22, but remain quite robust in Model 23. As the spatial dependence coefficient remains significant, the endogeneity issue is not severe and does not affect the findings in our analysis.

Given above findings, we propose a trading strategy that exploits the information in the locational comovements across the abnormal returns of all 115 U.S. REITs.¹⁷ Similar trading strategies have been previously used to exploit the information contained in the linkages between companies resulting from fire sales of stocks related to common institutional ownership (Chen et al. 2012; Anton and Polk 2014). Chen et al. (2012) develop a trading strategy that benefits from the divergence in the prices of a pair of similar stocks. Anton and Polk (2014) present an investment strategy that identifies stocks that temporarily move together and profits from their eventual divergence in price. Our trading strategy is similar to the one in Anton and Polk (2014), however, we use the proximity in the location of companies’ underlying assets to measure how similar or different stocks are. If geographic proximity between properties can cause comovement in the returns of the companies that own those assets, our trading strategy would use the return of a portfolio consisting of the connected stocks to benchmark against. The difference in the returns would be used as a signal of the mispricing of the respective company. The strategy exploits upward and downward real estate price pressures channelled through the spatial proximity of the companies’ underlying assets. We examine the buy-and-hold non-market returns on two portfolios. The first portfolio consists of an equally-weighted portfolio of the worst performing (low-return) companies (33% lowest returns in the past 12 months). In addition, these companies also invest in properties which locate close to properties of other bad-performing companies (33% lowest returns in the past 12-months). We call this the LL portfolio which stays for for low own return (L) and low connected portfolio return (L). Respectively, the second portfolio consists of an equally-weighted portfolio of companies with high returns (33% highest returns in the last 12 months) that invest in properties located close to properties of other high-performing firms (33% highest returns in the past 12 months). We call this the HH portfolio, for high own return (H) and high connected portfolio return (H). We then show in Fig. 4 the non-market buy-and-hold portfolio cumulative returns over a period of 12 months. The alphas of the buy-and-hold portfolios are calculated by regressing the returns of each of the two portfolios on the common factors and extending the sample period by one period each time (Anton and Polk 2014). Figure 4 shows the cumulative non-market returns of the two portfolios. The cumulative alpha of the LL portfolio is negative and increasing in absolute value over time. The LL portfolio is consistently underperforming, delivering increasing negative alphas in every subsequent period. These findings suggest that stock prices of real estate companies are pushed away from fundamentals by the spatial dynamics of the underlying property portfolio and this pattern is persistent. We can see that using a portfolio of spatially connected stocks helps to detect mispriced companies. Similar to the findings in Anton and Polk (2014), we show that the misvaluation seems to be much larger in value and more prolonged in time than the standard short-term reversal effect.

Given above findings, we construct composite portfolios that take into account the predictability of cross-sectional variation using the spatial linkages across the underlying assets. We construct 12-month buy-and-hold portfolios. We independently sort stocks according to their returns (own return) and the returns of a portfolio of companies which are connected to the stock of interest through their properties (connected portfolio return) over the past year. The connected portfolio return is calculated as \( {\sum}_{j=1}^N{w}_{i,j,t}{\overset{\sim }{r}}_{j,t} \). In total, we construct nine portfolios based on groupings at the 33th and 67th percentiles.¹⁸ We have the following portfolios which consist of the own return and the connected portfolio return as follows: HH, HM, HL, MH, MM, ML, LH, LM, and LL. H, M and L stay for high, medium (or median), and low returns in the own or the connected portfolio domain respectively. We construct the high (low) portfolio using the 33% highest (lowest) returns of each category. HL for example means a portfolio consisting of the companies with the stocks having the 33% highest returns and the stocks of the firms they are connected to which fall within the bottom of the 33th return percentile. We then use the time series of returns of each of those nine composite portfolios to estimate the alphas in a four-factor model using the constructed factors. Table 9 shows the results.

We can see that the locational proximity of properties pushes the returns of portfolios of connected companies away from their fundamental values on both ends of the performance spectrum. The connected portfolio return is a good measure of the extent of mispricing. Similar to the observations in Fig. 4, the alphas of the nine composite portfolios decrease as we move from high to low connected portfolio returns within the own-return quantile and as we move from high to low own-return portfolios within the connected-return quantile. This is to show that the information contained in the connected portfolio returns is a useful predictor of the own return. Those results remain robust when we use different return quantiles and different factor model specifications. Therefore, we propose a long-short trading strategy (HH-LL strategy) which exploits the information in the connected returns. This HH-LL strategy buys the composite HH portfolio and sells the composite LL portfolio yielding a significantly positive alpha of about 0.8% per month. We can use above long-short hedge to exacerbate excess comovement stemming from the spatial linkages across the property holdings. An investment strategy which buys the stocks that experience an increase in their price if their connected stocks have also gone up and sells the stocks that experience a drop in their price if their connected stocks have also gone down can earn an average non-market return of 9.7% per year. This is in line with previous findings who report non-market returns from a long-short strategy extracting information from institutional ownership of 9% (Anton and Polk 2014) and 10% (Coval and Stafford 2007) per year.

This table reports the non-market returns of composite portfolios as well as of trading strategies exploiting the connectedness across property holdings of 115 real estate companies. We independently sort stocks into quantiles based on firm’s own return over the last year and the return on firm’s connected portfolio over the last year. We first measure the degree of connectedness using the distance of the properties as formalized in Eqs. (3)–(6). We then calculate the connected return as \( {\sum}_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{w}}_{\mathrm{i},\mathrm{j},\mathrm{t}}{\overset{\sim }{\mathrm{r}}}_{\mathrm{j},\mathrm{t}} \). Each composite portfolio is an equal-weighted average of the corresponding simple strategies. We also report the average returns on an HH-LL trading strategy that buys the high own-return and high connected-return composite portfolio (HH) and sells the low own-return and low connected-return composite portfolio (LL).