Spatial Dependence in Commercial Real Estate

Transcription

1 Spatial Dependence in Commercial Real Estate Andrea M. Chegut a, Piet M. A. Eichholtz a, Paulo Rodrigues a, Ruud Weerts a Maastricht University School of Business and Economics, P.O. Box 616, 6200 MD, Maastricht, The Netherlands Abstract Omitting the contemporaneous spatial dependence between transaction prices in a commercial real estate environment may lead to mis-pricing building attributes in real estate valuation. This paper investigates whether spatial dependence is a significant pricing component of the largest commercial real estate markets globally, London, Paris and Frankfurt. Using cross-sectional transaction data for these markets we test for spatial dependence using two distinct spatial autoregressive model frameworks, across three different weight matrix specifications and two different estimation procedures. Compared to previous work in the commercial real estate literature, we find a very small spatial dependence parameter for the London, Paris and Frankfurt markets. Importantly, these results are based on markets where buildings are very heterogeneous and traded in a global environments where institutional investors are able to make comparisons across markets. Consequently, in older more heterogeneous markets spatial dependence may not play as significant a role for commercial real estate asset pricing. Keywords: Spatial dependence, Commercial real estate, Spatial autoregressive model, Hedonic model Please contact author before citing manuscript. Corresponding author: address: [email protected] (Andrea M. Chegut) 1

2 1. Introduction The market value of a commercial real estate asset may depend on the value of comparable real estate assets within the same markets. This may be a result of asset managers looking to their local competitors to incorporate similar building technologies; building codes may mandate homogeneous requirements; real estate growth during a particular cohort may lead to an expansive commercial building stock of one generation. Thus, when conventional models aim to price commercial real estate assets omitting spatial dependence may fail to represent the extent to which the building s prices are already correlated (Anselin, 1988; LeSage and Pace, 2010). Statistically, the t-statistics and F-statistics of these pricing models could be biased and economic inferences based on a building s characteristics may be erroneous (Downs and Slade, 1999). Moreover, any index derived from explaining commercial real estate attributes, like in the hedonic model, may be biased, especially if time effects are correlated with spatial dependence. Consequently, this paper investigates whether there is economically significant spatial dependence in commercial real estate pricing models. Importantly, spatial analysis in this literature will have consequences for valuation of commercial real estate. In turn, future theoretical and empirical models for commercial real estate markets may require further consideration of the extent of spatial dependence. Spatial dependence has been tested in commercial real estate markets in the past, but the extent of the work is limited. Tu et al. (2004) investigated spatial dependence for the Singapore office market using a Spatial Temporal Autoregressive model. Their results indicate that the spatial dependence parameter is large, positive and statistically significant at the five percent level. Moreover, these results are reported for strata-tiled office properties in Singapore, which are a more homogenous and new office product in Asian markets. (Nappi-Choulet and Maury, 2009) studied the spatial dependence of the Paris office market where they found evidence of large, positive and statistically significant spatial dependence. Paris is an old office market with a large number of heterogenous assets geographically distributed across a very large mega-city. The methodology for both analysis is derived from 2

3 (Pace et al., 1998) housing study implementing the first Spatial Temporal Autoregressive model. However, there is some debate within the spatial econometrics literature as to what is the correct functional form for incorporating the spatial dependence parameter. In the above literature, the spatial-temporal model estimated via a bayesian procedure with heteroskedastic errors. However, other spatial literature does not incorporate a temporal dependence parameter, where the transactions in one period would be correlated with the transactions in previous periods. Moreover, there is an additional filter for a spatial temporal dependence parameter; a so-called interaction of time and space. Thus, it is unclear whether the spatial dependence parameter is significant alone or only in addition to time. Consequently, there has been little application of this model in the commercial real estate literature. Moreover, there may be reasons grounded in the real estate economics literature that question the significance of the spatial dependence parameter altogether. (Geltner and Pollakowski, 2007) noted that spatial dependence may not be a significant factor in commercial real estate indices as the level of heterogeneity between buildings in commercial real estate markets is very high. In turn, we question whether spatial dependence is an important asset pricing variable for commercial real estate. Spatial dependence may arise in both the model and the disturbance structure. Spatial dependence empirically appears in the form of a spatial lag, which captures the spatial autocorrelation. And can be thought of as analogous to autocorrelation in time series, where the spatial matrix depicts the relationships between objects in space. In this case, the metric is physical space, but can be measured through other exogenous factors like time or money (Baltagi and Li, 2001; Baltagi et al., 2003). The spatial autoregressive (SAR) model captures spatial dependence by interacting the dependent variable with its spatial relations (LeSage and Pace, 2010). Additionally, spatial dependence could arise in the disturbance structure in the form of heteroskedasticity, caused by the heterogeneous nature of real estate assets and spatially correlated unobserved variables. The implementation of the spatial dependence in the disturbance structure is new in the application of spatial econometrics in commercial real estate. Due to the heterogeneous nature of commercial real estate assets, this form 3

4 of dependence is likely to appear in the residuals. The spatial autoregressive model with autoregressive disturbances (SARAR) captures the spatial dependence in both the model and the disturbance structure (Kelejian and Prucha, 2010). Thus, to test for economically significant spatial dependence, we employ a Cliff-Ord type SAR model and a Kelejian-Prucha type SARAR model. Our empirical specifications are applied to the Frankfurt, London and Paris commercial office markets using transaction data provided by Real Capital Analytics. In Europe, these three markets are ranked in the top 6 of markets with the lowest yields and in the top 5 of markets with the highest investment volume 1. And according to Real Capital Analytics, London and Paris are in the top five of total transaction volume Elhorst et al. (2012) documents that the specification of the weight matrix has significance in identifying spatial dependence. Consequently, the spatial weight matrix is specified in three ways. First, the spatial matrix is specified as a contiguity matrix. The application of this specification results in a spatial effect that looks only at the spatial dependence between neighbouring assets. Secondly, the matrix is specified as an inverse distance matrix. In this specification the distance is computed between all observations. The inverse distance ensures that the spatial dependence becomes smaller when the distance becomes larger. Thirdly, the matrix is an inverse distance matrix with a cut-off. This specification is similar to the second, but the distance equals zero when the distance is larger than a specified threshold. Finally, we benchmark these models and spatial matrices to a standard OLS hedonic framework employed in the commercial real estate literature. Results indicate that spatial dependence in these markets as estimated by two spatial auto-regressive models is limited and the spatial dependence in the disturbance structure is not always significant. However, the coefficients and t-statistics of other parameters are affected by modeling the spatial dependence in the disturbance structure. Importantly, compared to previous work in the commercial real estate literature, we find a very small spatial dependence parameter for the London, Paris and Frankfurt markets. Importantly, 1 The market research in the report European Office Market 2012 published by BNP Paribas reveals new insights regarding the latest developments of the office markets in Europe. 4

5 these results are based on markets where buildings are very heterogeneous and traded in a global environments where institutional investors are able to make comparisons across markets. It appears that the current literature in commercial real estate already prefers a spatio-temporal based autoregressive model. This may be due to the findings of a strong and significant spatial dependence parameter in these models. However, it appears for the Paris, London and Frankfurt empirical specifications that the spatial dependence parameter does not have the magnitude previously expected. Thus, going forward, it may not be models that incorporate spatial-dependence alone, which implies that for understanding the spatial dependence in European commercial real estate markets more work in space and time interactions should be done. The remainder of this paper is structured as follows. Section two reviews the literature on spatial analysis and commercial real estate with an emphasis on understanding the motivation including spatial dependence in real estate analysis. Section three contains the estimation strategy, SAR and SARAR operational models and estimation procedures. Part four covers the results of the analysis, including the regression outputs of the SAR, SARAR and OLS models and the transaction price indices of each market. Section six contains robustness checks. Section seven provides a discussion and and concludes. 2. Spatial Dependence in Commercial Real Estate On the one hand, commercial real estate is characterized by its high level of heterogeneity (Miles and McCue, 1984). Construction processes are time varying. New modes of travel and commerce are dynamic. Both construction and economic development are influencing where consumption, business and productivity agglomerate in cities. On the other hand, commercial real estate can have features of homogeneity similar to that of housing. Fisher et al. (1994) find significant factors that the functionality of the property and the quality to be a standard pricing characteristic. Colwell et al. (1998) find significant locational factors and conclude that the transaction prices of office buildings located in the official employment area of the city, the change in population, the net average income per square foot and the average annual capital improvement are significantly and positively related to 5

6 the transaction price. Moreover, neighborhood effects like train station proximity, a nearby subway system or a connection to major roads and interstate highway system has been shown to influence price(debrezion et al., 2007). Finally, in addition to neighborhood elements there are structural properties of commercial real estate assets that determine value. Nappi- Choulet et al. (2007) show that the size is the most important structural factor. Besides, size it appears that the performance of a renovation clearly affects the transaction price positively (Nappi-Choulet and Maury, 2009). There are numerous hedonic properties such as age, quality, number of floors, etc. that could affect the transaction price Empirical Tests of Spatial Dependence in Commercial Real Estate Spatial econometric models are an extension of conventional regression models. The models are extended by including a spatial component that is able to capture the potential dependence caused by the location of the observation and the interaction with neighbouring observations. This form of dependence is often found in data regarding observations that are characterized by their location (Anselin, 1988; LeSage and Pace, 2010). There are multiple motivations for using this type of model. First, if spatial dependence is present, the outcomes of a conventional regression model are biased. Hence, it is important to take the potential dependence into account. Pace et al. (2000) finds that the correlation between the housing assets and the reference group decreases substantially, almost to zero, while there is still correlation present when the non-spatial model regression model is applied. Additionally, the standard error is smaller (Pace et al., 2000). Second, se Can and Megbolugbe (1997) show that there is spatial dependence in the housing market and determine that this is potentially a structural parameter driven by spatial externalities or locational effects. Third, LeSage and Pace (2008) find an econometric motivation that is twofold for the use of spatial regression models that incorporate spatial lags of the dependent variable. On the one hand spatial dependence can be viewed as the long-run equilibrium of an underlying spatio-temporal process. This is explained by the definition of the cross-sectional spatial temporal autoregressive models. These have no explicit role for the passage of time an the models therefore reflect an equilibrium outcome or a steady state 6

7 (LeSage and Pace, 2008). Lastly, there is an omitted variable bias which arises when the data exhibits spatial dependence that is not captured by the model. Thus far, there has been very little academic or practical evidence on the use of traditional Cliff-Ord type spatial models in commercial real estate.tu et al. (2004) apply the spatial regression methodology to the construction of a commercial real estate index in Singapore. They argue that the previous literature based on non-spatial models encounters three major problems. Firstly, there is no uniform proxy to measure location attributes, factors related to the discounted cash-flow and/or lease structure. Secondly, spatial and temporal correlation can impair the power of the traditional hedonistic model due to omitted variable bias. Thirdly, office properties are less frequently traded than residential properties. The use of a spatial component in the regression could capture these anomalies. There results indicated that by allowing for spatial dependence in their hedonic model the spatial based index of the office market in Singapore captures standard hedonic properties as well as spatial dependence for the market. Nappi-Choulet and Maury (2009) apply this methodology on the office market in Paris. This study focuses on the two main business district in Paris and finds significant spatial dependence. A limitation of this study that is addressed by this study is that the potential heteroskedasticity is poorly modelled (Nappi-Choulet and Maury, 2009). Still, the spatial dependence tends to be more significant than the temporal dependence and the resulting index differs substantially form the traditional hedonic-based index Spatial Econometric Methods for Empirical Tests in Commercial Real Estate In addition to minimal to evidence in the commercial real estate literature there is some debate on the functional form of commercial real estate. Developments within the spatial econometrics literature regarding the spatial dependence in the disturbances make it interesting to compare the models to test the application on commercial real estate. So far the models have not been compared for the commercial real estate market and the effect of the allowance of known form heteroskedasticity is unclear. 7

8 LeSage and Pace (2010) extend the conventional regression model by implementing a spatial weight matrix. The matrix is defined so that each element in row i of the matrix W contains values of zero for regions that are not neighbour to i. By multiplying the spatial matrix with the dependent variable, the spatial correlation coefficient can be estimated (LeSage and Pace, 2010, p. 357). This extension of the regression model is called the spatial lag. By including the spatial lag in the regression model the SAR model is constructed. This model allows for spatial autocorrelation. Spatial autocorrelation is likely to be present in the commercial real estate market when the transaction price of one building is related to the transaction price of buildings that are close. The estimation of the parameters of the regression problem can be achieved via maximum likelihood estimation (MLE). The MLE method is a consistent estimator for the estimation of spatial autoregressive models (Lee, 2004). Moreover, LeSage and Pace (2010) stress that the use of maximum likelihood is advisable as other models suffer from deficiencies regarding the estimation of the correlation parameter and the sensitivity of the model to the dependence between the parameters. Thus, LeSage and Pace (2010) clearly argue for the use of maximum likelihood. The maximum likelihood estimation assumes, however, that the error term is normally distributed. If spatial dependence is present in the error structure, the outcomes of the model are biased. Kelejian and Prucha (1997) show the derivation of a spatial econometric model based on the generalised least square (GLS) method. A similar approach was chosen to deal with the potential asymmetric spatial weight matrix which makes the estimation via maximum likelihood not feasible (Kelejian and Prucha, 1999). Additionally, there could be spatial dependence in the disturbances (se Can and Megbolugbe, 1997). This would cause a violation of the assumption regarding the normally distributed error structure. More specifically, spatial dependence in the disturbance structure implies that there is spatial dependence in the unobserved variables such as amenities and other factors that are not captured by the hedonic vector. This form of dependence may result in heteroskedasticity. This model is estimated via a GLS estimator which leads to consistent estimates of the parameters (Kelejian and Prucha, 2010). 8

9 Thus, from a modelling point of view, there is a clear motivation for the use of spatial econometrics. Firstly, commercial real estate assets are characterised by their location. Secondly, spatial dependence can solve many problems regarding missing variables in the data such as data regarding amenities. Thirdly, the F- and t-statistics are biased if spatial dependence is present in the data. Fourthly, the spatial models reflect an equilibrium state of the market. Lastly, there is an omitted variable bias if the spatial dependence is present. Spatial dependence can, however, appear in different forms (Anselin, 1988; LeSage and Pace, 2010; Kelejian and Prucha, 2010). Therefore it is necessary to know what kind of spatial dependence can arise and which estimation methods are available to capture such dependence. 3. Estimation Strategy From our review of the academic literature on transaction based commercial property valuation with spatial dependence we can start by applying a hedonic estimation strategy. The method uses multi-variate controls for hedonic, location or neighborhood characteristics in a transaction event. In addition, we operationalize a mixed regressive-spatial autoregressive model, which is denoted as the SAR-model and the mixed-regressive-spatial autoregressive model with a spatial autoregressive disturbance structure, which is specified as the SARARmodel Hedonic Analysis The hedonic technique is a multi-variate cross-sectional analysis of transaction prices, which relates prices of goods to their bundle of components. For commercial real estate prices, a buildings fundamental characteristics and the services it provides, e.g., size, age, location, etc.. It is also customary to add controls for time and neighborhood effects that can accrue cross-sectionally. The standard hedonic framework as originally specified by Rosen (1974) is as follows: 9

10 log P i,t = X i,t β + T t δ t + ɛ i,t (1) where P is an nx1 vector of logged property transaction prices, X i is an nxk matrix of (exogenous) hedonic property characteristics; β i is a kx1 parameter vector; ɛ i is the nx1 vector of regression disturbances. Anti-loged parameter estimates from the time effect dummies are used to form the base of the index values Spatial Analysis The spatial approach is a multi-variate cross-sectional analysis of transaction prices. Unlike the standard hedonic approach, the spatial temporal function also incorporates a spatial weights matrix into a standard multi-variate analysis of transaction prices. Thus, the transaction prices are a function of building hedonic characteristics, spatial dependence. There is a simple breakdown relating to the spatial autoregressive process to correct for spatial autocorrelation of the error term. The standard spatial autoregressive SAR-model is specified as follows: n log P i,t = α + X i,t β + λ W i,j P i,t + ɛ i,t (2) where P is an nx1 vector of logged property transaction prices, X is an nxk matrix of (exogenous) hedonic property characteristics; and β is a kx1 vector of parameters; W is an nxn spatial weight matrix with nonnegative spatial characteristics on the off diagonal and zero elements on the diagonal and W =W ; ɛ is the nx1 of regression disturbances, λ is the spatial autoregressive parameter The Spatial Weight Matrix The spatial weight matrix specifies the spatial relation in the econometric model. The three matrices specified as M1, M2 and M3 are implemented in the spatial regression models 10 i=1

11 to test whether the spatial effect is robust to the specification of the spatial weight matrix We employ three different specifications of the weight matrix, M1, Binary contiguity matrix (normalized), M2, Symmetric normalized inverse distance matrix, M3, Symmetric normalized inverse distance matrix with a cut-off. Generally, a spatial weight matrix is an N N matrix that describes the spatial arrangement of the spatial units (Elhorst, 2001). More specifically, the elements contain the spatial relation between observation i and j. relationship can be an exogenous specification of distance between two observations or simply specify whether the observations are neighbours. For robustness, we will test various specifications of the spatial weight matrix (Elhorst, 2010a). This The binary contiguity matrix distinguishes only between neighbouring observations and non-neighbouring observations. The binary contiguity matrix is constructed so that each neighbouring observation is defined by w ij = 1. Consequently, the remaining observations are zero because they are not assumed to be neighbours and so are the elements on the diagonal because one observation cannot be a neighbour of itself (Elhorst, 2010a). specification results in the spatial weight matrix W 0 as shown in Equation 3. The matrix is named W 0 to emphasise that it is not standardised. This 0 w 1,2 w 1,N w W 0 = 2,1 0 w 2,N...., were (3).. w N,1 w N,2 0 1 if i is a neighbour of j w ij = 0 otherwise The standardised contiguity matrix is similar to the non-standardised matrix. However, the difference is that each row sums up to 1 which ensures the normalisation. An example of this procedure when N = 3 is shown in Equation 4, which shows the non standardised 11

12 matrix and Equation 5, which shows the standardised matrix. W 0 = W = /2 0 1 / (4) (5) The symmetric normalized inverse distance matrix is based on actual distances between observation i and j in the sample. The diagonal elements are equal to zero because the observation cannot be a neighbour of its own and consequently the distance is zero. The normal procedure is that the rows of the matrix are standardised so that they sum up to 1. However, this is not possible in this specification as the scaling causes this matrix to lose its economic interpretation of distance decay (Elhorst, 2001, p. 134). Therefore, the standardisation is achieved by setting its largest characteristic root equal to 1 (w max = 1). Consequently the spatial response parameters lie between 1/w min and 1. The result of this specification is shown in Equation 6. w ij = 1/w ij max(min rc ), i j (6) The variable w ij represents the euclidean space measured in actual distance between observation i and j. The inverse relation shows that the closest observations are given more weight in the spatial matrix. Furthermore, the inverse distance is divided by the maximum of the minimum of the sum of the rows and the sum of the columns in the normalisation procedure. The economic intuition is that this is largest direct distance between observation i and j in a region (Elhorst, 2010b). Thus, although the normalisation cannot be achieved by setting the sum of N-rows equal to 1, it is still possible to normalise the matrix. The symmetric normalized inverse distance matrix with a cut-off is similar as the one 12

13 shown above. The key difference is that not only the diagonals are zero but also the observations that are located further than the specified distance. This can be seen as a cut-off in the matrix as it does not allow for spatial dependence between all observations Estimation Procedure Different estimation strategies have been proposed in the spatial literature. Pace et al. (1998) employ a maximum-likelihood procedure and due to their sparse matrices technique the procedure becomes feasible for even large sample sizes. While there are numerous estimation methods available, LeSage and Pace (2010) choose to apply the maximum likelihood because this is a consistent estimation method for data that contain spatial dependence (Lee, 2004). The implementation of the maximum likelihood estimation seems to be appropriate for the spatial regression model. However, the implementation of maximum likelihood requires the assumption regarding the disturbance structure. More specifically, the disturbance vector is assumed to be normally distributed. Kelejian and Prucha (1999) propose this approach because of its flexibility regarding the specification of the spatial weight matrix. In addition, this estimation method does not require the assumption that the error term is normally distributed. This makes it possible to allow for heteroskedasticity (Kelejian and Prucha, 2010).Thus, we employ a socalled Generalized Spatial Two Stage Least Squares (GS2SLS) estimation procedure, with corrections for autoregressive and heteroskedastic disturbances (Kelejian and Prucha, 2010). Thus, the regression disturbances in Equation 2 are modeled as follows: n ɛ i,t = ρ W i,j ɛ i,t + γ i,t (7) i=1 where again W is an nxn spatial weight matrices with nonnegative spatial characteristics on the off diagonal and zero elements on the diagonal and W =W ; ɛ is the nx1 vector of regression disturbances, γ is a vector of innovations; ρ is the spatial autoregressive parameter in the residuals. 13

14 The second estimation procedure has three steps: the model is first estimated by two stage least squares using the instruments H n, which are a subset of the linearaly independent columns of (X,W X,W 2 X 2,...), in the second step a GM estimator for the autoregressive parameter ρ i is estimated using the 2SLS residuals ɛ i from the first step, and lastly the regression model is re-estimated by 2SLS after transforming the model through a Cochrane- Orcutt procedure to account for spatial correlation Expectations Ex- ante we have the following expectations for the comparison of the two models. Table 1 outlines the hypothesis where the parameter ρ captures the spatial dependence in the econometric model. The hypothesis is rejected when ρ > 0. This implies that the spatial lag exists and there is spatial dependence present in the model. The parameter λ captures the spatial dependence in the disturbance structure. Again, the hypothesis is rejected when λ > 0, which implies that there is spatial dependence in the disturbance structure. [Table 1 about here.] In addition, the model specification must consider varying relationships of spatial dependence. Firstly, the symmetric normalized inverse distance matrix (spatial matrix A), the symmetric normalized inverse distance matrix with n neighbours (spatial matrix B) and the normalized binary contiguity matrix (spatial matrix C). We anticipate that the results of the estimation shows to what extent the estimation is sensitive to the specification of the spatial weight matrix. 4. The London, Paris and Frankfurt Commercial Office Markets To execute our empirical analysis of spatial dependence we use data from Real Capital Analytics (RCA). 2 We examine three international property markets, London, Paris and Frankfurt. 2 Real Capital AnalyticsRCA is an international company that is engaged in property research with the main focus on the investment market for commercial real estate. RCA started to track the transactions of commercial real estate in the US. Yet, they expanded rapidly and start to track the transactions all over the world and by 2007 they achieved to cover all markets globally. 14

15 4.1. The Office Market in Frankfurt The office market in Frankfurt is a major investment location. This market is known for its high turnover. The vacancy rates remained relatively constant over the period and so did the rents. The office market in Frankfurt is a very stable market. The highest demand is driven by banks, other services and consultancy firms. This follows from the fact that the service sector is traditionally strong in Frankfurt. Compared to London and Paris, Frankfurt contains the smallest office market, which is divided in the city center, the proper market and the suburbs. The city center contains all the buildings located in the center of Frankfurt. The proper market contains the high quality buildings. The suburbs contain the buildings located around the city center of Frankfurt. After the exclusion of the missing properties, there are 242 observations. The map of these properties is shown in Figure 1. [Figure 1 about here.] Most of the office buildings are situated in the center. The remaining office buildings are located in small clusters around the city center and only few buildings are is isolated from the others. It is therefore likely to expect that most spatial dependence arises in the city center. The office buildings in the suburbs are unlikely to show much spatial dependence as the distance between the center and the suburbs is relatively large. The descriptive statistics for the office market in Frankfurt are shown in Table 2. [Table 2 about here.] The average price of an office building in Frankfurt is e85.5 M. The most expensive buildings are located in the proper market, where also the largest transaction took place. The transaction of almost 1 billion Euro is a large huge multi-tenant building sold by Morgan Stanley in The largest transaction in 2011 was the Deutsche Bank Twin Towers. The average value of an office building in the suburbs is with e41.7 M substantially less than the value of the office buildings in both the city center and the proper market. Moreover is the average size relatively smaller in this area. Most notably are the number of transactions over the years. RCA registered the most transactions in 2006 and 2007, a period that was known 15

16 at the top of the market. The number of transactions decreased to 7 in 2009, which can be directly related to the consequences of the financial crisis. The number of transactions increases from Besides this, the average value of a transacted asset increases as well in 2010, implying that only the most valuable assets tended to transact in this period of uncertainty The Office Market in London The London office market is characterized by its dynamics. This can be seen by the year to year differences in take-up, supply, prime rent, investment yield, etc. The office market relies heavily on foreign investors and is therefore heavily exposed to global economic conditions. In 2012, the yields are at an all time low 3. This implies that the London office market is considered safe during these times of financial distress. The financial sector dominated the office market until 2011, the year that the TMT-sector 4 became a larger occupier of the London office buildings. The office market in London is the largest market under scrutiny. The London market is divided into 10 sub-markets. The immense size of the office market in London makes it harder to define the actual city center as there are more important centralized locations that contain many office buildings. In fact, the city center in London contains three central points where multiple office buildings tend to cluster. This is shown in Figure 2. [Figure 2 about here.] Clearly, most buildings in the database are located in West End. However, there is another cluster in the market City Core. The remaining assets are more or less equally divided in distance around the city center. Based on this map, it is likely to expect that the spatial dependence arises in the city center. The buildings are situated close to each other and therefore the value of the assets are likely to be influenced by the value of neighbouring assets. Also, the value of the buildings around the city center are not likely to affect the value of the 3 Market data can be found in European Office Market 2012, BNP Parisbas 4 Technology, Media, Telecommunication sector 16

17 buildings in the city center for two reasons. Firstly, the distance is relatively large. Spatial models are based on, among others, the assumption that the spatial dependence shows a decay when the distance becomes larger. Secondly, they are not located in clusters which implies that they do not face direct competition from their neighbouring assets. Hence, the spatial dependence is more likely to stay in the city center itself. The descriptive statistics are shown in Table 3. [Table 3 about here.] The market in London shows a lot of variation in the observations. The smallest transaction value is approximately e750k while the largest transaction value is almost e1.8 Bln, which is a large shopping center in the UK partly owned by the Dutch pension fund APG. West End is by farthest the market and contains 528 transactions. City Core contains 338 transactions, which is the second largest amount. Most notably is the average transaction value in the Docklands. This market contains only 26 transaction but the average value is close to e270 Mln. This can be explained by the fact that this was once the largest dock in the world. The main docks are now no longer situated in the city center and consequently this area emerged as the central business area after the immense redevelopment of the area. The first peak of the number of transactions is in After 2007 the number of transactions decreases substantially but the market shows a quick recovery. The number of transaction becomes even larger in 2010 and The market in London shows signs of a dynamic market and there is likely to be spatial dependence in the central areas. The main activity takes place in the City Core and West End The Office Market in Paris The office market in Paris is known for its dynamics. Yet, the prime rents remain relatively stable. The take up follows the economic business cycle naturally, which is a sign that the market is relatively liquid. Lately, the market has become a safe haven for cash rich investors such as insurance companies and pension funds. The decline after the crisis hit Europe is less severe than the other two markets of interest. This might imply that the 17

18 Parisian office market is less exposed to the financial sector. Moreover, the take-up volume is the highest of Europe in 2010 and Furthermore, the Parisian office market shows a cluster in the city center and next to the center. For this market, the hedonic properties age and floors are excluded from the analysis because it would lead to a substantial decrease in observations. This would result in problems in applying the spatial models. Fortunately, size is the most important determinant for the value. The plot of the locations of the office buildings is shown in Figure 3. [Figure 3 about here.] The plot shows that the Parisian office market contains assets all over Paris. While there are clearly two clusters, the remaining assets are all located in the city. This is an excellent opportunity to test whether there is spatial dependence between observations in the clusters and the surrounding assets. The descriptive statistics are shown in Table 4. [Table 4 about here.] The exclusion of the two hedonic properties leads to 1025 observations in total. It is immediately evident that the database contains only the most valuable assets as the minimum is e2.7 billion. This building is called Cour Dé fense, which is one of the largest assets in the office district La Defence. The asset contains two towers that both have a height of 161 meter and are situated 25 meter from each other. The number of transactions tends to be larger from However, this could be explained by the increase in coverage by RCA. Yet, the decrease in the number of transactions in 2008 seems to be present in all three markets. The Parisian market, however, shows a an increase of the number of transactions in 2009 but this increase is accompanied by a decrease in the average transaction value. Most notably is that the average value of the assets in the suburbs is only tightly smaller than the average value in the other Parisian markets. When these statistics are related to the map, it is can be seen that this market shows signs of spatial dependence as the value of neighbouring assets tend to affect the value of each other. 18

19 5. Results Results for the transaction based SAR model and the SARAR model for the Frankfurt, London and Paris office markets are reported below. We operationalize the models using three different specifications of the spatial weight matrix Frankfurt The dataset of the office market in Frankfurt is the smallest sample of the three markets under scrutiny. Due to the small amount of observations in 2000, 2001 and 2002 and the lack of observations in 2003 and 2004 it is only possible to construct an index from 2005 till Table 5 contains the results of the office market in Frankfurt. [Table 5 about here.] The results in Table 5 show that all the models have a relatively high explanatory power according to the R 2 measure. Most notably are the results of the SARAR model with the spatial weight matrix specified as the contiguity matrix. The R 2 is substantially smaller and the variance is higher than the other specifications, which indicates that this combination performs worse in explaining the transaction price. Not all the time dummies are significant, which is against the expectations. Especially the years 2007 and 2008 are problematic. The remaining years have a higher explanatory power, implying that the models are able to explain the transaction prices better for these years.the location dummies are only significant and negative for the suburbs. The spatial dependence is limited in the present results. In the SAR-model with the inverse distance matrix with no cut off the spatial dependence is slightly negative and significant. This is also true for the SARAR-model with the similar matrix. This finding implies that the spatial dependence is related to the distance. When the distance becomes larger, the transaction value is positively affected. This makes intuitively sense as the value is likely to depend on the buildings that are relatively close. The SARAR-model with the contiguity matrix finds spatial dependence in the disturbance structure, which is both significant and 19

20 unstable. This implies that the unobserved variables affect the value of neighbouring observations but do not converge to a certain constant distribution. Consequently the results cannot be used for either forecasting purposes or the construction of an index. The other specifications of the spatial weight matrices are able to deal with this dependence in such a way that the dependence arises in the spatial lag, which implies that there is weak evidence of spatial autocorrelation. Compared to the estimation via OLS, only the SARAR model seems to be substantially different. Yet, the difference is mainly in the t-statistics and not in the coefficients. Based on these results it is clear that, considering the explanatory power, the variance and the coefficients that the spatial weight matrix, when specified as the inverse distance matrix with no cut-off, performs better than the other specifications London The sample of the office market in London is the largest of all three markets. This sample is sufficient to construct an index that starts in 2001 and ends in The results in Table 6 show the outcomes of the regression models as laid out in the methodology. [Table 6 about here.] The results in Table 6 indicate that all the models with the respective specification of the spatial weight matrix have similar explanatory power according to the R 2 measure and the variances. Yet, the estimation via OLS seems to suffer from the fact that only one hedonic property has been taken into account. This results in insignificant parameters, even for the time dummies. The spatial models, however, seem to find that all the year dummies are significant.the locational dummies behave as expected, considering that the city center is chosen as the reference location. The output shows that the Docklands is the most expensive location while the outer region of London is the cheapest. The spatial dependence in London is limited. Only the SAR model in combination with the spatial weight matrix with no cut-off finds significant spatial dependence. The parameter is, however, close to zero. The combination of the SARAR-model and the contiguity matrix 20

21 results again in significant spatial dependence in the disturbance structure Paris Table 7 shows the results of the Parisian market from 2001 till The sample of the Parisian market is a little bit smaller than the sample of the London market. Yet, this market is convenient to analyse the differences of the models in their behaviour in the analysis of larger samples. [Table 7 about here.] The results in Table 7 indicate that all the models have a relatively low R 2 around the This is due to the inclusion of only one hedonic property, the size. Again, the coefficients for the time dummies are considerably larger in the spatial models, implying that these models handle the small quantity of hedonic properties better. The locational dummies behave as expected. The assets in the suburbs have a significant lower transaction value than the assets in the suburbs. The proper market is again not significant. The transaction value is as expected lower in the suburbs. The spatial dependence is present in both models. In the SAR-model with the spatial weight matrix, specified as the inverse distance matrix with no cut-off, the spatial lag is significant and negative. This implies that there is a slight dispersion between the spatial component and the transaction price. In other words, when the distance becomes larger the value increases as well. The negative sign is due to the inverse distance matrix. The spatial effect in the disturbance structure of the SARAR-model with the matrix M2 is significant and unstable. This again reveals the problem regarding the high level of heterogeneity in the real estate market. 5 When the estimation is performed using only the office buildings in the database, the coefficient becomes larger than 2 and thereby violating the stability condition. This again indicates that the variance does not converge to a constant variance and the model is unstable. This is consistent with the findings in Frankfurt and indicates a large extent of heterogeneity. 21

22 6. Comparative Model Specifications Both models capture spatial dependence in two distinct ways. First, the SAR has a spatial lag. In contrast, the SARAR model contains not only the spatial lag but also a spatial component in the disturbance structure. The significance of the parameters of these components determines to what extent the components affect the estimation of the parameters of the explanatory variables. When for example the parameter of the spatial component in the disturbance structure is insignificant but the coefficient of the spatial lag is, the SARAR model is essentially similar to the SAR model. In the case where coefficients of the spatial lag are similar in both models, the parameters should be equal as the underlying data is exactly the same Spatial Weight Matrices The specification of the spatial weight matrices seems to affect the estimation. First, the contiguity matrix is not able to capture the spatial dependence in the form of a spatial lag very well. The parameters are either insignificant or the parameter is significant but has a value close to zero. The contiguity matrix is, however, able to capture significant spatial dependence in the disturbance structure. In fact, the use of the contiguity matrix not only results in significant spatial dependence in the disturbance structure, but also is significant for the spatial dependence in the spatial lag. Yet, the parameter is greater than 1 in absolute value and therefore not stable. Second, the inverse distance matrix with no cut-off seems to perform consistently in the analysis. Both models find parameters that are quite similar to each other. Only the parameter of the spatial lag in the model used for the market in Frankfurt varies slightly. This seems to affect the coefficients for the explanatory variables in the model as well. The coefficients for the explanatory variables are slightly smaller when the SARAR model is used. Overall, the inverse distance matrix with no cut-off performs well in capturing the spatial dependence. Thirdly, the inverse distance matrix with a cut-off performs different than the other two specifications. The cut-off only takes the office buildings within a circle of 5 miles into account. The parameter for the spatial lag is insignificant for the market in Frankfurt, but is significant for the markets in 22

23 London and Paris. The negative sign implies that the value of an asset increases when the distance to other assets becomes larger, which is a relevant effect as it is robust with the other specifications The robustness of the econometric specifications To compare the two model specifications it is convenient to look at the explanatory power, the variances and the signs of the parameters. Comparing the results of the three different markets it is shown that the explanatory power is approximately equal for both models in the respective market. Additionally, the variances are approximately similar as well. The coefficients are not only similar in sign but also in magnitude. This implies that both models perform similar in capturing the affect of the explanatory variables. Thus, considering the results in the preceding section it is shown that the models are robust to each other. However, if the models perform exactly the same they would result in exactly the same coefficients and consequently in similar indices as well. Yet, besides the small differences, all the models seem to result in a similar path. Thus, based on the coefficients and the indices it can be concluded that all the models are able to give some credence for incorporating spatial dependence, but keeping in mind that the parameters are very small for these markets. 7. Conclusion Compared to previous work in the commercial real estate literature, we find a very small spatial dependence parameter for the London, Paris and Frankfurt markets. Importantly, these results are based on markets where buildings are very heterogeneous and traded in a global environments where institutional investors are able to make comparisons across markets. Consequently, in older more heterogeneous markets spatial dependence may not play as significant a role for commercial real estate asset pricing. However, consistent with LeSage and Pace (2008); Tu et al. (2004) the findings here indicate that the coefficients in particular, are enhanced by the inclusion of the spatial dependence parameter. Indicating that spatial dependence parameter is potentially an omitted 23

24 variable. However, the spatial dependence parameter has a very small coefficient in all markets, across all models and weight matrix specifications. Furthermore, Tu et al. (2004) acknowledge that the lack of transactions can be problematic. Within this sample, data problems became evident in the application of the spatial models on the office market in Frankfurt in two forms. Firstly, the lack of transactions required the exclusion of the years Secondly, the results regarding the temporal effects were weak. The problem was also evident in the Parisian office market but only in the earlier years The remaining data was sufficient to analyse the remaining years. Nappi-Choulet and Maury (2009) find strong evidence of spatial dependence in the Parisian office market. Their research was, however, limited to the central business district and La Defence. By taking the whole Parisian office market into account, the evidence of spatial dependence was limited here. The analysis using the SARAR-model addresses this issue by modelling spatial dependence in the error structure. Although this appears to be a problem when the contiguity matrix is used, it cannot be responsible for the huge differences between the findings of Nappi-Choulet and Maury (2009) and these results because the parameter of the spatial lag is similar in the SAR model. The largest difference is from the focusnappi-choulet and Maury (2009) on the two main business districts while this empirical work focuses on the whole Parisian office market. In addition, Tu et al. (2004) find differences in the indices between an OLS estimation and the spatial model are not constant. This is consistent with the findings here and implies that the spatial bias is not constant. This implies that the transaction value of office buildings is different affected by the location in space across time. It is possible to allow for time-varying parameter to take this phenomenon into account (Kuethe et al., 2008). Additionally, small samples can impact the estimation as well. This is in line with the findings of Wong et al. (2012) who show that spatial dependence varies with trading volume. Future research should test an additional model, it appears that the current literature in commercial real estate already prefers a spatio-temporal based autoregressive model. This may be due to the findings of a strong and significant spatial dependence parameter in these models. However, it appears for these empirical specifications that spatial dependence 24

25 does not have the magnitude previously expected. Thus, going forward comparisons across three models may be important for understanding the spatial dependence in European commercial real estate markets. 25

26 References Anselin, L., Spatial Econometrics: Methods and Models. Studies in Operational Regional Science. Springer. Baltagi, B. H., Li, D., Lm tests for functional form and spatial error correlation. International Regional Science Review 24 (2), Baltagi, B. H., Song, S. H., Koh, W., Testing panel data regression models with spatial error correlation. Journal of econometrics 117 (1), Colwell, P. F., Munneke, H. J., Trefzger, J. W., Chicago s office market: Price indices, location and time. Real Estate Economics 26 (1), Debrezion, G., Pels, E., Rietveld, P., The impact of railway stations on residential and commercial property value: A meta-analysis. The Journal of Real Estate Finance and Economics 35 (2), URL Downs, D. H., Slade, B. A., Characteristics of a full-disclosure, transaction-based index of commercial real estate. Journal of Real Estate Portfolio Management 5 (1), Elhorst, J. P., Dynamic models in space and time. Geographical Analysis 33 (2), Elhorst, J. P., 2010a. Applied spatial econometrics: Raising the bar. Spatial Economic Analysis 5 (1), Elhorst, J. P., 2010b. Spatial panel data models. In: Fischer, M. M., Getis, A. (Eds.), Handbook of Applied Spatial Analysis. Springer Berlin Heidelberg, pp Elhorst, J. P., Lacombe, D. J., Piras, G., On model specification and parameter space definitions in higher order spatial econometric models. Regional Science and Urban Economics 42 (1), Fisher, J. D., Geltner, D. M., Webb, R. B., Value indices of commercial real estate: A comparison of index construction methods. The Journal of Real Estate Finance and Economics 9, Geltner, D., Pollakowski, H., September A set of indexes for trading commercial real estate based on the real capital analytics transaction prices database, release 2. Kelejian, H. H., Prucha, I. R., Estimation of spatial regression models with autoregressive errors by two stage least squares procedures: A serious problem. Electronic working papers, University of Maryland, Department of Economics. Kelejian, H. H., Prucha, I. R., A generalized moments estimator for the autoregressive parameter in a spatial model. Electronic Working Papers , University of Maryland, Department of Economics. Kelejian, H. H., Prucha, I. R., Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics 157 (1), Kuethe, T. H., Foster, K. A., Florax, R. J., A spatial hedonic model with time-varying parameters: A new method using flexible least squares Annual Meeting, July 27-29, 2008, Orlando, Florida 6306, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics 26

27 Association). Lee, L.-F., November Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72 (6), LeSage, J. P., Pace, R. K., Spatial econometric modeling of origin-destination flows*. Journal of Regional Science 48 (5), LeSage, J. P., Pace, R. K., Spatial econometric models. In: Fischer, M. M., Getis, A. (Eds.), Handbook of Applied Spatial Analysis. Springer Berlin Heidelberg, pp Miles, M., McCue, T., Commercial real estate returns. American Real Estate and Urban Economics Association Journal 12 (3), Nappi-Choulet, I., Maleyre, I., Maury, T., A hedonic model of office prices in paris and its immediate suburbs. Journal of Property Research 24 (3), Nappi-Choulet, I., Maury, T.-P., A spatiotemporal autoregressive price index for the paris office property market. Real Estate Economics 37 (2), Pace, R. K., Barry, R., Clapp, J. M., Rodriquez, M., Spatiotemporal autoregressive models of neighborhood effects. The Journal of Real Estate Finance and Economics 17 (1), URL Pace, R. K., Barry, R., Gilley, W., Sirmans, C., A method for spatial-temporal forecasting with an application to real estate prices. International Journal of Forecasting 16 (2), Rosen, S., Hedonic prices and implicit markets: Product differentiation in pure competition. Journal of Political Economy 82 (1), URL se Can, A., Megbolugbe, I., Spatial dependence and house price index construction. The Journal of Real Estate Finance and Economics 14, , /A: Tu, Y., Yu, S.-M., Sun, H., Transaction-based office price indexes: A spatiotemporal modeling approach. Real Estate Economics 32 (2), Wong, S., Yiu, C., Chau, K., Trading volume-induced spatial autocorrelation in real estate prices. The Journal of Real Estate Finance and Economics,

28 List of Figures 1 Frankfurt office market London office market Parisian office market

29 Figure 1: Frankfurt office market 29

30 Figure 2: London office market 30

31 Figure 3: Parisian office market 31

32 List of Tables 1 The general null-hypotheses The office market in Frankfurt The office market in London The office market in Paris The office market in Frankfurt The office market in London The office market in Paris

33 Table 1: The general null-hypotheses Hypothesis H 1 H 2 H 3 Formulation The parameter ρ is zero. The parameter λ is zero. The parameters β n are zero. 33

34 Table 2: The office market in Frankfurt Variable Mean Sigma Min Max n Total Price 85,455, ,089,430 2,925, ,422, Size(sq ft) 188, ,204 11,291 1,959, Submarkets (Prices) City center 97,959, ,139,507 5,468, ,833, Proper 113,547, ,125,953 2,960, ,422, Suburbs 42,704,019 63,799,781 2,925, ,196, Time (prices) ,436,296 85,792,849 2,960, ,267, ,410, ,188,315 2,925, ,422, ,079,942 45,525,376 8,705, ,937, ,739,976 76,150,549 3,318, ,248, ,415, ,481,934 5,691, ,833, ,911, ,649,845 6,488, ,591, ,971,816 57,982,689 12,483, ,928,

35 Table 3: The office market in London Variable Mean Sigma Min Max n Total Price 62,056, ,375, ,776 1,742,999,999 1,284 Size 102, , ,899,800 1,284 Sub-markets (Prices) City Core 81,409,140 87,654,824 1,236, ,175, City Fringe 93,723, ,209,472 4,073,505 1,742,999, Docklands 270,339, ,873,170 8,659,347 1,111,900, Midtown 61,755,465 68,028,395 1,450, ,500, North Central 32,682,227 54,131,527 1,100, ,000, Outer 25,266,281 33,100,810 3,000, ,000, South central 55,511,333 56,001,551 5,075, ,000, Southern Fringe 27,949,914 17,831,471 6,400,000 60,000,000 8 West Central 34,608,680 65,480,137 2,800, ,000, West End 47,454,035 68,222, , ,693, Time (prices) ,712,539 42,856,098 5,100, ,500, ,464,379 65,499,265 4,753, ,000, ,201,811 61,455,520 4,400, ,500, ,512, ,057,052 4,461,841 1,111,900, ,660,728 57,740,281 1,471, ,666, ,959,575 95,158,610 2,800, ,000, ,334, ,170,046 1,236,286 1,093,452, ,383, ,157, , ,000, ,845,491 70,304,067 2,465, ,000, ,745,253 87,779,661 1,780, ,693, ,651, ,651,733 3,250,000 1,742,999, ,403,342 82,820,197 4,410, ,000,

36 Table 4: The office market in Paris Variable Mean Sigma Min Max n Total Price 72,823, ,964,036 73,134 2,747,359,885 1,025 Size 109, , ,713,580 1,025 Submarkets (Prices) City center 68,599,658 92,263, , ,727, Proper 78,241, ,305, ,271 2,747,359, Suburbs 72,896, ,312,559 73, ,999, Time (prices) ,589,965 94,031,137 2,730, ,630, ,826,770 95,866, , ,350, ,113, ,809, ,387 2,747,359, ,986,118 65,429, , ,225, ,462,413 89,268, , ,360, ,862, ,873,616 73, ,999, ,981,764 92,702,938 73, ,076, ,460, ,651,659 3,704,001 44,4480,

37 Table 5: The office market in Frankfurt SAR SARAR OLS M1 M2 M3 M1 M2 M3 - R-squared Rbar-squared σ Nobs Nvars log-likelihood Variable Coefficients (t-stat) Coefficients (t-stat) Coeff.(t-st.) M1 M2 M3 M1 M2 M3 - constant (10.87) (11.99) (11.79) (9.78) (10.04) (9.43) (11.71) Size (25.18) (25.64) (25.31) (21.88) (21.64) (20.57) (24.82) (1.34) (0.29) (1.36) (2.03) (0.22) (1.61) (1.31) (-0.32) (-0.43) (-0.30) (-1.01) (-0.36) (-0.21) (-0.32) (1.05) (1.12) (1.06) (-0.05) (0.74) (0.70) (1.01) (2.07) (2.06) (1.89) (0.81) (2.65) (2.11) (2.02) (1.68) (1.74) (1.71) (-0.16) (2.05) (1.95) (1.65) (1.96) (2.02) (1.98) (0.83) (1.83) (1.69) (1.91) Proper (-0.91) (-1.12) (-0.91) (-1.38) (-1.19) (-0.94) (-0.91) Suburbs (-6.13) (-6.39) (-5.95) (-4.34) (-6.89) (-6.21) (-6.00) ρ (0.19) (2.20) (0.55) (0.19) (4.14) (0.59) - λ (4.88) (-0.52) (-0.18) - 37

38 Table 6: The office market in London SAR SARAR OLS M1 M2 M3 M1 M2 M3 - R-squared Rbar-squared σ Nobs Nvars log-likelihood Variable Coefficients (t-stat) Coefficients (t-stat) Coeff. (t-st.) M1 M2 M3 M1 M2 M3 - constant (67.89) (68.40) (69.20) (46.37) (52.02) (47.20) (87.13) Size (35.77) (35.26) (35.79) (20.97) (21.05) (20.80) (35.12) (4.95) (4.98) (5.07) (5.00) (5.12) (5.26) (-2.05) (5.31) (5.49) (5.49) (5.56) (5.94) (5.79) (-1.34) (6.37) (6.85) (6.75) (6.45) (7.10) (6.94) (-1.32) (7.94) (8.12) (7.94) (7.54) (8.22) (8.09) (-0.17) (7.94) (8.79) (8.50) (7.76) (8.51) (8.34) (-0.12) (8.10) (9.08) (8.72) (7.76) (8.70) (8.54) (-0.15) (6.38) (7.14) (6.85) (6.50) (7.32) (6.90) (-1.20) (7.19) (7.77) (7.41) (7.34) (8.16) (7.58) (-1.22) (11.23) (10.91) (11.09) (9.66) (9.53) (9.59) (-2.18) (11.41) (10.77) (11.28) (10.05) (9.59) (10.05) (2.40) (8.89) (9.14) (8.74) (8.94) (8.94) (8.65) (-2.04) City Fringe (-1.00) (-1.05) (-1.08) (-0.92) (-1.95) (-1.07) (-2.50) Docklands (2.50) (2.47) (2.47) (2.15) (2.01) (2.44) (-8.09) Midtown (-0.21) (-0.35) (-0.36) (-0.06) (-1.55) (-0.42) (-1.10) North Central (-2.26) (-2.39) (-2.43) (-1.48) (-2.42) (-1.93) (-2.36) Outer (-7.86) (-7.72) (-7.92) (-6.10) (-7.43) (-6.55) (-1.54) South Central (-0.13) (-0.20) (-0.09) (-0.33) (-0.98 ) (-0.11) (-0.11) Southern Fringe (-2.43) (-2.44) (-2.24) (-3.75) (-4.09) (-3.05) (-3.05 ) West Central (-1.03) (-1.24) (-1.21) (-1.56) (-1.96) (-1.04) (-1.04) West End (2.84) (2.73) (2.72) (2.52) (-2.69) (2.93) (2.93) ρ (-1.75) (-2.64) (-0.65) (-1.74) (-2.67) (-0.91) - λ (4.16) (0.83) (0.45) - 38

39 Table 7: The office market in Paris SAR SARAR OLS M1 M2 M3 M1 M2 M3 - R-squared Rbar-squared σ Nobs Nvars log-likelihood Variable Coefficients (t-stat) Coefficients (t-stat) Coeff.(t-st.) M1 M2 M3 M1 M2 M3 - constant (30.89) (34.67) (34.06) (26.34) (24.59) (27.16) (34.24) Size (21.97) (22.14) (22.11) (16.01) (15.86) (16.19) (21.96) (7.88) (7.85) (7.91) (7.35) (6.61) (7.41) (7.84) (13.15) (13.65) (13.20) (12.29) (11.14) (12.53) (13.09) (7.22) (7.78) (7.26) (6.76) (6.83) (6.82) (7.19)) (7.82) (8.16) (7.92) (7.65) (7.30) (7.88) (7.80) (6.57) (6.96) (6.62) (6.25) (6.49) (6.40) (6.59) (10.01) (10.68) (10.08) (9.34) (8.87) (9.42) (10.01) (7.26) (7.80) (7.27) (7.43) (7.31) (7.47) (7.24) Proper (-0.44) (-0.45) (-0.23) (-0.53) (-0.69) (-0.24) (-0.44) Suburbs (-3.55) (-3.49) (-3.58) (-3.70) (-3.98) (-3.74) (-3.53) ρ (0.00) (-3.53) (-1.44) (-0.74) (-2.03) (-1.93) - λ (0.10) (3.75) (-0.44) - 39