Spatial Dependence in International Office Markets

Abstract

This paper investigates spatial dependence in the prices of office buildings in Hong Kong, London, Los Angeles, New York City, Paris, and Tokyo for 2007 to 2013. Compared to prior literature, we find low economic impact from spatial dependence in all six markets, and spatial and spatial-temporal dependence do not moderate the effects of hedonic characteristics statistically or economically. However, investor and seller types as well as neighborhood location have a significant impact on the economic and statistical significance of the spatial and spatial-temporal parameters. Spatial office price indices for London, Paris and Tokyo decline somewhat more than do hedonic indices during the crisis.

Appendices

Appendix A: Location Dummies and CBD

The market and sub-market breakdown for the six office markets is constructed as follows. For London, the first two to three letter pre-fix of their postal codes is used to delineate sub-markets, e.g., EC to represent London City. We estimate relative to the South bank. The CBD is denoted as Westminster and London City.

For Paris, the markets have been divided into the Parisian arrondissements and greater suburbs, where the CBD is transactions occurring in the Quartier Central des Affaires (8th Arrondissement), and La Defense (15th Arrondissement). The 1st, 8th, and 15th Arrondissements are used as the base case.

In Tokyo, the city is broken down into administrative wards. The CBD is designated as the combination of the Chiyoda, Chuo, Minato and Central Wards. The wards included in the CBD as well as the Shinjuku Ward are used as the base case.

For Hong Kong, the Admiralty and Central districts form the CBD. These districts as well as the West district are used as the base for relative location value estimates.

New York City’s metro area is broken down into boroughs and then further disaggregated on Manhattan into Downtown, Midtown and Uptown. Downtown and Midtown Manhattan are used as the CBD for the New York City Metro area.

Finally, Los Angeles is defined by its zip-codes with ”Old Downtown” and ”New Downtown”, e.g., zip codes 90017 and 90071 as the locations for the CBD. These zip-codes plus 90210 are designated as the base case.

Appendix B: Spatial Estimation Procedure

Estimation of the spatial models is carried out by using the efficient version of the Generalized Method of Moments estimator developed in (Kelejian and Prucha 2010). We refer the reader to (Kelejian and Prucha 2010) for details on the estimation procedure. In a first step, we use a two stage least squares (2SLS) approach to estimate the model parameters. Since the spatial and temporal components contain the log price per square foot as an argument they are endogenous regressors. To find suitable instruments for the spatial and temporal components we follow the discussion given in (Kelejian and Prucha 2010), where for the SARAR model we note that the expected value of the endogenous regressors, i.e., the spatial and temporal dependence regressors, is a function of the exogenous components as shown in the following equation:

$$ E(W \log{P}) = (I - \rho_{1} W)^{-1} X \beta = \sum\limits_{i=1}^{\infty} {\rho_{1}^{i}} W^{i} X \beta. $$

(6)

Equation (6) shows that we can use the linearly independent columns (X, W X, W ² X, T) as instruments, and (Kelejian et al. 2004) has shown that a limit of two polynomial expansions are sufficient. To avoid over-identification issues we also restrict the columns of the hedonic matrix used in the multiplication with the power series of the spatial weights to the log size and age. For the STARAR model, we use the same idea and use the first two elements of the power series of L and (W ⊙ L) when determining the instruments. Collecting the regressor variables (WlogP, X, T) into the matrix Z and the instruments into the matrix H and use a 2SLS estimator given by:

$$ \widetilde{\theta} = \left(\hat{Z}^{\prime} Z \right)^{-1} \hat{Z}^{\prime} \log{P}, $$

(7)

where in the first step, with vector $\widetilde {\theta }$ collecting the model parameters ρ, β, and δ, and $\hat {Z} = H(H^{\prime }H)^{-1}H^{\prime }Z$. In the second step, the errors estimated using the first step 2SLS estimate are used to construct a nonlinear moments estimator for the spatial error parameter. The non-linear estimator for the spatial error parameter λ is found by minimizing the following criterion function:

$$ \min\limits_{\lambda} \left\{ (\gamma - \Gamma \alpha)^{\prime} \Psi (\gamma - \Gamma \alpha) \right\}, $$

(8)

where γ and Γ are a 2 × 1 vector and a 2 × 2 matrix, respectively. The vector α contains λ and λ ² as elements. The elements of γ and Γ are given by:

$$\begin{array}{@{}rcl@{}} \Gamma_{11} &=& \frac{2}{N} u^{\prime}W^{\prime}(W^{\prime}W - diag(w_{i}^{\prime}w_{i}))u, {\kern4pc} \Gamma_{21} = \frac{1}{N} u^{\prime}W^{\prime}(W - W^{\prime})u, \\ \Gamma_{12} &=& \frac{-1}{N} u^{\prime}W^{\prime}(W^{\prime}W - diag(w_{i}^{\prime}w_{i}))Wu, {\kern3pc} \Gamma_{22} = \frac{-1}{N} u^{\prime}W^{\prime}WWu, \\ \gamma_{1} &=& \frac{1}{N} u^{\prime}(W^{\prime}W - diag(w_{i}^{\prime}w_{i}))u, {\kern4pc} \gamma_{2} = \frac{1}{N} u^{\prime}Wu, \end{array} $$

and Ψ denotes the inverse of the covariance matrix of the two moment conditions which depends on the spatial error parameter λ. We use a first inefficient estimator for λ where we set the matrix Ψ equal to the identity matrix. A second efficient estimator for λ is found by using the matrix Ψ in the function given in Eq. 8. To determine the elements of Ψ we define the following:

$$\begin{array}{@{}rcl@{}} \epsilon & =& (I - \lambda W)u, \\ \Sigma &=& diag({\epsilon_{i}^{2}}), \\ \tilde{ \alpha}_{r} &=& \frac{1}{N} \left[ -Z^{\prime}(I - \lambda W^{\prime}) (A_{r} + A_{r}^{\prime}) (I - \lambda W)u \right], \\ \tilde{T}_{r} &=& (I - \lambda W^{\prime})^{-1}H \left( \frac{1}{N} H^{\prime}H \right)^{-1} \left( \frac{1}{N} H^{\prime}Z \right) \left[ \left( \frac{1}{N} Z^{\prime}H \right) \left( \frac{1}{N} H^{\prime}H \right)^{-1} \left( \frac{1}{N} H^{\prime}Z \right) \right]^{-1}, \\ \tilde{a}_{r} &=& T \tilde{ \alpha}_{r}, \end{array} $$

where the index r = 1, 2, $A_{1} = W^{\prime }W - diag(w_{i}^{\prime }w_{i})$ and A ₂ = W. Given these relations we have an estimator for the matrix Ψ given by:

$$ \Psi_{rs} = \frac{1}{2N} trace \left[ (A_{r} + A_{r}^{\prime}) \Sigma (A_{s} + A_{s}^{\prime}) \Sigma \right] + \frac{1}{N} \tilde{a}_{r}^{\prime} \Sigma \tilde{a}_{s}, $$

(9)

with r, s = 1, 2. A second efficient estimator for λ is found by minimizing the function given in Eq. 8 using Ψ as weighting matrix. The standard deviation of the efficient estimator of the spatial error parameter is given by:

$$ \Sigma_{\lambda} = \frac{1}{N}(J^{\prime} \Psi J)^{-1}, $$

(10)

where $J = \Gamma \left [ \begin {array}{ll} 1 \\ 2 \lambda \\ \end {array} \right ]$. In the third step, we use the estimated spatial error parameter to perform a Cochrane-Orcutt type transformation, i.e., we use $\widetilde {\log {P}} = \log {P} - \lambda W \log {P}$ and $\widetilde {Z} = Z - \lambda W Z$ instead of logP and Z in the 2SLS procedure given in Eq. 7.