Estimating Commercial Property Fundamentals from REIT data

Abstract

In this paper we propose a new methodology for the estimation of fundamental property asset-level investment time series performance and operating data based on real estate investment trusts (REITs). The methodology is particularly useful to develop publicly accessible operating statistics for investment real estate, such as income or expenses per square foot. Commercial property operating statistics are relatively under-studied from an investment perspective. We show how the methodology can be used to estimate the time series of property values, net operating income, cap rates, operating expenses and capital expenditures, per square foot of building area, by property type (sector) at the quarterly frequency for multiple specific geographic markets from 2004 through 2018. We show illustrative empirical results for Los Angeles offices and Atlanta apartments. The methodology allows estimation of actual quantity levels, not just dimensionless index numbers (longitudinal relative). It allows for an “additive” model structure that is more parsimonious, thereby addressing the need for granular market segmentation. We also introduce a Bayesian framework that allows the estimation of reliable time series even in small markets.

Appendix. Bayesian Model Details

In this Appendix we will detail the Bayesian model estimation. In order to estimate the model efficiently (i.e. with fast convergence) we log transform the left-hand side variable in all models before running the No-U-Turn-Sampler (NUTS). Logging the dependent variable ensures that the estimated parameters are closer to normally distributed without any large values. To ensure that we do not lose observations due to log transforming zero entries, we add a very small (1E-6) number to our yields.¹³ Unfortunately, this does mean that the estimated parameters cannot be easily interpreted. (Note that we do not log transform the right-hand side.) However, in this study, we do not focus on said parameters, but on the fitted values of our representative property. The trends are calculated by looking at (the exponentiated) fitted values over time of a “representative property”. As explained in “Methodology” section, we also add a common trend to the models. Log transforming the left-hand side, and adding a common trend (κ) results in the following Bayesian models;

$$\begin{array}{l}\begin{array}{c}\begin{array}{cc}\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}+\mathbf{p}\mathbf{t}+\mathbf{l}\mathbf{o}\mathbf{c}:& {y}_{kt}= {\kappa }_{t}{D}_{kt}^{T}+ {\delta }_{i}{D}_{kt}^{I}+ {\gamma }_{m}{D}_{kt}^{M}+ \beta {X}_{kt}+ {\varepsilon }_{kt};\end{array}\\ \begin{array}{cc}\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\times \mathbf{p}\mathbf{t}+\mathbf{l}\mathbf{o}\mathbf{c}:& {y}_{kt}= {\kappa }_{t} + {\mu }_{it}({D}_{kt}^{T}\times {D}_{kt}^{I})+ {\gamma }_{m}{D}_{kt}^{M}+ \beta {X}_{kt}+ {\varepsilon }_{kt};\end{array}\end{array}\\ \begin{array}{cc}\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\times \mathbf{p}\mathbf{t}+\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\times \mathbf{l}\mathbf{o}\mathbf{c}:& {y}_{kt}= {{\kappa }_{t}+ \mu }_{it}({D}_{kt}^{T}\times {D}_{kt}^{I})+ {\mu }_{mt}({D}_{kt}^{T}\times {D}_{kt}^{M})+ \beta {X}_{kt}+ {\varepsilon }_{kt};\end{array}\\ \begin{array}{cc}\mathbf{t}\mathbf{i}\mathbf{m}\mathbf{e}\times \mathbf{p}\mathbf{t}\times \mathbf{l}\mathbf{o}\mathbf{c}:& {y}_{kt}={ \kappa }_{t}+ {\mu }_{imt}\left({D}_{kt}^{T}\times {D}_{kt}^{I} \times {D}_{kt}^{M}\right)+ \beta {X}_{kt}+ {\varepsilon }_{kt}.\end{array}\end{array}$$

where;

$$\begin{array}{c}{\Delta }_{{\kappa }_{t}}\sim {\varvec{N}}\left(0, {\sigma }_{\kappa }^{2}\right);\\\Delta {\mu }_{t}\sim {\varvec{N}}\left(0, {\sigma }_{\mu }^{2}\right).\end{array}$$

Within each cluster the estimate 1 variance parameter. Another pro of estimating the model with the log on the left-hand side, is that we can keep the same priors irrespective of the left-hand side variable. Indeed, the average values (and thus priors) are vastly different for enterprise value per square foot, compared to for example cap rates in levels, but not so much in logs. We thus draw parameter κ_t=1 from a very uninformative prior (i.e. N (0,10)), whereas the other parameters are largely uninformative (N (0,1)) (Gelman, 2006). The random walks in state Eq. (4), are modelled in first differences, in line with Betancourt and Girolami (2015). The innovations of the random walks have a prior of N (0,1).

We have 2,000 iterations, of which the first half are used as warm-up, over 3 chains. All convergence statistics ($\overline{R }$ and effective sample sizes, not presented here, but available upon request) are satisfactory.