Cost drivers of operating charges and variation over time: An analysis based on semiparametric SUR models

Energy and Buildings 43 (2011) 1189–1199

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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild

Cost drivers of operating charges and variation over time: An analysis based on semiparametric SUR models Wolfgang A. Brunauer a , Sebastian Keiler b,∗ , Stefan Lang b a b

Immobilien Rating GmbH, Taborstr. 1-3, 1020 Wien, Austria University of Innsbruck, Universitätsstr. 15, A-6020 Innsbruck, Austria

a r t i c l e

i n f o

Article history: Received 18 May 2010 Received in revised form 21 November 2010 Accepted 15 December 2010 Keywords: Benchmarking Operating charges P-splines Random effects Seemingly unrelated regression Structured additive regression

a b s t r a c t Although building operating charges have turned out to be a major determinant of profitability for real estate investments, there is a noticeable lack of reports or studies that analyze these costs with stateof-the-art statistical techniques. Specifically, past studies usually assume linear relationships between costs and building attributes, they do not control for cluster-specific or longitudinal effects and do not account for the simultaneous structure of cost categories. Therefore, in this study we provide a novel approach to real estate cost benchmarking: we analyze the effects of building attributes on electricity, heating and maintenance costs for office buildings in Germany in a multivariate structured additive regression (STAR) model simultaneously, modeling potentially nonlinear effects as P(enalized)-splines and controlling for cluster-specific and individual heterogeneity in a three-way random effects structure. This way, we gain insights into how building attributes influence costs, and how cost levels vary across cities, companies and buildings. We furthermore construct quality-adjusted time indexes for the two major German submarkets. The results obtained can be used to derive portfolio allocation strategies and for planning, constructing, operating and redeveloping real estate. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In an increasingly more competitive market environment, building operating charges (the costs of building provision and management) have turned out to be a major determinant of profitability of real estate investment. This has drawn the attention to the need of identifying suboptimal cost structures in order to exploit potentials for optimization in asset and portfolio management. For this purpose, real estate professionals in the German speaking area largely rely on benchmarking reports such as the annual Office Service Charges Analysis Report (OSCAR) [12]. However, most of these reports evaluate operating costs over single building characteristics or regions without controlling for other systematic influences, which is likely to lead to biased results. Linear multiple regression models, where the cost drivers are estimated in a regression analysis of costs against building characteristics, provide a way to cope with these problems: amongst other publications on this topic (e.g. Stoy and Kytzia [19]), Stoy and Kytzia [18] explain office electricity consumption mainly by characteristics of technical installation, while Chung and Hui [6] regress energy use intensity in

∗ Corresponding author. Tel.: +43 6504505522. E-mail addresses: [email protected], [email protected] (S. Keiler). 0378-7788/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2010.12.014

office buildings on a set of predictors such as the age of the building, energy system parameters and floor space. However, the assumption of linearity seems to be too restrictive in many cases, which advocates the use of more flexible, namely non- and semiparametric regression models. A particularly broad and rich framework of semiparametric models is provided by structured additive regression (STAR) models introduced by Fahrmeir et al. [10] and Lang and Brezger [5]. In STAR-models, continuous covariate effects are modeled as P(enalized) splines as introduced by Eilers and Marx [9]. Furthermore, such models are also capable of accounting for (possibly correlated) random effects for spatial or cluster indexes, which makes them particularly suitable for our regression situation, where unobserved (a) building-, (b) company-, and (c) city-specific influences may affect cost structures considerably. Yet, some of the cost categories are likely to be subject to the same unobserved influences (e.g. due to the overall building condition, the market environment or public policy). Since neglecting existing association between response variables can lead to inefficient estimation of covariate effects, this article seeks to acquire information about the structure and causes of three major parts of building operating charges simultaneously using seemingly unrelated regression (SUR) models. For multivariate Gaussian response, the parametric SUR model (introduced by Zellner [23]) is a standard tool in econometrics. SUR models improve estimation results if disturbance terms are cor-

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related, and if the explanatory covariates differ to some extent between the equations. Astonishingly, there is a distinct lack of non- and semiparametric regression models for SUR models. Notable exceptions are the approaches in Smith and Kohn [17] and Lang et al. [13]. In this paper, we rely on the latter, which presents structured additive seemingly unrelated regression modeling. Thus, we allow for simultaneous nonparametric estimation of covariate effects as well as spatial and cluster specific effects, offering an efficient method of identifying the way and degree in which building attributes and cluster specific effects influence operating costs. As a basis of our study, we use a sample of 699 observations in 56 German cities, collected for benchmarking purposes by CREIS Real Estate Solutions from 2000 to 2005. The cost categories electricity, heating and maintenance are explained by various (continuous and categorical) building attributes as well as random effects for owning companies, cities and repeatedly observed buildings. The results provide building owners and operators with unbiased benchmarks for operating cost drivers and help them to identify potentials for optimization in their portfolio. Furthermore, we describe cost developments over time and geographical regions as quality-adjusted price indexes. The remainder of this paper is organized as follows: in Section 2, we describe our structured additive SUR model with respect to our application. In Section 3, the data set is described and we set up the working model. Furthermore, we give an intuition on interpreting the results of this study. Next, we present the results in Section 4, and draw some conclusions in the last part of this article. 2. Semiparametric SUR models As discussed in Section 1, we have to consider the following regression situation: we want to explain three response variables (i.e. cost categories) simultaneously in a multivariate regression framework with possibly nonlinear effects of continuous covariates. Furthermore, buildings are clustered in cities and companies, and some of them are repeatedly measured. We will discuss the methodological approaches applied for this situation in the following subsections. Note that we present for the statistically less involved reader a tutorial style introduction on how to interpret the models in Section 3.2 in the context of the estimated model.

of the cluster indicators, and vir r is the usual linear part of the model. The errors εi = (␧i1 , . . ., εik ), i = 1, . . ., n, are assumed to be i.i.d.  , multivariate Gaussian with mean zero and a covariance matrix i.e. εi |˙ ∼ N(0, ˙). This implies with i = (i1 , . . ., ik ) that yi |i , ˙∼N(i , ˙),

(3)

where responses yi are conditionally independent, given the predictors i . The nonlinear functions or random effects frj in (2) are modeled by a basis functions approach, i.e. a particular f of a covariate x is approximated by a linear combination of K basis or indicator functions f (x) =

K 

ˇk Bk (x).

k=1

The Bk are known basis functions and ˇ = (ˇ1 , . . ., ˇK ) is a vector of unknown regression coefficients to be estimated. Typically K is a large number to capture the variability of the data. Overfitting, i.e. highly variable estimators, are avoided by a roughness penalty applied to the regression coefficients. We use quadratic penalties of the form ˇ Kˇ where K is a penalty matrix. The penalty depends on a smoothing parameter  that governs the amount of smoothness imposed on the function f. From a Bayesian point of view the quadratic penalty ˇ Kˇ corresponds to a Gaussian (improper) prior for the regressions coefficients ˇ, i.e. p(ˇ| 2 ) ∝

 1 rk(K)/2 2



exp −



1  ˇ Kˇ . 2 2

(4)

Smoothness is now controlled by the variance parameter  2 which takes the role of an inverse smoothing parameter. The choice of basis functions B1 , . . ., BK and penalty matrix K depends on our prior assumptions about the smoothness of f as well as the type and dimension of x. In the next subsections we will give specific examples for modeling the frk or in other words for the choice of basis functions and penalty matrices. We restrict ourselves to examples used in the subsequent analysis. More examples can be found e.g. in Belitz and Lang [2].

2.1. Structured additive regression The dataset used for this article consists of observations yi = (yi1 , . . ., yik ) , i = 1, . . ., n, on a k-variate response y (in our case electricity, heating and maintenance costs) and on covariates. We distinguish between a vector xir = (xir1 , . . . , xirpr ) of continuous covariates (in our application e.g. net floor area, the age of the building or the time trend) or cluster indicators whose influence on the rth cost category of yi , will be modeled nonparametrically, and a further vector vir = (vir1 , . . . , virqr ) of categorical covariates (e.g. the existence of an elevator or air condition) whose effect is modeled in the usual linear form with parameters  r . We call a covariate a cluster indicator if it provides information in which city a building is located, which company a building belongs to or to which building an observation pertains to. For each cost category yir , r = 1, . . ., k, of the response we assume a semiparametric regression model yir = ir + εir ,

i = 1, . . . , n,

(1)

with structured additive predictors ir = fr1 (xir1 ) + · · · + frpr (xirpr ) + vir r ,

2.2. P-splines Suppose that a particular covariate x is continuous as is the case for the age of the building. We apply the P-splines approach for modeling nonlinear effects of continuous covariates. P-splines have been introduced in a frequentist setting by Eilers and Marx [9] and in a Bayesian framework by Lang and Brezger [14]. P-splines approximate the nonlinear function f by a polynomial spline of degree l with equally spaced knots xmin = 0 < 1 < · · · < m−1 < m = xmax over the domain of x. A spline has the following two properties: • in each of the intervals [j , j+1 ], j = 0, . . ., m − 1 the spline f is a polynomial of degree l, and • at the knots j (the interval boundaries) the spline is l − 1 times continuously differentiable.

(2)

where ir is the predictor of equation r and fr1 , . . . , frpr are possibly nonlinear functions of the continuous covariates or random effects

A spline can be written in terms of a linear combination of K = m + l basis functions, see DeBoor [7]. The most widely used bases

W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199

are the truncated power series basis and the B-spline basis. Using a truncated power series basis the function f is f (x) = ˇ0 + ˇ1 x + · · · + ˇl xl +

m−1 

ˇl+j tj (x, l),

(5)

j=1

where



tj (x, l) = (x − j )l+ =

(x − j )l 0

x > j else.

(6)

In a simple regression spline approach, the regression coefficients ˇk are estimated using standard inference techniques for linear models. The crucial problem with such regression splines is the choice of the number and the position of the knots. A small number of knots may result in a function space which is not flexible enough to capture the variability of the data. A large number may lead to overfitting. As a remedy Eilers and Marx [9] propose to define a fairly large number of knots (usually between 10 and 40) to ensure enough flexibility. Sufficient smoothness of the fitted curve is achieved through a roughness penalty on the regression coefficients. Using a truncated power series basis, overfitting is prevented using a quadratic ridge type penalty m−1 

P() = 

2.3. Varying coefficients Of course, we can also model interaction effects in this framework. Specifically, we are interested in heterogeneous nonlinear time trends for the two main German submarkets, the western and the eastern part of Germany (the former German Democratic Republic, henceforth GDR). Suppose that xi1 is a time index and vi1 is an indicator for the GDR in effect coding, i.e. vi1 = 1 if the ith observed building is located in the GDR and vi1 = −1 else. We use effect rather than dummy coding for technical reasons. Now we arrive at a model of the form yi = f1 (xi1 ) + · · · + fx1 |v1 (xi1 )vi1 + · · · + vi1 1 + · · ·,

(7)

(10)

where f1 (xi1 ) and 1 v1 are the main effects of the time index xi1 and the effect of being located in the GDR vi1 and fx1 |v1 is the smooth nonparametric function of the interaction term (we drop the equation index for simplicity). The function f1 can be interpreted as an average time trend of xi1 , and fx1 |v1 + 1 respectively −fx1 |v1 − 1 is now the deviation from f1 for vi1 = 1 (building is located in the GDR) and vi1 = −1 (building is located in West Germany). Summarizing, we obtain the following decomposition of the time trend:

 f (xi1 ) =

2 ˇl+j ,

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f1 (xi1 ) + fx1 |v1 (xi1 ) + 1 f1 (xi1 ) − fx1 |v1 (xi1 ) − 1

if building i is situated in the GDR if building i is situated in West Germany.

(11)

j=1

2.4. Cluster effects

leading to the penalized least squares criterion PLS(ˇ, ) =

n 

2

m−1 

(yi − f (xi )) + 

i=1

2 ˇl+j

(8)

j=1

to be minimized with respect to ˇ = (ˇ0 , . . ., ˇK−1 ) . Despite their simplicity P-splines based on a truncated power series basis in combination with penalty (7) are rarely used in practice, due to the numerical instability of the highly collinear basis functions. Instead a local B-splines basis is applied. There is a close relationship between B-splines and truncated polynomials as Bsplines can be computed as differences of truncated powers (see Eilers and Marx [9]). For instance, B-spline basis functions of degree one are computed as Bj (x, 1) = tj−2 (x, 1) − 2tj−1 (x, 1) + tj (x, 1) = 2 tj (x, 1), with tj defined in (6). Note that extra knots −l , . . ., −l left to 0 and m+1 , . . ., m+l right to m are required, so that the truncated polynomials in the above formula are properly defined to compute all basis functions Bj close to the left and right borders. Now the spline f may be written as f (x) =

K 

ˇx ∼(0,  2 )

(12)

is introduced. Here, x ∈ {1, . . ., K} is an index variable that denotes the cluster a particular observation pertains to. This is equivalent to a penalized least squares approach with function f(x) = ˇx and penalty matrix I or a Bayesian approach with prior (12). Note that more than one random intercept with respect to different cluster variables are possible, as in our application (we specify a company-, city- and building-specific random effect).

2.5. Inference and software ˇk Bk (x, l).

k=1

The local basis also gives rise to alternative penalization. The widely used approach by Eilers and Marx [9] penalizes the sum of squared dth order differences P() = 

Suppose now that x is a unit- or cluster index variable such as the company, city or building index in our analysis. There is a vast literature on modeling unit- or cluster specific heterogeneity, see e.g. Verbeke and Molenberghs [20]. An important special case arises for longitudinal data where individuals are repeatedly observed over time [8]. Typically, unit- or cluster specific random effects are introduced to account for heterogeneity. In its simplest form an i.i.d. random intercept ˇx with

K 

2

( d ˇk ) = ˇ Kˇ,

(9)

The semiparametric SUR model is estimated in a Bayesian framework using Markov chain Monte Carlo (MCMC) simulation techniques for inference. Details can be found in Lang et al. [13]. The approach is implemented in the software package BayesX which is publicly available at http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html

k=d+1

where d is the difference operator of order d. The penalty matrix is given by K = D D where D is a dth order difference matrix. The default for d in most implementations (e.g. mgcv in R or the software package BayesX, see below) is d = 2.

(see Brezger et al. [3,4]). The homepage of BayesX contains also a number of tutorials. The following code fragment exemplifies the usage of BayesX in the context of semiparametric SUR modeling:

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W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199

ing costs, while e.g. security, water and insurance have relatively small shares of total costs (see Fig. 1). • Large deviations from the expected costs point to inefficiencies in building equipment, operation or user behavior and are a starting point for an optimization process. In contrast, influencing the cost categories public charges and adminstration is not that straightforward and therefore of less practical interest.

Fig. 1. The ten observed cost categories and their share of total costs. The three categories analyzed in this paper (heating, electricity and maintenance) are highlighted.

The data were collected for the years 2000 to 2005, and some of the buildings have been observed repeatedly (Table 1). As Table 2 shows, one out of four buildings has been observed more than once, one building even over the whole period. This leads to an unbalanced panel structure of 501 buildings. Furthermore, observations are situated in cities on the one hand and owned by companies on the other hand, thus enabling the analysis of this classifications. In Table 2, the numbers of unique values (i.e. buildings) for

dataset costs; costs.infile using c:\data\costs.raw; bayesreg sur_model; sur_model.mregress ln_elect_psqm =

age(psplinerw2) + ... + company(random) + ... + quality_h + quality_m:

ln_heat_psqm = age(psplinerw2) + ... + company(random) + ... + quality_h + quality_m: ln_maintenance_psqm = age(psplinerw2) + ... + company(random) + ... + quality_h + quality_m, family=multgaussian iterations=12000 step=10 burnin=2000 using costs; Having defined the dataset object costs, we read the cost data from an ASCII file. We then define the bayesreg-object and apply the method mregress to fit our multivariate gaussian (multgaussian) model with structured additive predictor. In all of our three equations we have P-splines with second order difference penalties (psplinerw2), unstructured random company-, city- and buildingspecific effects (random) and linear effects. Furthermore, we define the number of MCMC-iterations and the number of burniniterations thereof as well as the thinning parameter for the MCMC simulation, step. See also the tutorial on Bayesian inference via MCMC simulation available at the BayesX homepage for details. 3. Data description and model specification 3.1. Data The data this research is based on has been collected for the annual OSCAR report [12]. For this report, data concerning floor areas, technical quality and equipment as well as operating charges of office buildings are carefully captured and presented in great detail for comparison purposes on a yearly basis. Total operating charges consist of 10 cost categories, which are depicted in Fig. 1. We have chosen to analyze electricity, heating and maintenance costs in this article (a total of 699 observations dispose of these cost categories) for the following reasons: • As these “technical” categories mainly depend on the same (possibly unobserved) influences, they are likely to be correlated, which makes a SUR model appropriate (which is not necessarily true for the other cost categories). The other cost categories do not depend on the same influences as electricity, heating and maintenance costs and therefore do not add to the quality of a SUR-model. Specifically, the costs for administration, security, janitors and cleaning are largely dependent on staff costs and driven by different shocks than the chosen categories. • Electricity, heating and maintenance costs are relevant in a sense that they account for approximately one third of total operat-

these cluster categories are displayed together with minimum, 25% quantile, median, 75% quantile and maximum of observations per cluster. Note that for both categories, the smallest group contains only one observation, while there is one city that contains 88 observations and one company which 277 observations (nearly half of all observations) pertain to. All buildings are characterized by continuous covariates such as the net floor area of the building (net), integer variables such as the age at the time of data collection (age) and the number of floors of the building (floors), as well as categorical variables such as the quality of the building at the time of measurement, the existence and quality of air condition, an indicator for a past refurbishment, identifying whether the building has been built in row with other buildings and the existence of a garage and an elevator. Furthermore, in order to capture location specific effects, the data has been matched with data on climate conditions (heating degree days, hdd),1 as well as data on economic strength (standardized purchasing power, pps).2 Table 3 gives an overview over the covariates employed in our multivariate regression model. 3.2. Model The model we set up using the covariates outlined in Section 3.1 can thus be described as follows: ln(elect) = f1 (age) + f2 (floors) + f3 (time) + f4 (time)e w + f5 (company) + f6 (city) + f7 (building) + v1 1 + ε1 (13)

1 Heating degree days are obtained from the German Weather Service; they are computed as the sum of days with an outside temperature below a threshold of 15 ◦ C, weighted with the difference between the actual outside temperature and this threshold; see the respective norm from the German Institute for Standardization, DIN 4108-6 [1] and the guideline from the Association of German Engineers, VDI 3807 [21]. 2 Standardized purchasing power is provided by the Michael Bauer Research GmbH.

W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199 Table 1 Multiply observed objects. Most of the buildings (73%) have been observed only once, while only one building has been observed over the whole period. No. of years

No. of buildings

1 2 3 4 5 6

365 94 29 7 5 1

Total

501

Freq. 72.85 18.76 5.79 1.4 1 0.2 100

Table 2 Description of cluster categories companies and cities: number of groups (unique values) per category along with minimum, 25% quantile, median, 75% quantile and maximum of observations per group that are observed in the respective category. The size of the groups varies considerably, from only one observation to 88 observations for cities and 277 observations for companies.

Cities Companies

Unique values

Min

p25

Median

p75

Max

56 21

1 1

2 4

4.5 9

11 31

88 277

ln(heat) = f1 (age) + f2 (floors) + f3 (net) + f4 (hdd) + f5 (time) + f6 (time)e w + f7 (company) + f8 (city) + f9 (building) + v2 2 + ε2

(14)

ln(maint) = f1 (age) + f2 (floors) + f3 (net) + f4 (pps) + f5 (time) + f6 (time)e w + f7 (company) + f8 (city) + f9 (building) + v3 3 + ε3 .

(15)

In all equations, continuous and integer covariates are modeled as P-splines, the categorical covariates describing the quality and condition of the building are encoded as dummy variables and subsumed in vr with estimated parameters  r . Furthermore, we included time trends with possible interactions with the two German submarkets (GDR and West Germany) as described in Section 2.3. We transform the response variables logarithmically, as we expect building characteristics to have multiplicative effects on operating cost categories (this topic has been discussed extensively in the context of hedonic price theory, see Malpezzi [15] for an overview). Due to these transformations, the estimated linear effects can be approximately interpreted as semi-elasticities (i.e. the percentage change of price by the absolute change of the

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covariate, see [11]). For large values, however, this approximation becomes too rough, so we calculate the percentage effect as ij (costs) = [exp(j × vij ) − 1] × 100, where j × vij is the estimated effect of covariate vj , evaluated for observation i. To enhance the readability for the statistically less experienced reader, we show how the model and the various terms therein can be interpreted using the example of the first equation (13). We may principally distinguish two types of terms: nonlinear effects of continuous variables and discrete cluster effects. Examples for nonlinear effects are the effect f1 of the age of the building (variable age), the effect f2 of the number of floors (floors) and the effect f3 for the time trend (time). The discrete cluster effects in the first equation are the company specific effect f5 , the city specific effect f6 and the building specific effect f7 . The nonlinear effects of continuous variables are best interpreted by visualization: an example can be found in the left panel of Fig. 2, which plots the estimated nonlinear function f2 of floors against the number of floors (solid black line). The plotted function shows how the average logarithmic electricity costs change as the number of floors increase and the other covariates are held constant at their mean level. This allows us to analyze an average building and varying only one parameter. This is generally referred to as the ceteris paribus assumption. The function (and with it logarithmic electricity costs) first increases almost linearly until the number of floors reaches 8, then stays constant for buildings up to 12 floors and finally increases linearly again until 20 floors. Between 2 and 8 floors the average logarithmic costs increase linearly from 0.25 to 1, denoting electricity costs ranging from exp(0.25) = 1.28 to exp(1) = 2.72 Euro per square meter on average. Between 8 and 12 floors average electricity costs are almost constant at about exp(1) = 2.72 Euro per square meter. Between 12 and 20 floors the average logarithmic electricity costs increase linearly again from 1 to almost 1.5, corresponding to electricity costs between exp(1) = 2.72 and exp(1.5) = 4.48 Euro per square meter. The shaded areas in Fig. 2 indicate 80% (dark grey) and 95% (light grey) confidence bands which should give an intuition for the uncertainty of the estimate. The wider the confidence band, the less accurate is the estimate. In our example uncertainty slightly increases for buildings with more than 15 floors. The reason for this is that there are only a few buildings with 15 or more floors in the available data set. Interpretation of the nonlinear effect f1 of age in Eq. (13) is in complete analogy, see Section 4. Interpretation of the time trend f3 (time) has to be done in combination with the nonlinear interaction effect f4 (time)e w. In fact, the term f3 (time) + f4 (time)e w models separate time trends in electricity costs for East and West Germany. Since e w is coded as

 ew=

1 −1

building located in GDR building located in West Germany

Table 3 Variable description. Name

Explanation

Electricity

Heating

Maintenance

Quality Air condition

Building’s quality in three categories: high (reference category), medium or low Building’s type air condition in three categories: no air condition, partially and fully (reference category) air conditioned Building has an elevator (binary) Building has reported vacancy (binary) Building has been refurbished (binary) Building has a garage (binary) Building is detached to another building (binary) Building is located in the GDR (binary) Number of floors Building’s age Net floor area Year of observation Heating degree days of the region Purchasing Power Standardized of the city

x x

x

x x

Elevator Vacancy Refurbished Garage Row e w dummy Floors Age Net Time hdd pps

x x x x x x x x

x x x x x x x x x

x x x

x x x x x x

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W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199 Effect of the covariate floors on electricity costs

Effect of the covariate age on electricity costs, with and without dummy

b 1.5 1 0

.5

Effect of Age

1.5 1 .5

−.5

−.5

0

Effect of floors

2

2

a

0

5

10

15

20

Number of floors

0

20

40

60

80

100

Age of the building, spline w/o dummy dashed

Fig. 2. Effects of continuous covariates for electricity costs, overlaid with partial residuals. Panel [a] variable floors and panel [b] variable age. The dashed lines in panel [b] indicate the posterior mean as well as the 95% credible interval for the case without the dummy for building number 58.

the nonlinear trend f3 (time) models the average time trend, whereas f4 (time) is the nonlinear deviation from the average for buildings in the GDR and −f4 (time) is the deviation for West Germany. Overall, f3 (time) + f4 (time) is the time trend for buildings in the GDR and f3 (time) − f4 (time) is the time trend in West Germany. These time trends are depicted in Fig. 6(a) and explained in full detail in Section 4.5. Interpretation of the company, city and building specific effects f5 , f6 and f7 is a bit more subtle. Consider for instance the city specific effect f6 . Overall, we have buildings from 56 cities indexed by 1, . . ., 56 in our dataset. Technically, we treat city as an unordered categorical variable and estimate (similar to the usual dummy coding for categorical covariates) one parameter for every city, i.e. we have parameters ˇ1 , . . ., ˇ56 and f6 (cityi ) = ˇcityi . If, e.g., the ith building in the dataset is from the city indexed by 12, i.e. cityi = 12, then f6 (cityi ) = ˇ12 . At first sight it seems heroic to estimate so many city parameters based on a comparably small database of roughly 700 observations. Indeed, if we estimated the parameters in an unrestricted way, the result would be highly variable and to a large extend unreliable estimates. However, by introducing a roughness penalty on the parameters (see the explanation in Section 2.4) the variability of estimates is considerably reduced. Compared to completely unrestricted estimates, in our model the estimated parameters are shrunk towards zero. The shrinkage effect is higher for cities with a small number of observations. On the contrary, if the number of observations is reasonable large, the shrinkage effect almost disappears. Inclusion of cluster specific effects in the model is necessary and important for the following reasons: • Control for unobserved heterogeneity. The primary reason for including cluster specific effects is to control for heterogeneity induced by cluster specific influential factors that are not included in the model. For the city effect possibly omitted factors could be specific climatic conditions, local legislation or the specific economic condition of the local energy market. Even though the model is misspecified due to the missing cluster specific variables, the cluster effect accounts for the heterogeneity and helps to avoid biased estimates of the remaining variables in the model. In that sense, the cluster specific effect may be seen as a surrogate for effects of covariates that have not been collected or have been omitted (for whatever reason) in the data analysis. • Reference to further influential variables. In many cases, close inspection of the cluster specific effects helps to identify further covariates that have been ignored in the analysis. In our case, the city specific random effect revealed in a preliminary analysis a difference in the market conditions of West Germany and the GDR (specifically the energy market), with in tendency higher city parameters in the GDR. This led to the inclusion of an east–west

dummy (Variable e w dummy) in the final model, which reduced the city specific cluster effect. • Improvement of estimation accuracy. Inclusion of the cluster specific effects helps to control for the correlation induced by repeated nonindependent observations within a cluster (e.g. a city). Obviously, efficiency is lost in the estimation when treating the observations as if they were independent, i.e. if no cluster effect is included. Standard errors of the corresponding estimates will be wrong, leading to potentially false conclusions when assessing significance of covariate effects. Often standard errors will be too small when ignoring cluster effects, but the reverse may also be observed. A final remark is concerned with the relationship of the three estimation equations (13)–(15). At first sight these equations are seemingly unrelated. However, due to common movements of markets and prices the equations, or more specifically the corresponding error terms, are correlated. Considering the correlation is not important for interpretation of the results but helps to increase the efficiency of estimates in the sense that their variability is reduced. 3.3. Expected effects Based on theoretical considerations, we expect the following results: for all of the cost categories examined, we expect the age of the building to have an increasing effect due to the decreasing technical and structural condition. Also an increasing number of floors is likely to lead to higher electricity and maintenance costs, considering the additional equipment for higher buildings (e.g. elevators, pumps and fans) and heating costs, as high buildings tend to have unfavorable ratios of envelope to volume. The net floor area (net) is only relevant for heating and maintenance costs (Eqs. (14) and (15)), where we assume that large buildings may benefit from economies of scale, leading to a negative effect in both equations. The heating degree days only appear in Eq. (14), where a larger number of heating degree days should lead to an increase in heating energy consumption. We employ the purchasing power as a predictor for maintenance costs in Eq. (15), as in cities with a high purchasing power labor costs are also high, which is likely to affect this labor intensive cost category. For all cost categories, we expect the time to have an increasing impact. Turning to the linear effects, the existence of air condition should be a major predictor both for energy and maintenance costs. Furthermore, refurbished buildings may tend to have lower costs in general. Buildings in a row should exhibit lower heating costs, as the outside envelope area is reduced, and the existence of an elevator most likely leads to higher electricity and maintenance costs.

W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199 Table 4 Estimated error correlations between the equations.

Electricity Heating Maintenance

Electricity

Heating

Maintenance

1 0.210 0.183

1 0.124

1

We do not have any prior assumptions concerning the quality of the building and the existence of a garage. From preliminary analyzes, we expect the indicator for the GDR to have a positive effect on costs. 4. Results In this section we present the estimation result for the model specified in Section 3.2. First, linear and nonlinear effects are presented for each cost category separately. Next, we will analyze cluster specific effects, and finally the variation over time is explored. Table 4 presents the estimated error correlation structure of our three equation model. The strongest correlation is observed for the energy dominated categories heating and electricity (although there is also a notable correlation between the other equations). This correlation indicates the potential usefulness of the SUR approach. 4.1. Electricity costs The linear effects for the electricity cost equation are displayed in the first column of Table 5. As expected, it becomes evident that building quality as well as air condition both have huge effects on electricity costs. Electricity costs per square meter of high quality buildings are expected to be 72% higher compared to electricity costs for low quality buildings and still 27% higher than for medium quality buildings; on average, electricity costs rise by 100% for buildings with a full air condition while partly air conditioned buildings still have 40% higher costs, both compared to buildings without air condition.3 This magnitude is largely in line with former publications described in the VDI 3807 [22]. The negative sign for the binary variable elevator may be counterintuitive at first, as additional technical equipment should lead to higher costs. Yet, considering the fact that all but nine buildings in the sample have an elevator, the lack of an elevator may rather be seen as a proxy for sub-standard objects with—in this case—clearly higher electricity costs. Vacancy within the building has an increasing effect on electricity costs per square meter, which is due to the fact that total electricity costs arising for building equipment are distributed over less rented space and thus increase operating charges (this is also true for heating and maintenance costs, see below). Neither is there any evidence that a previous refurbishment has an effect nor has the presence of a garage any influence on the electricity costs per square meter. It appears that this is due to the fact that the simple binary variable may be too insensitive for capturing possible effects on the electricity costs since effects from refurbishments are highly dependent on the type and decade of this measure. Note that the effect of the indicator for buildings in the GDR (e w dummy) is not significant. We will explore this issue further in Section 4.5. Fig. 2 depicts the effects of nonparametrically modeled continuous covariates on log-transformed electricity costs per square meter (shown with means of all other covariates held constant, which is what we call mean effect henceforth). The effect of the number of floors is strongly upward sloping with a consolidation phase

3 Note that we employed the ij (costs) = [exp(j × vij ) − 1] × 100 approximation for obtaining the percentage value of these effects.

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between the eighth and the twelfth floor; for buildings with more than twelve floors, there is again an increasing effect (interestingly, this pattern corresponds with the German building code definition of high-rise buildings). In total, electricity costs ceteris paribus differ by roughly 2.5 EUR per square meter between a two and a 20 storey building (the shape of this effect is in line with our prior assumptions, see Section 3.3).4 Remember that we observe a hypothetical average building and vary only one parameter (floors) at a time. The age of the building in the right panel of this figure has a negligible effect on electricity costs up to an age of approximately 40 years, followed by a steady increase up to 70 years. This suggests that old buildings tend to exhibit an inferior building equipment standard, resulting in sub-optimal energy efficiency and thus higher energy consumption. However, there is a notable decrease of this effect in the high range of this covariate. The reason for this surprising pattern is that the oldest building in the dataset, which happens to exhibit rather low energy costs, has been observed in three consecutive years. As data in this age group is rather sparse, this leads to a substantial drop in the age effect towards the end of the spline indicated by the dashed lines in Fig. 2 panel [b]. We control for this outlier beyond the random effect by introducing a dummy variable (for the effect of this outlier see the last row of Table 5; alternatively, we could have just dropped it from the dataset). This considerably changes the cost gradient and yields a more reliable estimate. 4.2. Heating costs Heating costs are largely driven by the building’s (technical and constructive) quality, which becomes obvious from the second column of Table 5, although the effect is considerably smaller than in the case of electricity costs (in this case minus 12% and minus 10% for low and medium quality, respectively). However, the negative sign seems counterintuitive (buildings with inferior quality are commonly associated with higher heating costs). This may be due the relatively broad definition of quality including construction variants contributing to thermal quality and factors counteracting thermal quality.5 Again, vacancy within the building has an increasing effect on costs. Neither a refurbishment nor a detached construction type (row) have a significant impact on heating costs. Again, observing an insignificant effect for the covariate refurbishment may be due to lack of granularity (we only have information wether a building has been refurbished or not). The effect of the indicator for buildings in the GDR is strong and highly significant: on average, buildings in the GDR have more than 20% higher heating costs. Fig. 3 again illustrates the effects of nonparametrically modeled covariates. Although the shape of the effect of covariate floor in panel [a] resembles the one in the case of electricity costs, its impact is smaller in this case. Panel [b] shows the effect of the variable age continuously increasing up to 45 years and a constant effect afterwards, indicating that new buildings ceteris paribus have approximately 60% lower heating costs than buildings built during the 1960s. This gradient portrays the changing technical standards and building codes: while improved construction methods allowed for thin walls in the 1960s and beyond, buildings before this time period needed relatively thick walls which, unintendedly, contribute to thermal performance. The net floor area (net), depicted in panel [c], has a negative effect on costs, suggesting

4 Since magnitude rather than precision is in the focus of this analysis, figures are rounded when referred to in the text. 5 A glass facade may for example be associated with a high quality building while potentially having a negative impact on thermal quality. Vice versa a thermal insulation with passive house standard may as well be associated with a high construction quality and actually contributing to thermal quality.

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Table 5 Results for binary covariates of the three equations. The table shows the posterior means along with the posterior standard deviation (in brackets). The reference is a high quality, fully air conditioned building. Electricity

Heating

Maintenance

Intercept Low quality Medium quality No air condition Partly air conditioned Elevator Vacancy Refurbished Garage Row e w dummy

2.315 (0.2535)*** −0.540 (0.1066)*** −0.236 (0.0704)*** −0.758 (0.0929)*** −0.347 (0.095)*** −0.364 (0.2096)** 0.081 (0.0419)*** −0.057 (0.0637) 0.032 (0.0612)

1.672 (0.0836)*** −0.134 (0.0580)*** −0.106 (0.0397)***

1.188 (0.2676)*** −0.352 (0.1092)*** −0.136 (0.0709)** −0.310 (0.1008)*** −0.214 (0.0951)*** 0.457 (0.2193)*** 0.125 (0.0447)*** 0.031 (0.0696) 0.014 (0.0676)

Outlier

−1.150 (0.5369)

0.151 (0.1974)

***

Indicate a credible interval of 80% not including zero. Indicate a credible interval of 95% not including zero.

4.3. Maintenance costs Also for maintenance costs, building quality as well as technical equipment (air condition) are highly relevant. A high building quality results in 29% higher maintenance costs compared to low quality

Effect of the covariate floors on heating costs

1

1.5

Effect of covariate age

2

b

Effect of the covariate age on heating costs

.5

.5

Effect of covariate floors

a

and 13% compared to medium quality. Maintenance costs rise by 25% for fully air conditioned buildings compared to no air condition and 18% compared to partial air condition. The significantly positive effect for the elevator dummy is in line with theoretical considerations. The effect of vacancy within the building is strong and significantly positive, while neither a previous refurbishment nor the presence of a garage has any significant influence on maintenance costs per square meter. The effect of the GDR indicator is insignificant. The nonparametric effects in Fig. 4 reveal the rather strong effect of the covariate floors (panel [a]) with about 1 EUR difference between the highest and the lowest building in the sample. Furthermore, almost a linear gradient for covariate net (panel [b]) is observable, again indicating considerable economies of scale (an increase of floor space by 10,000 m2 yields a reduction in main-

2

the presence of economies of scale (although with a decreasing marginal effect). This is again in line with theoretical assumptions, since larger heating systems usually work more efficiently. The climate related variable for heating degree days (hdd, panel [d]) is insignificant in the sense that pointwise confidence intervals cover the mean effect over the whole range of this covariate. This may be due to the fact that buildings in cooler areas are better insulated, thus endogenously removing any effect resulting in this observed pattern.

1.5

***

−0.042 (0.0374) 0.207 (0.0691)***

0.162 (0.1379)

1

**

0.035 (0.0252)** −0.039 (0.0372)

0

5

10

15

20

0

20

Number of floors

60

80

100

Age of the building

1.5

2

Effect of the covz HDD on heating costs

.5

1

1.5

Effect of covariate HDD

2

d

1

Effect of the covariate net on heating costs

.5

Effect of covariate net

c

40

0

10000

20000

30000

Net floor area

40000

1700

1900

2100

2300

2500

2700

Heating degree days

Fig. 3. Effects of continuous covariates of category heating costs, overlaid with partial residuals. Panel [a] shows the effect of the number of floors, panel [b] plots the effect of the variable age and panel [c] illustrates the gross internal floor area effect. Panel [d] shows the spline for the heating degree days.

W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199 Effect of the covariate net on maintenance costs

1.5

−.5

0

0

.5

1

Effect of net

1.5 1 .5

Effect of floors

2

c

2

Effect of the covariate floors on maintenance costs

a

1197

0

5

10

15

20

0

10000

20000

Number of floors

40000

d

Effect of the covariate pps on maintenance costs

−1

0

0

1

Effect of pps

2 1

Effect of age

2

3

Effect of the covariate age on maintenance costs 3

b

30000

Net floor area

0

20

40

60

80

100

90

100

Age of the building

110

120

130

140

Standardized purchasing power

Fig. 4. Effects of continuous covariates of category maintenance costs, overlaid with partial residuals. Panel [a] shows the effect of the number of floors, panel [b] plots the effect of the variable age and panel [c] illustrates the gross internal floor area effect. Panel [d] shows the spline for the standardized purchasing power.

a

b

8

Kernel densities for residuals and random effects for electricity

Kernel densities for residuals and random effects for heating

8

Residuals Building RE City RE Company RE

0

0

2

2

4

4

6

6

Residuals Building RE City RE Company RE

−1.6

−1.2

−.8

−.4

0

.4

x

1.2

1.6

−1.6

−1.2

−.8

−.4

0

x

.4

.8

1.2

1.6

Kernel densities for residuals and random effects for maintenance

8

c

.8

0

2

4

6

Residuals Building RE City RE Company RE

−1.6

−1.2

−.8

−.4

x

0

.4

.8

1.2

1.6

Fig. 5. Kernel density estimates for the models’ residuals and the random effects. The company specific random effects are illustrated as a histogram. The panels show the models electricity [a], heating [b] and maintenance [c].

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W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199

a

b

Time index for heating

.4

.4

Time index for electricity

east overall west

−.4

−.4

−.2

−.2

0

0

.2

.2

east overall west

2000

2001

2002

2003

2004

2005

2000

2001

2002

Year

d

Time index for maintenance

2005

Annual change in real Gross Domestic Product

−.4

0

−.2

0

.2

annual change in GDP

3

east overall west

2

.4

2004

1

c

2003

Year

2000

2001

2002

2003

2004

2005

Year

2000

2001

2002

2003

2004

2005

Source: destatis.de

Fig. 6. The variation over time split up for eastern (solid) and western states of Germany (dashed). The dotted line indicates the aggregated index. Panels [a–c] show the three cost categories, panel [d] shows the percentage growth of the German Gross Domestic Product (GDP) for the time span.

tenance costs per square meter of roughly 30 cents on average). In panel [c], the effect of covariate age is insignificant over the whole range of values. However, the shape suggests a (weak) quadratic cost gradient. Reasons for this may be that either there is a lesser amount of maintainable equipment in older buildings in general, or older buildings have already been equipped with new technique as the technical lifespan of the original equipment has already expired. In order to account for local market structures, the purchasing power (pps) has been introduced into this equation (differences in economic power affect wages and therefore maintenance costs). However, as can be seen in panel [d], this effect is insignificant as well, suggesting a common price level in all areas. This may be due to an increasing competition and a concentration of service suppliers (one notable example is the fusion of STRABAG and the Facility Management department of the German Telekom). 4.4. Random effects Recall that there are three types of random effects in each of the three models, the effect of the city, the company and a longitudinal effect for repeatedly observed buildings. Fig. 5 shows the results graphically: each panel depicts the effects for one cost category, where the distributions of the random effects of cities, companies and buildings are shown as kernel densities. In order to get an impression of the dimension of these effects, they are contrasted with the residuals. In general, city specific random effects account for a relatively small difference, while the building specific effects are quite pronounced, indicating the large relevance of user behavior: recent analyses (e.g. Messerschmidt [16]) estimate user induced fluctuation in a building’s energy demand at around 50%. However, also the company specific random effects indicate optimization potentials on a company level. In total, the random effects for category

heating costs (panel [b]) are smallest (we also observe the smallest standard error of regression for this equation). 4.5. Variation over time One of the major goals of real estate benchmarking is the revelation of developments over time. Therefore, we present quality adjusted time indexes for the cost categories under consideration, reflecting the heterogeneous market structure in Germany by splitting the effect between western and eastern states (the GDR) as explained in Sections 2 and 3.2. Fig. 6 illustrates the results of this procedure. Note that up to 2002, electricity costs in West Germany started decreasing considerably (panel [a]), while they were increasing in the GDR. This was followed by a period of convergence between these two submarkets. The development of heating costs, displayed in panel [b], shows no strong variation over time, with a pronounced and significant difference in the base level (see Table 5). Maintenance costs rose considerably in both parts of Germany (panel [c]). Interestingly, the pattern of rather big increases and a breakpoint in 2002 for electricity costs is also present in the annual growth of the German Gross Domestic Product (GDP, panel [d]). The correspondence to the GDP may provide evidence for systematic co-movement in prices unveiled by this interaction effect: the fact that western electricity cost increases are less affected by changes in GDP growth (panel [a]) might indicate systematical differences in market structure between West Germany and the GDR. The pattern observed may give evidence for a higher price elasticity in the western states during phases of reduced growth. 5. Conclusions The main goal of this article is to provide a cutting edge approach to real estate cost benchmarking for electricity, heating and mainte-

W.A. Brunauer et al. / Energy and Buildings 43 (2011) 1189–1199

nance costs that still provides useful results for practitioners. There are three challenges that make this regression problem particularly difficult: first, possibly nonlinear covariate effects prevent the use of purely linear models. Second, unobserved cluster-specific or individual heterogeneity (specifically, for the companies owning the buildings, the cities where they are located and longitudinal effects) must be accounted for in order to get unbiased results. And finally, as the three cost categories analyzed are likely to be simultaneously influenced by common shocks, a Seemingly Unrelated Regression (SUR) model should be applied, providing more efficient estimation results. Therefore, we propose a novel approach based on three components: potentially nonlinear effects of cost drivers are modeled as P(enalized) splines; a three-way random effects approach accounts for unobserved cluster-specific heterogeneity, leading to unbiased results and providing the basis for further exploration of the data; and finally, contemporaneous building-specific correlation between cost categories is modeled by a SUR-approach. This concept proves to be superior to conventional benchmarking attempts, providing insights into cost structures and cost developments over time and between markets. Our findings may help to anticipate operating charges ex ante, which can help to derive investment strategies and identify opportunities. On the other hand, the costs of individual buildings can be benchmarked ex post, providing insights into potentials for cost reductions. Acknowledgements The authors gratefully acknowledge financial support by the Tiroler Wissenschaftsfonds (Tyrolean Science Fund). Parts of this study have been analyzed in the Haus der Zukunft (Future Buildings) project for Life Cycle Cost prediction at the Kufstein University of Applied Sciences. Last but not least we thank an anonymous referee for his comments that helped to improve the first version of the paper. References [1] Deutsches Institut für Normung eV, DIN 4108 T6 Wärmeschutz in Gebäuden.

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