Valuing Commercial Spaces in Multistory Buildings Using a Three-level Mixed-effects Modeling Approach

Abstract

The location of commercial spaces in multistory buildings affects their values. We developed a three-level mixed-effects spatial hedonic model in which the nested structure of individual commercial units in multistory buildings is explicitly taken into account in the price determination process. The proposed modeling framework is intuitive and flexible from both substantive and technical perspectives, as evidenced by its diverse extensions to accommodate both the spatial autocorrelation and the complex heteroscedasticity inherent in property values. A unique data set based on the assessed values of commercial properties and officetels, collected by the National Tax Service of Korea in 2012, is merged with structural and location information on buildings for a case study. We found both the floor level and building effects are important factors when valuing commercial units in multistory buildings. We show that model fit is significantly affected if the spatial autocorrelation at the individual level is not included in hedonic price models.

Keywords

mixed-effects spatial hedonic model nested structure multistory building spatial autocorrelation heteroscedasticity

Introduction

The rapid changes that modern society has experienced in recent years have reshaped the structure of South Korean real estate, as evidenced by the overwhelming concentration of high-rise buildings and spatially intensive and diversified land use patterns. More specifically, the general survey of housing from the Korea Statistical Yearbook (2011) shows that the total units in apartment buildings have increased from 89,248 in 1975 to 8,185,063 in 2010 to accommodate explosive population growth. Simultaneously, soaring demand for commercial spaces has led to the division of many existing parcels into subparcels, and the replacement of old traditional houses with high-rise buildings. Consequently, heavy concentrations of multistory buildings, including condominiums, apartment complexes, multistory retail shops, and officetels¹, have come to dominate the landscape of South Korea.

The popularity of multistory buildings across the country required new management and control of the vertical dimension of a property, because the existing housing act could not properly address the increasing number of disputes over property damage that take place. Accordingly, “The Act on the Ownership and Management of Multistory Buildings” was enacted in April 1985 and has been frequently amended since then. Still, a systematic approach to assess the price for the vertical location within multistory buildings, referred to as floor-level premium hereafter, is much needed. Floor-level premiums for commercial spaces are different from those for residential units in an apartment complex. Although individuals tend to be willing to pay greater amounts for residential units with the same structure that are located on higher floor levels (Baranzini and Ramirez 2005; Wong et al. 2011), lower floor levels, particularly the first floor level (or ground floor level), are preferred for most commercial units in multistory buildings (Park 2005). Due to visibility and accessibility to amenities, floor-level premiums in commercial buildings tend to fall as floor level increases upward or downward from the first floor.

The practical implications of floor-level premium are not limited to individual trades or leases of commercial units, because they are often used as a reference to resolve legal conflicts and disputes. For instance, when calculating damages of a multistory building incurred from a major subway construction under the building, the cost of business disruptions might be different from floor level to floor level. However, traditional property rights cannot fully address such issues mainly because their coverage is limited to the horizontal dimensions of a property. In addition, floor-level premium can provide a baseline to estimate the taxation standard for property tax levies. The current property tax levy for commercial multistory buildings is calculated based on both the land price announced by the Korean government and the assessed value of buildings, neither of which explicitly take into account floor-level premium. In consequence, formal oppositions and administrative proceedings are often invoked when a property tax is imposed on the owners of the units in a multistory building.

Despite the significance of floor-level premium in the trade or management of commercial multistory buildings, little empirical and theoretical research has been conducted to investigate the floor-level effects on the price determination process. To our best knowledge, the study by Wong et al. (2011) is the first hedonic study primarily focusing on the vertical dimension of property location and its variability within and between buildings. Their approach, however, ignores the nested structure of individual units in multistory buildings and takes no account of characteristics inherent in geographic data, such as spatial autocorrelation or heteroscedasticity. The nested structure of commercial spaces influences prices; at a higher level, the price of individual commercial spaces is influenced by building-specific amenities such as physical conditions, distance to public transit, and zoning classification, as evidenced by price differentials for comparable commercial units in different buildings. Within a building, the sale prices of individual units are influenced by the floor level on which they are located as Wong et al. (2011) and Baranzini and Ramirez (2005) have suggested. On a microscale, the characteristics of other units nearby impact individual unit prices. We may risk overlooking the importance of floor-level effects and building effects, and even render traditional statistical models invalid if we ignore the naturally occurring hierarchy and spatial autocorrelation of commercial property values (Can and Megbolugbe 1997; Dubin 1998; Kim, Phipps, and Anselin 2003; Páez, Long, and Farber 2008). The importance of multilevel structures of data and the consequences of ignoring such hierarchical structure are well documented in the mixed-effects modeling literature not only from statistical complexity or technical perspectives but also from its practical implications for research design (Bryk and Raudenbush 1992; Jones 2004; Subramanian 2004a; Snijders and Bosker 2011).

Our goal is to develop an improved description of the underlying valuing process of commercial spaces in multistory buildings using mixed-effects models (Pinheiro and Bates 2000), which are also referred to as random-effects models (Laird and Ware 1982), hierarchical linear models (Raudenbush 1988), or multilevel models (Goldstein 1987). In the proposed three-level mixed-effects spatial hedonic models, building effects and floor-level effects are simultaneously taken into account under the explicit consideration of the nested structure of the data set. The flexibility of mixed-effects models enables us to model varying relationships between structural attributes of individual units and their prices across floor levels and to accommodate spatial effects remaining in residuals after accounting for individual compositional effects (Subramanian 2004b). The similarity in prices of adjacent units located on the same floor level within a building is modeled by a spatially correlated structure of the within-group errors. Based on our observation that the variation in sale prices of commercial spaces differs substantially according to zoning classification type for each building, we further extend the mixed-effects hedonic model to incorporate the heteroscedasticity of the within-group errors as a function of covariates.

Background

Rapid economic development over the last half of the century has completely changed the cityscape of Seoul, the capital city of South Korea. Nearly a quarter of the entire population resides in Seoul whose landscape is often characterized by tightly packed apartment complexes, old brick buildings, and a scattering of single-family homes mixed with commercial multistory buildings. The study region, KwangJin-gu, is located in the east part of Seoul above the Han river with a population of 370,000 and a total area of 17.05 km² (see Figure 1). The income level of this autonomous district is comparable to the city-wide average (163,000 vs. 196,000 won).²

Figure 1.

Multistory buildings in the study area.

The main data source for this study is the survey results on commercial property prices, officially referred to as “assessed value of commercial properties and officetels³”, collected by the National Tax Service of Korea in 2012. Transactions of commercial properties in multistory buildings are less frequent than those of residential properties, and developing a spatial hedonic model based only on such limited transaction records could render the model invalid. As a practical solution, the assessed values (survey data) for selected commercial properties, which are not exhaustive but larger than actual transaction records, are obtained by synthesizing recent transaction prices of commercial units and the concurrent average transaction prices based on assessors’ surveys. The survey data contain official assessment records for individual commercial spaces in 66 multistory buildings. In some buildings, entire spaces are designated for commercial uses, but in other buildings, commercial units are blended with residential units. Figure 1 shows the locations and building heights of 66 multistory buildings overlaid with the zoning type designated for each building.

In the study area, building heights range from 39 stories above ground to 5 stories below (B5). In each floor level, the floor space is divided into multiple units that are typically used as retail shops, offices, parking facilities, or entertainment facilities such as theaters, theme parks, and public bathhouses.

Most buildings in the study region are mid-rise, containing five to ten floor levels, with the exception of a few high-rise buildings including the supersize shopping center “Techno-Mart.” This 39-story mega shopping center holds a shopping mall, various discount stores, and more than 2,000 electronic shops from the first to the eighth floor level. Most higher floor levels are used for entertainment facilities, while the underground floor levels are used for bookstores, clothing shops, groceries, and parking facilities. Except Techno-Mart, the top floor levels of other high-rise buildings are between fifteen and twenty stories. In the study area, only a small number of buildings have floor levels above ten or underground (B5–B2), which results in a very different number of observations per floor level. The unbalanced data (unequal number of observations per group) may affect the homogeneity of variance assumption. To alleviate this problem, we simplified the floor levels from the observed fourty-four levels (B5–39) to fifteen levels by aggregating some floor levels with a small number of observations into a single floor level so that we have enough number of units for each reclassified floor level. The total number of units per floor level across sixty-six buildings is summarized in the third column $(N_{u})$ of Table 1. The number of buildings per floor level, denoted by $N_{b}$ , is smaller than the total number of buildings sixty-six across floor levels, because floor levels designated for residential uses only in some buildings are excluded in the current study. As seen in Table 1, a large portion of units belongs to Techno-Mart—the only building that has floor levels above twenty (100 percent). The average number of commercial units on each floor level per building is between five and fifteen after excluding the influence of Techno-Mart.

Table 1.

Reclassified Floor Levels (FLs).

Floor level	Original FL	$N_{u}$	$N_{b}$	$T_{u}$
FL 1	B5 to B2	21	8	4 (19%)
FL 2	B1	490	41	273 (56%)
FL 3	1	700	56	295 (42%)
FL 4	2	1035	64	393 (38%)
FL 5	3	884	55	394 (45%)
FL 6	4	853	52	397 (47%)
FL 7	5	859	48	394 (46%)
FL 8	6	827	44	400 (48%)
FL 9	7	727	33	397 (55%)
FL 10	8	684	30	396 (58%)
FL 11	9	302	23	48 (16%)
FL 12	10	233	20	9 (4%)
FL 13	11–15	492	13	31 (6%)
FL 14	16–20	215	3	90 (42%)
FL 15	21–39	297	1	297 (100%)

Note: “B” denotes below ground. $N_{u}$ and $N_{b}$ , respectively, denote the number of commercial units per floor levels and buildings across the study area. $T_{u}$ denotes the number of units per floor level in Techno-Mart and its proportion to the overall units across the study area $N_{u}$ is denoted in paranthesis.

There are a total of 8,604 commercial units in the study region with per square meter sale prices ranging between 217,000 and 13,915,000 won. The unit size, measured by floor area in square meters, varies from 5.35 m² to 2,901 m², as specialized retail shops in a high-end department store can be as small as 5 m², but parking facilities or underground warehouse areas can be as large as 3,000 m². The summary statistics of the sale prices on commercial real estates in multistory buildings and covariates are presented in Table 2. The distribution of sale prices are right skewed due to the small number of high-end retail shops and service centers present in the study area. To meet distributional assumptions, sale prices are log transformed. The log-transformed sale prices in Figure 2A are used as a dependent variable in subsequent hedonic price models.

Table 2.

Descriptive Statistics of Variables.

Variables	Minimum	M	Maximum	SD
Dependent variable
Sale prices (1,000 won)	6,011	156,940	4,758,582	269,000
Log of sale prices	8.70	11.53	15.38	0.78
Per square meter price (1,000 won)	217	2,059	13,915	1,203
Log of per square meter price	5.38	7.51	9.54	0.49
Covariates
Level 1
Unit size (m²)	5.35	79.82	2,901.00	142.19
Level 3
Distance to subway (m)	30.00	235.00	900.00	284.07
Built year in decades (%)	1979–1990	1991–2000	2001–2011
Built year in decades (%)	3	59	38
Zoning classification (%)	Medium-density residential	High-density residential	Commercial overlay	Commercial
Zoning classification (%)	6	10	73	10

Figure 2.

Assessed values of commercial properties.

We consider a set of characteristics that determine the market value of commercial spaces in multistory buildings, including the structural and locational attributes of individual properties such as unit size and the floor level, but also the conditions that all units in a building share as a group. Generally speaking, the unit size (private floor area combined with the common area shared by other units on the same floor level) matters in the price determination of a commercial space, but we found that per square meter unit prices substantially vary even within the same building depending on the floor level that individual property units are located. The box plot of Figure 2B shows the variations in per square meter price across floor levels, which indicates the peaks on the second and the third floor level (FL 2 and FL 3), that is, underground first floor and the ground floor level, respectively.

The box plot supports our speculation that floor level affects price of commercial units in multistory buildings even after accounting for their sizes. Structural characteristics of a unit, such as the size and the accessibility to the stairs or elevator, may influence the sale price, but floor level is clearly a crucial factor to consider in the market value determination.

Other determinants of commercial property sale prices include the conditions of the building that they belong to, which we hypothesize to be associated with the built year of a building, the land use regulation designated by a zoning classification type, and the accessibility to the nearest public transit. The age of a building entails the physical conditions of the building and the facilities that a building offers to individual units. Preliminary investigation indicates that the building age in decades—before 1990s, between 1990 and 2000, and after 2000—influences the market price of properties: the buildings constructed before 1990 tend to be valued around the overall average price; the property values of buildings built 1990–2000 are highly varying; and the property values of buildings built after year 2000 are above the average. The high prices of some buildings built prior to 1990 might be due to their locational advantages, including easier access to major public transportation or a city center. The accessibility of a building to the nearest public transit is another key factor determining price of commercial properties (Orford 2000; Páez, Long, and Farber 2008), as customers are more likely to visit retail shops and stores when the store is accessible from public transit. The subway system in Seoul is extensive as well as well connected to other public transportation systems such as bus or train stations. As shown in Table 2, the distance to the nearest subway station from each building is relatively short, ranging from 30 to 900 m, but with substantial variability per building as evidenced by a large standard deviation (284.07). It is highly anticipated that the distance to the nearest subway station will have an inverse impact on commercial property values.

We also include a zoning classification designated for each building as a potential price determinant. Zoning is one of the most important methods of land use regulation undertaken by the Korean government. According to zoning classification codes, jurisdictions are divided into geographically contiguous “zones.” The local zoning ordinance⁴ shapes the size and the uses of buildings, where buildings are located, and how dense the city’s diverse neighborhoods are allowed to be. The study area is divided into four zones—medium density residential, medium- to high-density residential, commercial overlay, and commercial zones. The residential zones are designed to promote a variety housing choices and accommodate the diversity of residential building forms, ranging from single-family homes to soaring high-rise apartment complexes. The commercial zone is designated for promoting business activities; property values are relatively high in this zone. In some residential zones, commercial activities are allowed to serve local retail needs varying from basic needs such as grocery stores, dry cleaners, restaurants, to professional needs such as banks or special service centers. Areas where mixed land uses are officially allowed are referred to as a commercial overlay zone, and in general, property prices in this zone vary substantially depending on the size of the district and the quality of services available.

Hedonic Models of Commercial Unit Prices in Multistory Buildings

Commercial spaces in a multistory building have a naturally occurring hierarchy in that individual units are associated with a floor level on which they are located, and the floor levels are nested within a building. The structures of surveyed units for this study are categorized into three levels: level 1 (individual units), level 2 (floor levels), and level 3 (buildings). We expect that the sale prices of adjacent units on the same floor level in a building are more similar to each other than other units on different floor levels or those in other buildings, as they share common locational and structural conditions such as the accessibility to the stairs or building amenities. Conversely, we expect unit values to differ substantially depending on the zoning type designated for each building. Property values are generally homogeneous in the residential or commercial zones, whereas extremely low and high property values coexist in the commercial overlay zones due to the mixed land uses. Therefore, in the following hedonic price model specifications, we explicitly take into account the naturally occurring hierarchical structure of the data, the interdependence among adjacent unit values, and heteroscedasticity for a better understanding of the valuing process of commercial spaces in multistory buildings. Based on a review of previous research and the understanding of the survey data, three-level mixed-effects models and one fixed-effect model are calibrated as follows:

Model A: A single-level linear regression model with dummy indicator variables for each floor level (fixed-effects model).

Model B: Three-level mixed-effects model with a level 1 predictor in the fixed part and differential intercepts at level 2 and level 3, respectively.

Model C: A full three-level mixed-effects model with level 1 and level 3 predictors in the fixed part and differential intercept and slope at higher levels; this model permits an assessment of the contextual effects allowing for compositional effects, as sale prices depend on the characteristics of individual units but also higher-level predictors, such as building conditions (Jones 1991; Duncan, Jones, and Moon 1993).

Model D: Same as Model C, but correcting the heteroscedasticity and the spatially correlated structure of within-group errors.

In Model A, the log-transformed sale price of the ith unit located on the jth floor in building k, denoted by Y_ijk , is determined by a linear combination of the typical average sale price across the study region and a set of fixed effects. In model A, both compositional and contextual effects at higher levels are considered as potential predictors of sale prices as:

Y_{i j k} = β_{0} + β_{1} {A r e a}_{i j k} + β_{2} {D i s t}_{i j k} + β_{3} {Y e a r}_{i j k} + β_{4} {Z o n e}_{i j k} + β_{5} {F L}_{i j k} + ϵ_{i j k},

where

${A r e a}_{i j k}$ : log-transformed unit size for the ith unit on the jth floor for building k (mean centered);

${D i s t}_{i j k}$ : distance to the nearest subway station from the kth building (mean centered);

${Y e a r}_{i j k}$ : dummy indicator variables ${{Y e a r}_{1980}, {Y e a r}_{1990}, {Y e a r}_{2000}}$ for the year that the kth building is built. The year that a building is built categorized into three classes, that is, ${Y e a r}_{1980} = 1$ , if the kth building is built prior to 1990 (1979–1990), 0 otherwise. ${Y e a r}_{1990} = 1$ , If the kth building is built between 1991 and 2000, and 0 otherwise. ${Y e a r}_{2000} = 1$ if the building is built after year 2000, and 0 otherwise;

${Z o n e}_{i j k}$ : dummy indicator variables for the zoning classification at the address of the kth building. Four zoning types are considered, medium-density residential ${Z o n e}_{1}$ , medium- to high-density residential ${Z o n e}_{2}$ , commercial overlay ${Z o n e}_{3}$ , and commercial zone ${Z o n e}_{4}$ ;

${F L}_{i j k}$ : dummy indicator variable for the floor level on which the ith unit is located, that is, ${F L}_{i j k} = 1$ , if the unit is located on the jth floor level (j = 1, …, 15), and ${F L}_{i j k} = 0$ , otherwise. See Table 1 for the floor-level definition;

$ϵ_{i j k}$ : residual differential for each unit on a floor level in a building, which is assumed to be independent normal distribution with a zero mean and constant variance $ϵ_{i j k} ~ N (0, σ_{ϵ}^{2})$ .

The log of the sale price of an individual commercial unit is the dependent variable, whereas the unit size, the distance to the nearest subway station, the built year of the building in decades, the zoning classification type that the building is designated for, and the floor level are the predictor variables. The regression intercept gives a fixed estimate of the typical price of an average-sized commercial space throughout the region from which the sample has been drawn. The other fixed effects are the slope terms that estimate the entire sample; the additional price ${\hat{β}}_{1}$ of the square meter in unit size, the distance to the nearest subway station ${\hat{β}}_{2}$ , the extra cost ${\hat{β}}_{3} = {[{\hat{β}}_{3 l}, l = 2, 3]}^{T}$ of a building based on the age of the building in decades compared with the buildings built before 1990, the additional cost ${\hat{β}}_{4} = {[{\hat{β}}_{4 l}, l = 2, 3, 4]}^{T}$ of a building due to the zoning classification at which the building is located, and the premium ${\hat{β}}_{5} = {[{\hat{β}}_{5 l}, l = 2, \dots, 15]}^{T}$ for the floor level compared with the floor level 1 (B5–B2). The random term is the residual, the difference between the actual price and the model prediction, which represents the “idiosyncratic” variations in individual sale prices that have not been captured by the model (Jones and Bullen 1993).

The specification of model A corresponds to analysis of covariance (ANCOVA), which becomes inefficient as the number of buildings increases or as the number of individual units in each building gets smaller. Model A includes floor-level effects but not building effects because it causes overfitting problems as we include additional 65 dummy variables for building effects. Moreover, such a fixed-effects model may not be able to incorporate higher-level predictors because all degrees of freedom at the higher level have been consumed.

Table 3 presents the generalized least squares fit of model A by restricted maximum likelihood (REML). The log of the sale price estimate of an average-sized commercial unit is 10.52 (37,049,120 won). The effect of unit size on the sale price is significant: a 1 percent increase in the unit size is associated with a 0.94 percent increase in the sale price. Conditional on the size of each unit, the level 3 predictors, such as the distance to the nearest subway stations from each building, the age of each building in decades, and the zoning classification type designated for each building were each statistically significant except for one contrast of the built year variable (1990–2000). The built year in decades of a building matters in the individual unit price determination, because commercial spaces located in the recently built buildings (after 2000) were generally expensive. The estimated coefficients for each floor level are worth noting; the effects of each floor level on the sale prices consistently fall up to the fifth floor from the first floor (from FL 3 to FL 7), but no consistent pattern is found in the estimated coefficients above sixth floor (FL 8). This variability of unit sale prices at higher floor levels might be attributed to the limited number of observations at high floor levels in the study region but also to how the floor spaces are used. For example, the entire floor space of the sixth to the eighth floor levels in Techno-Mart are used for entertainment services, whereas the upper floor levels in other high-rise buildings are used for regular business. Such a diversity in the use of floor spaces in upper floor levels challenges the identification of a consistent relationship between sale prices and floor levels.

Table 3.

Results of Multiple Regression Analysis (Model A).^a

Model A
Intercept	10.52 (0.09)***
Log (floor area)	0.94 (0.01)***
Distance to subway	−0.0005 (0.00)***
Year (1990–2000)	0.04 (0.03)
Year (>2000)	0.37 (0.03)***
Zoning (med-high R.)	0.35 (0.02)***
Zoning (overlay C.)	0.17 (0.02)***
Zoning (C.)	0.11 (0.03)***
FL 2	0.79 (0.08)***
FL 3	1.21 (0.08)***
FL 4	0.78 (0.08)***
FL 5	0.63 (0.08)***
FL 6	0.57 (0.08)***
FL 7	0.54 (0.08)***
FL 8	0.93 (0.08)***
FL 9	0.52 (0.08)***
FL 10	0.55 (0.08)***
FL 11	0.75 (0.08)***
FL 12	0.66 (0.09)***
FL 13	0.59 (0.08)***
FL 14	0.32 (0.09)***
FL 15	−0.10 (0.08)
AIC	7,432.16
BIC	7,594.49
Log likelihood	−3,693.08
Number of observations	8,604

Note: FL = floor level; R.= residential; C. = commercial; AIC = Akaike information criterion; BIC = Bayesian information criterion.

^a The standard deviation of each estimated coefficient is presented within the parentheses.

*p < 0.1. **p < 0.05. ***p < 0.01.

Model B is a base three-level mixed-effects model with the goal of estimating the floor-level differentials, but with no consideration of confounding factors at higher levels. The price of the ith unit located on the jth floor for the building k is specified with a single individual-level covariate, that is, the unit size, and the overall intercept, as:

Y_{i j k} = β_{0 i j k} + β_{1 i j k} {A r e a}_{i j k} + ϵ_{i j k}, k = 1, \dots, K, j = 1, \dots, J_{k}, i = 1, \dots, n_{j k},

where the total number of buildings in the study area is denoted by K, the number of floor levels in the kth building

J_{k}

, and the number of units on the jth floor level of the kth building is denoted by

n_{j k}

. We assume that the floor-level variability in sale prices is modeled as independent and normally distributed random vectors with a zero mean and a constant variance

ϵ_{i j k} ~ N (0, σ_{ϵ}^{2} I)

. Here I denotes the identity matrix with zeros at all nondiagonal elements of the matrix and ones at diagonal terms. The price

β_{0 i j k}

of an average-sized unit on the jth floor of the kth building is determined by the overall average-sized unit sale price

β_{0}

across all floor levels and buildings in the study region, and two random terms at higher levels,

u_{0 j k}

and

v_{0 k}

, for floor level j and building level k, respectively, which is written as:

β_{0 i j k} = β_{0} + u_{0 j k} + v_{0 k},

where

u_{0 j k}

refers to the unique effects of floor levels within a building. A positive value of the floor-level effects signifies that a floor level was above the building average of sale prices, while a negative value signifies a below-average price for the floor level. Similarly, the building effects

v_{0 k}

are allowed to be different from the overall average price. Generally speaking, mixed-effects models can have more than one random term at each level, which is typically modeled by Gaussian deviates with a vector of zero means and a positive semidefinite and symmetric variance–covariance structure, that is,

u_{j k} ~ N (0, Ω_{u})

and

v_{k} ~ N (0, Ω_{v})

. The random effects of model B consist of a single term

u_{j k} = [u_{0 j k}]

for floor level and

v_{k} = [v_{0 k}]

for building level, and both terms are independent of the within-group errors

ϵ_{i j k}

, that is,

C o v (u_{0 j k}, e_{i j k}) = 0

and

C o v (v_{0 k}, e_{i j k}) = 0

for

k = 1, \dots, K, j = 1, \dots, J_{k}, i = 1, \dots, n_{j k}

Table 4 reports the model fitting results: the regional average price estimate ${\hat{β}}_{0}$ of an average-sized unit (83.34 m²) is 11.53, which is higher than the fixed-effect model (model A) estimate of 10.52. The mixed-effects model allows estimating the variance in the sale prices at different levels: between buildings ${\hat{σ}}_{v 0}^{2} = {0.35}^{2}$ and, within buildings, between floor levels ${\hat{σ}}_{u 0}^{2} = {0.34}^{2}$ . On the other hand, the variance between a pair of commercial units on the same floor level in the same building is smaller as ${\hat{σ}}_{ϵ}^{2} = {0.23}^{2}$ than the higher-level variances. The variance estimates obtained at different levels enable us to disentangle the different sources of variations in the sale prices of commercial units in a multistory building and to address the following research questions: how much variation in the sale prices is attributed to the floor-level effects after taking into account the compositional effect? and which random effects dominate the sale price variability between floor level and building levels?

Table 4.

Mixed-Effects Model Estimates.

	Model B	Model C	Model D
Fixed effects
Intercept	11.53 (0.05)***	11.26 (0.20)***	11.29 (0.20)***
Log (unit size)	0.96 (0.01)***	0.98 (0.01)***	0.99 (0.01)***
Distance to subway		−0.0003 (0.00) *	−0.0003 (0.00)**
Built year (1990–2000)		−0.13 (0.21)	−0.16 (0.21)
Built year (>2000)		0.28 (0.20)	0.25 (0.19)
High-density residential zone		0.25 (0.12)**	0.24 (0.12)**
Commercial-overlay zone		0.15 (0.11)	0.16 (0.11)
Commercial zone		0.29 (0.16)*	0.28 (0.16)*
Random effects
Level 3
Building effects ${1 Ω}_{\hat{v}}$	0.35²	0.26²	0.25²
Level 2
Floor-level effects ${1 Ω}_{\hat{u}}$	0.34²	$[\begin{matrix} {0.32}^{2} & 0.19 \\ {0.08}^{2} \end{matrix}]$	$[\begin{matrix} {0.31}^{2} & 0.11 \\ {0.12}^{2} \end{matrix}]$

	Level 1
Within-group errors $σ_{\hat{ϵ}}^{2}$	0.23²	0.22²	0.28²
$r_{2}, r_{3}, r_{4}$			0.17, 1.21, 0.21
( $\hat{a}$ , $\hat{b}$ )			(11, 0.01)
AIC	297.10	241.98	−9,888.24
BIC	332.40	333.75	−9,761.17
Log likelihood	−143.55	−107.99	4,962.12
Number of observations	8,604	8,604	8,604

Note: AIC = Akaike information criterion; BIC = Bayesian information criterion. $r_{2}, r_{3}, r_{4}$ , respectively, denotes the ratio of estimated variance per zoning type defined as

$r_{2} = {\hat{σ}}_{ϵ_{2}} / {\hat{σ}}_{ϵ_{1}}, r_{3} = {\hat{σ}}_{ϵ_{3}} / {\hat{σ}}_{ϵ_{1}}, r_{3} = {\hat{σ}}_{ϵ_{4}} / {\hat{σ}}_{ϵ_{1}}$ .

*p < 0.1. **p < 0.05. ***p < 0.01.

The variance partitioning coefficient (VPC) indicates that 43 percent of the total variance lies at the building level, and the 82 percent of the total variance lies at the building–floor level. That is, the prices of any two units randomly chosen from the same floor level within a building will be correlated with a coefficient of 0.82. The practical implications of this result are twofold: regardless of the characteristics of individual units used for commercial purposes, whether they are retail shops or offices, the basic sale prices of same-sized commercial units in multistory buildings are determined by the building they belong to and the floor level they are on. Floor-level and building effects are equally important as shown in the relative proportion in ${\hat{σ}}_{v 0}^{2} / ({\hat{σ}}_{u 0}^{2} + {\hat{σ}}_{v 0}^{2}) = 0.52$ . In summary, our findings suggest that both the floor-level premium and the characteristics of a building are primary determinants of commercial units in multistory buildings. As previous studies suggest (Goodman and Thibodeau 1998; Orford 2000; Zietz, Zietz, and Sirmans 2008), the locational and physical conditions of a building that individual commercial units belong to significantly affect their sales prices or rents, but our results show that floor-level effects might be equally important, particularly in the price determination of commercial spaces in multistory buildings.

Model B separately estimates floor-level effects, building effects, and differences in the sale prices through $u_{0 j k}, v_{0 k}$ , and $σ_{u 0}^{2}, σ_{v 0}^{2}$ . We speculate that the missing covariates at higher levels contributes to the strong building effects estimate ${\hat{σ}}_{v 0}$ of model B. Model C retains the same level of random terms as model B but incorporates higher-level predictors to model contextual effects. Furthermore, we hypothesize that the effects of extra square meter on sale prices differ from floor level to floor level such that the “price–unit size” relationship varies over floor levels in each building. The hedonic model specification of model C is similar to model A, but without the set of dummy variables for floor levels, and random effects are incorporated as in model B. The functional form of model C is written as:

Y_{i j k} = β_{0 i j k} + β_{1 j k} {A r e a}_{i j k} + β_{2} {D i s t}_{i j k} + β_{3} {Y e a r}_{i j k} + β_{4} {Z o n e}_{i j k} + ϵ_{i j k} .

β_{1 j k} = β_{1} + u_{1 j k} .

ϵ_{i j k} ~ N (0, σ_{ϵ}^{2}), v_{0 k} ~ N (0, σ_{v 0}^{2}), (\begin{matrix} u_{0 j k} \\ u_{1 j k} \end{matrix}) ~ N (0, Ω_{u}) = ([\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} σ_{u 0}^{2} \\ σ_{u 0 u 1} & σ_{u 1}^{2} \end{matrix}]) .

where the price

β_{0 i j k}

of average-sized unit is specified in the same way as model B (see equation 3), which includes two random-effects terms at higher levels. Model C also includes an additional random-effects term for slope

β_{1 j k}

. The random slope in equation (5) captures the effects of an extra square meter of the unit size

β_{1}

across the study area and its differential per floor-levels

u_{1 j k}

in each building. In summary, model C has two random terms at level 2, the differential

u_{0 j k}

for the price of an average-sized unit for each floor level in each building, and the differential

u_{1 j k}

in price of an extra square meters in unit size for each floor level within a building. Both terms are assumed to be Gaussian deviates with zero means and standard deviations,

σ_{u 0}

and

σ_{u 1}

, respectively, with covariance

σ_{u 0 u 1}

Model fit results are summarized in the third column of Table 4. The sale price estimate of 11.26 for an average-sized unit is similar to the estimate of model B, but the building-level variance substantially drops to ${\hat{σ}}_{v 0}^{2} = {0.26}^{2}$ in model C because higher-level predictors are included. In model C, the price of extra square meter is estimated as ${\hat{β}}_{1}$ = 0.98 with differential ${\hat{u}}_{1 j k}$ , which has very little variability ${(\hat{σ}}_{u 1}^{2} = {0.08}^{2})$ . On the other hand, the correlation between the two random terms is not negligible ${(\hat{σ}}_{u 0 u 1} = 0.19)$ . The high correlation between the two random terms indicates that the differential in the cost of an extra square meter for each floor level is higher when the differential for the price of an average-sized unit for each floor level is high. The result agrees with a common practice in that the price of extra square meter is higher on the first floor level whose average sale price is higher than that of other floor levels. The effect of the distance to the nearest subway station from each building is summarized by ${\hat{β}}_{2}$ . The negative value indicates that there is an inverse relationship between the travel distance and the sale prices if all other conditions are equal: on average, a 1 m increase in the distance to the nearest subway station corresponds to a 0.03 percent decrease in the sale price. The effects of the built year variable are relatively weak; the 95 percent confidence intervals for the built year 1990–2000 and after 2000 are bounded with zero as ${\hat{β}}_{32} \in [- 0 .54, 0 .29]$ and ${\hat{β}}_{33} \in [- 0 .11, 0 .67]$ . Unlike model A, the built year in decades of each building is no longer a significant predictor in the three-level mixed-effects model. We suspect that the building effects ${\hat{v}}_{0 k}$ in the mixed-effects model absorb the variation explained by the built year variable in the fixed-effect model (model A). Effects of the zoning indicators seem to be substantial on the sale price determination process in the three-level mixed-effects model, although the standard errors of estimators have increased. Before making any further inferences from model C, we examine the fitted model based on the following two assumptions underlying the mixed-effects models (Pinheiro and Bates 2000):

The within-group errors are independent and identically normally distributed, with a zero mean and constant variance, and they are independent of the random effects.

The random effects are normally distributed, with a zero mean and covariance matrix, and they are independent of different groups.

The within-group residuals are a measure of the difference between the observed sale price and the model fit. Figure 3A and B, respectively, show the box plots of the within-group residuals for model C by building and by floor level. Both plots show that the residuals are centered at zero but with considerably variable across groups. Outlying observations and large residuals are associated with high-rise buildings, in which a number of units share floor space. The distribution of residuals per floor level is displayed in Figure 3B. Large variability is found at the reclassified floor space (FL 3; ground floor level), which might be due to the location effects not accounted for by the fixed part of the model. Scatterplots in Figure 3C reveal that the variability in residuals for model C varies as a function of zoning classification; greater variabilities of residuals are found in the commercial overlay zone compared with the other zones. This large variability might be attributed to the mixed land use in a commercial overlay zone, which officially allows both residential and commercial buildings to coexist but also to the large proportion (73 percent) of commercial multistory buildings concentrated in this zone. Heteroscedasticity is a norm in spatial data modeling, and thus, traditional approaches that assume a constant variance can be overly simplistic and potentially misleading (Kutner et al. 1974; Zuur et al. 2009).

Figure 3.

Within-group residual analysis of model C.

We also investigated the correlation structure of the within-group residuals for model C. The sale prices of commercial units on the same floor level of a building are likely to be similar to each other because of similar exposure to customers. Considering that the distance from the store to the shopper’s location is the most important variable to determine the shopper’s consumption as asserted in traditional economic geography models (Cadwallader 1975; Lusch 1981), the retailer’s decision where to locate its store is affected not only by the location of customers but also by surrounding businesses. Consequently, similar retail stores or businesses tend to be located on the same floor level in a building, and their spatial clustering affects their sale prices.

Figure 3D shows the empirical semi-variogram of model C residuals, which reveals the presence of substantial spatial autocorrelation. It should be noted that the distance between any two commercial spaces is measured by the unit numbers, which implies that the measured separation vector is a relative measure of proximity between two commercial spaces. In the empirical variogram calculation, we limit our maximum separation vector to 15 distance units, because the average number of commercial units per floor level except Techno-Mart is between five and fifteen. The empirical variogram indicates that price valuing processes operate at different spatial scales, where one of the processes can be modeled by a nested variogram model. We speculate that the variability at a longer range (approximately h = 13) might be due to the large number of commercial units in Techno-Mart, while the shorter range (approximately h = 7) might be an area-wide separation distance at which spatial dependence in the within-group residuals exists.

We accounted for both the heteroscedasticity and the spatial dependence of the within-group errors for model D based on the residual analysis of model C. A general form of the variance–covariance structure of within-group errors can be written as $V a r (ϵ_{i j k}) = σ_{ϵ}^{2} [Λ_{i j k}]^{2}$ , where $Λ_{i j k}$ always being decomposed into a product of simpler matrices, that is, $Λ_{i j k} = V_{i j k} C_{i j k} V_{i j k}$ . Here, $V_{i j k}$ denotes a diagonal matrix and $C_{i j k}$ is a symmetric and positive definite correlation matrix (Pinheiro and Bates 2000). In both models B and C, we assume a constant variance, which can be summarized by a single parameter for the within-group residual structure and the absence of correlation. The model C residual plot in Figure 3C indicates that the variability in the within-group residuals differs substantially by zoning. Moreover, the empirical variogram in Figure 3D suggests the presence of a spatially correlated structure of the within-group residuals for model C.

Therefore, we allowed the variance in within-group residuals to differ by zoning type in model D and estimated a separate variance ${\hat{σ}}_{ϵ_{z o n e}}^{2}$ for each zone. For example, the variance estimate of the within-group residuals for medium-density residential zone is denoted by ${\hat{σ}}_{ϵ_{1}}^{2}$ . We also updated the correlation structure $C_{i j k}$ of the within-group residuals by adopting a stationary random function whose behavior is characterized by a variogram (Isaaks and Srivastava 1989). The pattern of spatial dependence underlying the within-group residuals is characterized by the choice of a variogram model and a set of parameters, such as nugget, range, and sill. The choice of a particular variogram model directly implies a belief in a certain type of spatial continuity, while the nugget effects and range are associated with sampling error and the maximum spatial distance at which the dependence exists. Sill represents the magnitude of variability at the distance of range. In the context of the three-level mixed model, the range corresponds to the number of adjacent units whose sales values are similar and the nugget effects quantify the portion of variability at short scale associated with sampling errors. The dependence remaining in the residuals is modeled as a function of separation vector $h_{i i^{'}} = | x_{i j k} - x_{i^{'} j k} |$ between any pair of units, denoted by $x_{i j k}$ and $x_{i^{'} j k}$ , located on the same floor (j) in a building (k) as:

c o r (ϵ_{i j k}, ϵ_{i^{'} j k}; θ) = b (1 - \frac{3}{2} \frac{| h_{i i^{'}} |}{a} + \frac{1}{2} \frac{| h_{i i^{'}} |^{3}}{a^{3}}), i, i^{'} = 1, \dots, n_{j k},

where

θ

denotes the set of parameters consisting of the range a and nugget effects b.

The results for model D can be found in the fourth column of Table 4. Model D is considerably better than the other mixed-effects models (models B and C) as the general model fit indices are considerably lower (Akaike information criterion = −9,888.24 and Bayesian information criterion = −9,761.17). However, the heteroscedastic model with a correlated structure does not have substantial impacts on fixed parameter estimates, as they are similar to those of model C. Although the change is not substantial, the random-effect estimates at the building level is slightly reduced to ${\hat{σ}}_{v 0}^{2} = {0.26}^{2}$ from that of model C. The heteroscedastic model accommodates the impact of the outlying observations within each zoning type and consequently reduces the estimated between-building variability. That is, the degree of shrinkage in the random-effect estimates increases in the commercial overlay zone.

The between floor-level variance estimate ${\hat{σ}}_{u 0}^{2}$ was similar to that of model C, but the variance in the differential of unit size at floor-level ${\hat{σ}}_{u 1}^{2}$ was greater and the correlation ${\hat{σ}}_{u 0 u 1}$ between the random terms for intercept and slope was lower. Furthermore, the differentials of an extra square meter on the sale price of a commercial unit ${(\hat{σ}}_{u 1}^{2} = {0.12}^{2})$ in model D were larger than those in model C. In other words, the impact of unit size change on the sale price determination process was considerably different from floor level to floor level within the same building. In model C the effects of unit size changes were confounded by the average sale prices of each floor level as evidenced by the correlation ${\hat{σ}}_{u 0 u 1} = 0.19$ , which was reduced to ${\hat{σ}}_{u 0 u 1} = 0.11$ in model D.

The results for model D include the parameters of the spatially correlated structure and the heteroscedasticity of the within-group residuals. As we speculated from the model C within-group residual analysis, the nugget effect and range parameter are estimated as $\hat{b} = 0.01$ and $\hat{a} = 11$ , respectively. In the study area, the sale prices of eleven average-sized consecutive units on the same floor level are similar, while the variability in sale prices are substantially different per zoning type. In order to avoid the identifiability problem, we consider medium-density residential zone as a baseline and calculate the ratio between the standard deviation of each zone and the standard deviation of baseline (Pinheiro and Bates 2000). The ratios for high-density residential, commercial overlay, and commercial zone to the medium-density zone are denoted by $r_{2}, r_{3}, r_{4}$ in Table 4. The results indicate that the variance of the within-group residuals for the commercial overlay zone $r_{3}$ is 1.21 times higher than that of the medium-density residential zone, which challenges the constant variance assumption underlying traditional modeling approaches. In summary, although the consideration of correlation and heteroscedasticity of within-group residuals did not have a direct impact on the fixed part of the model, it did for random-effect estimates.

The adequacy of model D specification was assessed by examining the heteroscedasticity and the spatial correlation remaining in the within-group residuals. We calculated the semi-variogram for the within-group residuals of model D as shown in Figure 4A. An explicit consideration of the correlated structure in the model D accounted for the spatial dependence in the within-group residuals. In order to facilitate the comparison, we examined the empirical variogram of model C residuals in Figure 4A. The correlation structure found in model C residuals disappeared in the variogram of model D, and only the low variance with small variability remained, which varies around $\hat{γ} (h) = 0.2$ across the separation vector h. Model D was updated for both spatial correlation and heteroscedastic structure of the within-group residuals. The variogram of a mixed-effects model, which has the same structure as that of model D, with no consideration for the heteroscedastic structure is shown in Figure 4A with the other two variograms of models C and D. Results indicated that both the heteroskedasticity and the correlation structure need to be accounted for to capture the spatial continuity at a short range as well as a long range. The variogram of model D indicated that the residuals were uncorrelated, and hence, the spherical spatial correlation model was adequate. The correction of heteroscedastic structure also improved the model fit, but not substantially.

Figure 4.

Assessing assumptions on model D.

The scatterplot displayed in Figure 4B reveals the pattern of variability remaining for the standardized residuals of model D by zoning type. Some improvements were made as evidenced by the reduced variability of within-group residuals for commercial overlay zone compared to that of model C (in Figure 3C). However, such a flexibility of model D allows the within-group residual variance at other zones, for example, high-density residential zone, to increase, although the increased variability in other zones might be associated with the small number of observations.

Lastly, normal plots of the estimated within-group residuals and random effects are presented in Figure 4C–E. Figure 4C and D suggests a departure from the normality assumption for the within-group residuals, as the distribution of within-group residuals has heavier tails than expected under normality, while it is symmetric around zero. The heavier tails tend to inflate the within-group residuals under the Gaussian model and consequently leads to more conservative tests for the fixed effects (Pinheiro and Bates 2000). However, the symmetric distribution of the heavy tails would not substantially change the fixed-effect estimates, and the fixed-effect parameters in model D were all statistically significant (see Table 4. In summary, we would not expect any changes to main conclusions as a result of the departure from the normal distributional assumption for the within-group residuals. The distributional assumption for random effects at building level was evaluated using a QQ plot. Figure 4E shows no significant deviation from normal distribution. The test of other random effects at floor level was not included due to space limitation, but both of them generally followed normal distributions.

Discussion and Conclusions

The results suggest that a mixed-effects modeling approach provides a flexible and comprehensive framework for the market value assessment of commercial spaces in multistory buildings. The nested structures of survey data are seamlessly incorporated in mixed-effects models, and the compositional attributes of individual units are jointly taken into account with contextual effects. As evidenced by random-effect estimates, both the floor-level premium and building effects are crucial price determinants of commercial units in multistory buildings across all three mixed-effect models (models B–D). Some may argue that the fixed-effect model (model A) estimates for floor-level premium or building effects using dummy variables are as good as those of mixed-effects models, but the former tends to ignore the the nested structure of commercial units in multistory buildings, and it is often unable to handle the complexity of the valuing processes of housing markets. The substantial difference in the applications of a spatial mixed-effects model (model D) and the fixed-effect model (model A) is in the improved precision of the estimator of parameters of the same type (Haining 2003), as floor-level effects ${\hat{u}}_{0 j k}$ are estimated by pooling data together and borrowing strength from other floor levels in a building.

Mixed-effects models are also better designed to accommodate endemic problems of geographic analyses. As evidenced by the recent development of spatial hedonic models (Dubin 1992; Basu and Thibodeau 1998; Orford 2000; Páez, Long, and Farber 2001; Kim, Phipps, and Anselin 2003; Yoo and Kyriakidis 2009), spatial autocorrelation is likely to be present in commercial property values. The interdependence of adjacent property values arises from their nested structures but also from inherent characteristics of spatial data. Therefore, the application of standard statistics, which are typically based on the assumption of data independence, to the assessed values of commercial units in multistory buildings is likely to yield a severely misleading estimation of model parameters. In contrast, mixed-effects models explicitly take the autocorrelation into account during the modeling process, as the interdependence among level 1 observations is one of the key model assumptions (Jones and Bullen 1994).

Similarly, geographic analysis often encounters heteroscedasticity problems. For example, the land use regulation in the study region centralizes the retail activities by constraining the quantity and location of available space. Consequently, property values in each zone are affected by such regulation and the structure of the variation in property values might differ from one zoning type to the other. The nonfulfillment of the homogeneous variance assumption in traditional statistics will lower the explanatory power of the fitted model and may raise questions about the validity of model parameter estimates. The flexibility of mixed-effects models allows researchers to extend the basic model to accommodate heteroscedasticity using various forms, for instance, as a function of covariates.

The proposed three-level mixed-effects models are intuitive and flexible from both substantive and technical perspectives. The mixed-effects models explicitly take into account the nested structure of the data, that is, the three levels of location information, and simultaneously incorporate both the contextual and compositional effects in the property valuing process. In all three-level mixed-effects models considered in the study (models B–D), we assess the relative importance of the different components of location in determining commercial space prices by separately estimating floor-level premium and building effects. As evidenced by VPC of model B, the nested structure of location information, both the building effects and the floor-level effects, is a key factor in valuing commercial spaces in multistory buildings. We also found that the implicit price of extra square meter substantially differs from floor level to floor level in model C. Lastly, we show that a three-level mixed-effects model can achieve considerable improvements by explicitly taking into account heteroscedasticity and spatial autocorrelation. In the current study, we focus on the correlation structure at level 1, though correlation structures at higher levels can be accounted for if nontrivial spatial autocorrelations are present.

Our findings have various practical implications for the Korean government’s taxation practices but also provide a platform for researchers to explore general issues related to the availability of rich data set. The proposed three-level mixed-effects model provides a flexible and systematic means of valuing commercial properties in multistory buildings. It is particularly worth noting that the proposed approach considers the vertical dimension of properties, that is, floor levels of commercial spaces, as a key price determinant. We found both the price differential of an average-sized commercial space for each floor level and the price differential of an extra square meter are substantial. In the proposed mixed-effects models, the floor-level premium is estimated conditioning on the building effects by explicitly modeling the nested structure of the data set. The floor-level premiums are different by building and their variability depends on the zoning designated for each building. Our findings raise questions about the validity of an oversimplified but commonly used property valuing approach in Korea—properties on the second floor level are half of the values on the first floor level if all conditions are equal.

The proposed three-level mixed-effects models achieved a considerable improvement from the traditional hedonic approach, although there are still potential improvements that may be worth exploring. Other structural variables in addition to the unit size, for example, the accessibility to the entrance or stairs/elevators, could improve the explanatory power of the proposed model. We currently limit our modeling efforts of spatial autocorrelation to the within-group residuals, but spatial autocorrelation at higher levels, such as among buildings or floor levels, need to be further explored. Lastly, we will expand our model to incorporate residential units in apartment complexes, which will allow developing a spatial hedonic price model for multistory buildings under the consideration of different, but closely linked, valuing processes of housing markets.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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