Hedonic Models Incorporating Environmental, Social, and Governance Factors for Time Series of Average Annual Home Prices

Abstract

Using data from 2000 through 2022, we analyze the predictive capability of the annual numbers of new home constructions and four available environmental, social, and governance (ESG) factors on the average annual price of homes sold in eight major U.S. cities. We contrast the predictive capability of a P-spline generalized additive model (GAM) against a strictly linear version of the commonly used generalized linear model (GLM). As the data for the annual price and predictor variables constitute non-stationary time series, we transform each time series appropriately to produce stationary series for use in the GAMs and GLMs in order to avoid spurious correlations in the analysis. While arithmetic returns or first differences are adequate transformations for the predictor variables, we utilize the series of innovations obtained from AR(q)-ARCH(1) fits for the average price response variable. Based on the GAM results, we find that the influence of ESG factors varies markedly by city and reflects geographic diversity. Notably, the presence of air conditioning emerges as a strong factor. Despite limitations on the length of available time series, this study represents a pivotal step toward integrating ESG considerations into predictive time series models for real estates.

1. Introduction

Hedonic models are employed to analyze and predict average real estate prices via intrinsic and extrinsic factors. The average home price in a city plays an important role in the calculations made by potential homebuyers, particularly for low- and fixed-income buyers. Undoubtedly, the impacts of climate change and extreme weather will also affect the decisions made by potential homebuyers (as well as current homeowners) as the century progresses. Much work has been carried out in quantifying and modeling residence-based (e.g., lot area and number of bedrooms) and neighborhood-based (e.g., school zoning and homeowners’ association fees) factors. The impact of recent developments in environmental, social, and governance (ESG) policies and factors on real estate prices have not been as well-analyzed. This paper contributes to that analysis.

We begin by briefly describing work that has been carried out to assess the impact of ESG factors on homebuilding and consumer decision-making. The environmental, social, and governance components of ESG represent the sustainability factors of a property. Resiliency to global warming, the risk of a natural disaster, and the installation of renewable energy systems are examples of environmental factors. Noise pollution, construction worker labor standards, and homeowner satisfaction are examples of social factors. Legal issues related to property owner practices, regulatory compliance with standards set at all governmental levels, and overall transparency are examples of governance factors.

Ma et al. (2019) analyzed the impact of governmental policymaking processes on residential green energy additions and constructions. Even with a consumer base open to adopting environmentally friendly technologies, the cost bases, measured relative to non-green energy prices, play a large role in the adoption of such technologies. In particular, governmental policies on residential green energy subsidies that are too stringent can have an adverse effect on household installations.

Lauper et al. (2013) analyzed the green home acquisition and installation process from the point of view of a homebuilder. Social factors (e.g., behavioral control and social norms) have meaningful impacts on the energy-relevant decisions made in homebuilding. Social policies and norms, such as low energy consumption building certificates and awareness of available green technologies, have been shown to heighten consumer interest and spending on environmentally friendly appliances.

In addition to qualitative analyses based on consumer behavior, quantitative indices have been developed to provide guidance to consumers in assessing home prices. Environmental factors (e.g., average maximum temperatures and flood risk) can be expected to play a role in homebuyer decisions (and, therefore, real estate pricing). Mahanama et al. (2021) developed a natural disasters index to assess the level of future systemic risk caused by natural disasters. Their index used decades of property losses from the NOAA Storm Data to assess the main contributors to property losses. Although a homeowner’s thought process can be very subjective, a quantification of the risk of extreme weather events represents an important step in translating subjective thought processes into quantitative factors for use in modeling. The results of a survey of research at the intersection of climate risks, housing, and mortgage markets revealed that natural disasters are expected to continue to weigh heavily on home prices (Contat et al. 2023). Specifically, the risks of flooding and wildfires were shown to correlate inversely with home prices, as higher risks of floods and wildfires result in discounts on said prices.

Intrinsic and extrinsic factors, including some ESG factors, have been used to describe the variance in (the logarithm of) the expected sales price of homes (Bailey et al. 2022). When ESG factors (accessibility for the elderly and disabled, presence of central air conditioning, “green home” rating, and waterfront location) commonly available on real estate vendor websites were included, minor improvements were observed in the model adjusted $R^2$ values. Although the model results were city dependent, the potential impact that such ESG factors had in assessing home prices was established.

Such factors have also been shown to have different impacts on home valuations depending on a home’s value in the context of the local housing price distribution. For example, an analysis of 1366 home sales in Orem and Provo, Utah, from mid-1999 to mid-2000 found that segmenting houses by quantile was important in identifying the effects of several input factors (Zeitz et al. 2007). The number of bedrooms was found to be more significant in lower-priced homes than higher-priced homes, whereas the number of bathrooms was more significant in higher-priced homes than lower-priced homes. As another example, an analysis of 136,000 single-family home sales in Jacksonville, Florida, from 1990 to 2006 found that square footage and lot size were found to be more significant at the upper level of home prices, whereas home age was more significant at lower home prices (Zeitz et al. 2008). The analysis also assessed the impact of home location relative to a body of water, such as whether or not the property was an oceanfront one or bordered the St. John’s River.

Furthermore, an analysis in a similar vein conducted in Changsha, China, found that homes near the lowest and highest quantiles of the price distribution were more affected by the prices of nearby properties than those in the middle (Liao and Wang 2012). Similarly to the Jacksonville analysis, square footage was found to weigh more heavily as the home price increased. With regard to green areas, the study found that lower-priced units were positively impacted by the presence of green spaces, whereas the opposite was true for higher-priced units. Another study in China focused on the capital financing patterns of A-listed Chinese companies as commercial enterprises also found the significance of ESG factors, particularly for creditworthiness (Zahid et al. 2023). The relationship between the combined ESG score and market-based financial leverage was found to be negative and significant, which indicates that stronger ESG disclosures may have resulted in higher investor confidence in those companies. In a similar vein, they also found that a positive correlation existed between a company’s ESG performance and the managerial skills of its chief executive officer (Zahid et al. 2024). Furthermore, companies experiencing financial difficulties prioritized ESG performance more significantly than companies not experiencing financial difficulties. Thus, these analyses reveal that ESG factors are important considerations in non-Western countries, too.

Other ESG factors not currently featured on real estate vendor sites, such as the impacts of air pollution, have been explored. For example, an analysis of the closure of a toxic site leading to changes in atmospheric pollution levels found that the corresponding drop in SO₂ levels correlated with an average house price increase of 6% (Lavaine 2019). However, the average price of flats decreased by 9%, suggesting that the impacts of refinery closures and changes in air pollution levels have heterogeneous effects on the subsamples.

The usual application of a hedonic home-pricing model is “cross-sectional”, consisting of a data set of response (price) and predictor (e.g., number of bedrooms, bathrooms, home size, etc.) variables for a sample of homes. Implicit in the cross-sectional analysis is the assumption that the data set represents independent and identically distributed random samples reflective of the pricing structure in a particular geographic area. By contrast, the application in this paper is to time series data. For a geographic area (specifically a city), the data set consists of the average annual home price, the number of homes sold per year, and yearly values for each of four available ESG factors. As we show (Appendix B), each time series is non-stationary and exhibits strong year-by-year trends, which can produce spurious correlations in fits by hedonic models.

This paper pursues three goals. The first is the determination of an appropriate transformation into a stationary form for each time series of annual data. The second is to evaluate the effectiveness and accuracy of the application, to these transformed series, of a P-spline-based generalized additive model (GAM) compared to a generalized linear model (GLM). The analysis used data from eight cities, and the cities are shown on the map in Figure 1 below.

These cities were chosen to represent variations in geography, primary economic activity, and population size and density. For example, the cities of Portland and Seattle represent the Pacific Coast, whereas the cities of Columbus and Oklahoma City represent the interior of the country. Furthermore, Austin has a strong economic base in the tech sector, whereas Nashville is a major center of the music industry. Finally, Atlanta has a high population density of over 1420 people per square kilometer, whereas Jacksonville has a much lower population density of about 490 people per square kilometer.

Each transformed time series is “de-trended”, as it represents values from a fixed-mean and fixed-variance random variable. Using principal component analysis, the third objective is to investigate the residual error time series (aggregated across cities) from each of the GAM and GLM fits to determine the presence of further latent random variables. As the model is deliberately parsimonious in terms of the number of predictor factors, we hypothesize the presence of additional latent variables.

2. Materials and Methods

2.1. Price and Factor Data

Price and factor data were acquired from Zillow1. The data set is composed of completed sale transactions of homes2 each year for the years 2000 through 2022 for eight cities3. For each year and city, the data set consists of the average home sale price (Av Price), the total number of homes constructed (New Homes), and four ESG factors: the number of homes with central air conditioning (Central AC), the number of green-rated homes (Green), the number of homes considered accessible to the elderly and disabled (Accessible), and the number of homes along a waterfront (Waterfront)4. The eight cities studied were Atlanta, GA (ATL), Austin, TX (AUS), Columbus, OH (COL), Jacksonville, FL (JAX), Nashville, TN (NAS), Oklahoma City, OK (OKC), Portland, OR (POR), and Seattle, WA (SEA). As an example,

Table A2. Price and factor data for ATL.

Year	Av Price	New Homes	Accessible	Central AC	Green	Water- front
2000	174,500	456	43	454	20	31
2001	187,800	718	85	716	38	38
2002	196,400	919	112	873	46	79
2003	203,400	649	38	615	26	43
2004	211,700	1267	139	1235	40	107
2005	222,000	1842	140	1815	55	142
2006	229,200	1775	171	1718	37	139
2007	233,800	1451	124	1386	36	99
2008	225,500	916	145	902	19	52
2009	212,000	427	41	401	30	74
2010	195,600	330	46	305	47	31
2011	180,500	89	2	88	7	5
2012	172,900	105	1	114	6	14
2013	183,400	168	2	174	7	11
2014	200,900	167	3	182	13	8
2015	216,600	283	5	277	16	14
2016	232,400	319	5	305	11	23
2017	249,100	404	6	394	12	26
2018	269,600	451	2	432	25	28
2019	286,400	629	11	823	31	26
2020	303,200	1225	31	968	32	75
2021	351,300	1091	54	1019	55	88
2022	430,000	740	29	760	51	67

in Appendix B summarizes the full data set for ATL.

2.2. Generalized Additive and Linear Models

A GAM relates a univariate response variable $Y_t$ to a set of predictor variables (factors) $x_{k,t}$, $k=1, \dots , m$. (Here, the subscript $t=1, \dots , \tau$ indicates the observed set of values of the response and predictor variables. In this application, $t$ indicates yearly time values.) Specifically, the GAM relates the expected value ${\mu }_t=E\left[Y_t\right]$ to the predictor values via

$$g\left({\mu }_t\right)={\mathsf{\beta }}_0+f_1\left(x_{1,t}\right)+f_2\left(x_{2,t}\right)+\cdots +f_m\left(x_{m,t}\right), t=1,\dots ,\tau .$$

(1)

The model assumes $Y_t~EF\left({\mu }_t,\theta \right)$, where $EF\left({\mu }_t, \theta \right)$ denotes the exponential family of distributions having mean ${\mu }_t$ and scale parameter $\theta$. The choice of the link function $g\left( · \right)$ relates expected values of the average ${\mu }_t$ to the factors via

$${\mu }_t=g^{-1}\left({\mathsf{\beta }}_0+f_1\left(x_{1,t}\right)+f_2\left(x_{2,t}\right)+\cdots +f_m\left(x_{m,t}\right)\right)+{\epsilon }_t,$$

(2)

where ${\epsilon }_t$ denotes the residual error that is not captured by the model. The identity function was used for $g\left( · \right)$, and P-splines (Eilers and Marx 1996) were used for the functions $f_j\left( · \right)$. Such P-splines minimize the penalized sum of squares

$$\sum _{i=1}^{\tau }{\left(Y_{\tau }-\sum _{j=1}^m{f}_j\left(x_{j,t}\right)\right)}^2+\sum _{j=1}^m{\lambda }_j\int {f_j}^{″}{\left(z\right)}^{2}dz,$$

(3)

where the tuning parameters ${\lambda }_j>0$ determine the weight assigned to the smoothness of each function. The values $x_{tj}$ are referred to as the knots for the function $f_j\left( · \right)$.

The results acquired from this GAM were compared to those from a standard GLM of the form

$$g\left(E_Y\left(\mathit{Y}|\mathit{X}\right)\right)={\beta }_0+{\beta }_1{\mathit{x}}_1+\cdots +{\beta }_m{\mathit{x}}_m+\mathit{\xi }\equiv \mathit{X}\mathit{\beta }+\mathit{\xi }.$$

(4)

In (4), matrix notation is used to represent the response and predictor values: $\mathit{Y}={\left[{\mathrm{Y}}_1, \dots , {\mathrm{Y}}_{\tau }\right]}^T$ is the column vector of response values, $\mathit{\beta }={\left[{\beta }_0, {\beta }_1, \dots , {\beta }_m\right]}^T$ is the column vector of unknown parameters, $\mathit{\xi }={\left[{\epsilon }_0, {\epsilon }_1, \dots , {\epsilon }_{\tau }\right]}^T$ is the column vector of residuals, and $\mathit{X}$ is a $\tau \times \left(m+1\right)$ matrix. The first column of $\mathit{X}$ is a vector of ones, while column $j, j=2, \dots , m+1$, is the vector ${\mathit{x}}_{\mathit{j}}={\left[x_{j,1}, \dots , x_{j,\tau }\right]}^T$ of values for factor $x_j$. As in the GAM, we used the identity function for $g\left( · \right)$, which reduced (4) to a pure linear model.

In the GAM (1), the use of a non-linear function $f_j\left( · \right)$ provides non-linear dependence on the time-dependent value of its argument $x_{j,t}$. In the GLM (4), the corresponding coefficient ${\beta }_j$ is constant, which results in linear dependence on the time-dependent value of $x_{j,t}$. Thus, a non-linear form for the $f_j\left( · \right)$ enables a greater fitting accuracy for the GAM compared to the GLM. On the other hand, the GLM provides a superior model for prediction. The form of the function $f_j\left( · \right)$ depends on the known values of its knots. Accurate prediction by GAM requires knowledge of future knot values to a much greater degree than the GLM5 does.

2.3. Transformation to Stationary Time Series

We wish to compare the accuracy of models (1) and (4) in predicting the time series of the expected value ${\mu }_t=E\left[Y_t\right]$ of average annual home price $Y_t$ in terms of the time series of the five factors $x_{j,t}$, $j=1, \dots , 5$ described earlier for each city. Each time series consists of 23 years of values. It is necessary that the time series for the response variable and each factor be stationary in order to avoid spurious correlations in the factor analyses. Stationarity of each time series was investigated via the augmented Dickey–Fuller (ADF) test applied to the random walk model6

$$\Delta y_t=\alpha y_{t-1}+\sum _{i=1}^q{\delta }_i\Delta y_{t-i}+{\mathsf{\epsilon }}_t.$$

(5)

The statistic ${\mathrm{D}\mathrm{F}}_{\alpha }=\widehat{\alpha }/\mathrm{S}\mathrm{E}\left(\widehat{\alpha }\right)$7 is used to test the hypotheses $H_0: \alpha =0$ (the existence of a unit root) against $H_A: \alpha <0$. The rejection of the null hypothesis through a sufficiently small $p$-value suggests that no unit root is present and that stationarity can be assumed.

As illustrated in Figure A1 (Appendix B) for ATL and as verified by the ADF test, stationarity could not be inferred at any reasonable level of statistical significance for any factor or price time series for any of the eight cities. As discussed in Appendix B and illustrated for ATL in Figure A2, the use of the arithmetic return series for each predictor factor produced transformed time series that were acceptable. There were five exceptions for which the arithmetic return time series had to be replaced by simple first differences in order to avoid division by zero: the Accessible factor for Seattle, the Green factor for Columbus, and the Waterfront factor for three cities. As the first-difference time series for the Waterfront factor for all eight cities had very acceptable $p$-values (below 1.5%), we used the first-difference time series for the Waterfront factor for all cities for consistency. Thus, the predictor variables used in (1) and (4) represent transformed series (with the transformation being either arithmetic returns or first differences). The specific transformation used for each factor is summarized in

Table A4. Type of transformed series used in the GAM and GLM fits.

ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
rtn a	rtn	rtn	rtn	rtn	rtn	rtn	rtn
rtn	rtn	rtn	rtn	rtn	rtn	rtn	rtn
rtn	rtn	fd b	rtn	rtn	rtn	rtn	rtn
rtn	rtn	rtn	rtn	rtn	rtn	rtn	fd
fd	fd	fd	fd	fd	fd	fd	fd
$\mathit{q}=2$ c	$q=1$d	$q=1$	$q=2$	$q=1$	$q=1$	$q=2$	$q=2$

^a rtn: arithmetic return ^b fd: first difference ^c AR(2)-ARCH(1) ^d AR(1)-ARCH(1).

in Appendix B.

Table 1. ADF test $p$-values for each transformed time series by city.

Factor	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
New Homes	0.074	**	0.017	0.034	**	0.021	**	0.014
Central AC	0.069	0.015	0.013	0.032	0.012	0.036	**	**
Green	**	**	**	**	**	**	**	**
Accessible	0.012	**	**	0.244	**	0.076	**	**
Waterfront	0.015	**	**	**	**	**	**	**
Av Price Innovations	0.020	**	0.036	0.053	0.095	0.055	0.020	0.097

** Indicates a $p$-value $< 0.01$.

provides the $p$-values for the ADF tests computed on each of the transformed predictive factor series. The transformed time series for each factor is assumed stationary at a 5% significance level with four exceptions: New Homes and Central AC for ATL and Accessible for JAX and OKC. Only one (Accessible for JAX) is not significant at the 10% level.

Neither arithmetic returns nor first (nor second) differences were sufficient to achieve stationarity for the average price time series. To obtain stationarity, we resorted8 to fitting an AR($q$)-ARCH(1)-Student’s-$t$ model

$$\begin{gathered} r_t-{\mu }_r=\sum _{i=1}^q{\phi }_i\left(r_{t-i}-{\mu }_r\right)+{ϵ}_t , \\ {ϵ}_t={\sigma }_t{z}_t , z_t~t_{\nu } , \\ {\sigma }_t^2=\omega +{\alpha }_1{ϵ}_{t-1}^2 \end{gathered}$$

(6)

to the arithmetic return series $r_t=\left(Y_t-Y_{t-1}\right)/Y_{t-1}$ of the yearly average price. In (6), $t_{\nu }$ denotes Student’s-t distribution with $\upsilon$ degrees of freedom. A fit was judged satisfactory if the innovation series $z_t$ was determined to be stationary (as verified by the ADF test). For each city, we chose the smallest value of $q$ that produced a stationary innovation series. As detailed in Table A4 in Appendix B, the value $q=1$ was sufficient for four cities, whereas $q=2$ was required for the remaining four. The $p$-values obtained for the price innovation time series are also listed in Table 1. Four are significant at the 5% level, and the remaining four are significant at the 10% level. The resulting innovation time series $z_t$ served as the response-variable time series in the GAM and GLM fits.

2.4. Principal Component Analysis for Additional Systematic Factors

As noted in the previous section, the GAMs and GLMs were applied to response variables consisting of “average-price innovation” time series and either arithmetic return or first-differenced transformed predictor variable time series. As a result of the transformations, each time series is reduced to 22 (rather than 23) years of observations (2001 through 2022). For each city, the difference between the AR($q$)-ARCH(1)-derived innovation and the regression model fit results in twenty-two residual error values (one per year). These residuals can be assembled in a matrix $R=\left\{{\epsilon }_{t,k}\right\}$, $t=1,\dots , 22$, $k=1,\dots , 8$ ($k$ is the city index9). We performed a principal component analysis by computing the eigenvalues and eigenvectors of the variance–covariance matrix $R^{T}R$ in order to determine whether systemic factors remained in the residuals (Rachev et al. 2007). The eigenvectors correspond to the principal components (ordered in descending order). We refer to extreme value theory (de Haan and Ferreira 2006) to analyze the type of decay exhibited by the explained variances10 associated with the principal components. Consider the decaying discrete exponential distribution

$$f_1\left(x\right)={\displaystyle \frac{1}{\beta }}{\left(1-\beta \right)}^{(x-1)}={\displaystyle \frac{1}{\beta \left(1-\beta \right)}} e^{x\mathrm{l}\mathrm{n}\left(1-\beta \right)}, x=0, 1, \dots , \beta \in \left(0, 1\right)$$

(7)

and the decaying power-law zeta distribution

$$f_2\left(x\right)={\displaystyle \frac{1}{\zeta \left(b\right)}}{x}^{-b}, x=1, 2, \dots , b>1,$$

(8)

where $\zeta \left(b\right)$ is the Riemann zeta function and $x$ is the index of the principal component. The relative changes with respect to $x$ in these two distributions are

$$R_1\left(x\right)={\displaystyle \frac{f_1\left(x+1\right)-f_1\left(x\right)}{f_1\left(x\right)}}=-\beta , R_2\left(x\right)={\displaystyle \frac{f_2\left(x+1\right)-f_2\left(x\right)}{f_2\left(x\right)}}={\left({\displaystyle \frac{x}{1+x}}\right)}^b-1.$$

(9)

As the magnitude of $R_1\left(x\right)$ is independent of $x$, each component of the exponential fit has the same relative drop in importance. On the other hand, the magnitude of $R_2\left(x\right)$ decreases as higher components are added for any $b>1$; thus, additional components add less value to the model. Power decay suggests that noise dominates the residuals, whereas exponential decay suggests that systemic factors continue to be unaccounted for (de Haan and Ferreira 2006). If $f\left(x\right)$ represents the observed distribution of proportion of variance, plots of $\mathrm{l}\mathrm{n}f\left(x\right)$ vs. $x$ compared with $\mathrm{l}\mathrm{n}f\left(x\right)$ vs. $\mathrm{l}\mathrm{n}\left(x\right)$ will distinguish between exponential and power-law tail behavior.

3. Results

3.1. GLM and GAM Results

GLM and GAM fits were obtained, respectively, using the R lm (linear model) function and the gam package (Hastie 2023).

Table 2. Significance ($p$-value) of the factors in the GLM and GAM fits.

Factor	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
	GLM
New Homes	0.747	0.189	0.184	0.103	0.025	0.515	0.176	0.632
Accessible	0.467	0.994	0.169	0.315	0.585	**	0.353	0.320
Central AC	0.594	0.234	0.169	0.117	0.024	0.700	0.550	0.879
Green	0.500	0.100	0.249	0.633	0.247	0.116	0.191	0.457
Waterfront	0.629	0.975	0.838	0.807	0.041	0.929	0.855	0.242
Adj. ${\mathit{R}}^2$	$-$0.167	$-$0.061	0.144	0.159	0.218	0.504	$-$0.525	0.226
	GAM
New Homes	0.747	0.151	0.100	0.017	0.091	0.152	0.017	0.555
Accessible	0.467	0.945	0.031	0.021	0.677	**	0.720	0.169
Central AC	0.594	0.240	0.063	0.027	0.085	0.015	0.032	0.997
Green	0.500	0.019	0.356	0.188	0.102	**	0.073	0.363
Waterfront	0.629	0.646	0.069	0.085	**	0.984	0.123	0.462
Adj. ${\mathit{R}}^2$	$-0.167$	0.388	0.518	0.560	0.703	0.855	0.468	0.349

** indicates $p$-value $<0.01$.

displays the $p$-values associated with the various factors for each city as fit by the GLM and GAM. Note that the $p$-values for ATL are identical under both the GAM and the GLM. For this city, the GAM P-splines simplified to linear terms and became identical to the GLM.

Because there is only a small set of factors, we evaluate the significance of each factor with a level of significance of 10%.

Table 3. Summary of marginal significances in Table 2 using a $p$-value threshold of 10%.

	Number of significant factors
Model	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
GLM	0	1	0	0	3	1	0	0
GAM	0	1	4	4	3	3	3	0
	Number of cities for which a factor is significant
Model	New Homes		Accessible	Central AC		Green	Water- front
GLM	1		1	1		1	1
GAM	4		3	5		3	3

summarizes the number of significant factors for each city as well as the number of cities for which each factor was found to be significant. In either marginal view (i.e., by city or by factor), the number of significant quantities under the GAM equaled or exceeded that under the GLM. Notable differences in the number of significant factors occurred for COL, JAX, OKC, and POR. Increases in the significant number occurred for all five factors and particularly for New Homes and Central AC.

Table 2 also presents the adjusted $R^2$ values obtained from the model fits. These values are reflective of the marginal significance numbers summarized in Table 3. Figure 2 presents a box-and-whisker summary of the spread of the adjusted $R^2$ values for each model. The non-linear GAM produced consistently better values. The fact that some values are negative, particularly for the GLM, indicates model inappropriateness. Given the small number of predictor variables, the large adjusted $R^2$ values for the GAM were unexpected and indicate a direction for future investigation.

The results illustrate the potential for using ESG return (or first-difference) factors in modeling average home price innovation time series for cities. The GAM results indicate that such relationships are nonlinear. The nonlinear nature of a GAM is able to distinguish the predictive capability of the ESG factors (particularly the influence of central air conditioning) on average home prices. The results show conformity with the geographic locations and demographic profiles of the cities. As an example, consider the Waterfront factor. Water-body percentage (the percentage of the city area that is of a body of water) is a proxy (though not always an accurate one11) for waterfront acreage. The percentage of each city’s area comprised of bodies of water is provided in

Table 4. Percentage of water area and percentage of seniors living alone, by city.

	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
Water Area a	0.7	2.0	2.6	14.5	4.2	2.3	7.9	40.9
Seniors b	3.8	4.6	7.2	7.9	8.2	13.4	9.0	4.1

^a source: US Census Gazetteer (2023) ^b source: US Census (2010).

.

Consider the scatterplot of the Waterfront GAM $p$-values versus water-body percentage shown in Figure 3. As the proxy is approximate, we look for a fuzzy relationship by dividing the plot into four quadrants. Significant occupancy in the (low, high) and (high, low) quadrants indicates a fuzzy inverse relationship between the water-body percentage and Waterfront $p$-value. Three cities (ATL, AUS, and OKC) occupy the (low, high) quadrant with three (COL, JAX, and NAS) occupying the (high, low) quadrant. Two cities (POR and SEA) occupy the (high, high) quadrant with POR lying very close to the (high, low) quadrant. SEA’s water-body percentage is a poor proxy for waterfront area since a large fraction of the water-body area (Puget Sound and Lake Washington) is distant from the shoreline.

Similarly, we consider the percentage of seniors living alone as a proxy for the Accessible factor. Table 4 also shows the 2010 census results on the percentage of seniors living alone in each of the eight cities, and Figure 3 shows the relevant scatter plot and quadrants. Again, three cities (ATL, AUS, and SEA) occupy the (low, high) quadrant with three (COL, JAX, and OKC) occupying the (high, low) quadrant. This indicates a fuzzy inverse relationship as established by six of the eight cities.

For comparison purposes, Figure 4 presents these quadrant plots for the GLM fits. Maintaining a $p$-value threshold of 0.01 leads to no meaningful inverse relationship between the Waterfront $p$-value and water-body percentage. One might argue that there is a fuzzy inverse relationship between the percentage of seniors living alone and the Accessible $p$-value in the GLM results, but this would be based upon the (high, low) quadrant occupancy of a single city (OKC).

3.2. Principal Component Analysis and Residuals Results

Table A5. Residual values (USD) for the GLM fit on each city by year.

Year	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
2001	0.068	−0.229	−0.082	0.050	−0.101	0.001	0.218	−0.025
2002	−0.793	−0.282	−0.086	0.292	−0.540	−0.207	−0.591	−0.655
2003	0.246	−0.608	0.394	0.244	−0.332	0.253	−0.100	0.773
2004	−0.191	0.057	−0.079	0.341	−0.088	−0.043	0.826	0.755
2005	−0.411	−0.034	−0.004	0.005	0.828	0.322	1.577	0.700
2006	−0.433	−0.047	−0.352	−0.271	0.112	0.206	−0.020	−0.815
2007	−0.053	0.251	−0.089	−0.198	−0.481	−0.577	−0.196	−0.780
2008	−0.887	−0.151	0.016	−0.056	−0.761	−0.089	0.259	−1.170
2009	−0.416	−0.354	−0.188	0.289	−0.243	−0.485	0.294	−0.067
2010	−0.292	−0.145	−0.239	−0.161	0.234	−0.253	−0.150	−0.036
2011	0.319	−0.039	−0.341	−0.171	0.319	−0.024	−0.699	−0.275
2012	0.348	0.452	0.192	−0.099	0.085	0.202	−0.054	0.710
2013	1.857	0.326	0.333	−0.018	0.137	−0.407	0.180	0.150
2014	−0.630	0.149	0.391	−0.119	0.246	0.675	−0.488	−1.296
2015	−0.948	0.072	−0.092	−0.300	0.616	−0.312	−0.718	0.347
2016	0.027	−0.077	−0.087	0.142	−0.003	0.036	0.214	−0.013
2017	−0.151	0.228	0.219	−0.158	0.406	0.070	−0.116	0.921
2018	−0.295	−0.158	−0.221	0.094	−0.031	0.117	−0.268	−0.511
2019	−0.387	−0.026	−0.156	−0.006	−0.498	0.273	−0.561	0.225
2020	0.340	−0.115	−0.086	−0.053	−0.164	−0.038	−0.022	0.131
2021	2.152	0.657	0.259	0.189	0.148	0.144	0.717	1.027
2022	0.530	0.073	0.298	−0.038	0.111	0.134	−0.302	−0.096

and

Table A6. Residual values (USD) for the GAM fit on each city by year.

Year	ATL	AUS	COL	JAX	NAS	OKC	POR	SEA
2001	0.068	−0.147	0.037	−0.048	−0.296	0.159	−0.006	0.184
2002	−0.793	−0.530	−0.232	0.119	−0.463	−0.369	−0.052	−0.686
2003	0.246	−0.321	0.141	0.116	−0.559	−0.220	−0.295	0.935
2004	−0.191	0.195	−0.126	0.601	−0.304	0.000	1.535	0.945
2005	−0.411	0.195	−0.186	−0.203	0.857	0.530	1.807	0.884
2006	−0.433	−0.019	−0.591	−0.293	0.045	−0.481	−0.784	−0.704
2007	−0.053	0.057	−0.454	−0.867	−0.606	−0.317	−0.926	−1.442
2008	−0.887	−0.190	−0.397	−0.025	−0.967	−0.607	−0.414	−1.151
2009	−0.416	−0.442	−0.035	0.345	−0.951	−0.430	−0.378	−0.255
2010	−0.292	−0.404	−0.195	−0.034	−0.164	−0.536	0.167	−0.122
2011	0.319	−0.103	−0.149	0.145	0.026	−0.513	−1.677	−0.200
2012	0.348	0.443	0.392	0.015	−0.324	0.654	0.363	0.707
2013	1.857	0.081	0.461	0.092	0.032	−0.606	0.299	0.463
2014	−0.630	0.172	0.516	−0.078	0.688	0.971	−0.090	−1.100
2015	−0.948	−0.063	0.094	−0.279	0.670	−0.017	0.061	0.452
2016	0.027	−0.220	−0.370	0.329	−0.141	−0.481	0.483	−0.051
2017	−0.151	−0.066	0.200	−0.200	0.331	−0.061	−0.271	0.853
2018	−0.295	−0.450	−0.053	0.009	−0.348	−0.383	−0.289	−0.469
2019	−0.387	0.130	−0.015	−0.127	−0.073	0.898	−0.006	0.475
2020	0.340	0.036	−0.113	−0.051	−0.308	0.032	−0.385	−0.006
2021	2.152	1.406	0.558	0.509	1.370	0.217	1.952	0.949
2022	0.530	0.240	0.517	−0.044	1.483	1.560	−1.092	−0.660

in Appendix C provide the residual matrices $R$ for each model.

Table 5. Proportion of explained variance by principal component (PC).

Model	PC 1	PC 2	PC 3	PC 4	PC 5	PC 6	PC 7	PC 8
GLM	0.453	0.205	0.111	0.087	0.061	0.041	0.028	0.014
GAM	0.319	0.209	0.144	0.138	0.075	0.050	0.039	0.028

displays the proportion of variance obtained for each of the identified components for each model. The fits of the exponential and power-law decays (7) and (8) to the proportions of variance data in Table 5 are presented in Figure 5 along with the $R^2$ and mean squared error (MSE) results for each.

Visually and through the quantitative $R^2$ and MSE values, it is clear that the exponential fit does a better job of describing the data. Thus, we conclude that systemic factors not included in the models exist in the residuals. This was fully anticipated, as there was no expectation that a model for average annual home prices based only on new home constructions and four ESG factors would encompass all significant factors. Moreover, a comparison of the $R^2$ and MSE for the GAM and GLM exponential fits further supports our findings from the adjusted $R^2$ numbers and $p$-values that the GAM provides a model superior to the GLM.

4. Discussion

Our results demonstrate that P-spline GAMs possess strong predictive capabilities for the expected value of the average annualized home sale price in major U.S. cities with ESG factors. The results stand in stark contrast to the GLMs. Although each factor in the GAM was significant for multiple cities, some factors (particularly central air conditioning) were especially prevalent. Overall, the results of the eight surveyed cities strongly suggest that the significance of ESG factors is very city-dependent.

As climate change continues to warm the planet, cities at more northerly latitudes which would otherwise not experience hotter temperatures will see higher rates of central air conditioning. Therefore, we expect that the significance of available central air conditioning will increase as the century progresses for both average annualized real estate prices and individual home prices. The cities of Columbus, Jacksonville, Nashville, Oklahoma City, and Portland each had significant $p$-values for the air conditioning factor. Four of these cities have annual temperature ranges in excess of forty degrees Fahrenheit and no more than two months out of the year with daily highs in excess of ninety degrees Fahrenheit. Because global temperatures have increased over the past few decades, residents in these cities will experience more days with excessive heat; thus, air conditioning units will continue to increase in demand.

Such a situation and its policy ramifications occurred in Portland and Seattle during the 2021 Western North America heat wave during late June and early July. Temperature anomalies well in excess of twenty degrees Fahrenheit above the historical average were present in these two cities. According to the U.S. Census Bureau, only 44% of homes in Seattle had air conditioning in 2019; that number increased to over 53% just two years later. Since the heat wave, the City of Seattle has enhanced its rebates and credits for air conditioning units, and these rebates and credits are particularly focused on energy-efficient and green energy implementations12. Combined with the green energy credits of the Inflation Reduction Act passed at the federal level in 2022, it is clear that governmental financial incentives for consumers to install air conditioning units and green energy sources have encouraged many homeowners to do so.

One weakness of the current data set is the length of each time series. Yearly data points over 23 years result in first-difference, return, and innovation time series of length 22, which reduces the effective sample size needed for fits by the ARFIMA-GARCH-based model, GAM, and GLM. It would be better to have access to monthly data over the same 23-year time period. Unfortunately, we had no access to such data for this study. Additionally, the four selected ESG factors were the ones readily accessible from Zillow, and the incorporation of more ESG factors would have enhanced the analysis.