Prices of residential property are difficult to measure because of their heterogeneous nature, where it can be observed that dwellings are not identical even by the sole virtue of occupying different locations (Hill 2011).

Different methodologies exist to develop property price indices. The Organisation for Cooperation and Economic Development outlines several methodologies to construct residential property price indices (De Haan & Erwin 2011). Simpler methods, like the average or median mix adjustment approach, group properties into homogenous strata, calculating a central measure of tendency for each stratum. A weighted average is then applied to roll up the indices into a single index. The repeated sales approach performs regression on pooled property data for properties that have been transacted more than once in the estimation period. Hedonic price modelling is pervasive in economic literature and has been employed to model property prices where the price of the property is valued according to its set of structural and locational attributes (Shulz & Werwatz 2004). Hedonic pricing is a mathematical technique used in economics that aims to measure the price of a good through its utility-bearing attributes, where a vector of attributes determines the price of the good (Rosen 1974). The hedonic characteristics price approach runs separate regression models for each time period and calculates price inflation using index number theory (De Haan & Erwin 2011). This makes hedonic pricing a suitable approach to produce price estimates for cross sectional residential property data where properties have not been transacted frequently.

Residential property is a single class of good or commodity in the eyes of individuals, households and investors; however, it is differentiated or heterogeneous in nature (Hill 2011). A residential property is a collection of attributes that each hold certain utility and value, which can be characterised as structural, like size and the number of bedrooms, relate to how accessible the property is to amenities like schools and may include location-specific attributes, such as being in a specific geographic area or suburb. Typically, hedonic pricing techniques model property prices as a function of a set of inherent structural attributes, neighbourhood or location characteristics and accessibility to amenities (Lyons 2015). Market forces regulate heterogeneous product prices, and these prices are contingent on the individual product’s set of attributes. Hedonic methods express that residential properties can be decomposed by the constituent attributes thereof, and although no market for the individual attributes exists, supply and demand forces in the property market can determine each attribute’s marginal contribution to the property’s price implicitly (De Haan & Erwin 2011). Market forces are responsible for the different prices of residential properties, which is contingent on each individual property’s set of attributes. Generally, the market will settle on a set of prices for the various combinations of residential properties that will clear the market through the reconciliation of supply and demand (Day 2003). Rosen (1974) propounds that economic agents can ascertain hedonic prices from the observed prices of heterogeneous products, where the hedonic prices equate to the implicit prices of the attributes of the heterogeneous products.

The hedonic pricing model describes each property by a vector of Z quantifiable and inseparable attributes that determine its price:

Zj=(Zj1,Zj2,Zj3,…Zjk)[Eqn 1]

Rosen (1974) defines hedonic pricing as the functional relationship between the price of a heterogeneous product and the associated attributes:

Pj=P(Zj)=P(Zj1,Zj2,Zj3,…,Zjk)[Eqn 2]

where Pj is the price of the product. Simply stated, Pj = f(Zj), where the price of a property is a function of a set of a smaller number of attributes (Goodman 1978). Notably, an increase in price is experienced by attributes that are more positive and a decrease in price is experienced by more negative attributes, ceteris paribus (Els & Von Fintel 2010). Regression analysis makes it possible to calculate the implicit price for attribute i of property j by taking the partial derivative, represented as follows:

Pi(Zj)=∂P∂Zi(i=1 to Z)[Eqn 3]

This function describes the additional amount to be paid to obtain a marginally higher level of attribute Zi, ceteris paribus (Day 2003). Hedonic prices can be measured with the use of regression, a statistical technique that aims to establish the relationship between a set of property attributes and property prices by regressing property price on a set of property attributes.

The omission of important attributes in hedonic price analysis has the propensity to bias estimates of the implicit prices measured; however, many models are subject to data availability. Model misspecification may arise in a hedonic analysis because of data availability constraints and subjective judgements by the researcher where important variables are not included in the analysis (Jiang, Phillips & Yu 2015). An important consideration in developing a model is the principle of parsimony, where the aim is to choose a parsimonious or simpler model that explains the data well and is more generalisable. Simplicity through parsimony of parameter selection is a desired feature of any model, as complexity is reduced and predictive accuracy increased (McCullagh & Nelder 1989).

Typically, international and local residential property hedonic price studies use ordinary least squares to derive hedonic pricing functions. Given a vector of a dependent variable and a matrix of independent variables, ordinary least squares make it possible to express the dependent variable as a linear combination of the independent variables (Greene 2003).

Day (2003) modelled house prices in Glasgow using hedonic pricing and ordinary least squares where a set of structural, accessibility, neighbourhood and environmental attributes were regressed on the selling price of properties sold. The natural logarithm of sales price was regressed on a linear combination of independent variables to derive the hedonic pricing function. He applied the natural logarithm transform to the floor area variable in his study. Day (2003) found that the inclusion of spatial data was an extremely important consideration in the estimation of the hedonic price function. A widely accepted tenet is that location is a significant determinant of a property’s price (Özyurt 2014).

Bourassa, Cantoni and Hoesli (2010) derived a hedonic price function for the Auckland housing market using several ordinary least squares models, applying the natural logarithm to the dependent variable in each model. They took cognisance of the fact that property prices are closely related to adjacent properties and effectively modelled the spatial dependence thereof. Broadly speaking, spatial autocorrelation can be defined as the dependence of observations across geographic locations, which has the propensity to render the standard errors of ordinary least squares models inefficient and biased (Liao & Wang 2012). Bourassa et al. (2010) found that property price predictions were more accurate when submarket dummy locational variables were used in contrast to using traditional statistical methods alone; however, incorporating both methods yielded the best results. Notably they argue that the use of submarket dummy locational variables in ordinary least squares is a far simpler technique than trying to model the structure of the errors using complicated statistical methods, and the benefit was evident in their results. Adding a dummy spatial variable to the combination of independent variables can remove the misspecification of the model, which can be seen in the ordinary least squares regression diagnostics, making the interpretation of the results straightforward (Thayn & Simanis 2013). In order to test for the presence of spatial autocorrelation, Borcard and Legendre (2012) found that the Mantel test was a reliable method, which they used on univariate and multivariate data in an ecological study that investigated the relationship between grain and spatial autocorrelation using various statistical tests. Despite recent criticism of the Mantel test, Diniz-Filho et al. (2013) found that it was a powerful technique to analyse the amount of spatial variation in multivariate data where the results were congruent with a priori knowledge.

Els and Von Fintel (2010) conducted a pooled cross sectional hedonic analysis in the housing market of the Western Cape province, including Stellenbosch, Somerset West, Strand and Gordon’s Bay, from 2004 to 2007, where they employed ordinary least squares and quantile regression techniques. Two models were derived using ordinary least squares, the first a standard approach not including location or neighbourhood effects and the second incorporating dummy variables for the area, thereby introducing neighbourhood effects. In both ordinary least squares hedonic models, the natural logarithm of sales price was used as the dependent variable. By taking the natural logarithm of the sale price variable, all the coefficients were interpreted as percentage effects. The results showed that by capturing neighbourhood effects through the inclusion of area dummy variables, bias was reduced and the presence of spatial autocorrelation was mitigated, thus improving the overall fit of the model and increasing the R-squared diagnostic. The study of Els and Von Fintel (2010) included many structural attributes and, interestingly, the results revealed that the number of bedrooms was not a statistically significant variable; however, the size of the residence and the number of bathrooms were. Moreover, the number of bedrooms coefficient in the ordinary least squares model without locational effects had a negative sign, whilst the same coefficient in the ordinary least squares model that included locational effects through dummy variables had a positive coefficient. The presence of the sign change could have been attributed to adding the locational dummy variables. Kennedy (2005) asserts that an omitted explanatory variable in a hedonic regression model can change the sign of one or more existing coefficients already specified in the model and to consider adding an independent variable to correct the misspecification. Els and Von Fintel (2010) were concerned over the presence of heteroscedasticity in their study using ordinary least squares and therefore endeavoured to develop a non-parametric quantile regression model that is typically more robust to heteroscedasticity. Heteroscedasticity is a common problem in econometric studies and is endemic to spatial studies (Anselin 2013). Using linear transformations such as taking the natural logarithm of the dependent variable often reduces the effects of heteroscedasticity and mitigates its presence by changing the variance of the error term or residuals (Malpezzi 2003). Heteroscedasticity violates one of the fundamental assumptions of ordinary least squares, namely that there is constant variance of the residuals (Stohldreier 2012). Formally stated, the error term must be independently and identically distributed (Rawlings, Pantula & Dickey 1998). The presence of heteroscedasticity may render the ordinary least squares coefficient estimates inefficient where standard errors and p-values may be biased or incorrect, making hypothesis testing or deriving confidence intervals problematic. However, heteroscedasticity does not affect the consistency nor impair the unbiasedness of the actual ordinary least squares coefficient estimates (Gujarati 2004).

Dodds (2011) conducted an analysis of residential properties that were sold in the Westrand area in the Gauteng province, where he aimed to predict property prices using an ordinary least squares hedonic pricing model based on statistically significant structural variables and location. Whilst the choice of structural attributes was contingent on the data, there was a total of 11 structural variables and 1 dummy location variable. An important observation made by Dodds (2011) was that the number of bedrooms and number of bathrooms had the highest positive correlations with the dependent variable, sale price. However, the output of the hedonic model revealed a negative coefficient for the number of bedrooms. This may have been because of an important omitted variable or multicollinearity as the model specified 12 independent variables in total. Multicollinearity is commonly experienced in ordinary least squares regression models which is caused by highly correlated independent variables. The variance inflation factor is a useful method for detecting the magnitude of multicollinearity (Chen & Rothschild 2010). Hedonic models are often subject to the presence of multicollinearity which can result in measurement errors and negative coefficients (Triplett 2005). Dodds (2011) found that heteroscedasticity was present in the linear hedonic model and applying a natural logarithm to the dependent variable made the error term less heteroscedastic.

The gamma distribution provides a possible alternative to the commonly used log-linear approach to derive hedonic price functions for residential properties. A generalised linear model based on the exponential, gamma distribution can be used to model a positive continuous dependent variable where the conditional variance of the dependent variable increases with the mean and the coefficient of variation is constant (McCullagh & Nelder 1989). Fu and Moncher (2004) propound that the log-normal and gamma distributions are both widely used to model non-negative data that is positively skewed. Bromideh and Valizadeh (2013) assert that similarities exist between log-normal and gamma exponential distributions in terms of fit on moderate data sizes and both can prove effective in analysing non-negative positively skewed data. In a study of household expenditure, Battese and Bonyhady (1981) found that that the gamma distribution proved more effective than the log-normal distribution in dealing with the heteroscedasticity. Moran, Solomon, Peisach and Martin (2007) in a study of patients’ intensive care units costs, found that cost models employing log-linear models were improved upon by the use of correctly specified generalised linear models, which more effectively modelled the error structure. Fu and Moncher (2004) conducted an analysis on insurance claims in an actuarial study where a generalised linear model was fit to the data. They found that the gamma distribution resulted in better predictive accuracy and efficiency than the log-normal distribution. Furthermore, they suggest that examining the residual plots is a good measure to gauge the distribution assumptions. The error structure of a model is an important consideration in modelling data. Examining the error structure through diagnostic plots provides guidance of how well a model fits the data (Murphy, Brockman & Lee 2000). An appealing feature of the use of generalised linear models is that estimates are kept in the natural units of measurement, producing estimations that are more attractive than transformed estimations through log-linear models (Jones 2010).

A flexible and general approach to statistical inference is bootstrapping where the sample is treated as the population and repeated samples are drawn from it. Bootstrapping builds a sampling distribution of a statistic by re-sampling from the data and is considered a general approach to statistical inference (Fox 2002). The flexibility is derived where asymptotic results cannot be relied upon or the assumptions made about the population are incorrect. Specifically, the non-parametric bootstrap facilitates a practical estimate of the sampling distribution of a statistic without knowing or deriving the explicit sampling distribution (Fox & Weisberg 2011). The bootstrap can be used as a general tool for assessing statistical accuracy (Hastie, Tibshirani, Friedman & Franklin 2005). Gandy and Kvaloy (2013) applied non-parametric bootstrapping to circumvent estimation errors of parameters in control charts where it was found that non-parametric bootstrapping was robust against model specification errors. Wilcox (2008) used bootstrapping as a strategy to determine the correctness of hypothesis tests of coefficients in a multiple regression analysis that was found to be highly effective. Bootstrapping is used to validate the generalised linear model developed in this study.

The main objective of this study was to develop a hedonic pricing model for apartments located within three metropolitan KwaZulu-Natal coastal regions. The sub-objectives of this study were as follows:

The location of the study involved three metropolitan coastal regions within KwaZulu-Natal, South Africa, namely Ballito, Umhlanga, and Durban Central which are subsets of the eThekweni Municipality. Within these regions, thirty-six suburbs were present in the data. Private Property (Pty) Ltd provided the data for the research. The data was a snapshot of all the apartment listings for sale in Ballito, Durban Central and Umhlanga as at 23 February 2016. Rows with missing values were removed and rows with incorrect geographic coordinates were identified and removed by plotting the data on a map. Duplicate properties were removed based on having the same residential street address to avoid biasing the results. This resulted in a data set of 1314 observations for the study. The variables used this study are presented in Table 1 with a brief description and their respective use.

The presence of spatial autocorrelation in the residuals of a statistical model has the propensity to increase Type I errors for parameter estimates, falsely rejecting the null hypothesis of no effect (Dormann et al. 2007). Constructing a spatial autocorrelation function can be achieved with a Mantel test, which produces a standardised Mantel statistic similar to the Pearson correlation coefficient (Borcard & Legendre 2012). The Mantel test is formulated as follows:

Zm=Σi=1nΣi=1n gij×dij[Eqn 6]

where g_ij and d_ij are the respective variable and geographic distances between the distributions i and j, and where Z_m is the sum of products of distances, which is compared to a null distribution (Diniz-Filho et al. 2013). This technique was used to formally test for the presence of spatial autocorrelation.

Multicollinearity has the potential to produce parameter estimates of the incorrect sign and magnitude by increasing parameter variance (O’brien 2007). The variance inflation factor is a suitable measure for detecting the effects of multicollinearity, where a value greater than 10 is indicative of multicollinearity (Kennedy 1985). By testing for the presence of multicollinearity, correct model specification and results can be obtained, which was a primary initiative of all the modelling done in this study. The variance inflation factor is calculated as follows:

VIF=1r(x1)2[Eqn 7]

Whereas r(x1)2 tends towards 1, the variance inflation factor (VIF) approaches infinity. This means that the variance of an estimator increases as the extent of collinearity increases, and a score of 1 indicates no multicollinearity between X2 and X3 (Gujarati 2004).

Bootstrapping accounts for variance in the parameters estimated by drawing many repeated samples (Greene 2003). This technique was used as a general tool for assessing statistical accuracy in this study. The notation for bootstrapping is as follows:

T¯*=E*^(T*)=∑b=1RTb*R[Eqn 8]

where T¯* is the estimator or averaged bootstrapped estimate derived by ∑b=1RTb*R, where R is the number of bootstraps applied (Fox & Weisberg 2011).

The gamma distribution is suitable for non-negative continuous data. Figure 1 illustrates the kernel density estimator and cumulative density function.

The moment-generating function facilitates the accurate identification of a distribution and computes the respective moments (Kerns 2010). To ascertain whether the distribution of the listing prices is indeed gamma, the shape and scale parameters presented in Table 2 are determined using the matching moments method.

To test that the apartment prices were gamma distributed, the Villaseñor and González-Estrada (2015) test was applied using the shape and rate parameters obtained from the matching moments function. The p-value was 0.3857, indicating that there was sufficient evidence not to reject the null hypothesis that the apartment prices follow a gamma distribution.

The VIF presented in Table 3 was computed to test for the presence of multicollinearity in the independent variables which will adversely affect the model results. Evident from the VIF results is that there was no multicollinearity between the set of independent variables used in this study with the highest score being approximately 3.59.

The results of the generalised linear model based on the gamma distribution and log-link function are presented in Table 4. The results show that all the coefficients are statistically significant including all the levels of the dummy locational variable (suburbs). Moreover, the residual deviance is less than null deviance indicating that the saturated model is a better fit than the null model.

Interpretation of the coefficients of the generalised linear model:

The signs for all the coefficients are as expected based on a priori expectations, unlike what was experienced in the study by Dodds (2011). The number of bedrooms is statistically significant, which was not the case in the study of Els and Von Fintel (2010). The generalised linear model results show that there are 35 suburb coefficients, yet there is a total of 36 in the data. This is because one of the suburbs was withheld from the model output that all the other suburbs were compared to. The suburb that was withheld is Albert Park, in Durban Central, a comparatively low-priced suburb. The suburb coefficients presented in the results of the model will always be in comparison with the Albert Park suburb.

The first plot in Figure 2 is the jacknife standardised deviance residuals against the fitted values, indicating homogeneous variance and no curvilinear pattern of the standardised deviance residuals. The second plot is of the standardised deviance residuals, indicating normality thereof. The bottom two plots relate to observations with high influence. Upon inspection of these data points, no clear outliers are evident, indicating that these points may have high degrees of leverage or influence on the model.

Bourassa et al. (2010) found that property price predictions were more accurate when submarket dummy locational variables were used and the presence of spatial autocorrelation was mitigated. In order to formally test for the presence of spatial autocorrelation the Mantel test was computed. Distances between the apartments were computed in kilometres and saved as a distance matrix. The distance matrix was then passed to the Mantel test function along with a matrix of the residuals from the generalised linear model. The Mantel test then tested the null hypothesis of no relationship between the residuals and the distance matrix, providing a test of spatial autocorrelation. The correlation between the distance matrix and the residual matrix was −0.02411 and the p-value was 0.999, providing sufficient evidence not to reject the null hypothesis of no spatial autocorrelation. The use of the Mantel test to detect the presence of spatial autocorrelation in this study was consistent with the assertion of Borcard and Legendre (2012) and Diniz-Filho et al. (2013).

Bootstrapping was used as a means of model validation where 5000 random samples were drawn with replacement. A model was then developed for each sample and the average of the 5000 models computed was performed. The results indicate that bootstrapped coefficients are similar to the generalised linear model coefficient which is evident by comparing the column headed ‘original’ with the column headed ‘bootMed’ in Table 5. Simulating many random sampling distributions provides a measure of reliability for the generalised linear model parameter estimates where the results are consistent and represent a valid reflection of the data. These results coincide with the views of Carruthers et al. (2008) and Hastie et al. (2005) where the bootstrap methodology can be used as a general tool for assessing statistical accuracy.

A software application was built as the final objective of this study to render the results of the generalised linear model through a user interface, named the listing price calculator. The input parameters on the listing price calculator are dynamic so that the user can select the size of the apartment, the number of bedrooms, the number of bathrooms and the suburb. Each of these input parameters allows for values to be selected that are within the range of the independent variables used to build the model. The software application then calls the generalised linear model and calculates the average listing price and the 95% confidence interval. Figure 3 details the average price for a flat in Morningside in the Durban Central region that is 115 square metres in size (floor area) with two bedrooms and two bathrooms as R1 385 000. The lower and upper limits of the 95% confidence interval are R1 293 000 and R1 484 000, respectively, which means that we can be 95% confident that this range includes the true average listing price.

Location is indeed an important determinant of listing prices of residential apartments. For example, using the listing price calculator one can estimate the listing price of an apartment in Umhlanga Rocks, within the Umhlanga submarket, that is 115 square metres in size (floor area) with two bedrooms and two bathrooms as R3 744 000; this is 170% more than an apartment with similar attributes located in Morningside (R1 385 000). This listing price calculator can add value to households wishing to sell their apartments, where they can obtain an understanding of market pricing dynamics and obtain listing price estimates. Furthermore, credit providers such as banks can use this tool to assess how an apartment is priced relative to the market to determine the fair market value of the asset. Real estate agencies could use this software application as a tool to value new apartments and determine listing prices that are congruent to the general market consensus.