**4. Results**

After different tests confirmed that using a linear functional form for the HPM model was the best option, the independent variables considered too insignificant to include in the model were excluded from it. For all of these, the probability of error if the null hypothesis is rejected using the Student's *t*-statistic is greater than 1% (*p* > 0.01). Next, 5 atypical cases were eliminated from the system in the Seville model and 12 in the Porto model because they were considered to be nonrepresentative of the entire dataset, and their inclusion would cause significant distortions in the models. The final set of variables analysed in the two datasets and their coefficients is presented in Table 5.

**Table 5.** Variables and coefficients of HPM model for Seville and Porto.



**Table 5.** *Cont.*

Note: Prob. = probability; VIF = variance inflation factor; UF = União de Freguesias (Joint Parishes).

The coefficients represent the marginal price variations (i.e., endogenous variable) produced by each exogenous variable. Thus, an HR in Porto that is located within the Union of Parishes of Aldoar, Foz do Douro and Nevogilde increases its daily price of a one-night stay by EUR 57.63 compared to another facility that does not (see Table 5 above). Concurrently, every extra minute spent walking from a HR to the Plaza del Triunfo in Seville (i.e., the city's main tourist attraction) reduces accommodations' price by EUR 1.48. In contrast, if guests walk from an HR to Praça da Liberdade in Porto (i.e., the city's main tourist attraction) the reduction in price is only EUR 0.30. Finally, each extra bed that a holiday rental offers in Seville increases its price by EUR 14.85, compared to EUR 12.22 in Porto.

Regarding the variables related to seasonality (see Table 5 above), in the Seville model, the HWD variable was the basis on which the price was estimated, so this variable was excluded from the model to avoid the problem of exact multicollinearity. In Porto, both HWD and HWE proved to be irrelevant to the model, so the same price was estimated for the high season without a distinction being made between weekend or weekday prices. Additional tests were performed to rule out multicollinearity between independent variables using the variance inflation factor (see Table 5 above). No independent variables exceeded the tolerance level (i.e., set at 10), thereby implying that no multicollinearity was present.

A comparison of the models (see Table 5 above) highlighted the main similarities and differences. Similarities include the variables referring to the accommodations' size (BEDS and M2), distance to the centre (MIN) or visual attractiveness (VSAT). Special events are also decisive for both Seville (HW, FAIR) and Porto (SJ). The models diverge regarding the vacation rentals' amenities. Pool availability (POOL) is a key feature for Seville's HRs but irrelevant for Porto's HR establishments. Conversely, courtyard or patio availability (CRT) is significant in Porto but extraneous in Seville.

Table 6 includes an assessment of the models' overall goodness of fit. The coefficient of determination (*R* 2 ) represents the total percentage of each endogenous variable's variation that is explained by the model's full set of exogenous variables. The Seville model has a significantly higher *R* 2 than the Porto one does, that is, the former model's exogenous variables explain 19.2% more of the estimated price than the Porto model does.

**Table 6.** Adjustment measurements of Seville and Porto HPM models.


The mean relative error (see Table 6 above) shows the differences in percentage between each model's predicted prices and its actual values. The Porto model has a slightly

higher goodness of fit than that of Seville since the absolute average of errors committed is approximately 2% lower. The Theil index of inequality represents a given model's predictive power, namely a greater accuracy the closer this index gets to zero. Both models have values that indicate a good ability to predict prices. Finally, the Chow test was run to check the models' stability, which produced results indicating no structural changes occurred in both models' parameters. higher goodness of fit than that of Seville since the absolute average of errors committed is approximately 2% lower. The Theil index of inequality represents a given model's predictive power, namely a greater accuracy the closer this index gets to zero. Both models have values that indicate a good ability to predict prices. Finally, the Chow test was run to check the models' stability, which produced results indicating no structural changes occurred in both models' parameters. Figure 3 presents graphs comparing the real price with the price estimated by the

**Variables Seville Porto** 

**Mean relative error** 22.97% 21.09% **Theil inequality index** 0.139 0.129

**Coefficient of determination (***R***²)** 0.732 0.54

The mean relative error (see Table 6 above) shows the differences in percentage between each model's predicted prices and its actual values. The Porto model has a slightly

Figure 3 presents graphs comparing the real price with the price estimated by the Seville and Porto models. The former model shows a significantly higher price range than that of Porto. An outlier above EUR 400 appears in the Porto model in the real price range, but that price's exclusion would mean a lower goodness of fit. The models' degree of fit, if perfect, should appear as point clouds in a diagonal line, as seen in Figure 3. Both models' estimated values thus suggest that the linear form is a good fit. Seville and Porto models. The former model shows a significantly higher price range than that of Porto. An outlier above EUR 400 appears in the Porto model in the real price range, but that price's exclusion would mean a lower goodness of fit. The models' degree of fit, if perfect, should appear as point clouds in a diagonal line, as seen in Figure 3. Both models' estimated values thus suggest that the linear form is a good fit.

*Economies* **2021**, *9*, x FOR PEER REVIEW 12 of 17

**Table 6.** Adjustment measurements of Seville and Porto HPM models.

**Figure 3.** Comparison of real vs estimated price for Seville and Porto models. **Figure 3.** Comparison of real vs estimated price for Seville and Porto models.

#### **5. Discussion 5. Discussion**

The dependent variables found to be relevant to the models are in agreement with previous studies in terms of distance to the city centre or tourist attractions of greatest interest. Comparable results have been reported by, among others, Soler-García and Gémar-Castillo (2017), Gyódi (2017) (i.e., a Booking.com model), Gibbs et al. (2018), Soler-García and Gémar-Castillo (2018) and Tong and Gunter (2020) (i.e., a Seville case study). However, Voltes-Dorta and Sánchez-Medina's (2020) research did not confirm any significant relevance, and Gyódi and Nawaro's (2021) results vary depending on the city ana-The dependent variables found to be relevant to the models are in agreement with previous studies in terms of distance to the city centre or tourist attractions of greatest interest. Comparable results have been reported by, among others, Soler-García and Gémar-Castillo (2017), Gyódi (2017) (i.e., a Booking.com model), Gibbs et al. (2018), Soler-García and Gémar-Castillo (2018) and Tong and Gunter (2020) (i.e., a Seville case study). However, Voltes-Dorta and Sánchez-Medina's (2020) research did not confirm any significant relevance, and Gyódi and Nawaro's (2021) results vary depending on the city analysed.

lysed. More specifically, the number of beds appears as an important variable in Gibbs et al. (2018), Tong and Gunter (2020), Voltes-Dorta and Sánchez-Medina (2020), Fearne (2021) and Gyódi and Nawaro's (2021) findings. The m2 of accommodations is also significant in the present study's two models, as reported by Chen and Rothschild (2010), but this variable is rarely present in other tourism accommodation pricing models. In addi-More specifically, the number of beds appears as an important variable in Gibbs et al. (2018), Tong and Gunter (2020), Voltes-Dorta and Sánchez-Medina (2020), Fearne (2021) and Gyódi and Nawaro's (2021) findings. The m<sup>2</sup> of accommodations is also significant in the present study's two models, as reported by Chen and Rothschild (2010), but this variable is rarely present in other tourism accommodation pricing models. In addition, the date on which the price is recorded is seldom mentioned in the literature. However, variables related to this factor are similarly treated as important in work done by Coenders et al. (2003) and Rigall i Torrent et al. (2011) on seasonality and Soler-García and Gémar-Castillo (2017) on special events such as Seville's April Fair.
