Valuation of ticket prices for first-division football matches in the Spanish league

This paper measures the optimal prices of football tickets and investigates the pricing strategy of the first-division teams in the Spanish league during the 2018/2019 season. The paper develops a dual hybrid model of supply and demand based on a hedonic price approach. Fans have multiple motivations to attend the stadium, such as the quality of the opposing teams, the pre-match qualifying position, the schedule, the day of the match, the stadium facilities, and the atmosphere. Their final decision will be conditioned by the price set by the clubs. The data show a difference of almost 300% in ticket prices among clubs. The estimation results from a hedonic price equation reveal that an optimal pricing strategy is followed by only five out of 20 clubs in the league. We also quantify the percentage of overvaluation or underval-uation of ticket prices.


| INTRODUCTION
Although we firmly believe that the future of professional sports management lies in the implementation of new management systems based fundamentally on technology and science, we cannot reject the traditional approach since it is necessary and complementary to the strategy of modern comprehensive management. We are convinced that the big data system has become one of the new strategic monitoring alternatives and is essential to modernize the sports industry; however, it is still vital to monitor the matches on site in the stadiums since the fusion of the two strategies is the key to the sustained growth of content production. The rise of technology is facilitating the continuous evolution of the sport of football, allowing us to enjoy football matches without having to attend stadiums live and with an excellent level of monitoring and detail. The technological progress, undertaken by companies specializing in football management, has been especially significant and has facilitated alliances between these companies and clubs, with investments in new audio-visual equipment, the development of new computer systems, monitoring applications, and the development of modern telecommunications and electronic devices. Naturally, this technological investment has translated into higher prices of football matches, either for attendance at stadiums or for viewing on television platforms. It is not possible to compare the sensations and emotions engendered by watching a live match with those produced on a screen, despite the fact that match broadcasts today offer us an excellent level of monitoring thanks to numerous television cameras, surround sound, and multiple replays, with a perception of detail that sometimes does not correspond to that of live viewing.
From this point of view, sports economists have contemplated the best option for the fans and the clubs, naturally considering the budget restriction of the cost of the match induced either by paying for television rights or by purchasing tickets. On the one hand, we find the modern approach of television networks that base their strategy on the use of technology as a means of attracting fans. On the other hand, we see the traditional approach of clubs that need the live assistance of fans, with the double purpose of the fans cheering on their teams in addition to buying tickets and thus providing an important source of income. The next question concerns the price impact of the two approaches. Authors such as Wang et al. (2018) have argued in their studies about the strategies of the Belgian television networks, the holders of the retransmission rights of the Belgian Pro League, and the different strategies to persuade fans to watch the matches on television to the detriment of attendance at the stadium. Obviously, attending a match live and watching it on television are completely heterogeneous products with different characteristics, so a comparative study of the prices of the two products would be an option for analysis but is not the subject of this study.
Considering this idea from all angles, it would be worth analysing the pricing strategy of football clubs as a favourable situation in which to achieve efficiency in professional sports management. Within this analysis, we will find a market or league competition in which 20 participating teams with different economic and structural characteristics will compete for the same product, victory in a football match. "Different flavours, same price" is how McMillan (2002) described this market model, which compares the prices of carbonated drinks, such as Coca Cola, Fanta, and Seven-Up, but with the particularity on this occasion of finding different teams but with similar prices. The local teams set similar prices for all the games of the season so that their fans experience little variation, regardless of the visiting team, within a pricing strategy. Nevertheless, within the same league market, we observe a 300% difference in the price of tickets between the cheapest and the most expensive within the framework of the first division of the Spanish league during the 2018/2019 season.
Focusing on our objective, we observe that, in the league competition, the local team is in charge of organizing the match between the two teams, so it is also in charge of setting the price of its tickets to maximize its income. Following an efficiency criterion, it is understood that ticket prices will be set as close as possible to a unit rate income elasticity of demand, with which the price and attendance ratio is the most optimal. García et al. (2020) analysed the product "football match" in the Spanish League, using the calculation of the elasticity of demand; they concluded that football is considered to be a luxury sport. Therefore, in this market, we find the same product (football match) for all the teams but each team sets the price of its tickets based on its structural costs. These costs will be conditioned by variables related to the quality of the match to be played and explicit variables linked to the evaluation of the aspects of the environment, atmosphere, accessibility, and equipment of the stadiums. If we take the example of FC Barcelona, the fixing of its ticket prices will be conditioned on the salary costs of its squad, so its tickets will be more expensive than those of other teams with lower-rated players. In the same way, the maintenance of its stadium, with almost 99,000 spectators, is not comparable with another with a capacity 10 times smaller, as is the case of Huesca. For our similar study, we incorporated the idea of Corts (1998), who argued that products with different production prices cause different prices to exist within the same market. This analogy occurs in the league football competition, in which, within the same market or competition, each team sets its price according to its cost structure, showing great disparity in the price of tickets between teams.
In the aforementioned study by McMillan (2002), he advocated setting uniform prices for the different carbonated drinks (Coca-Cola, Fanta, and Seven-up) as the cost of production is different for each drink. This approach, transferred to our study, makes a team like Huesca set the same average price for the entire season, regardless of the rival teams' value and classification position. McMillan defended his idea through the development of a hybrid model, subject to supply and demand. On the supply side, it tries to maximize the utility of a good, in the same way that football teams aim to maximize their revenue from ticket sales. However, we also want to incorporate the demand into this model, calculating its curve from the residual price, that is, the price at which the demand to purchase the good is satisfied. It considers consumers, in our case football fans, as accepting the price through learning or habit, as mentioned in the study by Rhee and Bell (2002), in which prices do not influence consumption when they are inelastic, as stated by Hoch et al. (1994).
To obtain the most up-to-date and homogeneous data, the period of this study was the last season before the COVID-19 pandemic, the 2018/2019 season, during which there were no attendance restrictions at the stadiums in Spain before they would remain empty of the public from March 2020. This paper quantitatively measures the pricing strategy on the supply side, that is, football clubs, considering the behaviour of the demand, which depends on fans' level of satisfaction with the ticket price. We propose a dual and hybrid model, in which the supply and demand participates, within the framework of study in the search for efficient sites in the football industry. Considering that the sale of tickets is a line of business that probably has room for improvement in terms of efficiency, we develop a hedonic price model for football tickets, which divides their value between the intrinsic and the extrinsic characteristics of a football match, according to the valuation of the fans. We can obtain the target or predicted ticket price from the residual values by performing a multiple linear regression of the calculation variables mentioned above. The most important contribution of this study is to determine the overvaluation or undervaluation in pricing as well as identifying the clubs that apply the optimal strategy. Previous studies, such as the one by Welki and Zlatoper (1994), have explained the influence of the overvaluation of ticket prices in the NFL, considering it to be harmful in terms of the attendance ratio at the stadium due to an inelastic demand, which is only compensated for when local fans see to win to their team the match.
Our study has important implications for decision making in the financial management of clubs, especially regarding pricing and its acceptance by fans. Specifically, our analysis can be used to calculate the optimal ticket price, its current deviations through residuals, the ticketing revenue, and the impact if the price was to be adjusted, both per match and during the season.
Starting the analysis of the study variables within the model, it is important to point out that the clubs are responsible for ticket pricing and that, in addition to determining the price according to their structural costs, they should consider the variables that the fans value most when buying a ticket. From here, the concept of a dual hybrid model arises. We consider it important to split all the variables studied into two large groups. We call the first group intrinsic variables, those related to the quality of the matches, with which we will analyse the quality of the match for the fans, taking on the value of the squads.
The better the players participating in a match are, the more value fans will give it and therefore the ticket price that they are willing to pay will be higher. Naturally, the team with the best players will have a higher salary cost, which will determine higher pricing. Therefore, it will be necessary to analyse the squad's value, the fit moment, and the table position related to the values of the local and the visiting team. The second group of variables is defined as extrinsic and consists of those related to the environment and the surroundings of the match and the time and day of the week variables, which are essential for the decision making of a fan when buying a ticket.
The data incorporated into our study on the squad's value were obtained from the www.TransferMarket.com website. This website is one of the most popular and reputable data sources of football statistics. The squad's value, which has been incorporated into this study, comes from the sum of the individual valuations of each player in the possible event of a transfer. Many authors, such as Müller et al. (2017), have considered this database to be very reliable in terms of research literature when calculating these valuations through "selective decisions" on various indicators, such as salary, match performance, and the possible transfer or interest of other teams. Other authors have been interested in the value given by Transfermarket because this value and its weighting are influenced by comments from specialized magazines and newspapers (e.g., Bryson et al., 2012). In addition, it incorporates into its evaluation indices the correlation between the variables of the salary of the players and the weighting of the performance estimated by football experts, as in the studies by Franck and Nüesch (2011) and Torgler and Schmidt (2007). According to Müller et al. (2017), www.transfermarket.com is one of the best websites for estimating market value, offering great utility.
Within the set of motivations to attend a match, the previous classification of a match between the two opposing teams is recurrent. A fan will prefer to watch teams that are in a high-ranking position than teams that are in a lower-ranking position. Previous studies, such as the one by Bitran and Wadhwa (1996), have incorporated the analysis according to the qualifying position of the opposing teams as a weighted variable for pricing in a football match. García et al. (2020), in their study of ticket prices, adopted the value of the local team, the uncertainty of the result, and the day of the week of the match as the main motivations for attending a match. They grouped all the teams in the Spanish first division into four groups of teams based on the historical average in the league rankings. The existence of a correlation between the series studied and the historical one was concluded. In our study, to improve the motivational variables of the fans, we included other variables, such as the influence of the time and day of the match, following Cisyk and Courty (2021).
These authors, in their studies, developed a pricing model through the so-called hedonic pricing approach. This method consists of determining the price of a product from an econometric model, evaluating its characteristics individually. The modern hedonic approach, in which the price of an item can be defined using a vector of attributes, is based on the early papers by Lancaster (1966) and Rosen (1974). This argument has been widely applied in different economic fields to various product lists in many research articles, such as the article by Griliches (1961) on market pricing in the automobile industry. Hence, Stewart and Jones (1998), studying professional sports, evaluated baseball players in MLB before a possible transfer according to performance statistics. Wilhelmsson (2002) followed the same approach in the real estate market, and Gustafson et al. (2016) priced the wine market based on vintage attributes. In addition, the hedonic approach to pricing has been used in articles on the economics of sport. This article adopts an alternative approach to the hedonic price method. Its main contribution is that it can quantitatively measure the variables in fans' decision making, identifying those with the most influence when make purchasing decisions and comparing them with the club's pricing strategy in a hybrid dual model of both demand and supply.

| TICKET PRICE MEASUREMENT: THE HEDONIC PRICE APPROACH
In our analysis of football ticket prices, based on a hedonic price methodology, we perform a decomposition of the set of attributes that can be associated with the product called "football match." Kemper and Breuer (2016), following the hedonic pricing method, per- Baseball, Drayer and Shapiro (2009) and Mondello and Rishe (2005) studied ticket prices in the NFL playoffs. The former conducted a study of tickets in the Colorado Rockies team with a methodology called variable ticket pricing (VTP), taking the opposing team and the date of the match as study variables. In addition, they developed computer software for pricing using the DTP system, as reflected in the studies of the San Francisco Giants by Dunne (2012) and Moore (2010). Authors such as Shapiro and Drayer (2014) wanted to expand this subject through a similar study within a framework of other professional sports, such as Major League Baseball. Splitting attributes or characteristics for calculating the ticket prices for sporting events is a very common methodology in pricing strategy. Bitran and Mondschein (1993), Gönsch et al. (2009), and Klein and Steinhardt (2008) used the dynamic pricing model (DPM) to create a multiple pricing model based on a multitude of variables, such as match day, seat location, opponent evaluation, and qualifying position.
The hedonic method uses a multiple regression model to calculate the objective or forecast price, and this price is calculated using a set of variables that make up the characteristics of a football match in terms of its quality and environment. The theoretical foundations of this approach come from Lancaster (1966), who developed the consumer theory of the differentiated product. The key to Lancaster's approach is that the utility function of consumers is defined according to attributes. Rosen (1974) also applied the hypothesis-based hedonic approach to the calculation of attributes' utility value. A product can be defined as a group of attributes that are valued positively by consumers. The price of a good is calculated as the sum of the implicit economic value of each attribute.
The most important contribution of this study is the creation of a dual model for supply and demand, for that reason, we consider it to be a hybrid model. On the supply side, the main objective is to maximize the utility of the good, in this case football tickets. On the demand side, the aim is to maximize the level of satisfaction regarding the purchase of tickets and find the break-even point at which the price paid is similar to the pricing by clubs. At this point, using a statistic based on building a multivariable linear regression model allows us to calculate the objective ticket price for each match, that is, the price at which the supply and demand find their balance. The calculation of this objective or forecast price for each match is made from the residual records derived from the regression model. Chan (2006) and Kim et al. (2002) carried out studies based on hybrid supply and demand models, in which they used the logarithm as a tool to reduce the dimensionless variables.
The objective of the hedonic approach allows us to analyse the factors that make up a football match individually and to determine the fans' level of satisfaction. From the extraction of these values, we can create a time series of cross-sectional data and calculate the objective or forecast ticket price for each match. This price may be compared with the real price set by clubs, the difference being the residual records. Previous authors, such as Griliches (1971), based on the calculation of hedonic price regressions, have shown in a reduced way how the representation of consumer and producer behaviour is optimally and efficiently developed. The hedonic price approach allows us to learn how the price of a product changes when there are variations in its attributes. The price of a good can be split into a first component, which reflects the real price change, and a second component, which reflects the variation in terms of a change in its characteristics. The hedonic approach postulates that each good is defined by the set of its attributes. For any good, we can explain this by means of a vector of attributes, X, as X = (X 1 , X 2 … X n ), where X k (k = 1, 2 … k) defines each of the characteristics of the product. The key is to establish that, for any product, there is a functional relationship between the price, P, and its vector of attributes, X, such as Implicit prices can be defined from the function described below, which determines how much the price of a product changes depending on the variation of its attributes. Hence, the regression model can be represented by where P k is the vector of prices of variety k, X k is the vector of characteristics of each variety, β k is a vector of coefficients, and ε k is an error term. In empirical analysis, the price of a good is predicted from its attributes and dummy variables. From the regression model, we can calculate the importance of each attribute, called the implicit price, and the estimate of the quality-adjusted price, which we call the real price. If we look at these prices from a statistical point of view, the first conclusion is that the current price quantification changes a product with respect to its quality because prices must be forecast for quality to be compared through variables. The main objective of measuring the satisfaction of football fans who attend stadiums is to determine the difference between the real paid price and the expected price. Therefore, the smaller the difference is between the two prices, the higher the level of satisfaction fans will have and the higher revenues the clubs will receive.
We can obtain a double measurement from Equation (2). One is the level of satisfaction of the fans, and the other is the optimization of the revenues from clubs' ticket sales. Both measures are defined as the change in the quality of the characteristics that has occurred with respect to ticket prices. In other words, the level of satisfaction with both can be represented as the difference between the change in pricing not forecasted by the variable match quality and the change in clubs' pricing. This methodological approach can be applied using multiple regressions, following, for example, Berndt (1996) The dummy variables are coded with 0 and 1 to include them in the regression, which shows the percentage trend of consumer behaviour from the coefficient when the ticket price changes. We use this method in our study because we seek to measure the true price change when some of the variables influenced by fans change. Therefore, the logarithm simplifies as follows: Many authors have used the hedonic price approach to investigate pricing behaviour in a variety of markets. Vaugh (1928) studied the factors that explain the prices of tomatoes, cucumbers, and asparagus, presenting the first hedonic price analysis. Court (1939) then used this method to study prices in the automobile industry, developing the methodology later used by Griliches (1971). The contribution of Rosen (1974) was decisive in defining "hedonic prices as the implicit price of a good, which is made up of a series of attributes where economic agents associate an economic value to each variable or attribute." Rosen developed two stages in the study of hedonic prices.
The first involves a regression of the prices on the characteristics of the goods. The coefficients of this regression are often interpreted as implicit prices or as the consumer's marginal willingness to pay for each characteristic. Rosen's second stage includes the regression of the marginal prices of each characteristic of a good in relation to the demographic variables of the good and the consumer. This second stage aims to recover a demand function for each characteristic. However, it was later found (Bartik, 1987;Brown & Rosen, 1982;Epple, 1987) that the second-stage regression suffered from a simultaneity problem because consumers with a strong preference for a given feature naturally purchased large amounts of that feature. This simultaneity problem caused inconsistent estimates in this second stage. Epple (1987) suggested that this problem can be solved if the data obtained allow research to conclude that the tastes of the consumers are the same. However, data of this type have proven difficult to find; therefore, Rosen's second stage is not as widely used today.
Because we found many criticisms of the second stage of Rosen's hedonic price evolution in previous works, we preferred to develop only the first stage for the article and not to include the second stage.
There was a high possibility of finding inconsistent data due to the considerable heterogeneity between the different fans of the La Liga teams and the inconsistency of being able to consume high amounts of the characteristics analysed in our model because they are intangible variables. This led us to reject their incorporation.
Similarly, Triplett (1973) considered the final price of a good as the individual sum of the value of each attribute that makes up said good. Hedonic prices have been the subject of many previous research articles in various economic fields, such as the aforementioned study by Griliches (1961) on market prices in the automobile sector. Stewart and Jones (1998), in the professional sports sector, assessed the valuation of the players before a possible transfer according to the statistics of Major League Baseball players. Wilhelmsson (2002) followed the same approach in the real estate market, and Gustafson et al. (2016) and Nerlove (1995) set prices in the wine market according to crop attributes.
Hence, the importance of this study is due to its analysis of the fans' preferences when consuming this intangible product, football, and the deviation between the prices set by the clubs and the prices desired by the fans. This aspect was included as a line of research by Drayer et al. (2012) and Kemper and Breuer (2016), who analysed the satisfaction of fans with the price of football tickets, considering that there are variable multiples depending on the day of purchase of the tickets: the same day as the game, the day before, or 2 weeks before.
Another variable included in our analysis is the importance to the fans of the opposing team's qualifying position prior to the match. A fan will prefer to watch teams that are ranked higher than teams that are ranked lower. In their study, Bitran and Wadhwa (1996) incorporated the analysis according to the qualifying position as a weighted variable for pricing tickets in a football match. García et al. (2020), in their study of football ticket prices, according to the value of the players in the squad, classified the 20 teams of the first division of the Spanish league, distinguishing them into four large groups based on the correlation between historical rankings and ticket pricing when acting as away teams. In this way, it was concluded that the tickets were more expensive the higher the historical qualifying coefficient a visiting team had.
The main contribution of this study is the inclusion of the different and most representative variables in the previous literature that best represent the characteristics of the product (X i ) with a two-dimensional model approach, oriented towards supply and demand.
To this end, regarding the product, we based the analysis on two groups of variables: some implicit in football matches, related to the quality of the match; and other explicit ones regarding the setting and atmosphere of the match. From previous studies, we adopted the six main recurring variables in the comparative analysis to set the objectives or forecast prices.

| DATA
The data originate from various statistical sources in relation to player ratings, ticket prices, ratios of public attendance at matches, schedules, match dates, team classifications, and stadium ratings, if we appreciate that there is a correlation between the salary cost of the players and the ticket price, we can affirm that the ticket pricing policy is conditioned on the level of the players of each team. If a club wants to have better-quality players on its team, it must apply higher ticket prices and fans must assume higher costs.
On the demand side, the objective was to compare the different attributes that make up a good related to the predicted ticket price. In our case, the dependent variable is the average price of the tickets for each match. We took the lowest and the highest price in each club, and the dependent variable was the average between them. In this way, we incorporated it into the model though its natural logarithm.
As independent variables that indicate the purchase of a ticket, we considered the most sensitive for a football fan to be those referring to the quality of the opposing team and the factors around a match.
We refer to the quality variables as intrinsic variables and those related to the stadium and its environment as extrinsic variables. covered stands, and entertainment shows affect the variation in price.
We incorporated the classification with which UEFA has already rated the stadiums into our study. All the stadiums in the Spanish first division are rated with a three-or four-star typology. We incorporated this into the model through a dummy variable, assigning a value of 1 to those with four stars and a value of 0 to the rest. Continuing with variables sensitive to price, without a doubt, fans consider match schedules to have a great influence on match attendance. During the season, the disparity of schedules due to the sale of television rights establishes that they can be played every day of the week and in extended hours in the morning, in the afternoon, and at night. In the same way, both variables were treated as dummy variables. Regarding the day of the week, a value of 1 was assigned to the matches played on Saturdays and Sundays, as they are considered more traditional days, and a value of 0 was assigned to the rest of the days of the week. Similarly, with respect to the schedule, a value of 1 was assigned to the matches played in the afternoon and evening, with a value of 0 being given to the matches played in the morning and at midday. Finally, the attendance ratio variable also influences the motivation of certain fans who are hesitant to attend the stadium. If a team tends to have a high rate of attendance in a regular season, it directly influences the purchasing ticket. If the attendance drops, clubs must set prices according to the rate of attendance to avoid having empty seats. In this way, we used the data referring to attendance at each match from the Besoccer website and integrated the value into the model through the percentage attendance rate, calculated according to the maximum capacity of the stadium and the number of spectators.
In our hedonic price approach, we had already defined the six variables considered to be sensitive for the fans, which will cause variations in ticket prices, and they were incorporated into the model though the regression system to calculate the forecast price for each match in the 2018/2019 season. variable for our analysis since it reflects the variation in the forecast price according to the quality of the matches and their environment when any of the variables to which fans are sensitive vary. Berndt (1996) presented a description of the variations in variables due to a change in the quality of a good in addition to constructing a hedonic price index using a multiple regression method. As main results, we can highlight the regression coefficient of 0.75, considered significant as it is within the range (0.70-1), with an R 2 of 0.578. In the analysis of variance, as the most significant variables, we underline direct confrontation and the type of stadium as they have a probability lower than 0.05. The coefficients that weigh the most in the model are the type of stadium and the classification of the teams prior to the match ( Table 2).
The most important result gained from the analysis was the measurement of the predicted ticket price using the hedonic method, so the implicit prices were obtained from the different characteristics that make up a football match. The estimated coefficients of the characteristics reflect the variation in the price of a certain percentage when there is a variation in the value of the characteristics. Hence, we analyse our coefficients, expressed in logarithmic form, which represent the increase in face value adjusted for price quality. The estimated value for the qualifying position variable is 0.358, which implies that the growth of 35.8% in the forecast price is due to a variation in the qualifying position of the opposing teams, keeping the rest of the characteristics constant. We can say that, for fans, the position of the teams before the game is decisive for the ticket price. If we focus on the regression results presented in Table 2, we find that all the estimated parameters are statistically significant at conventional levels except for the schedule and match day.
The estimation of our model also included three dummy variables: schedule, match day, and UEFA stadium rating. We consider that the estimated coefficients for the schedules and match days are not significant because they present negative indicators. This means that a variation in any of these characteristics does not significantly modify the ticket prices. The stadium rating shows a high level of importance, with a coefficient of 0.566. In other words, a variation of 56.6% in the forecast price is due to the type of stadium, the rest of the characteristics remaining constant. These data reveal that the price of the tickets is correlated with the size of the club; therefore, the clubs that have larger budgets and the best sporting results over a long history have modern stadiums.
In general, the model explains, with a confidence level close to  Table 3. In our analysis, we regrouped all the residual records by the team that plays at home, so we had 19 residual records equivalent to the 19 matches as the home team. Each residual record allowed us to determine the difference between the objective forecast price and the average fixed price, enabling us to state whether the prices set are optimal or whether they are overvalued or undervalued. A positive value indicates that the ticket price of a match is higher than the predicted price obtained from the estimated hedonic equation, and the indicators of deviation can be classified as "expensive" or "overvalued". Similarly, if a value is negative, we can consider it to be "cheap" or "undervalued" compared with the average price set by the clubs. However, in earlier literature on professional sports pricing, Fort (2004)

| CONCLUSIONS
The application of microeconomics in the field of sports economics This study is based on previous studies following the hedonic price approach, such as those by Lancaster (1966) and Rosen (1974), which analysed the demand in terms of preferences according to established prices. New models and methodologies related to the calculation of prices in professional football, such as the DTP model developed by Kemper and Breuer (2016) as well as the VTP model by Drayer et al. (2012), formed the basis of this article. A study very similar to this one but with important differences in terms of objective and model design was carried out by García et al. (2020) This article has no limitations. In the first place, it has demonstrated the contribution of the economy to the sporting field and especially to the financial management of clubs. Analysing characteristics such as the schedule, day of the game, type of stadium, qualifying position, value of the confrontation between the teams, and attendance rate at the stadiums enabled the study to contribute to building an empirical tool with the aim of optimizing clubs' income statement.
Therefore, as a future application, this hedonic approach with the contribution of multiple linear regressions could be very useful for sports managers in the predicted assessment of players in the case of transfer, considering their own endogenous and exogenous variables in the model.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.