Prediction of risk: Decoding the Serial Dependence of Stock Return Volatility with Copula

Abstract

In this article, a new approach based on the copula theory is employed in the analysis and forecasting of hospitality and tourism-related stock return volatility (HTSRV). The application of copula-based models for univariate time series is state-of the-art methodologies with new perspectives for economic analysis in tourism and hospitality. This flexible method provides numerous functions for serial dependence specification of volatility series. Eight hospitality–tourism stocks are analyzed for their serial dependence structures to provide insight for forecasting stock return volatility. While the forecasting performance between our chosen copulas and benchmark models is inconclusive, the empirical results show that copulas well specify both linear and nonlinear serial dependence structures, which lead to forecasting results as good as or even better than those of the benchmark models. This property allows us to use copulas for HTSRV forecasting without the concern of model misspecification.

Keywords

hospitality and tourism-related stock return volatility copula-based model serial dependence risk prediction

Introduction

Application of statistical models using autoregressive integrated moving average (ARIMA) and artificial neural network (ANN) for forecasting has been proven to be successful in tourism demand research (Song et al., 2019). We are not aware of any research that compares ARIMA and ANN with the copula-based approach. While the copula-based model is a relatively new methodology in the tourism and hospitality literature, can it be a strong contender to conventional approaches from a forecasting perspective? We intend to answer this in our study by focusing on the application of copula-based models for univariate time series, state-of the-art methodologies with new perspectives for economic analysis in tourism and hospitality.

The hospitality industry has become a major sector and a catalyst for national development in the world. With improving hotel operating performance driven by global GDP growth and strong tourism demand, hotel investment worldwide has increased from US$66.4 billion in 2017 to US$67.7 billion in 2018 (Jones Lang Lasalle, 2019).

As cited in Zhu and Lim (2018), the extant hospitality finance literature has revealed several factors affecting hotel stock performance. M. H. Chen et al. (2005) discuss the underlying macroeconomic and nonmacroeconomic factors that have significantly affected the stock performance of Taiwanese hotel. In M. H. Chen (2007), macro and nonmacro variables are used to explain hotel stock returns in China. Although these studies provide valuable information of hotel stock trading, they have not contributed to the understanding of stock volatility.

Various empirical studies (Tehrani et al., 2015) have demonstrated that volatility is a fundamental stock market risk. Concern about large swings of price movements in the stock market has motivated researchers’ interests in examining the level and stationarity of volatility over time, or specifically, volatility forecasts (Brailsford & Faff, 1996). Forecasting stock return volatility is of great importance in portfolio management, risk management, and derivatives pricing (Sokolinskiy & Van Dijk, 2011), and the same is true for forecasting hospitality and tourism-related stock return volatility (HTSRV), which can provide foresight to investors in financial risk management. While the forecast of stock return volatility is informative for investment and financial risk management using copula-based approach, such research is rare in the hospitality and tourism literature.

Researchers have proposed different techniques for forecasting in the tourism and hospitality literature (Song et al., 2019), and they are mainly the conventionally used linear models which are computationally straightforward and can be easily applied in practice. However, linear relationship is just one of the numerous dependence structures between the dependent variable and its predictors. In this study, the copula method is used to forecast HTSRV of several listed hospitality and tourism-related companies. The copulas are one of the advanced methodologies with some attractive features. This method is flexible and allows for an arbitrary distribution of the dependent variable. In particular, copulas can serve for modelling dependence structures where a different choice of marginal distributions is possible for capturing both linear and nonlinear associations, and extreme values. Our research aims to shed light on the copula method for forecasting HTSRV, and deepen the understanding of serial dependence structure of stock return volatility.

The investigation of risk prediction using copula is still in the initial phase, and copulas have been used mainly for multivariate dependence modelling between several series of financial assets (in a given portfolio) of different markets. Moreover, previous research tends to use two-dimensional copulas to study paired series. We intend to contribute to the hospitality and tourism research in new methods and perspectives by applying copulas to dependence analysis of individual HTSRV series. Our contribution includes the discussion of high dimensional nonparametric method, namely the Bernstein copula, in forecasting HTSRV.

The hospitality industry as defined in Zhu and Lim (2018), is “a collection of business providing accommodation, catering (food and beverage) and leisure facilities and services, which includes but not limited to hotels, resorts, restaurants, clubs, casinos, cruise ships and theme parks” (p. 615). Given the diverse nature of the industry, our research will focus on the travel, accommodation and foodservice sectors. Singapore equity is our subject of interest for a variety of reasons. Investments in the hospitality industry in general, the hotel industry in particular, are low risk within alternative form of investment in the Consumer Discretionary sector. This is because Singapore tourism is vibrant. In 2018, for the 17th consecutive year, Singapore was ranked Asia Pacific’s top convention and meeting city by the International Congress and Convention Association. And Singapore is a “creative” global city in which international tourists are attracted by the cosmopolitan ambience of the city state (Oswin & Yeoh, 2010).

The outline of this article is as follows. Section 2 reviews previous forecasting research in finance, hospitality, and tourism studies. Section 3 demonstrates the copula method. Section 4 describes the data used and discusses the empirical results. Some concluding remarks are provided in Section 5.

Literature Review

Forecasting is an important topic in hospitality and tourism research, and this topic has been widely discussed in the related literature. Remarkable progress has been made in the development of tourism demand forecasting methods as evident in the recent extensive review of the literature spanning 50 years (Song et al., 2019). The authors argued that the three types of quantitative models used in past studies include time-series, structural/econometric, and AI-based models. Structural models need the predicted values of independent variables as inputs to perform forecast of the dependent variable of interest. The former involves extra task and probably increases the forecasting error. On the contrary, univariate time-series models only require the historical information of the dependent variable, and are more commonly used in forecasting.

According to Zhu and Lim (2018), Olsen and Jose (1982) were recognized among the pioneers to introduce time-series modeling in hospitality studies. Different time-series methods to forecast room occupancy have been employed in successive research (Ellero & Pellegrini, 2014; Koupriouchina et al., 2014; Law, 1998; Pereira, 2016; Schwartz et al., 2016). According to Zhu and Lim (2018), research in predicting turning point/s for industry (Choi et al., 1999) and guest room nights (Lim et al., 2009) used mainly linear univariate methods, applying Box–Jenkins models (Box & Jenkins, 1970). They include among others, the autoregressive moving average, ARIMA and seasonal autoregressive integrated moving average models. Although they are popular, their application is not effective in estimating possible nonlinear relationship between current and past demand. As there are numerous possible functions for nonlinear relationships (Claveria et al., 2015a), the specification of a nonlinear model to a particular data set is a difficult task. As for the ANNs, a priori knowledge about the relationship between variables is not necessary. While this may seem appealing, the ANN model suffers from “black box” issues related to network structure determination difficulty, thus restricting the applicability of ANNs (Kisi, 2011). The forecasting literature has provided evidence that ANN models have outperformed ARIMA, and vice versa (Song et al., 2019).

While the various models have been used for forecasting in the hospitality literature, the application of these models to hospitality finance is very rare. In contrast, forecasting stock market volatility in general has been of interest for decades. With the focus on more regulations since the1987 international stock market crash, research on volatility has increased (Brailsford & Faff, 1996). Subsequently, forecasting practice using GARCH models is applied to predict stock market volatility (Chong et al., 1999; Franses & Van Dijk, 1996; Kambouroudis et al., 2016; Marcucci, 2005).

The copula approach is more flexible compared with the conventional forecasting models. The former incorporates linear, nonlinear, symmetric, asymmetric, and the range of dependencies from perfectly negative to perfectly positive dependence (Vaz de Melo Mendes & Aíube, 2011). Moreover, the copula method does not require the stock volatility variable to follow any given distribution. On the contrary, stock volatility is assumed to follow the normal distribution pattern in the conventional models. Copulas have been used considerably to examine the dependence structures between two or more time series in/across financial markets. As Zhu and Lim (2018) explained,

Relevant studies shed light on the co-movement of stock returns, especially the dependence structures of stock return pairs. For instance, Fortin and Kuzmics (2002) demonstrate that the joint distribution of return pairs of European stock indices displays strong tail dependence and should not be represented by elliptical copulas; and Rodriguez (2007) indicates that dependence structure between stock market returns of countries in Asia and Latin America changed during the Asian and Mexican crises. (p. 613)

The copula method has gradually attracted attention in different areas of forecasting research. For instance, copulas are applied to construct forecasting models in the research of natural resources phenomena (e.g., Allamehzadeh et al., 2015; Bessa et al., 2012; L. Chen et al., 2013; He et al., 2017; Khedun et al., 2014). Researchers in economics also considered the use of copulas to predict macroeconomic variables such as GDP growth, inflation, unemployment, interest rate, industrial production, and price indices (Bianchi et al., 2010; Liu et al., 2014; Loaiza-Maya & Smith, 2019; Smith & Vahey, 2016). In the finance literature, the copula method is an attractive statistical tool for risk prediction estimation (Sahamkhadam et al., 2018; Segnon & Trede, 2018; Weiß, 2013). In most of these studies, copulas have been used to model the dependence structures between two or among several series. For example, empirical studies in Finance apply copulas to forecast returns and their volatilities with broad applications for a portfolio of asset (bivariate portfolios or multivariate portfolios consisting of more than three stocks using stock market indexes or portfolio of individual stocks). Patton (2009) called these copula-based multivariate time-series models.

According to X. Chen and Fan (2006), copulas can also be applied to depict the dependence structures of univariate time series. While copula-based models become popular in the forecasting literature, the application of copulas for univariate time series (or univariate copulas) is still limited. Among the very rare attempts of using univariate copulas for forecasting, Gupta and Majumdar (2015) only consider the two-dimensional case. They compare the house price returns forecasts of elliptical and Archimedean copulas with those generated by the Naive model and AR(1). Zhu et al. (2017) also focus on the two-dimensional copulas, and their study shows that copulas can outperform ARIMA and seasonal ARIMA models in forecasting tourism demand. Arya and Zhang (2017) use the nested elliptical and Archimedean copula functions in the vine copula framework for high dimensional serial dependence of water quality time series. The aforementioned studies have only compared copulas with linear time-series models, but not with any nonlinear forecasting methods. Furthermore, the copula functions under consideration are all parametric, while the nonparametric copulas have not been covered. To address these gaps, our research is also motivated by the rare use of the univariate copula approach for forecasting hospitality and tourism stock volatility. We do so by applying univariate copula to eight publicly listed companies in Singapore, and the results will be compared with ARIMA and ANN as benchmark models.

Methodology

According to Parsa and Klugman (2011), copulas are useful for describing the dependence structure between variables. The foundation of copulas is Sklar’s Theorem, which connects marginal distributions with their joint distribution (Sklar, 1959). Random variables follow a valid joint distribution which is the combination of the given marginal distributions function and a copula function. Specifically, the random variables $X_{1}, X_{2}, . . ., X_{n}$ have continuous marginal distribution functions $M_{1}, M_{2}, . . ., M_{n}$ an n-dimensional joint distribution function $F (.)$ . Based on Sklar’s Theorem, $F (.)$ can be represented by a unique copula function $C (.)$ as follows:

F (x_{1}, x_{2}, \dots, x_{n}) = C {M_{1} (x_{1}), M_{2} (x_{2}), \dots, M_{n} (x_{n}); ϒ}

(1)

where $x_{i}$ is the observation of $X_{i}$ and i = 1, . . ., n; $ϒ$ is a vector of parameters labeling the dependence among $X_{i}$ .

Two-Dimensional Copula

It has been shown that the copula method can be used to model serial dependence of stationary Markov processes (X. Chen et al., 2009; X. Chen & Fan, 2006; Ibragimov, 2009). For univariate time series, $X_{t}$ and $X_{t - 1}$ have a joint distribution function $F (.)$ and continuous marginal distribution function $M_{j}$ . $H (.)$ is thus represented by a copula function $C (.)$ as follows:

F (x_{t}, x_{t - 1}) = C {M_{j} (x_{t}), M_{j} (x_{t - 1}); γ}

(2)

We can rewrite Equation (2) as

F (x_{t}, x_{t - 1}) = C (u_{t}, u_{t - 1}; γ)

(3)

where the cumulative density $u_{t - i} = M_{j} (x_{t - i})$ for $i = 0, 1$ ; γ is the association parameter between $u_{t}$ and $u_{t - 1}$ .

The maximum likelihood method is employed for the estimation of γ in Equation (3). We define $c (.)$ as the probability density function for $C (.)$ . The log-likelihood function is as follows:

L = \sum_{t = 1}^{T} \ln c (u_{t}, u_{t - 1}; γ)

(4)

where

c (.) = \frac{\partial^{2} C}{\partial u_{t} \partial u_{t - 1}}

(5)

Several copula functions have been employed for the specification of dependence structure. Of all the different copula families, we consider Archimedean and elliptical families, and select five copulas from them. Two main copulas in the elliptical family, which are widely applied in statistics and econometrics, are the Gaussian and Student-t copulas (Frahm et al., 2003). Therefore, these two copulas are chosen for empirical illustration. By using Clayton, Frank, and Gumbel copulas from the Archimedean family, we can describe different tail dependence structures.

The elliptical and symmetric dependence for the Gaussian copula is given in Equation (6):

F (x_{t}, x_{t - 1}) = Φ_{G} [Φ^{- 1} (u_{t}), Φ^{- 1} (u_{t - 1}); γ]

(6)

where $Φ^{- 1}$ refers to the inverse of the standard normal cumulative distribution; $Φ_{G}$ is the standard bivariate normal distribution with a dependence parameter γ. Another option for elliptical structure is the Student-t copula. However, the Student-t copula is different from the Gaussian, as the probability of joint extreme events is higher in Student-t copula (Aas, 2004). The Student-t copula function is as follows:

F (x_{t}, x_{t - 1}) = T_{γ, ν} [T_{ν}^{- 1} (u_{t}), T_{ν}^{- 1} (u_{t - 1}); γ, ν]

(7)

where $T_{γ, ν}$ is the bivariate Student’s t distribution with the degree of freedom $ν > 2$ and dependence parameter $γ; {andT}_{ν}^{- 1}$ is the inverse of the scalar standard Student’s t distribution.

The Frank copula is an alternative function to specify the symmetric dependence structure with the strongest association in the center of the joint distribution:

F (x_{t}, x_{t - 1}) = - \frac{1}{γ} \ln (1 + \frac{(e^{- γ u_{t}} - 1) (e^{- γ u_{t - 1}} - 1)}{e^{- γ} - 1})

(8)

Nonetheless, the dependence could be asymmetric. We consider the Clayton copula for the asymmetric dependence which exhibits strongest negative or left tail dependence:

F (x_{t}, x_{t - 1}) = {[\max ({(u_{t})}^{- γ} + {(u_{t - 1})}^{- γ} - 1; 0)]}^{\frac{- 1}{γ}}

(9)

If the asymmetric dependence exhibits strongest positive or right tail, the Gumbel copula is an appropriate option to capture the dependence characteristics:

F (x_{t}, x_{t - 1}) = \exp [- {({(- \log (u_{t}))}^{γ} + {(- \log (u_{t - 1}))}^{γ})}^{\frac{1}{γ}}]

(10)

Previous studies usually use these two families and five copulas as illustrations to analyze dependence structures, especially in hospitality and tourism research area (Pérez-Rodríguez et al., 2015; Tang et al., 2014; Tang et al., 2016; Zhu et al., 2017; Zhu & Lim, 2018). We follow this pattern and analyze the dependence structures of HTSRV in our article. It should be noted that both Clayton and Gumbel copulas can only be used to describe a positive association.

High Dimensional Copula

The elliptical and Archimedean copulas are parametric and provide association parameters to visualize the dependence structures of interest. While the two families are popular, they are not always applicable for risk modeling, as they are restricted to a certain degree of symmetry and certain correlation structures (Diers et al., 2012). We thus take into account the Bernstein copula as an alternative in forecasting HTSRV. This method is nonparametric and has no association parameter. However, it is flexible and can be used to approximate any copulas for both known and unknown dependence structures (Diers et al., 2012; Sancetta & Satchell, 2004). In addition, the Bernstein copula is capable of modeling high dimensional dependence of high-order Markov processes.

Following the procedures by Sancetta and Satchell (2004), we define $α (\frac{v_{t}}{m_{t}}, \dots, \frac{v_{t - k}}{m_{t - k}})$ as a real-valued constant with $v_{t - i} \in N$ , i=0, 1, . . . k, and $0 \leq v_{t - i} \leq m_{t - i}$ . The Bernstein copula function for k + 1 dimensional Markov process can be written as

C_{B} (u_{t}, \dots, u_{t - k}) = \sum_{v_{t} = 0}^{m_{t}} \dots \sum_{v_{t - k} = 0}^{m_{t - k}} α (\frac{v_{t}}{m_{t}}, \dots, \frac{v_{t - k}}{m_{t - k}}) P_{v_{t}, m_{t}} (u_{t}) \dots P_{v_{t - k}, m_{t - k}} (u_{t - k})

(11)

and

P_{v_{t - i}, m_{t - i}} (u_{t - i}) \equiv (\begin{matrix} m_{t - i} \\ v_{t - i} \end{matrix}) {u_{t - i}}^{v_{t - i}} {(1 - u_{t - i})}^{m_{t - i} - v_{t - i}},

(12)

where for the ease of notation, we can set $m_{t} = m_{t - 1} = \dots = m_{t - k} = m$ , such that

α (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m}) = C_{n} (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m}) = \frac{1}{n} \sum_{s = 1}^{n} I {\cap_{i = 0}^{k} [u_{t - i, s} \leq \frac{v_{t - i}}{m}]}

(13)

where $C_{n} (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m})$ is the empirical copula; n is the sample size; I{A}is the indicator of set A.

If we take the partial derivatives of Equation (11) with respect to $u_{t - 1}, u_{t - 2}, \dots, u_{t - k}$ , we have

\begin{array}{l} \frac{\partial C (u_{t}, u_{t - 1}, \dots, u_{t - k})}{\partial u_{t - 1} \dots \partial u_{t - k}} = \sum_{v_{t} = 0}^{m} \dots \sum_{v_{t - k} = 0}^{m} α (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m}) \\ P_{v_{t}, m} (u_{t}) D_{v_{t - 1}, m} (u_{t - 1}) \dots D_{v_{t - k}, m} (u_{t - k}), \end{array}

(14)

and

D_{v_{t - i}, m} (u_{t - i}) = (\begin{matrix} m \\ v_{t - i} \end{matrix}) [\begin{array}{l} v_{t - i} {u_{t - i}}^{v_{t - i} - 1} {(1 - u_{t - i})}^{m - v_{t - i}} \\ - (m - v_{t - i}) {u_{t - i}}^{v_{t - i}} {(1 - u_{t - i})}^{m - v_{t - i} - 1} \end{array}],

(15)

We can then rely on the conditional cumulative density function (CDF) of Bernstein copula to generate the forecasting value of $u_{t}$ . The conditional CDF of Bernstein copula is

G (u_{t} {| u}_{t - 1}, \dots, u_{t - k}) = \frac{\begin{array}{l} \sum_{v_{t} = 0}^{m} \dots \sum_{v_{t - k} = 0}^{m} α (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m}) \\ P_{v_{t}, m} (u_{t}) D_{v_{t - 1}, m} (u_{t - 1}) \dots D_{v_{t - k}, m} (u_{t - k}) \end{array}}{\begin{array}{l} \sum_{v_{t} = 0}^{m} \dots \sum_{v_{t - k} = 0}^{m} α (\frac{v_{t}}{m}, \dots, \frac{v_{t - k}}{m}) \\ P_{v_{t}, m} (1) D_{v_{t - 1}, m} (u_{t - 1}) \dots D_{v_{t - k}, m} (u_{t - k}) \end{array}},

(16)

Empirical Results

Data and Company Description

Stock price volatility measures the stock returns over a period of time. Essentially, there are two types of stock volatility measures. Realized volatility is extracted from the actual historical data and calculated by taking the standard deviation of past stock returns over a period of time. An alternative measure is to use market-based forecasts of “implied volatility” of the underlying assets; it is a model-dependent measure of volatility implied by the observed option prices (Andersen & Benzoni, 2009; Kambouroudis et al., 2016). To our knowledge, traded options are not available on the Singapore Stock Exchange (SGX) equity. Hence, this study computes the historical/realized volatility (RV) using 1 month of daily returns. Since volatility is persistent, this simple procedure can provide information for current and future volatility, and with no measurement error since it is based on realized return. RV is calculated from the daily log returns of the closing price during a month (Zhang et al., 2013), such that

Log return = Y_{i} = \ln \frac{C_{i} + 1}{C_{i}}

where C_i is the daily closing price.

RV = \sqrt{n} * \sqrt{\frac{1}{n} * \sum_{i = 1}^{n} {(Y_{i} - \bar{Y})}^{2}}

(17)

where n represents the trading days in a particular month and $\bar{Y}$ = drift = Average ( $Y_{i}$ )

To identify all the relevant stock prices for the study, initially a thorough search is conducted based on the company description provided in the SGX. Companies with exposure to the tourism, hospitality and leisure industry are listed mainly in the Consumer Staples and Consumer Discretionary, and Real Estate sectors of Singapore equity. Among them are hotel, foodservice, and transport operators, as well as companies that run gaming, leisure, and recreational facilities. We have identified these companies for our sample which extends from January 2000 to December 2017. The daily closing prices for each stock are subsequently obtained from Thomson Reuters Datastream to calculate their daily stock returns before applying Equation (17) to generate monthly return volatility time series for each stock. Due to the availability of data, our data set comprised volatility series of eight stocks, each of which is Markov process, with further discussion provided in the following section.

The eight stocks are AF Global, Amara Holdings, Genting Singapore, HL Global Enterprises, Hotel Grand Central, Hotel Properties, OUE and Singapore Airlines. Among the eight companies, six are industry peers in the Consumer Discretionary sector. AF Global Limited is an investment holding company, which owns and operates hotels and serviced residences in Singapore, United Kingdom and Thailand. Amara is also a Singapore-based investment holding company which operates hotels, properties, specialty restaurants and food services businesses in Singapore, China, and Thailand. Genting Singapore Ltd., which is an investment holding company involved in the development and operation of resort and casino, has the largest market capitalization. It owns Singapore Resorts World Sentosa, one of two integrated resorts in the city state, which is designed with a Universal Studios theme park and Marine Life Park, the world’s second largest oceanarium to attract leisure and business visitors.

HL Global has the smallest market capitalization amongst the six peers; it is an investment holding company which operates and manages hotels (in Malaysia and China) and restaurants. The company also engages in property devtelopment activities. Hotel Grand Central principal activities are those of a hotel owner and operator in five countries but most of its revenue is generated from Australia and Singapore. Hotel Properties Limited is a hotelier and investment holding company. The principal business segments are hotels (in nine countries), properties (rental & sales operations) and others (include distribution & retail operations).

OUE Limited is a diversified real estate owner, developer, and operator. The company’s main activities include hospitality services (hotel operation and management, restaurant operation), and property investment in Asia, the United States, and China. Singapore Airlines provides passenger and cargo air transportation services globally. Since 2000, the national carrier has won the “World’s Best Airline” in the Skytrax awards four times (in 2004, 2007, 2008, and 2018).

For each volatility time series, we split the sample into two parts: the estimation period containing data from 2000M1 to 2014M12, and data from 2015M1 to 2017M12 are used in the validation period. The data for the estimation period are used to specify the copula-based univariate models, as well as the benchmark models. Subsequently, the recursive approach is employed to generate forecasts for 1, 6, 12, 18, and 24 months. The forecast results will be compared with the validation data set.

The copulas described in Section 3 are used to estimate the dependence structure of each stock return volatility series. The benchmark models include ARIMA and ANN, to see how well the copula performs. While the ARIMA models are the basic and commonly used linear models for forecasting, the ANNs are popular nonlinear methods applied to generate tourism forecasts. An ANN contains a number of neurons that process and transform information from the input side into the different signals in the output side (Claveria et al., 2015b). The ANN methods can be divided into two categories based on the connecting patterns of layers; namely, the feed-forward networks and recurrent networks. While the former can just statically map the input vectors, the latter models a high dimensional nonlinear dynamical system (Doya, 1995). Therefore, we employ the recurrent ANN with supervised learning in this article.

Preliminary Test

Table 1 shows the preliminary test results. According to the Augmented Dickey Fuller statistics, Genting Singapore and HL Global are stationary in level. AF Global, Amara Hotel Grand Central, Hotel Properties, OUE, and Singapore Airlines become stationary after first differenced. Hence, we use the original series of Genting Singapore and HL Global, and first differenced series of the others in the subsequent analyses.

Table 1

Preliminary Test Results

Stock Return Volatility Series	ADF Statistics		Chi-Square Test Statistics				Serial Correlation Test
Stock Return Volatility Series	Level	First Differenced	Normal	Cauchy	Triangular	Critical Value α = .01	Lags Included	LM(12)^a	p
AF Global	−2.181	−17.180***	29.939	8.335	203.750	18.475	AR(1)	9.823	.632
Amara Holdings	−2.457	−13.171***	53.358	2.484	243.900		AR(1)	11.329	.501
Genting Singapore	−2.927*	—	33.311	51.776	14.911		AR(1) AR(3)	19.328	.081
HL Global	−2.905*	—	4.686	11.411	10.043		AR(1)	16.942	.152
Hotel Grand Central	−2.558	−18.456***	28.627	3.143	144.810		AR(1)	13.071	.364
Hotel Properties	−2.227	−12.910***	51.125	16.032	277.680		AR(1)	10.104	.607
OUE	−2.682	−23.358***	17.337	10.082	96.742		AR(1)	19.165	.085
Singapore Airlines	−2.516	−20.277***	34.683	4.815	141.260		AR(1)	20.296	.062

Note: ADF = Augmented Dickey–Fuller test.

Include 12 lags for monthly data.

Indicates results are statistically significant at 5% level. ***indicates results are statistically significant at 1‰ level.

The chi-square test for normality is applied to the volatility series under study. However, test for the Cauchy or Triangular distribution may be more applicable. Thus Y ~ Normal, Y ~ Cauchy or Y ~ Triangular are the null hypotheses for the chi-square test. One can compare the chi-square test statistics with the critical value; and from the viewpoint of hypothesis testing, the null is accepted if the test statistics is less than the critical value. We conclude that the volatility series probably follows the corresponding distribution. As shown in Table 1, the chi-square test statistics for all the series are greater than the critical value at the 1% significance level, with the exception of HL Global and OUE. Thus, it is evident that most of the volatility series are not normally distributed using traditional forecasting methods. On the other hand, the volatility series for all the stocks follow a Cauchy distribution except Genting Singapore, and Triangular distribution is appropriate for Genting Singapore. In addition, Cauchy distribution fits OUE better than normal distribution with smaller test statistics. Thus, we select normal distribution for HL Global, Triangular distribution for Genting Singapore, and Cauchy distribution for the others to generate the cumulative density $u_{t}$ .

Markov property test of the volatility series is conducted using the Breusch–Godfrey LM test for autocorrelation in the residuals. The results are also available in Table 1. The AR(1) residuals for all series are not serially correlated, except Genting Singapore, because the p values of the Breusch–Godfrey LM test statistics are greater than the critical value 0.05. We further test the AR(3) residuals of Genting Singapore, and the residuals become uncorrelated. This indicates that the volatility series of Genting Singapore is third-order Markov process and the others are first-order Markov processes.

Dependence Analysis

Scatter plots of the cumulative densities is the first assessment of the dependence structures of the stock return volatilities under study (see Figure 1). They display diverse dependence structures between the eight volatilities series investigated. In general, symmetric dependence structures with apparent tail dependence is found for the AF Global, Amara, Hotel Grand Central, Hotel Properties, OUE, and Singapore Airlines series; while HL Global exhibits strong left tail dependence asymmetric joint distribution. Since the residuals of Genting Singapore with AR(1) and AR(3) are uncorrelated, we illustrate a 3D scatter plot for this stock return volatility. The 3D scatter plot shows an asymmetric pattern similar to HL Global. Thus, it is incorrect to only specify the linear dependence structure for forecasting stock return volatilities.

Figure 1

Scatter Plots of Cumulative Densities for Singapore Hospitality Stock Return Volatilities

Several goodness-of-fit testing methods for copulas can be used as selection criteria for the determination of appropriate dependence structure for the volatility series. We rely on the statistic $S_{n}^{(C)}$ recommended by Genest et al. (2009) in this study. For a p value of $S_{n}^{(C)}$ higher than the significance level (5% in our study), we argue that the corresponding copula can specify the serial dependence. Table 2 reports the goodness-of-fit test results. According to the p value of $S_{n}^{(C)}$ , the Student-t copula fits all the series which are the first order Markov process.

Table 2

Selection of Optimal Copula

	Gaussian	Student-t	Frank	Clayton	Gumbel	Optimal Copula	Association Parameter
p Value of SnC Statistics (Log-likelihood Value)
AF Global	0.001	0.079 (27.11)	0.021	—	—	Student-t	−0.455***
Amara Holdings	0.027	0.146 (37.14)	0.079 (24.21)	—	—	Student-t	−0.534***
HL Global	0.064 (113.7)	0.299 (123.2)	0.0334	0.002	0.022	Student-t	0.864***
Hotel Grand Central	0.162 (15.28)	0.206 (17.40)	0.117 (13.75)	—	—	Student-t	−0.434***
Hotel Properties	0.233 (28.73)	0.219 (35.32)	0.511 (28.51)	—	—	Student-t	−0.581***
OUE	0.676 (25.70)	0.429 (29.27)	0.387 (21.60)	—	—	Student-t	−0.511***
Singapore Airlines	0.014	0.058 (30.44)	0.011	—	—	Student-t	−0.468***

Note: ***Indicates results are statistically significant at 1‰ level.

The values in bold indicate that the optimal copula is determined by: 1) the p value of $S_{n}^{(C)}$ Statistics higher than 0.05; and 2) the largest log-likelihood value.

The Gaussian copula can be applied for the serial dependence structures of HL Global, Hotel Grand Central Hotel Properties, and OUE, while the Frank copula is also a reasonable option for these series except HL Global. Additionally, the Frank copula is suitable for Amara Holdings.

Besides using the statistic $S_{n}^{(C)}$ , comparison of the log-likelihood (LL) values of the copula models shows us the optimal copula model which should be selected. Table 2 presents the LL values of the copula models (in the bracket) and the largest LL value indicates that the corresponding copula provides the best fit. Evidently, the Student-t copula is optimal for all the first-order Markov processes. As for Genting Singapore which is third-order Markov process, the Bernstein copula should be used.

The association parameters ( $γ$ ) generated from the optimal copulas (see Equation [7]), are also listed in Table 2. It can be seen that all parameters are statistically significant at 1‰ level. The negative parameter of AF Global, Amara Holdings, Hotel Grand Central, Hotel Properties, OUE and Singapore Airlines, reveals negative tail dependence. The Student-t copula describes the negative relation between the changes of current and future return volatilities is more pronounced in the tails than in the center. Similar association pattern is also evident for HL Global.

Comparison of Forecasting Results

Volatility forecasts for the eight stocks are generated using the selected copulas and they are compared with those using ARIMA models. The optimal ARIMA model with uncorrelated residuals is chosen based on the Schwarz’s Bayesian information criterion. ANN is another benchmark model commonly selected in tourism forecasting research and practice (Song et al., 2019).

To evaluate volatility forecast performance, we consider the usual measures of forecast error, namely the root mean square error (RMSE) and mean absolute error (MAE). They are computed for different forecast horizons (1, 6, 12, 18, and 24 months). We have reported the results of our forecast in Table 3. When comparing the forecasting performance, we found that no one particular model produces the best forecasts for all the volatility series. Generally speaking, the nonlinear models (copulas and ANN) yield better forecasting performance than the linear ones (ARIMAs) for AF Global, AMARA, HL Global, Singapore Airlines and Genting Singapore. According to the MAE results of these five series shown in Table 3, the ARIMA model only outperforms the two nonlinear models for the 1-step ahead forecast of volatility change of Singapore Airlines. If we compare the copulas with ARIMAs, it is obvious that the copula models produce more accurate forecasts for the five volatility series. Further comparison between the copula models and recurrent ANN shows that the copulas are better choices for HL Global, Singapore Airlines and Genting Singapore. However, the recurrent ANN performs slightly better than copulas in forecasting the changes of volatilities for AF Global. In terms of Amara, while the recurrent ANN has better MAE results, the Student-t copula leads to better RMSE results.

Table 3

Comparison of MAEs and RMSEs for Five Time Horizons

Stock Return Volatility Series	Model	MAE					RMSE
Stock Return Volatility Series	Model	i = 1	i = 6	i = 12	i = 18	i = 24	i = 1	i = 6	i = 12	i = 18	i = 24
AF Global	Student-t (AR[1])	0.135	0.116	0.113	0.132	0.130	0.220	0.136	0.136	0.151	0.151
	ARIMA (2,1,3)	0.156	0.127	0.118	0.130	0.130	0.237	0.152	0.146	0.153	0.150
	ANN	0.133	0.119	0.106	0.124	0.129	0.199	0.139	0.127	0.146	0.154
Amara Holdings	Student-t (AR[1])	0.096	0.059	0.063	0.069	0.089	0.146	0.098	0.107	0.117	0.139
	ARIMA(2,1,3)	0.076	0.076	0.080	0.082	0.100	0.102	0.107	0.116	0.124	0.149
	ANN	0.071	0.056	0.064	0.068	0.091	0.102	0.099	0.112	0.116	0.144
HL Global	Student-t (AR[1])	0.111	0.753	0.996	1.102	1.483	0.242	1.080	1.282	1.408	1.705
	ARIMA (1,0,1)	0.563	0.763	1.337	1.181	1.287	0.875	1.137	1.570	1.459	1.545
	ANN	1.095	0.761	0.872	1.028	1.592	1.380	0.976	1.050	1.415	1.816
Hotel Grand Central	Student-t (AR[1])	0.070	0.047	0.034	0.030	0.033	0.095	0.067	0.054	0.054	0.062
	ARIMA (0,1,1)	0.045	0.047	0.034	0.030	0.033	0.060	0.067	0.054	0.052	0.060
	ANN	0.054	0.051	0.052	0.039	0.042	0.070	0.072	0.062	0.056	0.064
Hotel Properties	Student-t (AR[1])	0.064	0.029	0.031	0.029	0.030	0.139	0.036	0.037	0.032	0.034
	ARIMA (0,1,1)	0.043	0.029	0.030	0.029	0.030	0.096	0.036	0.037	0.032	0.034
	ANN	0.040	0.028	0.044	0.038	0.050	0.067	0.038	0.058	0.047	0.064
OUE	Student-t (AR[1])	0.072	0.049	0.047	0.038	0.040	0.092	0.064	0.064	0.051	0.055
	ARIMA (2,1,3)	0.041	0.046	0.047	0.039	0.041	0.059	0.063	0.064	0.050	0.055
	ANN	0.049	0.049	0.049	0.051	0.043	0.063	0.064	0.066	0.059	0.060
Singapore Airlines	Student-t (AR[1])	0.046	0.027	0.028	0.025	0.029	0.061	0.037	0.038	0.036	0.041
	ARIMA (2,1,2)	0.036	0.032	0.030	0.028	0.033	0.047	0.041	0.040	0.038	0.043
	ANN	0.046	0.029	0.041	0.024	0.040	0.053	0.040	0.057	0.036	0.046
Genting Singapore	Bernstein (AR[1], AR[3])	0.073	0.084	0.089	0.088	0.075	0.101	0.102	0.107	0.102	0.091
	ARIMA (4,0,2)	0.079	0.155	0.261	0.319	0.374	0.093	0.180	0.278	0.327	0.378
	ANN	0.140	0.162	0.332	0.174	0.158	0.191	0.190	0.351

The RMSE and MAE results endorse consistent findings. The linear model performs the worst for AF Global, Amara, HL Global, Singapore Airlines and Genting Singapore. The selected copulas are superior in forecasting hotel stock volatility for HL Global and Genting Singapore and changes of volatilities for Amara and Singapore Airlines. As for AF Global, the recurrent ANN has relatively smaller forecasting errors than the other two methods.

Nonetheless, it is difficult to tell which method provides better forecasts for Hotel Grand Central, Hotel Properties, and OUE. The MAE results demonstrate that the student copula and ARIMA have very similar performance in forecasting the changes of volatility for Hotel Grand Central and OUE. This makes sense as the dependence structures of the volatility changes for these two stocks are probably linear. In this case, the copula approximates a linear dependence structure as the ARIMA, which results in similar forecasting performance. However, the ARIMA, which is a linear model, can perform slightly better for the linear serial association. This is evident in the MAE results for Hotel Properties and the RMSE results of Hotel Grand Central and OUE. When the dependence structure is linear, the nonlinear model should not be a good choice. This could be the reason why the recurrent ANN performs the worst for Hotel Grand Central, Hotel Properties and OUE.

Conclusion

In the current study, copula, an alternative measure of dependence is proposed to overcome some of the limitations of current practices in quantitative time-series prediction modeling. Copula-based modeling provides different perspectives as it applies a procedure based on the marginal distributions of random variables. With the use of copulas, we can model both linear and nonlinear dependence. Moreover, the approach is capable of modeling extreme endpoints.

We examined eight hospitality- and tourism-related stocks listed in the SGX and analyzed the serial association of volatility. The task of illustrating empirically how relatively new methods like the copulas can be systematically applied to evaluate HTSRT, is a step in the right direction for forecasting in hospitality and tourism research. The current article introduces the very commonly used copula functions, including the elliptical, Archimedean, and Bernstein copulas. These copulas can contribute to new methods and perspectives in HTSRT research with economic/financial applications.

This study makes invaluable contributions in modeling dependence structure with various types of copula and prediction of stock return volatility. By selecting the best fit copulas, the study also compares the forecast performance of ARIMA and ANN models with that of the selected copulas. Moreover, our empirical results show that nonlinear models, such as the recurrent ANN, may suffer from model misspecification when forecasting the series with a linear serial dependence structure, which leads to poor forecasting performance. Since the copula approach can be used for both linear and nonlinear dependence structures, it is more convenient than the nonlinear methods. In addition to providing insight for the financial decision making to corporations in the hospitality industry, the copula can also benefit institutional and retail investors in hospitality stock investments.

While it is not immediately clear to what extent copulas can outperform conventional approaches, this study is a first attempt to apply a state-of-the-art methodology with new perspectives for economic analysis in tourism and hospitality. We have done so by comparing the forecast performance of copulas with ARIMA and ANN. As the copula technique is developing, future research in hospitality and tourism should consider the application of more advanced copulas to improve forecasting accuracy. Although our analysis covers eight stock volatility series, the study suggests a potential novel benefit of using copula in forecasting. Admittedly, the relatively small number of series used limits the generalization of our findings. Thus, future studies should apply copulas to hospitality-related financial assets in other stock exchanges for risk prediction. In addition, the lack of theoretical justification to determine the copula used, remains a limitation for extrapolation and forecasting.

Footnotes

Authors’ Note:

The authors are grateful to the anonymous reviewers and the Guest Editors for helpful comments and suggestions. The first author would like to acknowledge the financial support of the National Natural Science Foundation of China Fund (Grant 71803135). The authors also want to thank Zhen Huang, for technical assistance; Finance colleagues at the University of Macau and Singapore Nanyang Technological University for their invaluable advice.

ORCID iD

Christine Lim

Liang Zhu (e-mail: lzhu111@szu.edu.cn) is an assistant in the Division of International Economics and Trade, College of Economics, Shenzhen University, China. Christine Lim (corresponding author; e-mail: christinelim@um.edu.mo) is an associate professor in integrated resort and tourism management, Faculty of Business Administration, University of Macau, China. Jianlun Zhang (e-mail: zjl1130@foxmail.com) is a PhD student in Faculty of Business Administration, University of Macau, China.

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