Forecasting daily visits in Shanghai with Model combination and Telco big data

Abstract

Analyzing and forecasting visitor flows has significant potential to enhance policy decisions, facilitate long- and short-term planning of local resources, and optimize tourism offerings. Leveraging Big Data in this context provides distinct advantages, particularly regarding real-time predictions. This paper proposes an econometric approach to account for uncertainty in the prediction model, a key source of forecasting errors. A description and prediction of daily visits in Shanghai City are provided using models commonly employed in tourist flows analysis. We show that model combination and selection procedures can serve as valuable tools for policymakers, equipping them with accurate forecasts to support tourist flow management.

Keywords

tourism flows mobile positioning Smart cities Shanghai forecasting

Introduction

Managing the volatility of tourism dynamics has become increasingly important as destinations experience greater variability in visitor numbers across regions and over time. These oscillations, whether a spike that exceeds a destination’s capacity or a rapid decline in visitor numbers, can trigger multidimensional socio-economic and environmental impacts. For example, when visitation exceeds acceptable sustainability limits, overtourism can detract from the tourism experience, overstress infrastructure and public services, and threaten environmental sustainability (Lin, 2024; Mussoni and Vici, 2025). However, prolonged low visitation can lead to weakened local economies, diminished corporate profit, and inefficient use of public resources (Bisht et al. 2025; Blázquez-Salom et al., 2023). Both extremes expose the structural fragility of tourism systems, exacerbate regional disparities, and threaten balanced and equitable development (Song et al., 2019; Zheng et al., 2024).

In this context, accurate demand forecasting is essential for anticipating demand fluctuations and supporting evidence-based policy-making. Reliable projections provide government officials and key tourism stakeholders with a basis for balancing the economic advantages of tourism against the destination’s environmental and infrastructural conditions, thereby minimizing risk from either excessive or insufficient demand for tourism space. Forecasting insights also support strategic planning by helping decision-makers and policymakers determine how and when to allocate limited resources, adjust operational capacities, and, if needed, use adaptive management processes to enhance resilience and sustainability in the long term (Duan et al., 2022; Fan et al. 2025; Prayag et al., 2024).

The integration of real-time big data into forecasting is a new area of research in tourism and hospitality. Using Big Data in policy-making offers numerous advantages, particularly in providing real-time predictive insights. Access to real-time data enables the identification of daily patterns in urban systems and facilitates optimal data processing to formulate fundamental responses for policymakers, as highlighted by Kandt and Batty (2021). The extensive collection of big data enables the analysis of tourists’ preferences and characteristics, empowering decision-makers in tourism planning. The advantages of leveraging big data can help mitigate risks associated with extreme scenarios, as discussed in studies by Ardito et al. (2019) and Li et al. (2018). In this paper, we use a Big Data dataset provided by Shanghai Unicom, the second-largest mobile phone provider in Shanghai. This dataset contains real-time tourists’ positions in the Shanghai metropolis from 1 December 2020 to 17 January 2022. It has been successfully used in Camatti et al. (2024) to study tourism spillover effects within the Shanghai districts. We consider a period during which a significant event, namely the Chinese New Year, occurs and study the forecasting ability of standard models in periods with an extremely large number of visits. Some strategies for addressing forecasting errors and model uncertainty are proposed.

Although the growing availability of high-frequency and behaviorally rich data has significantly enhanced the empirical basis of tourism forecasting, improved data alone do not automatically translate into more reliable predictions. Even with detailed real-time information on visitor movements, researchers and policymakers must still decide how to model these dynamics, which variables to include, how to specify temporal structures, and which assumptions to impose on the data-generating process. In this sense, a fundamental methodological challenge persists in the form of model uncertainty. Tourism demand forecasting has evolved from classical time-series approaches – such as Autoregressive Integrated Moving Average (ARIMA) and Seasonal ARIMA (SARIMA), designed to capture seasonality and persistence – to more flexible systems that integrate statistical and computational techniques (Chu, 2009; Önder and Gunter, 2016; Song and Li, 2008; Song and Zhang, 2025). However, empirical evidence consistently shows that no single model systematically outperforms others across destinations, time horizons, or data environments. Forecast accuracy in tourism analysis depends not only on the richness of the available data, but also on model specification, parameterization, and the structural assumptions underlying the forecasting framework. Tourism systems are inherently dynamic and volatile, and forecasting errors may arise as much from specification risk as from stochastic variability in the data itself. This challenge has become increasingly relevant as the diversity of forecasting methodologies available to researchers and practitioners has grown. Traditional econometric approaches now coexist alongside machine learning techniques and hybrid frameworks that combine statistical rigor with computational flexibility. Each approach captures different dimensions of tourism dynamics, and is characterized by distinct assumptions, strengths, and limitations. Consequently, no single model can be expected to consistently outperform others across all destinations, time periods, or forecasting horizons. These considerations underscore the importance of forecasting frameworks that explicitly recognize and manage model uncertainty rather than implicitly ignoring it. By acknowledging that different models may provide complementary insights, researchers can develop more adaptive and robust forecasting systems that better capture the evolving nature of tourism demand.

An original contribution of this paper is to address model uncertainty through forecast combination. Instead of selecting a single preferred specification, we implement a Bayesian Model Averaging (BMA) framework that combines predictions from alternative models estimated on the same dataset. In the seminal work of Bates and Granger (1969) and its subsequent extensions (Billio et al., 2013; Fawcett et al., 2015; Geweke and Amisano, 2011; Gneiting and Ranjan, 2011; McAlinn and West, 2019), BMA assigns dynamic weights to competing models based on predictive performance. In our application, SARIMA-type models serve as structured and interpretable core specifications, and their forecasts are combined probabilistically to account for time variation in model performance. Following Billio et al. (2013) and Costantini et al. (2016), we use common forecast performance metrics to guide model dynamics and also incorporate economic evaluation to illustrate how model selection can be aligned with specific policy objectives. By integrating high-frequency mobile positioning data on Shanghai’s tourist flows into a coherent probabilistic averaging framework, our study demonstrates how forecast robustness can be enhanced, specification uncertainty explicitly quantified, and predictive performance improved in periods of extreme demand. The resulting approach provides a scalable, decision-oriented tool for managing tourism volatility in complex urban environments such as Shanghai.

The remainder of the paper is organized as follows. Section 3 introduces the forecasting models and the model combination strategy. Section 4 presents the empirical application to forecasting visits to Shanghai. Section 5 concludes and outlines future research directions.

Literature review

Tourism has become a key driver of economic growth in many countries and regions, stimulating the development of a wide range of related sectors such as transportation, hospitality, and culture (Lin, 2024). While tourism generates significant development opportunities, its strong interdependence with other industries increases the complexity of its management and planning processes (Chen et al., 2025; Duan et al., 2022; Mussoni and Vici, 2025). These structural characteristics require forward-looking strategies to ensure balanced and sustainable growth.

In recent years, evolving behavioral changes and market conditions in response to major events, such as the COVID-19 pandemic, naturally called for flexible, resilient, and sustainable tourism management and governance (Fan et al., 2025; Prayag et al., 2024). Rather than focusing solely on recovery from shocks, emerging paradigms feature structural strengthening of tourism systems and their ability to adapt to demand changes and long-term sustainability challenges. Within this framework, reliable demand forecasts are crucial for effective data-driven policymaking and tourism governance. Accurate forecasting enables public authorities to anticipate fluctuations in visitor flows and align medium- and long-term investments with expected demand, thereby strengthening transport systems, optimizing environmental carrying capacity, and improving service provision (Ghalehkhondabi et al., 2019; Guizzardi et al., 2021; Wang et al., 2019). Linking infrastructure planning and resource allocation to tourism dynamics predictions enhances policy coherence and reduces the risks of over- and under-investment at the aggregate and regional levels. Forecasting provides a critical foundation for informed decision-making in budgeting, staffing, and destination development (Jakovlev and Kitanov, 2024; Zhou, 2021), while also guiding marketing and promotional strategies through alignment with anticipated demand trends (He and Qian, 2025). Beyond these operational applications, demand forecasting serves a dual role as both a management tool and a strategic instrument for driving tourism toward a more sustainable path (Liang et al., 2025). Accurate projections favor balancing economic gains with environmental and operational limits, risk mitigation, efficient resource use, and sustainable development (Song et al., 2019; Zheng et al., 2024). They further support the identification of ecological carrying capacities and the design interventions to safeguard fragile ecosystems (Lu and Chen, 2023; Sejati et al., 2025). Finally, by anticipating variations in visitor flows, destination managers can improve efficiency, resilience, and competitiveness (Hewapathirana, 2025; Zheng et al., 2024).

Many forecasting approaches have been developed over the last few decades, ranging from traditional time-series and econometric models to state-of-the-art artificial intelligence (AI) and hybrid approaches, with the aim of improving predictive accuracy (Bi et al., 2022; Song et al., 2019). Although many of these methods perform well in specific empirical contexts, no single forecasting approach consistently demonstrates superior accuracy across all settings (Höpken et al., 2021; Pan et al., 2025). Consequently, enhancing the generalizability, robustness, and adaptability of forecasting models has become a central focus of contemporary research. The historical evolution of tourism demand forecasting reflects a broader methodological and epistemological shift: from static, theory-driven econometric models to adaptive, data-integrated systems. Advances in computational power and the increasing availability of high-frequency and behaviorally rich data have fostered the convergence of statistical inference, economic reasoning, and machine learning techniques, fundamentally reshaping how researchers analyze the complexity of tourism dynamics (Song and Li, 2008; Song and Zhang, 2025).

Time-series models, particularly within the Box–Jenkins framework (Box and Jenkins, 1976; Box et al., 2015), have long dominated tourism demand forecasting. Basic time-series approaches include autoregressive (AR), moving average (MA), single exponential smoothing (ES), and historical average (HA) models (Wan and Song, 2018). These models have been widely employed over the past five decades due to their straightforward implementation and their capacity to capture historical patterns effectively. Among them, ARIMA models are frequently applied (Lim and McAleer, 2002), and their seasonal extension, SARIMA, is extensively used in tourism time-series analysis (Chu, 2009). Early forecasting studies relied heavily on ARIMA and SARIMA models to capture persistence, cyclicality, and temporal dependence in tourism demand, and the empirical evidence has consistently confirmed their predictive reliability (Chu, 2009; Önder and Gunter, 2016). Nevertheless, the linear structure of ARIMA-type models limits their ability to account for structural variability, nonlinear patterns, and volatility clustering that increasingly characterize tourism demand. Despite these limitations, ARIMA and SARIMA remain essential and frequently serve as interpretable cores within more flexible hybrid frameworks (Gil-Alana et al., 2004; Koc and Altinay, 2007). They have been extended along several directions. For example, the ARFIMA model introduces fractional integration (Chu, 2009), while ARIMAX and SARIMAX incorporate exogenous variables to increase explanatory power. For example, Li et al., 2018 used an ARX model to explore the impact of relative climate variability on tourism demand. Beyond ARIMA-type specifications, additional econometric frameworks have been developed to capture inter-temporal relationships and structural dynamics in tourism demand. These include the Autoregressive Distributed Lag Model (ADLM) (Huang et al., 2017; Huang et al., 2017), the Error Correction Model (ECM) (Vanegas Sr, 2013), the Vector Autoregressive (VAR) model (Gunter and Önder, 2016), and Time-Varying Parameter (TVP) models (Song et al, 2011). Moreover, time-varying volatility has been incorporated into tourism forecasting by assuming Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models for the error variance. ARIMA-GARCH and SARIMA-GARCH specifications are particularly appropriate when tourism demand exhibits conditional heteroskedasticity, allowing for seasonality and time-varying variance (Araya et al., 2024; Tsui and Balli, 2017), and have been shown to be well-suited for forecasting high-frequency flows (Chan et al., 2005). Divino and McAleer, 2010 applied ARIMA-GARCH to model and forecast daily tourist arrivals in Peru, while Liang (2014) employed SARIMA-GARCH to forecast tourism demand in Taiwan. In this paper, we build on this literature and provide a suitable framework for incorporating uncertainty in SARIMA-GARCH specifications into tourism forecasts.

In parallel with these developments, the extensive use of large-scale, high-frequency, and behaviorally rich datasets has accelerated the need for adaptive forecasting techniques in tourism studies that combine econometric rigor with machine-learning flexibility (e.g., see Camatti et al., 2024). Machine learning techniques such as support vector machines (Pai et al., 2014), random forests (He and Qian, 2025), and hybrid systems that incorporate search engine data (Sun et al., 2019) have shown strong potential for capturing nonlinear relationships and complex dependencies (Li et al., 2018). Ensemble and hybrid frameworks, including ARIMAX artificial neural networks (NN) (Wen et al., 2019), gradient boosting (Liu et al., 2019), and recurrent NN (Bi et al., 2020), combine linear and nonlinear components to improve predictive performance. However, empirical evidence shows that these approaches do not consistently outperform traditional models across all settings (Volchek et al., 2019), and their increasing complexity raises concerns about interpretability, overfitting, and parameter sensitivity (He and Qian, 2025; Wei and Mukherjee, 2024). Thus, standard ARIMA and SARIMA models continue to play a central role, often providing an interpretable baseline for hybrid and ensemble mechanisms (Coshall and Charlesworth, 2011; Hassani et al., 2017; Shen et al., 2011).

Model uncertainty arises from the coexistence of competing prediction approaches and from the recognition that the forecasting performance depends not only on data quality but also on model specification and selection, an aspect often neglected in tourism economics (Peng et al., 2014). No model can be expected to consistently outperform all others across different time periods, destinations, or datasets (Song et al., 2019; Yang et al., 2015). Forecast Errors may arise from structural uncertainty in specification choices, parameter selection, or omitted variables (Armstrong et al., 2015). This awareness has redirected methodological research toward forecast combination strategies. In tourism econometrics, it has been shown that forecast combination outperforms individual models, producing more robust and accurate predictions (Claveria et al., 2015; Clements and Hendry, 2004; Wang et al., 2023). In this study, we adopt a BMA strategy instead of relying on a single selected model. The seminal contribution of Bates and Granger (1969) demonstrated that combining forecasts from different models can reduce forecast variance and enhance reliability. BMA has become standard in many fields of economics, including regional and urban economics, for addressing uncertainty in specification and variable selection (Crespo Cuaresma and Feldkircher, 2013; Fernández et al., 2001; LeSage and Parent, 2007; Moral-Benito, 2015). In tourism econometrics, BMA has been applied by Oh and Morzuch (2005) via an equally weighted combination of the best-performing time-series models. An application to travelers’ arrivals to Singapore between 1977 and 1990 shows that BMA forecasts always outperform the poorest individual model, and perform better than the best model. Wong et al. (2007) applied BMA to Hong Kong inbound tourism forecasting and found that combined forecasts can outperform the worst individual models. Compared to the two studies, we extend the analysis from point to density forecasts, from static to dynamic weighting, and from mean squared error and mean absolute error measures to general performance measures. See also Song and Li (2008), Song et al. (2019), Song and Zhang (2025) for a review. Outside the BMA, a frequentist approach based on Weighted-Average Least Squares (WALS) has been applied by Wan and Zhang (2009) to combine time-series models for visitor arrivals in Barbados between 1956 and 1992. The authors found that WALS forecasts outperform the worst model in the combination. The advantage of WALS over BMA is its lower computational cost; however, its optimality is not guaranteed when used with ARIMA models, which are standard in tourism econometrics. A summary of the main approaches to tourism forecasting (Time Series Models, Time Series with Stochastic Volatility, Machine Learning, Hybrid Machine Learning, Ensemble Machine Learning, Forecast Combination), including the references discussed in this paper, is given in Table 1. Rather than selecting a single “best” model, BMA assigns posterior probabilities to models based on their predictive performance and computes a weighted average of individual model predictions. By combining predictive distributions, BMA explicitly quantifies uncertainty and produces forecasts that are statistically consistent and operationally robust (Billio et al., 2013; Geweke and Amisano, 2011; Gneiting and Ranjan, 2013). In tourism forecasting models, uncertainty has been recognized as a central challenge, motivating the use of forecast combination (Coshall and Charlesworth, 2011; Peng et al. 2014; Song and Li, 2008).

Table 1.

Summary of the literature review on tourism demand forecasting models. Approaches: Time Series Models (TSM), TSM with Dynamic Volatility (TSM-SV), Machine Learning (ML), Hybrid ML (H-ML), Ensemble ML (E-ML), Forecast Combination (FC).

Approach	Model/Algorithm	References	Description
TSM	AR, MA, ES, HA	Wan and Song (2018)	Basic models used to capture historical patterns; simple and widely applied.
	ARIMA	Chu (2009), Lim and McAleer (2002), Önder and Gunter (2016)	Standard model for capturing autocorrelation and temporal dynamics; strong empirical reliability.
	SARIMA	Chu (2009), Önder and Gunter (2016)	Seasonal extension of ARIMA, suitable for tourism data with seasonality.
	ARFIMA	Chu (2009)	Introduces fractional integration to capture long-memory processes.
	ARIMAX/SARIMAX/ARX	Li et al. (2018)	Incorporates exogenous variables to enhance explanatory power.
	ADLM	Huang et al. (2017)	Dynamic model with distributed lags for intertemporal relationships.
	ECM	Vanegas Sr (2013)	Error correction model capturing long-run equilibrium relationships.
	VAR	Gunter and Önder (2016)	Multivariate framework capturing inter-dependencies among variables.
	TVP models	Song et al. (2011)	Time-varying parameters to account for structural changes.
TSM-V	ARIMA-GARCH	Araya et al. (2024), Divino and McAleer (2010), Tsui and Balli (2017)	Combines mean dynamics with conditional volatility modeling.
TSM-V	SARIMA-GARCH	Araya et al. (2024), Liang (2014), Tsui and Balli (2017)	Incorporates both seasonality and heteroskedasticity.
ML	Support vector machines	Pai et al. (2014)	Captures complex nonlinear relationships.
	Random forest	He and Qian (2025)	Ensemble method robust to nonlinearities and interactions.
	Search data-based models	Sun et al. (2019)	Integrates search engine data to improve forecasting performance.
H-ML, E-ML	ARIMAX + neural networks	Wen et al. (2019)	Combines linear and nonlinear components.
	Gradient boosting	Liu et al. (2019)	Boosting ensemble approach enhancing predictive accuracy.
	Recurrent neural networks	Bi et al. (2020)	Suitable for sequential data and dynamic temporal dependencies.
FC	Bayesian model averaging	Oh and Morzuch (2005), Wong et al. (2007)	Applied classical BMA to linear models, including ARIMA and ADL, with static and standard combination weights.
FC	Frequentist model averaging	Wan and Zhang (2009)	Applied weighted average least square to linear regression and ARIMA models and provide combination confidence bounds.

Among existing approaches to BMA, a modeling framework known as Bayesian agent and expert opinion analysis was introduced by Lindley et al. (1979) and subsequently discussed in later works (Genest and Schervish, 1985; West, 1984, 1988, 1992). Recently, this approach has been further extended through Bayesian predictive synthesis (Johnson and West, 2025; McAlinn et al., 2020; McAlinn and West, 2019), which provides a probabilistic and coherent framework. Recent advances extend BMA through time- and feature-varying model averaging (Sun et al., 2023), hybrid Bayesian–bootstrap approaches (Liu et al., 2025), and adaptive weighting schemes incorporating alternative data sources (Llewellyn et al., 2023), while broader Bayesian Model Combination frameworks such as Dynamic Model Averaging allow model weights to evolve with predictive performance (Billio et al., 2013; Geweke and Amisano, 2011; Raftery et al., 1997). A summary of the main approaches to forecast combination (Forecast Combination, BMA, Bayesian Predictive Synthesis, Expert Opinion Models, Dynamic Model Averaging, BMA Extensions), including the references discussed in this paper, is given in Table 2.

Table 2.

Summary of the literature review on model and forecast combination. Methods: Forecast Combination (FC), Bayesian Model Averaging (BMA), Bayesian Predictive Synthesis (BPS), Expert Opinion Models (EOM), Dynamic Model Averaging (DMA), BMA Extensions (BMA-E).

Method	References	Description
FC	Bates and Granger (1969), Claveria et al. (2015), Clements and Hendry (2004), Wang et al. (2023)	Combines multiple models to reduce variance and improve robustness.
BMA	Crespo Cuaresma and Feldkircher (2013), Fernández et al. (2001), LeSage and Parent (2007), Moral-Benito (2015)	Weighted averaging of models based on posterior probabilities.
EOM	Genest and Schervish (1985), Lindley et al. (1979), West (1984, 1988, 1992)	Modeling framework combining judgmental probability assessments.
BPS	Aastveit et al. (2023), Johnson and West (2025), McAlinn (2021), McAlinn et al. (2020), McAlinn and West (2019), Tallman and West (2024)	Probabilistic framework combining predictive distributions.
DMA	Billio et al. (2013), Geweke and Amisano (2011), Raftery et al. (1997)	Model weights evolve over time based on performance.
BMA-E	Liu et al. (2025), Llewellyn et al. (2023), Sun et al. (2023)	Advanced approaches with adaptive weights and alternative data.

In this paper, we apply BMA to a set of alternative SARIMA specifications with and without GARCH errors. Within our BMA framework, we assume that the combination weights depend on the model’s predictive performance. Standard predictive performance metrics are considered (e.g., see Costantini et al., 2016) together with cost functions in order to account for the economic evaluation of the forecast and to align statistical relevance and policy and tourism management objectives (Geweke and Amisano, 2011; Gneiting and Ranjan, 2011, 2013; McAlinn and West, 2019).

Expanding on this methodological rationale, the empirical phase of this research utilizes mobile positioning data to implement the proposed forecasting framework. These data, obtained from mobile network capabilities, provide detailed, real-time data on tourist presence and mobility behaviors (Chen et al., 2025; Metulini and Carpita, 2024). Mobile data provide dynamic measures of how people move through space and time, mitigating some of the limitations of conventional surveys and administrative data sources. In addition, mobile data have been used to understand the spatial and temporal impacts of social and cultural events, thereby further expanding their relevance to tourism research. More recently, the development of big data—high-frequency, complex information from mobile devices, social media, and digital transactions—has transformed our understanding of tourist behavior (Wu et al., 2025). These data sources can identify travel pathways, peak engagement periods, and popular attractions, yielding useful insights for destination and infrastructure management. Predictive analytics will further extend this capacity to understand mobility patterns and data-informed tourism policymaking (Ramos et al., 2021).

Nevertheless, predicting mobility is inherently challenging due to nonlinear interactions among behavioral, environmental, and demographic variables. Classic models (ARIMA models) rely on linearity and stationarity assumptions, while machine learning models (methods) can better handle (model nonlinearity) nonlinear functions but frequently do not capture temporal dependence. To address these issues, recent studies have employed ensemble frameworks to combine existing models, leveraging the strengths of both traditional and machine-learning approaches, thereby improving flexibility and predictive performance while remaining cautious about overfitting and parameter selection (Punia and Shankar, 2022; Wei and Mukherjee, 2024). The current study put forward the combination of mobile positioning data within a BMA framework – a probabilistic approach that weights multiple model specifications dynamically based on predictive performance. This controls for uncertainty while also capturing the temporal and structural variability of tourist mobility. With the addition of mobile data, the new approach improves forecasting accuracy and resilience while leveraging BMA’s flexibility to produce predictive outcomes, offering a scalable, generalizable tool for adaptive and sustainable tourism management.

Methodology

The temporal evolution of visitation rates across statistical areas reflects the inherently dynamic nature of tourism demand. The temporal patterns in observed visitation rates across statistical areas exhibit both periodicity and trend. Understanding these temporal dynamics is essential for tourism planning, destination management, and the efficient allocation of resources and services. Rather than fluctuating randomly, visitation counts exhibit clear periodic behavior alongside persistent changes over time, highlighting the need for modeling approaches that capture both regular cycles and stochastic variation. To capture these complex patterns, this study adopts dynamic stochastic models that represent the persistent, cyclical, and irregular components embedded in tourism visitation counts. Such models are particularly valuable in tourism analytics because they not only provide accurate forecasts of future demand but also help identify underlying temporal structures critical to policy design, infrastructure planning, and market responsiveness. Their flexibility and robustness have made them widely applicable in tourism forecasting research, and they can be readily implemented using modern analytical environments such as Python, MATLAB, and R. In this study, the empirical analysis was conducted in MATLAB using the Econometrics Toolbox. Specifically, we used the arima, estimate, forecast, garch, lbqtest, mae, mape, and rmse libraries within the econometric toolbox in Matlab, for inference and model diagnostics.

Bayesian predictive synthesis

Bayesian predictive synthesis (BPS) and various generalizations of the framework have been introduced in particular for forecasting in economics and finance by Aastveit et al. (2023), Bassetti et al. (2018), McAlinn (2021), McAlinn et al. (2020), Johnson and West (2025) and McAlinn and West (2019) introduced the mixture-BPS, which includes Bayesian Model Averaging as a special case.

Let $M_{j}$ , with j = 1, …, J, be a set of J models, where each $M_{j}$ defines a predictive density function $p_{j} (y | M_{j})$ , which set is defined as $P = {p_{1} (y | M_{1}), \dots, p_{J} (y | M_{J})}$ . Let $π_{j} = P (M_{j})$ be the model probabilities. Under mixture-BPS, the predictive distribution, conditional on learning $P$ , can be written as

f (y | P) \propto \sum_{j = 0}^{J} π_{j} α_{j} (y) p_{j} (y | M_{j}),

(1)

where α_j(⋅) j = 0, 1, …, J are non-negative weight functions and π_j j = 0, 1, …, J are the initial model probabilities. The probability density function of the baseline model

M_{0}

is denoted with

p_{0} (y | M_{0})

. Equation (1) can be rewritten so that models can be explored for various choices of α(y). The predictive distribution

f (y | P)

becomes:

f (y | P) = \sum_{j = 0}^{J} {\tilde{π}}_{j} f_{j} (y | M_{j}), f_{j} (y | M_{j}) = \frac{α_{j} (y) p_{j} (y | M_{j})}{α_{j}}, j = 0,1, \dots, J,

(2)

where

α_{j} = \int_{y} α_{j} (y) p_{j} (y | M_{j}) d y, {\tilde{π}}_{j} = k π_{j} α_{j}, k^{- 1} = \sum_{j = 0}^{J} π_{j} α_{j} .

(3)

Bayesian Model Averaging is a special case of the mixture BPS when π₀ = 0 and α_j(y) = 1 for j = 1, …, J (e.g., see Tallman and West, 2024).

Model classes

The stochastic model, coupled with an inference procedure, facilitates generating daily predictions of the variable of interest over a specified time horizon. In the subsequent section, we illustrate a 60-day forecast for Shanghai City, using the aggregated statistical area. This methodology is versatile and can be extended to provide disaggregated forecasting analyses, including area-specific predictions for various statistical areas of interest. Furthermore, it enables predictions of visitor origin and type, as well as forecasts of flows between and within statistical areas.

Empirical tourism analysis must explicitly model trend and seasonal behaviour and ignoring these components can lead to biased conclusions. Beyond weather conditions – particularly extremes such as humid summers that shape leisure demand – seasonal patterns in Shanghai visits are also influenced by pronounced domestic tourism peaks and recurring business travel cycles. We employ SARIMA models, initially proposed by Box and Jenkins (1976), which are particularly well suited to our dataset. Differencing is used to address the trend, while seasonal differencing captures seasonal dynamics. Let us introduce the lag polynomials ϕ(L) = 1 − ϕ₁L − … − ϕ_pL^p and θ(L) = 1 − θ₁L − … − θ_qL^q for the non-seasonal dynamics and Φ(L^s) = 1 − Φ₁L^s − … − Φ_PL^sP and Θ(L^s) = 1 − Θ₁L^s − … − Θ_QL^sQ for the seasonal dynamics, with (1 − L)^dz_t the dth order differencing operator, ${(1 - L^{s})}^{D} z_{t}$ the Dth order seasonal differencing operator, p the autoregressive order, q the moving average order, d the number of differencing operations, P, D and Q the seasonal orders, and s the seasonality. The SARIMA(p, d, q) × (P,D,Q)_s model takes the following form:

ϕ (L) Φ (L^{s}) {(1 - L^{s})}^{D} {(1 - L)}^{d} z_{t} = θ (L) Θ (L^{s}) u_{t} .

(4)

See also Box et al. (2015) for further details.

Time–varying volatility, that is heteroschedasticity, in tourist visits arises from the persistence or clustering of exogenous shocks, such as pandemics, exchange rate movements, or global income fluctuations. Other factors, such as mega-events, expos, and sudden travel booms, can generate bursts of variability that are not constant over time. These features are well–captured by GARCH models. In the GARCH(p, q) specification Bollerslev (1987), it is assumed that $E (u_{t} | F_{t - 1}) = 0$ and $V (u_{t} | F_{t - 1}) = σ_{t}^{2}$ , where:

σ_{t}^{2} = α (L) u_{t}^{2} + β (L) σ_{t}^{2},

(5)

where

F_{t - 1}

is the information set including past observations

F_{t - 1} = σ ({z_{s}}_{s \leq t - 1})

with lag polynomials α(L) = α₀ + α₁L + …, α_qL^q for the ARCH component and β(L) = β₁L + ⋯ + β_pL^p for the GARCH component. The SARIMA and SARIMA-GARCH models are estimated by maximum likelihood (Ling and McAleer, 2003a, 2003b).

Combination weight specification

We consider Bayesian model averaging (e.g., see Bates and Granger, 1969) and assume time-varying combination weights. Following Costantini et al. (2016), we consider some of the most used forecasting performance measures to build the combination weights. The weights are specified based on a rolling estimate of the model’s performance measures (e.g., see Billio et al., 2013). Three groups of measures are considered: i) point forecast ability, ii) density forecast ability, and iii) economic evaluation. In the following, ${\hat{y}}_{ℓ, s}$ denotes the forecast made at time s − 1 for time s using model $M_{ℓ}$ , ℓ = 1, …, N with N the number of models, and τ denotes the window size.

The first group of measures includes: Mean Squared Forecast Error (MSFE), Exponential Mean Squared Forecast Error (EMSFE), Mean Squared Root Forecast Error (MSRFE), Discount Mean Squared Forecast Error (DMSFE), Mean Absolute Error (MAE), the Mean Absolute Percentage Error (MAPE), Hit/Success Rates (HR), and Exponential of Hit/Success Rates (EHR). The MSFE and EMSFE weights are:

\begin{matrix} w_{j, t}^{(M S F E)} = \frac{M S F E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} M S F E_{ℓ, t}^{- 1}}, w_{j, t}^{(E M S F E)} = \frac{\exp (- M S F E_{j, t})}{\sum_{ℓ = 1}^{N} \exp (- M S F E_{ℓ, t})}, \end{matrix}

(6)

\begin{matrix} {M S F E}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} {({\hat{y}}_{ℓ, s} - y_{s})}^{2}, \end{matrix}

(7)

where k_j denotes the number of parameters of model

M_{j}

. The exponential transform is commonly used in this literature to ensure numerical stability and smoothness of the weights when the MSFE is close to zero. The MSRFE and DMSFE weights are defined as:

\begin{matrix} w_{j, t}^{(M S R F E)} = \frac{M S R F E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} M S R F E_{ℓ, t}^{- 1}}, M S R F E_{ℓ, t} & = \sqrt{\frac{1}{τ} \sum_{s = t - τ + 1}^{t} {({\hat{y}}_{ℓ, s} - y_{s})}^{2}} \end{matrix}

(8)

\begin{matrix} w_{j, t}^{(D M S F E)} = \frac{D M S F E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} D M S F E_{ℓ, t}^{- 1}}, D M S F E_{ℓ, t} & = \sum_{s = t - τ + 1}^{t} θ^{t - s} {({\hat{y}}_{ℓ, s} - y_{s})}^{2}, \end{matrix}

(9)

where 0 ≤ w_j,t ≤ 1 are the combination weights, such that w_1,t + ⋯ + w_N,t = 1. Similarly, MAE and MAPE weights are defined as follows:

\begin{matrix} w_{j, t}^{(M A E)} = \frac{M A E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} M A E_{ℓ, t}^{- 1}}, {M A E}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} | {\hat{y}}_{ℓ, s} - y_{s} |, \end{matrix}

(10)

\begin{matrix} w_{j, t}^{(M A P E)} = \frac{M A P E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} M A P E_{ℓ, t}^{- 1}}, {M A P E}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} \frac{| {\hat{y}}_{ℓ, s} - y_{s} |}{y_{s}} . \end{matrix}

(11)

HR and EHR are built using the DA measure:

\begin{matrix} w_{j, t}^{(H R)} = \frac{D A_{j, t}}{\sum_{ℓ = 1}^{N} D A_{ℓ, t}}, w_{j, t}^{(E H R)} = \frac{\exp (D A_{j, t})}{\sum_{ℓ = 1}^{N} \exp (D A_{ℓ, t})}, \end{matrix}

(12)

\begin{matrix} {D A}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} I (sgn ({\hat{y}}_{ℓ, s} - {\hat{y}}_{ℓ, s - 1}) = sgn (y_{s} - y_{s - 1})), \end{matrix}

(13)

where

I (x = y)

is the indicator function which take values 1 if x = y and 0 otherwise and sgn(x) returns −1 if x is negative, 0 if x is null, and 1 if x is positive.

The second group comprises measures of density-forecast ability, such as Log-Score (LS), Continuous Ranked Probability Score (CRPS), and Frequentist Model Averaging (FMA). Let IC_ℓ,s be the predictive information criterion of model $M_{ℓ}$ at time s evaluated on the point forecast ${\hat{y}}_{ℓ, s}$ , $F_{ℓ, s} (\cdot) = F (\cdot | M_{ℓ}, y_{1}, \dots, y_{s - 1})$ the forecasting distribution at time s − 1 for time s, and f_ℓ,t(⋅) its probability density function. The CRPS, LS, and FMA weights are defined as:

\begin{matrix} w_{j, t}^{(L S)} = \frac{L S_{j, t}}{\sum_{ℓ = 1}^{N} L S_{ℓ, t}}, {L S}_{ℓ, t} = - \frac{1}{τ} \sum_{s = t - τ + 1}^{t} \log f_{ℓ, s} (y_{s}), \end{matrix}

(14)

\begin{matrix} w_{j, t}^{(C R P S)} = \frac{C R P S_{j, t}}{\sum_{ℓ = 1}^{N} C R P S_{ℓ, t}}, {C R P S}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} \int_{- \infty}^{\infty} {(F_{ℓ, s} ({\hat{y}}_{s}) - I {{\hat{y}}_{s} \geq y_{s}})}^{2} d {\hat{y}}_{s}, \end{matrix}

(15)

\begin{matrix} w_{j, t}^{(F M A)} = \frac{I C j, t}{\sum_{ℓ = 1}^{N} I C ℓ, t}, {I C}_{ℓ, t} = - 2 \sum_{s = t - τ + 1}^{t} \log (f_{ℓ, s} ({\hat{y}}_{ℓ, t})) + w (τ, k), \end{matrix}

(16)

where w(τ, k) = 2k for the Akaike Information Criterion (AIC), w(τ, k) = log(τ)k for the Bayesian Information Criterion (BIC) and w(τ, k) = 2 log(log(τ))k for the Hannan and Quinn Criterion (HQ), with k the number of parameters of a given model.

The third group comprises two measures of the economic impact of forecast errors: the Economic Evaluation of Directional Forecasts (EEDF) and the Weighted MSFE (WMSFE). The EEDF is built on the Directional Value (DV) measure

w_{j, t}^{(E E D F)} = \frac{D V_{j, t}}{\sum_{ℓ = 1}^{N} D V_{ℓ, t}}, {D V}_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} | y_{s} - y_{s - 1} | I (sgn ({\hat{y}}_{ℓ, s} - {\hat{y}}_{ℓ, s - 1}) = sgn (y_{s} - y_{s - 1}))

(17)

which indicates the model’s ability to capture the correct change in direction of the predicted variable, accounting for the magnitude of the realized changes. The WMSFE exploits the economic relevance of forecasting, incorporating the costs incurred by the municipality related to tourism. The idea is to measure how prediction errors vary across days of the year, accounting for cost asymmetries in over- and underestimating flows across holiday and non-holiday periods. WMSFE is computed for each model as follows:

{\tilde{w}}_{j, t} = \frac{W M S F E_{j, t}^{- 1}}{\sum_{ℓ = 1}^{N} W M S F E_{ℓ, t}^{- 1}}, W M S F E_{ℓ, t} = \frac{1}{τ} \sum_{s = t - τ + 1}^{t} {({\hat{y}}_{ℓ, s} - y_{s})}^{2} c_{s}

(18)

such that

0 \leq {\tilde{w}}_{j, t} \leq 1

and

{\tilde{w}}_{1, t} + \dots + {\tilde{w}}_{N, t} = 1

. The weights c_s are chosen based on policy analysis, such as the extra costs associated with overtourism during certain periods of the year.

Empirical analysis

Data description

The dataset comprises daily tourist visits to Shanghai from 1 December 2020 to 17 January 2022, covering all 16 administrative districts—Huangpu, Xuhui, Changning, Jing’an, Putuo, Hongkou, Yangpu, Minhang, Baoshan, Jiading, Pudong, Jinshan, Songjiang, Qingpu, Fengxian, and Chongming. It is derived from real-time positional data of mobile phone signals provided by Shanghai Unicom, the city’s second-largest mobile service provider. The dataset encompasses users from diverse telephone companies in proportions reflecting their market shares, and it is relied upon by the Shanghai municipal government for tourism flow prediction, underscoring its accuracy, inclusiveness, and institutional relevance. The final panel contains 409 daily observations per district (e.g., see Camatti et al., 2024). The target population comprises tourists visiting Shanghai’s districts during the study period. Operationally, a “tourist” is defined as a mobile phone user who records an overnight stay in a district either on the night prior to or following the visit, and whose activity is not attributable to other categories of users. China Unicom provided pre-filtered data to exclude residents, hikers, workers, students, and commuters. Under the intra-destination flow framework, a tourist is counted once per district per day, although multiple districts may be visited on the same day. Based on SIM card activation records, tourists are classified as either Shanghai locals or tourists from outside Shanghai. These clearly defined behavioral criteria ensure that the analytical population closely corresponds to visitors engaging with districts across Shanghai, thereby supporting data representativeness and comprehensive coverage of the defined tourist population.

The top plot in Figure 1 represents the trends in visits to Shanghai during the period from December 1, 2020, to January 16, 2022. We conducted a preliminary analysis of the dataset by examining the temporal dependence structure. The autocorrelation functions in Panel (b) allow for identifying persistence and seasonal effects. In addition, Phillips–Perron and augmented Dickey–Fuller tests reject the null hypothesis of a unit root at the 5% level for all series. Together, these results provide a compelling rationale for employing seasonal ARMA models to capture the dynamic structure of tourism flows across Shanghai’s districts.

Figure 1.

Shanghai visits data. Top: Millions of visits to Shanghai in the period from 1^st December 2020 to 16^th January 2022. The vertical dashed line highlights an outlier. Bottom: Autocorrelation (left) and partial autocorrelation functions (right), which are two measures of dependence of the current number of visits on the numbers in the days before (from 1 to 50).

Model selection

We estimate the following 10 different SARIMA specifications. Following the literature, low-order AR and MA terms are adopted for the non-seasonal component, with first-order integration (e.g., Song et al., 2019; Wan and Zhang, 2009), while the seasonal specification is selected after a preliminary AIC-based analysis. Of the 20 models initially specified, we report results for the eight with the lowest AIC. GARCH components with lags of 1 and 2 are specified in line with the standard tourism time-series literature (e.g., Chan et al., 2005; Divino and McAleer, 2010; Tsui and Balli, 2017). Of the 7 GARCH models initially specified, we report results for the 2 with the lowest AIC.

The first three models in our set are two SARIMA(2, 0, 2)(24,0,0)₅ without and with restrictions and a SARIMA(2, 0, 2)(24,0,0)₅ without intercept:

• $M_{1}$ : (1 − ϕ₂L²)(1 − Φ₅L⁵ − Φ₁₀ L¹⁰ − Φ₅₀ L⁵⁰ − Φ₈₀ L⁸⁰ − Φ₁₂₀ L¹²⁰)y_t = μ + (1 + θ₂L²)u_t,

• $M_{2}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Φ₅L⁵ − Φ₁₀ L¹⁰ − Φ₅₀ L⁵⁰ − Φ₈₀ L⁸⁰ − Φ₁₂₀ L¹²⁰)y_t = μ + (1 + θL + θ₂L²)u_t,

• $M_{3}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Φ₅L⁵ − Φ₁₀ L¹⁰ − Φ₅₀ L⁵⁰ − Φ₈₀ L⁸⁰ − Φ₁₂₀ L¹²⁰)y_t = (1 + θL + θ₂L²)u_t.

The second block of models includes a SARIMA(2, 1, 0)(16,0,0)₅ and a SARIMA(0,1, 2)(16,0,0)₅:

• $M_{4}$ : (1 − ϕ₁L − ϕ₂L²)(1 − L)(1 − Θ₅L⁵ − Θ₄₀ L⁴⁰ − Θ₈₀ L⁸⁰)y_t = μ + u_t,

• $M_{5}$ : (1 − L)(1 − Θ₅L⁵ − Θ₄₀ L⁴⁰ − Θ₈₀ L⁸⁰)y_t = μ + (1 + θL + θ₂L²)u_t.

The third block includes three models without MA components, that are a SARIMA(1,1, 0)(16,0,0)₅, a SARIMA(2,0, 0)(16,0,0)₅ and a SARIMA(2,0,0)(24,0,0)₅:

• $M_{6}$ : (1 − ϕ₁L)(1 − L)(1 − Θ₅L⁵ − Θ₄₀L⁴⁰ − Θ₈₀L⁸⁰)y_t = μ + u_t,

• $M_{7}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Θ₅L⁵ − Θ₄₀L⁴⁰ − Θ₈₀L⁸⁰)y_t = μ + u_t,

• $M_{8}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Θ₅L⁵ − Θ₄₀L⁴⁰ − Θ₈₀L⁸⁰ − Θ₁₂₀L¹²⁰)y_t = μ + u_t.

The last block includes heteroskedasticity in the different model specifications. Namely, a SARIMA(2,0,2)(24,0,0)₅ with GARCH(2,2) errors model and a SARIMA(2,0,0)(24,0,0)₅ with GARCH(2,2) errors model:

• $M_{9}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Φ₅L⁵ − Φ₁₀ L¹⁰ − Φ₅₀ L⁵⁰ − Φ₈₀ L⁸⁰ − Φ₁₂₀L¹²⁰)y_t = μ + (1 + θL + θ₂L²)u_t, u_t = σ_tz_t,

• $M_{10}$ : (1 − ϕ₁L − ϕ₂L²)(1 − Θ₅L⁵ − Θ₄₀L⁴⁰ − Θ₈₀L⁸⁰ − Θ₁₂₀L¹²⁰)y_t = μ + u_t, u_t = σ_tz_t,

Where for both models the GARCH(2,2) component writes as $σ_{t}^{2} = γ_{0} + α_{1} σ_{t - 1}^{2} + α_{1} σ_{t - 2}^{2} + β_{1} z_{t - 1}^{2} + β_{2} z_{t - 2}^{2}$ . We conducted various diagnostic checks on the residuals for each model specification, employing autocorrelation and periodogram analyses. We compared the residuals’ quantiles to theoretical quantiles of a normal distribution and applied the Ljung-Box Q-test to assess residual autocorrelation. Figures A.5-A.14 in the Appendix present the results of these diagnostic checks for each model.

The optimal model was selected by minimizing the absolute values of both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The values for AIC and BIC are presented in Table 3. Diagnostic results indicate that the preferred model is

M_{10}

– that is the SARIMA(2, 0, 0)(24,0,0)₅ with GARCH(2,2) errors, followed by

M_{9}

– that is a SARIMA(2, 0, 2)(24,0,0)₅ with GARCH(2,2) errors. In conclusion, the diagnostics favor models with a seasonal component, more lags than models

M_{4}

M_{7}

, and heteroscedastic effects. In the next section, we show that out-of-sample forecasting performance differs from in-sample performance and varies over time, which naturally calls for either sequential selection of the best model or dynamic combination techniques.

Table 3.

In-sample performance for each model $M_{ℓ}$ , ℓ = 1, …, 10. Mean values over 60 days of the Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Square Forecasting Error (RMSFE), Log-Score value (LS), Continuous Ranked Probability Score (CRPS), Directional Accuracy (DA), and Directional Value (DV). Boldface indicates the best-performing model for each measure.

Model	AIC	BIC	MAE	MAPE	RMSFE	LS	CRPS	DA	DV
$M_{1}$	123.56	158.26	2.35	101.52	2.52	−8.44	2.01	0.51	1.10
$M_{2}$	−102.60	−60.19	2.37	103.42	2.49	−16.00	2.13	0.47	1.02
$M_{3}$	−104.51	−65.96	2.36	102.47	2.51	−16.44	2.11	0.48	1.01
$M_{4}$	102.48	129.47	9.84	455.98	11.14	−42.38	9.51	0.48	1.04
$M_{5}$	103.06	130.05	1.81	86.98	2.54	−7.59	1.52	0.51	1.11
$M_{6}$	103.53	126.66	1.06	45.99	1.28	−2.40	0.79	0.49	1.06
$M_{7}$	84.10	111.09	2.14	92.96	2.34	−8.06	1.82	0.48	1.04
$M_{8}$	−105.23	−70.53	2.21	95.21	2.33	−13.86	1.96	0.53	1.12
$M_{9}$	−491.97	−434.14	12.13	549.91	14.60	−37.72	11.43	0.45	0.97
$M_{10}$	−1690	−1650	2.89	126.85	2.95	−50.20	2.87	0.49	1.06

Prediction and combination

Panel (a) of Figure 2 shows the number of visits for the statistical area of Shanghai City from 01-Dec-2020 to 17-Jan-2022 (black solid line). In the same picture, the red dashed line shows the predicted daily number of visits (in Millions) from 18-Jan-2021 to 18-Mar-2022. The top plot shows forecasts for the model with the best in-sample and out-of-sample performance, that is, SARIMA(2,0,0)(24,0,0)₅-GARCH(2,2) $(M_{10})$ . The bottom plot in Panel (a) shows the forecasts for the BMA-MSFE model combination.

Figure 2.

Panel (a): Forecasted visits (dashed red) using the best model $M_{10}$ (top) and the BMA-MSFE combination (bottom). Actual number of visits from 01-Dec-2020 to 17-Jan-2022 in the Shanghai City area (black solid), and forecast confidence intervals (CI) at the 80%, 90%, and 95% levels (gray areas). Panel (b): Mean Square Forecasting Errors (top), MSFE_ℓ,t for each model $M_{ℓ}$ , ℓ = 1, …, 10.

When integrating predicted values into a city management process, careful consideration of various scenarios and acknowledgment of prediction uncertainty are paramount. Recognizing the dynamic nature of visits, it is imperative to address potential variations in anticipated outcomes. To address this, our approach goes beyond mere point prediction based on a single model. We propose two strategies to deal with model uncertainty. The first one is based on using density prediction, where the density incorporates the uncertainty around the point prediction. The second strategy is based on dynamically combining predictions from different models, thus accounting for both possible model specifications and temporal variation in the data-generating process.

Regarding forecast uncertainty, we provide a more comprehensive depiction by including confidence intervals at different levels. For each model $M_{j}$ , F_j,s+1|s(y) is the cumulative distribution of the number of visits at time s + 1, forecasted at time s associated with the predictive density $f_{j, s + 1 | s} (y | M_{j})$ . For each model it is a normal distribution with mean and variance given by conditional mean μ_j,s+1|s and conditional variance $σ_{j, s + 1 | s}^{2}$ of the SARIMAs and SARIMA-GARCHs, whereas for the BMAs is a mixture of normal distributions, F_s+1|s(y) = w_1,sF_1,s+1|s(y) + ⋯ + w_J,sF_J,s+1|s(y). The conditional mean is the point forecast and the forecast confidence intervals (CIs) at time s are the number of visits ${\underset{̲}{y}}_{j, s + 1 | s}$ , and ${\bar{y}}_{j, s + 1 | s}$ , such that the probability that ${\underset{̲}{y}}_{j, s + 1 | s} \leq y_{s + 1} \leq {\bar{y}}_{j, s + 1 | s}$ is equal to α% with α = 80%, 90%, 95%, where ${\underset{̲}{y}}_{j, s + 1 | s}$ and ${\bar{y}}_{j, s + 1 | s}$ are computed using the inverse cumulative distribution. For normal distribution ${\underset{̲}{y}}_{j, s + 1 | s}$ and ${\bar{y}}_{j, s + 1 | s}$ can be evaluated using standard normal quantile function, while for the mixture ${\underset{̲}{y}}_{j, s + 1 | s}$ is the smallest y such that F_j,s+1|s(y) ≥ α/2 and ${\bar{y}}_{j, s + 1 | s}$ is the smallest y such that F_j,s+1|s(y) ≥ 1 − α/2, which can be computed numerically using bisection or a fine grid. These intervals provide insights into expected values and indicate the potential deviation of future observations from the point forecast, contributing to a nuanced understanding of the model’s predictive uncertainty. The grey areas represent the uncertainty around the point prediction, measured by CIs at 80% (dark grey), 90% (grey), and 95% (light grey). If future prediction errors are stable, unbiased, and normally distributed, approximately 80% of future outcomes are expected to fall within the 80% uncertainty bands shown in dark grey, with the remaining outcomes falling symmetrically above and below the bands. In formulas, the coverage probability is computed as $τ^{- 1} \sum_{s = t - τ + 1}^{t} I (μ_{j, s + 1 | s} \in ({\underset{̲}{y}}_{j, s + 1 | s}, {\bar{y}}_{j, s + 1 | s}))$ . This comprehensive perspective ensures decision-makers a more robust and informed response to potential scenarios for a better city management. The point and density forecasts obtained with the 10 models are in Figure B.15 of the Appendix.

Regarding model uncertainty, we evaluated the out-of-sample predictive performance of each model described in Section 3. Point-forecast measures such as MAE, MAPE, RMSFE, and density-forecast measures such as Log-Score (LS), and the Continuous Ranked Probability Score (CRPS), indicate that models without the GARCH effects outperform the ones with GARCH effects (see Table 4). The results reported in the top plot of Figure 3 show the greater predictive ability (larger weights on average) of Model 7, the SARIMA(2,0,0)(16,0,0)₅ without the MA component, followed by Model 8, the SARIMA(2,0,0)(24,0,0)₅ without restrictions and Model 2, the SARIMA(2,0,2)(24,0,0)₅ with restrictions (see Figure B.15 of the Appendix). Nevertheless, when model parsimony is taken into consideration, as indicated by the AIC and BIC performance measures, GARCH models

M_{9}

and

M_{10}

exhibit better performance. Compared to the other models, they are able to capture the reduced variability in that last part of the sample (see last row in Figure B.15 of the Appendix) and the decrease in the flows as confirmed by the directional accuracy measures, DA and DV, favor models in the GARCH family.

Table 4.

Out-of-sample forecast accuracy evaluated sequentially for each model $M_{ℓ}$ , ℓ = 1, …, 10. Mean values over 60 days, respectively of the Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Square Forecasting Error (RMSFE), Log-Score value (LS), Continuous Ranked Probability Score (CRPS), Directional Accuracy (DA), and Directional Value (DV). Boldface indicates the best-performing model for each measure.

Model	AIC	BIC	MAE	MAPE	RMSFE	LS	CRPS	DA	DV
$M_{1}$	76.83	112.96	0.44	22.33	0.60	−1.39	0.31	0.52	0.97
$M_{2}$	−188.91	−144.76	0.36	18.14	0.49	−1.34	0.30	0.42	0.79
$M_{3}$	−190.80	−150.66	0.37	18.72	0.50	−1.35	0.31	0.42	0.79
$M_{4}$	52.81	80.90	0.67	34.97	0.77	−2.15	0.57	0.52	0.98
$M_{5}$	53.49	81.58	0.67	34.73	0.76	−2.12	0.55	0.52	0.98
$M_{6}$	54.38	78.46	0.67	34.84	0.77	−2.23	0.65	0.5	0.95
$M_{7}$	31.27	59.36	0.33	17.43	0.44	−1.41	0.25	0.5	0.94
$M_{8}$	−191.30	−155.18	0.35	17.87	0.48	−1.34	0.27	0.43	0.83
$M_{9}$	−1200	−1140	0.59	29.85	0.66	−1.37	0.43	0.57	1.08
$M_{10}$	−2480	−2440	0.67	33.22	1.01	−1.56	0.45	0.43	0.82

Figure 3.

Combination weights $w_{ℓ, t}^{MSFE}$ for each model $M_{ℓ}$ , ℓ = 1, …, 10 for t = 1, …, 60, that is the period from 17^th January 2022 to 17^th March 2022.

We apply BMA and use time-varying combination weights based on rolling estimates of the model’s forecasting performance measures. Panel (b) of Figure 2 shows the model MSFE over time, and Figure 3 shows BMA weights based on the MSFEs. There are periods, e.g., the beginning of January and February, when Models 1 and 4 outperformed, and other periods when Models 6 and 7 performed better. Models without GARCH effects outperform GARCH models, except during periods of higher series variability. The time variations in predictive ability are confirmed by other measures, which induce time variations in the BMA weights (see Figures C.18-C.19 in the Appendix). The time-varying nature of the problem naturally calls for either the sequential selection of the best model or the combination of models. We apply the different BMA procedures presented in the previous section and obtain the out-of-sample point and density predictions reported in Figures C.16-C.17 of the Appendix. The BMA specifications exhibit different levels of forecasting errors (see Panel (a) of Figure 4). In our results, BMA based on MAE, MAPE, and RMSFE (top left) yields nearly identical errors; similarly, FMA based on AIC, BIC, and HQ (bottom right) yields similar errors. BMA-LS and BMA-CRPS exhibit different performance, with BMA-CRPS usually performing better (bottom right). The BMA based on DA, performs worse than its exponential version, that is, the BMA-EHR (middle-right). On average, the point-forecast performance of BMA-MSFE is superior to that of the other weighting strategies. In particular, it also outperforms the best individual model, as shown by comparing the black lines in the left and right plots of Panel (b) in Figure 4. This indicates that the combination based on MSFE weights can exploit the information across the different models more effectively than relying on a single specification. The out-of-sample point and density forecast of the BMA-MSFE is reported for comparison purposes in the second plot of Panel (a) of Figure 4. The best model combination provides a better fit to seasonality and variability than the best individual model (compare with the first plot in the same panel). Regarding the forecast CIs coverage, there are three groups of models. The predictive intervals of the models with the best density forecasts (large CRPS), $M_{4}$ , $M_{5}$ , and $M_{6}$ , are wider and include all realizations, indicating over-coverage. Models $M_{1}$ , $M_{2}$ , $M_{3}$ , $M_{7}$ , and $M_{8}$ have better coverage and point forecast performances, but the lowest density forecast performances (e.g, low CRPS). The third group includes the SARIMA-GARCHs $M_{9}$ and $M_{10}$ , which have narrower intervals, good coverage, and intermediate density- and point-forecasting abilities. The combined forecasts provide a compromise among the three model classes and better capture the dynamics and variability of the tourist flows.

Figure 4.

Forecasting errors (in logarithms). Panel (a) Alternative BMA specifications. Panel (b) BMA with MSFE weights (left) and the best model (right). Model weights w_ℓ,t are based on MSFEs (black) and ${\tilde{w}}_{ℓ, t}$ are based on cost–weighted MSFEs (red), ℓ = 1, …, 10, t = 1, …, 60, that is the period from 17^th January 2022 to 17^th March 2022. Tourism costs are the ones incurred by the municipality of Shanghai for cleaning services.

Among the various urban dimensions affected by tourism, waste management is among the most significantly affected, as increased visitor flows often strain existing collection and disposal infrastructure (e.g., see Xiao et al., 2020; Yuxi et al., 2023). Thus, one can expect that the economic implications of a forecast error differ depending on whether the cleaning and waste management system is operating under high or low pressure. Indeed, large forecasting errors tend to occur during holiday periods when visit volumes spike. Cost-based weighting schemes reduce the influence of these large errors and tend to favor models that perform more reliably during non-holiday periods. To exploit the economic relevance of forecasting error, we use the average daily and per-visit cleaning costs. We assumed an average cost of 178 (in 10,000 Yuan/Day) and 2.3 million visits per day during the non-holiday periods and 440 (in 10,000 Yuan/Day) during holidays, with 12 million trips (Culture and Bureau, 2024). The relative cost per day and visit, c_s, is equal to 0.7 during non-holiday periods and 0.3 during holiday periods. With these weights, the WMSFE is computed for each model as in equation (18).

The two plots in Panel (b) of Figure 4 show the errors (in logarithms) of forecasts based on equally weighted errors (black line) and on cost-weighted errors (red line). The forecasting errors for prediction averaging and the best model are shown in the top and bottom plots, respectively. In the Appendix, Figure C.20 shows the MSFE-based and cost-based weights for each model. Comparing the results in the two plots shows that a model combination strategy yields smaller forecasting errors than a prediction strategy that selects the best model. In addition, comparing the red and black lines shows that using a cost-weighted combination or selection strategy can reduce forecasting errors. Finally, among measures that incorporate an economic value component, BMA-WMFSE performs better than BMA specifications based on more standard default measures, such as EEDF (see the middle–right plot of Panel (a)). This result suggests that the proposed cost-weighted measure is more effective at capturing the relevant economic trade-offs embedded in the forecasting exercise and leads to improved forecast-combination performance.

Conclusion

Using Big Data and predictive models is crucial when making timely policy decisions for tourism management. Usually, when predicting tourist flows, multiple models are applied, which adds uncertainty and poses challenges for their use in supporting policy-making. We propose a flexible yet straightforward procedure based on forecast combination models to address model uncertainty in decision-making. First, using Telcom Big Data to forecast daily visits to Shanghai, we show that model uncertainty arises when using prediction models from classes commonly used in practice, such as ARIMA, ARIMA with seasonal and GARCH effects.

The empirical analysis identifies SARIMA with seasonal dynamics and heteroskedastic errors (GARCH) as the most suitable model class for forecasting tourism flows. This model class performs consistently well both in-sample and out-of-sample, demonstrating robustness in estimation and forecasting. However, forecasting accuracy varies over time, suggesting that reliance on a single model may not always be optimal. For this reason, forecast combination strategies prove particularly valuable. By integrating information from multiple models rather than relying exclusively on a single best specification, combined forecasts achieve lower prediction errors and greater stability over time. This improvement becomes even more pronounced when cost-weighted combination methods are employed, as they allow the forecasting framework to place greater emphasis on models that perform better under specific conditions. These findings underscore the value of flexible and adaptive forecasting frameworks in tourism analysis, where uncertainty and rapid changes in demand patterns are intrinsic features of the sector.

From a policy perspective, these findings carry important implications for tourism destinations. The results provide practical evidence that more accurate forecasting tools can significantly support public decision-making. Improved predictions of tourism flows allow policymakers and destination management organizations to anticipate demand fluctuations, mitigate congestion during peak periods, and better plan infrastructure, mobility systems, and visitor services. More reliable forecasts also help optimize the allocation of public resources, enabling authorities to design and implement policies that balance tourism growth with sustainability objectives and the well-being of local communities. Moreover, the results highlight the strategic importance of integrating diverse data sources and analytical approaches into tourism governance. The increasing availability of Big Data—particularly high-frequency mobility data—creates opportunities to expand operational management tools and develop more flexible analytical frameworks. In this context, the ability to combine different modelling approaches becomes particularly valuable, as it allows policymakers to continuously incorporate new data streams and methodological advances. Such an adaptive analytical environment supports more responsive, evidence-based tourism management and strengthens destinations’ capacity to address complex challenges related to demand volatility, resource management, and sustainable development.

Despite these encouraging results, some limitations remain. The modelling framework relies on traditional time-series specifications and could be further improved by exploring hybrid approaches that combine statistical and machine learning methods. Moreover, the analysis focuses on a specific dataset and context; future research should test the proposed models across different destinations and time horizons to assess their generalizability and to support the development of more adaptive forecasting tools. Future research should also focus on improving data availability at the individual point-of-interest level, with the aim of testing the applicability of the models developed in this study at a finer spatial scale and analysing tourism dynamics across individual attractions within the same city. Such developments could provide policymakers with more detailed information to support the planning and management of tourism services.

Supplemental material

Supplemental Material - Forecasting daily visits in Shanghai with Model combination and Telco big data

Supplemental Material for Forecasting daily visits in Shanghai with Model combination and Telco big data by Nicola Camatti, Giulia Carallo, Roberto Casarin, Xiang Feng in Tourism Economics

Footnotes

Acknowledgments

The computation has been performed using MATLAB through the HPC cluster at the Venice Centre for Risk Analytics at Ca’ Foscari University of Venice.

Nicola Camatti

Giulia Carallo

Roberto Casarin

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: RC acknowledges the support from the European Union - Next Generation EU - Project ‘GRINS - Growing Resilient, INclusive and Sustainable’; the National Recovery and Resilience Plan (NRRP) PE00000018.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The complete reproducibility package can be accessed at: https://doi.org/10.24433/CO.3148292.v1. (Giulia et al., 2026)

Supplemental material

Supplemental material for this article is available online

Author biographies

Nicola Camatti is an Associate Professor of Applied Economics at Ca’ Foscari University of Venice. His research explores tourism business ecosystems, sustainable tourism planning, and regional development. He focuses on developing tourism decision support systems (DSS) and applying big data analytics for destination management and promotion.

Roberto Casarin is a Full Professor of Econometrics at Ca’ Foscari University of Venice, Director of the Venice Centre in Economic and Risk Analytics (VERA) and Fellow of the International Society for Bayesian Analysis (ISBA). His publications and current research focus on Bayesian inference and modeling, and computational methods, with applications to time series analysis.

Giulia Carallo is a Researcher at University of Salento, specializing in econometrics, Bayesian analysis, and time series modeling. Her research investigates the statistical dynamics of tourism systems and economic processes. She collaborates on several national and international projects focused on monitoring tourism ecosystems and data-driven destination management.

Xiang Feng is an Associate Professor at the Institute of Tourism, Shanghai Normal University, China. Her research focuses on urban tourism, intercity tourist flows, and tourism geography. She is a Fellow of EURICUR (EU) and combines academic expertise with extensive professional experience as a planner and consultant for major Chinese tourism organization.

References

Aastveit

Cross

van Dijk

(2023) Quantifying time-varying forecast uncertainty and risk for the real price of oil. Journal of Business & Economic Statistics 41(2): 523–537. https://doi.org/10.1080/07350015.2022.2039159

Araya

Aduda

Berhane

(2024) A hybrid GARCH and deep learning method for volatility prediction. Journal of Applied Mathematics 2024(1): 6305525. https://doi.org/10.1155/2024/6305525

Ardito

Cerchione

Del Vecchio

, et al. (2019) Big data in smart tourism: challenges, issues and opportunities. Current Issues in Tourism 22(15): 1805–1809. https://doi.org/10.1080/13683500.2019.1612860

Armstrong

Green

Graefe

(2015) Golden rule of forecasting: be conservative. Journal of Business Research 68(8): 1717–1731. https://doi.org/10.1016/j.jbusres.2015.03.031

Bassetti

Casarin

Ravazzolo

(2018) Bayesian nonparametric calibration and combination of predictive distributions. Journal of the American Statistical Association 113(522): 675–685. https://doi.org/10.1080/01621459.2016.1273117

Bates

Granger

(1969) The combination of forecasts. Journal of the Operational Research Society 20(4): 451–468. https://doi.org/10.1057/jors.1969.103

J-W

Liu

(2020) Daily tourism volume forecasting for tourist attractions. Annals of Tourism Research 83: 102923. https://doi.org/10.1016/j.annals.2020.102923

J-W

Han

T-Y

(2022) International tourism demand forecasting with machine learning models: the power of the number of lagged inputs. Tourism Economics 28(3): 621–645. https://doi.org/10.1177/1354816620976954

Billio

Casarin

Ravazzolo

, et al. (2013) Time-varying combinations of predictive densities using nonlinear filtering. Journal of Econometrics 177(2): 213–232. https://doi.org/10.1016/j.jeconom.2013.04.009

10.

Bisht

Kumar

Nag

, et al. (2025) Managing seasonality and overtourism for sustainable tourism development: challenges and strategies. International Journal of Research and Review 12(3): 443–453. https://doi.org/10.52403/ijrr.20250355

11.

Blázquez-Salom

Cladera

Sard

(2023) Identifying the sustainability indicators of overtourism and undertourism in majorca. Journal of Sustainable Tourism 31(7): 1694–1718. https://doi.org/10.1080/09669582.2021.1942478

12.

Bollerslev

(1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics 69(3): 542–547. https://doi.org/10.2307/1925546

13.

Box

Jenkins

(1976) Analysis: Forecasting and Control, Vol. 10.

14.

Box

Jenkins

Reinsel

, et al. (2015) Time Series Analysis: Forecasting and Control. John Wiley & Sons.

15.

Camatti

Carallo

Casarin

, et al. (2024) Measuring daily tourism mobility spillover at the intra-metropolis level with mobile positioning data. Regional Studies & Regional Science 11(1): 701–723. https://doi.org/10.1080/21681376.2024.2412015

16.

Chan

Lim

McAleer

(2005) Modelling multivariate international tourism demand and volatility. Tourism Management 26(3): 459–471. https://doi.org/10.1016/j.tourman.2004.02.013

17.

Chen

Law

(2025) Tourism demand forecasting: a novel multi-channel imaging model. Journal of Tourism Futures. https://doi.org/10.1108/jtf-12-2024-0263

18.

Chu

F-L

(2009) Forecasting tourism demand with arma-based methods. Tourism Management 30(5): 740–751. https://doi.org/10.1016/j.tourman.2008.10.016

19.

Claveria

Monte

Torra

(2015) Tourism demand forecasting with neural network models: different ways of treating information. International Journal of Tourism Research 17(5): 492–500. https://doi.org/10.1002/jtr.2016

20.

Clements

Hendry

(2004) Pooling of forecasts. The Econometrics Journal 7(1): 1–31. https://doi.org/10.1111/j.1368-423x.2004.00119.x

21.

Coshall

Charlesworth

(2011) A management orientated approach to combination forecasting of tourism demand. Tourism Management 32(4): 759–769. https://doi.org/10.1016/j.tourman.2010.06.011

22.

Costantini

Cuaresma

Hlouskova

(2016) Forecasting errors, directional accuracy and profitability of currency trading: the case of eur/usd exchange rate. Journal of Forecasting 35(7): 652–668. https://doi.org/10.1002/for.2398

23.

Crespo Cuaresma

Feldkircher

(2013) Spatial filtering, model uncertainty and the speed of income convergence in Europe. Journal of Applied Econometrics 28(4): 720–741. https://doi.org/10.1002/jae.2277

24.

Culture

Bureau

(2024) Shanghai Monthly Tourism Data Report. https://whlyj.sh.gov.cn/tjzl/20250527/4468038bff1a454497366fcc99e91941.html

25.

Divino

McAleer

(2010) Modelling and forecasting daily international mass tourism to Peru. Tourism Management 31(6): 846–854. https://doi.org/10.1016/j.tourman.2009.09.002

26.

Duan

Xie

Morrison

(2022) Tourism crises and impacts on destinations: a systematic review of the tourism and hospitality literature. Journal of Hospitality & Tourism Research 46(4): 667–695. https://doi.org/10.1177/1096348021994194

27.

Fan

Uzunboylu

Efthymiou

, et al. (2025) Innovation and value creation amidst global challenges and crises in tourism and hospitality industry. In: Global Challenges and Uncertainty in Tourism and Hospitality, Volume II: Marketing Advancements and Value Creation. Springer, pp. 1–25.

28.

Fawcett

Kapetanios

Mitchell

, et al. (2015) Generalised density forecast combinations. Journal of Econometrics 188: 150–165. https://doi.org/10.1016/j.jeconom.2015.02.047

29.

Fernández

Ley

Steel

MFJ

(2001) Model uncertainty in cross-country growth regressions. Journal of Applied Econometrics 16(5): 563–576. https://doi.org/10.1002/jae.623

30.

Genest

Schervish

(1985) Modeling expert judgments for Bayesian updating. Annals of Statistics 13: 1198–1212. https://doi.org/10.1214/aos/1176349664

31.

Geweke

Amisano

(2011) Optimal prediction pools. Journal of Econometrics 164(1): 130–141. https://doi.org/10.1016/j.jeconom.2011.02.017

32.

Ghalehkhondabi

Ardjmand

Young

, et al. (2019) A review of demand forecasting models and methodological developments within tourism and passenger transportation industry. Journal of Tourism Futures 5(1): 75–93. https://doi.org/10.1108/jtf-10-2018-0061

33.

Gil-Alana

Gracia

FPD

Cunado

(2004) Seasonal fractional integration in the Spanish tourism quarterly time series. Journal of Travel Research 42(4): 408–414. https://doi.org/10.1177/0047287503258843

34.

Giulia

Roberto

Nicola

, et al. (2026) Forecasting daily visits in Shanghai with model combination and telco big data [Source Code]. https://doi.org/10.24433/CO.3148292.v1

35.

Gneiting

Ranjan

(2011) Comparing density forecasts using threshold and quantile weighted scoring rules. Journal of Business and Economic Statistics 29: 411–422. https://doi.org/10.1198/jbes.2010.08110

36.

Gneiting

Ranjan

(2013) Combining predictive distributions. Electronic Journal of Statistics 7: 1747–1782. https://doi.org/10.1214/13-ejs823

37.

Guizzardi

Pons

FME

Angelini

, et al. (2021) Big data from dynamic pricing: a smart approach to tourism demand forecasting. International Journal of Forecasting 37(3): 1049–1060. https://doi.org/10.1016/j.ijforecast.2020.11.006

38.

Gunter

Önder

(2016) Forecasting city arrivals with Google Analytics. Annals of Tourism Research 61: 199–212. https://doi.org/10.1016/j.annals.2016.10.007

39.

Hassani

Silva

Antonakakis

, et al. (2017) Forecasting accuracy evaluation of tourist arrivals. Annals of Tourism Research 63: 112–127. https://doi.org/10.1016/j.annals.2017.01.008

40.

Qian

(2025) Forecasting tourist arrivals using stl-xgboost method. Tourism Economics 32(2): 408–436. https://doi.org/10.1177/13548166241313411

41.

Hewapathirana

(2025) Advancing tourism demand forecasting in Sri Lanka: evaluating the performance of machine learning models and the impact of social media data integration. Journal of Tourism Futures 11(2): 261–285. https://doi.org/10.1108/jtf-06-2023-0149

42.

Höpken

Eberle

Fuchs

, et al. (2021) Improving tourist arrival prediction: a big data and artificial neural network approach. Journal of Travel Research 60(5): 998–1017. https://doi.org/10.1177/0047287520921244

43.

Huang

Zhang

Ding

(2017) The baidu index: uses in predicting tourism flows–a case study of the forbidden city. Tourism Management 58: 301–306. https://doi.org/10.1016/j.tourman.2016.03.015

44.

Jakovlev

Kitanov

(2024) The importance of the planning function in the business financial decision-making process in tourism enterprises. International Journal of Economics, Management and Tourism 4(2): 109–116. https://doi.org/10.46763/ijemt2442111j

45.

Johnson

West

(2025) Bayesian predictive synthesis with outcome-dependent pools. Statistical Science 40(1): 109–127. https://doi.org/10.1214/24-sts954

46.

Kandt

Batty

(2021) Smart cities, big data and urban policy: towards urban analytics for the long run. Cities 109: 102992. https://doi.org/10.1016/j.cities.2020.102992

47.

Koc

Altinay

(2007) An analysis of seasonality in monthly per-person tourist spending in Turkish inbound tourism from a market segmentation perspective. Tourism Management 28(1): 227–237. https://doi.org/10.1016/j.tourman.2006.01.003

48.

LeSage

Parent

(2007) Bayesian model averaging for spatial econometric models. Geographical Analysis 39(3): 241–267. https://doi.org/10.1111/j.1538-4632.2007.00703.x

49.

Goh

Hung

, et al. (2018) Relative climate index and its effect on seasonal tourism demand. Journal of Travel Research 57(2): 178–192. https://doi.org/10.1177/0047287516687409

50.

Liang

Y-H

(2014) Forecasting models for Taiwanese tourism demand after allowance for Mainland China tourists visiting Taiwan. Computers & Industrial Engineering 74: 111–119. https://doi.org/10.1016/j.cie.2014.04.005

51.

Liang

Shu

, et al. (2025) Tourism demand forecasting using graph neural network. Current Issues in Tourism 28(6): 982–1001. https://doi.org/10.1080/13683500.2024.2320851

52.

Lim

McAleer

(2002) Time series forecasts of international travel demand for Australia. Tourism Management 23(4): 389–396. https://doi.org/10.1016/s0261-5177(01)00098-x

53.

Lin

(2024) Tourism and economic growth: assessing the significance of sustainable competitiveness using a dynamic panel data approach. International Journal of Tourism Research 26(5): e2760. https://doi.org/10.1002/jtr.2760

54.

Lindley

Tversky

Brown

(1979) On the reconciliation of probability assessments. Journal of the Royal Statistical Society 142(2): 146–180, Series A (General). https://doi.org/10.2307/2345078

55.

Ling

McAleer

(2003a) Asymptotic theory for a vector arma-garch model. Econometric Theory 19(2): 280–310. https://doi.org/10.1017/s0266466603192092

56.

Ling

McAleer

(2003b) On adaptive estimation in nonstationary arma models with garch errors. Annals of Statistics 31(2): 642–674. https://doi.org/10.1214/aos/1051027884

57.

Liu

Chen

Y-n.

Qiu

, et al. (2019) Forecast of the tourist volume of Sanya city by XGBoost model and GM model. In: 2019 International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery (CyberC), pp. 166–173.

58.

Liu

Chen

, et al. (2025) Tourism demand forecasting in normal and crisis times: combining bootstrap-aggregating and Bayesian approaches. Journal of Hospitality & Tourism Research 50(3): 323–340. https://doi.org/10.1177/10963480251313492

59.

Llewellyn

Ross

Ryan-Saha

(2023) Covid-era forecasting: Google trends and window and model averaging. Annals of Tourism Research 103: 103660. https://doi.org/10.1016/j.annals.2023.103660

60.

Chen

(2023) Dynamic evaluation and forecasting analysis of touristic ecological carrying capacity of forest parks in China. Forests 15(1): 38. https://doi.org/10.3390/f15010038

61.

McAlinn

(2021) Mixed-frequency Bayesian predictive synthesis for economic nowcasting. Journal of the Royal Statistical Society - Series C: Applied Statistics 70(5): 1143–1163. https://doi.org/10.1111/rssc.12500

62.

McAlinn

West

(2019) Dynamic Bayesian predictive synthesis in time series forecasting. Journal of Econometrics 210(1): 155–169. https://doi.org/10.1016/j.jeconom.2018.11.010

63.

McAlinn

Aastveit

Nakajima

, et al. (2020) Multivariate bayesian predictive synthesis in macroeconomic forecasting. Journal of the American Statistical Association 115(531): 1092–1110. https://doi.org/10.1080/01621459.2019.1660171

64.

Metulini

Carpita

(2024) Modeling and forecasting traffic flows with mobile phone big data in flooding risk areas to support a data-driven decision making. Annals of Operations Research 342(3): 1629–1654. https://doi.org/10.1007/s10479-023-05195-8

65.

Moral-Benito

(2015) Model averaging in economics: an overview. Journal of Economic Surveys 29(1): 46–75. https://doi.org/10.1111/joes.12044

66.

Mussoni

Vici

(2025) Exogenous shocks and resilience in tourism. a regional science perspective. Scienze Regionali 24(3): 297–309.

67.

C-O

Morzuch

(2005) Evaluating time-series models to forecast the demand for tourism in Singapore: comparing within-sample and postsample results. Journal of Travel Research 43(4): 404–413. https://doi.org/10.1177/0047287505274653

68.

Önder

Gunter

(2016) Forecasting tourism demand with Google Trends for a major European city destination. Tourism Analysis 21(2-3): 203–220. https://doi.org/10.3727/108354216x14559233984773

69.

Pai

P-F

Hung

K-C

Lin

K-P

(2014) Tourism demand forecasting using novel hybrid system. Expert Systems with Applications 41(8): 3691–3702. https://doi.org/10.1016/j.eswa.2013.12.007

70.

Pan

Liao

Wang

, et al. (2025) Tourism forecasting: a dynamic spatiotemporal model. Annals of Tourism Research 110: 103871. https://doi.org/10.1016/j.annals.2024.103871

71.

Peng

Song

Crouch

(2014) A meta-analysis of international tourism demand forecasting and implications for practice. Tourism Management 45: 181–193. https://doi.org/10.1016/j.tourman.2014.04.005

72.

Prayag

Jiang

Chowdhury

, et al. (2024) Building dynamic capabilities and organizational resilience in tourism firms during covid-19: a staged approach. Journal of Travel Research 63(3): 713–740. https://doi.org/10.1177/00472875231164976

73.

Punia

Shankar

(2022) Predictive analytics for demand forecasting: a deep learning-based decision support system. Knowledge-Based Systems 258: 109956. https://doi.org/10.1016/j.knosys.2022.109956

74.

Raftery

Madigan

Hoeting

(1997) Bayesian model averaging for linear regression models. Journal of the American Statistical Association 92(437): 179–191. https://doi.org/10.1080/01621459.1997.10473615

75.

Ramos

Yamaka

Alorda

, et al. (2021) High-frequency forecasting from mobile devices’ bigdata: an application to tourism destinations’ crowdedness. International Journal of Contemporary Hospitality Management 33(6): 1977–2000. https://doi.org/10.1108/ijchm-10-2020-1170

76.

Sejati

Putri

Tyas

, et al. (2025) Predicting urban carrying capacity to support sustainable tourism using gis. Journal of Policy Research in Tourism, Leisure and Events 17(3): 632–655. https://doi.org/10.1080/19407963.2023.2279065

77.

Shen

Song

(2011) Combination forecasts of international tourism demand. Annals of Tourism Research 38(1): 72–89. https://doi.org/10.1016/j.annals.2010.05.003

78.

Song

(2008) Tourism demand modelling and forecasting—a review of recent research. Tourism Management 29(2): 203–220. https://doi.org/10.1016/j.tourman.2007.07.016

79.

Song

Zhang

(2025) Tourism demand modelling and forecasting: a horizon 2050 paper. Tourism Review 80(1): 8–27. https://doi.org/10.1108/tr-12-2023-0882

80.

Song

Witt

, et al. (2011) Forecasting tourist arrivals using time-varying parameter structural time series models. International Journal of Forecasting 27(3): 855–869. https://doi.org/10.1016/j.ijforecast.2010.06.001

81.

Song

Qiu

Park

(2019) A review of research on tourism demand forecasting: launching the annals of tourism research curated collection on tourism demand forecasting. Annals of Tourism Research 75: 338–362. https://doi.org/10.1016/j.annals.2018.12.001

82.

Sun

Wei

Tsui

K-L

, et al. (2019) Forecasting tourist arrivals with machine learning and internet search index. Tourism Management 70: 1–10. https://doi.org/10.1016/j.tourman.2018.07.010

83.

Sun

Zhang

, et al. (2023) Forecasting tourism demand with a new time-varying forecast averaging approach. Journal of Travel Research 62(2): 305–323. https://doi.org/10.1177/00472875211061206

84.

Tallman

West

(2024) Bayesian predictive decision synthesis. Journal of the Royal Statistical Society Series B: Statistical Methodology 86(2): 340–363. https://doi.org/10.1093/jrsssb/qkad109

85.

Tsui

WHK

Balli

(2017) International arrivals forecasting for Australian airports and the impact of tourism marketing expenditure. Tourism Economics 23(2): 403–428. https://doi.org/10.5367/te.2015.0507

86.

Vanegas Sr

(2013) Co-integration and error correction estimation to forecast tourism in El Salvador. Journal of Travel & Tourism Marketing 30(6): 523–537. https://doi.org/10.1080/10548408.2013.810992

87.

Volchek

Liu

Song

, et al. (2019) Forecasting tourist arrivals at attractions: search engine empowered methodologies. Tourism Economics 25(3): 425–447. https://doi.org/10.1177/1354816618811558

88.

Wan

Song

(2018) Forecasting turning points in tourism growth. Annals of Tourism Research 72: 156–167. https://doi.org/10.1016/j.annals.2018.07.010

89.

Wan

Zhang

(2009) On the use of model averaging in tourism research. Annals of Tourism Research 36(3): 525–532. https://doi.org/10.1016/j.annals.2009.03.004

90.

Wang

Xie

Jie

(2019) A mathematical analysis for the forecast research on tourism carrying capacity to promote the effective and sustainable development of tourism. Discrete and Continuous Dynamical Systems - Series S 12: 837–847. https://doi.org/10.3934/dcdss.2019056

91.

Wang

Hyndman

, et al. (2023) Forecast combinations: an over 50-year review. International Journal of Forecasting 39(4): 1518–1547. https://doi.org/10.1016/j.ijforecast.2022.11.005

92.

Wei

Mukherjee

(2024) Analyzing and forecasting service demands using human mobility data: a two-stage predictive framework with decomposition and multivariate analysis. Expert Systems with Applications 238: 121698. https://doi.org/10.1016/j.eswa.2023.121698

93.

Wen

Liu

Song

(2019) Forecasting tourism demand using search query data: a hybrid modelling approach. Tourism Economics 25(3): 309–329. https://doi.org/10.1177/1354816618768317

94.

West

(1984) Bayesian aggregation. Journal of the Royal Statistical Society: Series A 147(4): 600–607. https://doi.org/10.2307/2981847

95.

West

(1988) Modelling expert opinion (with discussion). Bayesian Statistics 3: 493–508.

96.

West

(1992) Modelling agent forecast distributions. Journal of the Royal Statistical Society Series B: Statistical Methodology 54(2): 553–567. https://doi.org/10.1111/j.2517-6161.1992.tb01896.x

97.

Wong

Song

Witt

, et al. (2007) Tourism forecasting: to combine or not to combine? Tourism Management 28(4): 1068–1078. https://doi.org/10.1016/j.tourman.2006.08.003

98.

Zhong

, et al. (2025) Tourism and hospitality forecasting with big data: a systematic review of the literature. Journal of Hospitality & Tourism Research 49(3): 615–634. https://doi.org/10.1177/10963480231223151

99.

Xiao

Dong

Geng

, et al. (2020) An overview of the municipal solid waste management modes and innovations in Shanghai, China. Environmental Science and Pollution Research 27(24): 29943–29953. https://doi.org/10.1007/s11356-020-09398-5

100.

Yang

Pan

Evans

, et al. (2015) Forecasting Chinese tourist volume with search engine data. Tourism Management 46: 386–397. https://doi.org/10.1016/j.tourman.2014.07.019

101.

Yuxi

Filimonau

Ling-en

, et al. (2023) The impact of tourism on municipal solid waste generation in China. Journal of Cleaner Production 427: 139255. https://doi.org/10.1016/j.jclepro.2023.139255

102.

Zheng

(2024) Tourism demand forecasting using complex network theory. Asia Pacific Journal of Tourism Research 29(3): 302–320. https://doi.org/10.1080/10941665.2024.2324536

103.

Zhou

(2021) Prediction of urban and rural tourism economic forecast based on machine learning. Scientific Programming 2021(1): 4072499. https://doi.org/10.1155/2021/4072499

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