#
Bernoulli Time Series Modelling with Application to Accommodation Tourism Demand^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

## 2. Methods

#### 2.1. Bernoulli Time Series Modeling

#### 2.2. Log-Log Modeling BeTSUF: Estimated by Generalized Method Moments HAC-Newey-West (GMM + HAC-Newey-West)

#### 2.3. Accuracy of the Predictive Capacity of the Models

## 3. A Case Study in the Social Sciences: The Dichotomy of Choice between Hotels and Tourist Apartments

#### 3.1. Data and Correlations

#### 3.2. Empirical Results

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Maasoumi, E.; Medeiros, M.C. The link between statistical learning theory and econometrics: Applications in economics, finance, and marketing. Econ. Rev.
**2010**, 29, 470–475. [Google Scholar] [CrossRef] - Shoja, M.; Soofi, E.S. Uncertainty, information, and disagreement of economic forecasters. Econ. Rev.
**2017**, 36, 796–817. [Google Scholar] [CrossRef] - Maasoumi, E. A compendium to information theory in economics and econometrics. Econ. Rev.
**1993**, 12, 137–181. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef][Green Version] - Lehmann, E.L.; Casella, G. Theory of Point Estimation, 2nd ed.; Springer: Berlin, Germany, 1998. [Google Scholar]
- Harvey, A. Chapter 7 forecasting with unobserved components time series models. Handb. Econ. Forecast.
**2006**. [Google Scholar] [CrossRef] - Ruiz-Reina, M.Á. Entropy of tourism: The unseen side of tourism accommodation. In Proceedings of the International Conference on Applied Research in Business, Management and Economics, Barcelona, Spain, 12–14 December 2019. [Google Scholar]
- Box, G.E.P.; Jenkins, G.M.; Reinsel, G.C. Time Series Analysis: Forecasting and Control, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Ruiz-Reina, M.Á. Big data: Does it really improve forecasting techniques for tourism demand in spain? In International Conference on Time Series and Forecasting; Godel Impresiones Digitales S.L.: Granada, Spain, 2019; pp. 694–706. [Google Scholar]
- Li, G.; Song, H.; Witt, S.F. Recent developments in econometric modeling and forecasting. J. Travel Res.
**2005**. [Google Scholar] [CrossRef][Green Version] - Song, H.; Li, G. Tourism demand modelling and forecasting-A review of recent research. Tour. Manag.
**2008**, 29, 203–220. [Google Scholar] [CrossRef][Green Version] - Peng, B.; Song, H.; Crouch, G.I. A meta-analysis of international tourism demand forecasting and implications for practice. Tour. Manag.
**2014**. [Google Scholar] [CrossRef] - Jiao, E.X.; Chen, J.L. Tourism forecasting: A review of methodological developments over the last decade. Tour. Econ.
**2019**. [Google Scholar] [CrossRef] - Wu, D.C.; Song, H.; Shen, S. New developments in tourism and hotel demand modeling and forecasting. Int. J. Contemp. Hosp. Manag.
**2017**, 29, 507–529. [Google Scholar] [CrossRef] - Mariani, M.; Baggio, R.; Fuchs, M.; Höepken, W. Business intelligence and big data in hospitality and tourism: A systematic literature review. Int. J. Contemp. Hosp. Manag.
**2018**. [Google Scholar] [CrossRef][Green Version] - Li, J.; Xu, L.; Tang, L.; Wang, S.; Li, L. Big data in tourism research: A literature review. Tour. Manag.
**2018**, 68, 301–323. [Google Scholar] [CrossRef] - Zeger, S.L.; Qaqish, B. Markov regression models for time series: A quasi-likelihood approach. Biometrics
**1988**, 44, 1019–1031. [Google Scholar] [CrossRef] [PubMed] - Peduzzi, P.; Holford, T.; Detre, K.; Chan, Y.K. Comparison of the logistic and Cox regression models when outcome is determined in all patients after a fixed period of time. J. Chronic Dis.
**1987**, 40, 761–767. [Google Scholar] [CrossRef] - Hung, Y.; Zarnitsyna, V.; Zhang, Y.; Zhu, C.; Wu, C.F.J. Binary time series modeling with application to adhesion frequency experiments. J. Am. Stat. Assoc.
**2008**, 103, 1248–1259. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bakker, M.; Twining-Ward, L. Tourism and the Sharing Economy: Policy and Potential of Sustainable Peer-to-Peer Accommodation; The World Bank: Washington, DC, USA, 2018. [Google Scholar]
- Portolan, A. The impacts of private accomodation attributes on tourism demand. In DIEM: Dubrovnik International Economic Meeting; Sveučilište u Dubrovniku: Dubrovnik, Croatia, 2013. [Google Scholar]
- Juaneda, C.; Raya, J.M.; Sastre, F. Pricing the time and location of a stay at a hotel or apartment. Tour. Econ.
**2011**, 17, 321–338. [Google Scholar] [CrossRef] - Ert, E.; Fleischer, A.; Magen, N. Trust and reputation in the sharing economy: The role of personal photos in Airbnb. Tour. Manag.
**2016**, 62–73. [Google Scholar] [CrossRef] - Hyndman, R.J.; Koehler, A.B. Another look at measures of forecast accuracy. Int. J. Forecast.
**2006**, 22, 679–688. [Google Scholar] [CrossRef][Green Version] - Hyndman, R.J.; Khandakar, Y. Automatic time series forecasting: The forecast package for R. J. Stat. Softw.
**2008**, 27. [Google Scholar] [CrossRef][Green Version] - Ruiz-Reina, M.Á. Entropy of Tourism: The Unseen Side of Tourism Accommodation. Available online: https://www.dpublication.com/wp-content/uploads/2019/12/424.pdf (accessed on 22 June 2021).
- Hayashi, F. Econometrics; Princeton University Press: Princeton, NJ, USA, 2000. [Google Scholar]
- Chow, G. Econometrics; McGraw-Hill Book Company: New York, NY, USA, 1983. [Google Scholar]
- Sargan, J.D. The estimation of relationships with autocorrelated residuals by the use of instrumental variables. J. R. Stat. Soc. Ser. B
**1959**, 21, 91–105. [Google Scholar] [CrossRef] - Hall, A.R. Advanced Texts in Econometrics: Generalized Method of Moments; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
- Christ, C.F.; Theil, H. Economic forecasts and policy. Econometrica
**1962**. [Google Scholar] [CrossRef] - Gill, P.E.; Murray, W.; Saunders, M.A.; Tomlin, J.A.; Wright, M.H.; George, B. Dantzig and systems optimization. Discret. Optim.
**2008**, 151–158. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Out-sample forecast hotel accommodation h = 12 (January 2019 to December 2019). Own Elaboration.

${\mathit{y}}_{\mathit{t}}$ | ${\mathit{x}}_{\mathit{t}}$ | $\mathit{B}\mathit{e}\mathit{T}\mathit{S}\mathit{U}{\mathit{F}}_{\mathit{t}}$ | |
---|---|---|---|

Mean | 22,934,393 | 5,891,936 | 0.163905 |

Median | 21,721.214 | 4,858,973 | 0.162965 |

Maximum | 46,657,187 | 12,520,497 | 0.221206 |

Minimum | 9,797,644 | 3,302,242 | 0.118740 |

Std. Dev. | 9,463,750 | 2,433,807 | 0.023312 |

Skewness | 0.565925 | 1,234,384 | 0.312166 |

Kurtosis | 2.330 | 3.408 | 2.441 |

Observations | 216 | 216 | 216 |

**Table 2.**Cross correlations for explanatory and instruments variables. Sample January 2001–December 2018. Own Elaboration.

${\mathit{x}}_{\mathit{t}}$ | $\mathit{B}\mathit{T}\mathit{S}\mathit{U}{\mathit{F}}_{\mathit{t}}$ | ${\mathit{z}}_{1\mathit{t}}$ | ${\mathit{z}}_{2\mathit{t}}$ | ${\mathit{z}}_{1\mathit{t}-1}$ | ${\mathit{z}}_{2\mathit{t}-1}$ | |
---|---|---|---|---|---|---|

${x}_{t}$ | 1.00 (----) | |||||

$BTSU{F}_{t}$ | 0.15 (0.03) | 1.00 (----) | ||||

${z}_{1t}$ | 0.74 (0.00) | −0.24 (0.00) | 1.00 (----) | |||

${z}_{2t}$ | 0.94 (0.00) | 0.03 (0.61) | 0.84 (0.00) | 1.00 (----) | ||

${z}_{1t-1}$ | 0.38 (0.00) | −0.39 (0.00) | 0.49 (0.00) | 0.33 (0.00) | 1.00 (----) | |

${z}_{2t-1}$ | 0.66 (0.00) | −0.13 (0.06) | 0.53 (0.00) | 0.56 (0.00) | 0.84 (0.00) | 1.00 (----) |

**Table 3.**Summary of forecasting accuracy (RMSE). Out-Sample training January 2019–December 2019. Own Elaboration.

$\mathbf{log}\text{-}\mathbf{log}\text{}\mathit{B}\mathit{e}\mathit{T}\mathit{S}\mathit{U}\mathit{F}$ | $\mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}$ | $\mathit{A}\mathit{D}\mathit{R}\mathit{L}+\mathit{S}\mathit{e}\mathit{a}\mathit{s}\mathit{o}\mathit{n}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{y}$ | $\mathit{S}\mathit{A}\mathit{R}\mathit{I}\mathit{M}\mathit{A}$ | |
---|---|---|---|---|

$RMSE\text{}(h=12)$ | 78,507.36 | 107,581 | 1,524,295 | 1,528,357 |

**Table 4.**Summary of forecasting accuracy ($R{T}^{\prime}s\text{}{U}_{1}$). Out-Sample training January 2019–December 2019. Own Elaboration.

$\mathbf{log}\text{-}\mathbf{log}\text{}\mathit{B}\mathit{e}\mathit{T}\mathit{S}\mathit{U}\mathit{F}$ | $\mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}$ | $\mathit{A}\mathit{D}\mathit{R}\mathit{L}+\mathit{S}\mathit{e}\mathit{a}\mathit{s}\mathit{o}\mathit{n}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{y}$ | $\mathit{S}\mathit{A}\mathit{R}\mathit{I}\mathit{M}\mathit{A}$ | |
---|---|---|---|---|

$R{T}^{\prime}s\text{}{U}_{1}\text{}(h=12)$ | 1 | 1.3696 | 19.0210 | 19.0699 |

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Ruiz Reina, M.Á.
Bernoulli Time Series Modelling with Application to Accommodation Tourism Demand. *Eng. Proc.* **2021**, *5*, 17.
https://doi.org/10.3390/engproc2021005017

**AMA Style**

Ruiz Reina MÁ.
Bernoulli Time Series Modelling with Application to Accommodation Tourism Demand. *Engineering Proceedings*. 2021; 5(1):17.
https://doi.org/10.3390/engproc2021005017

**Chicago/Turabian Style**

Ruiz Reina, Miguel Ángel.
2021. "Bernoulli Time Series Modelling with Application to Accommodation Tourism Demand" *Engineering Proceedings* 5, no. 1: 17.
https://doi.org/10.3390/engproc2021005017