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Forecasting foreign tourist visits to Bali using Bayesian Vector Autoregression with Normal-inverse-Wishart Prior

I Wayan Sumarjaya1 and Ni Ketut Tari Tastrawati2

  • 1Department of Mathematics

Faculty of Mathematics and Natural Sciences, Udayana University

Bali, Indonesia

sumarjaya@unud.ac.id 2Department of Mathematics

Faculty of Mathematics and Natural Sciences, Udayana University

Bali, Indonesia

tastrawati@unud.ac.id

ABSTRACT

As a major tourist destination, Bali has become an icon for tourism in Indonesia. In general, the number of foreign tourist visits shows an increasing trend. However, there is considerable fluctuation in the number of visits which is affected by season. In another word, there is a stochastic trend in the number of tourist visits. Policy makers need a method to predict this tourist visits. A commonly used method to predict tourist visits is time series analysis. Time series analysis has been used in various fields such as finance, business, engineering, meteorology, geophysics, and tourism, to name but a few. Research on forecasting tourist visits usually uses univariate data. This research aims to forecast the number of foreign tourist visits from four major countries such as Asia Pacific, ASEAN, America, and Europe simultaneously using Bayesian vector autoregression with normal-inverse-Wishart prior. First data is plotted to see its characteristics. Then the data is modeled using Bayesian vector autoregression. In this stage normal-inverse-Wishart prior is used. Next, Markov chain Monte Carlo is conducted to make a prediction from the posterior distribution. The forecast suggests that the number of tourist visits in general increased, albeit some fluctuation in some months.

Keywords: autoregression, Bayesian vector autoregression, time series analysis, tourist visits forecasting

  • I.    Introduction1

As a major tourist destination, Bali has become an icon or role model for tourism in Indonesia. In general, the number of tourist visits shows an increasing trend. However, these visits vary from months to months or fluctuated due to seasonal effect. This means that there is a stochastic trend on the tourist visits data. Policymaker, for instance, the Bali tourism board, needs to predict these tourist visits so that a proper marketing strategy can be made. One way to forecast this visits is by using time series analysis.

Research on forecasting tourist visits to Bali using time series analysis has been carried out by [1], [2], and [3]. However, these authors mainly use univariate time series model. Univariate time series has few shortcomings. First, it cannot measure interdependence structure between the time series. Second, long-term relationship cannot be measured by using single series alone. These shortcomings can be overcome by using multivariate time series model.

This research aims to forecast the number of tourist visits of four major countries using Bayesian vector autoregression. Bayesian vector autoregression (BVAR) is different from classical vector autoregression in that the former uses prior information. This research uses foreign tourist visits data from four major countries: Asia Pacific, ASEAN, America (including Canada and Latin America), and Europe. The classification of these countries is based on grouping as mentioned in Bali Tourism Board.

This article is organized as follows. Section one contains motivation and background for forecasting foreign tourist visits to Bali. Section two discusses the brief concept of multivariate time series using Bayesian vector autoregression and prior selection. Forecasting and its analysis are discussed in section three. Section four concludes the article.

  • II.    Research Method

This section discusses basic concept of vector autoregression, Bayesian vector autoregression, and normalinverse Wishart prior. The following discussion is adapted from [4]. Let yt for t = 1,...,T be M ×1 vector of observation on M time series variables, εt be M × 1 vector of errors, β0 be M ×1 vector of intercepts, and Bj be M × M. matrix of coefficients. Vector autoregression of order p is defined as

p

yt = βo + ∑ Bjyt - j + εt (1)

j=1

where vector εt is assumed to be independent and identically distributed N (0, Σ).

Equation (1) can be modified further. Define Y as T × M matrix which stack T observation of response variables, xt = (1, yt ∣,., y∖-p) , and define

x1

x2

xT

(2)


Next, define B = (β0,B1,.,Bp)' and β = vec(B). Then (1)

can be written in form of normal distribution variate matrix as

Y = XB + E

or


yt = ( Im ® X) β + ε

where ε ~ N(0, Σ ® It ).

The likelihood function that corresponds to (4) is

L (β, ∑I y, X)

= (2π)-tm /2 ∑8 Im   2

× exp{-1(y - (Im ® X)β)'(Σ-1 ® Im )(y - (Im ® X)β) I 2

and the log-likelihood has the form [5]

ln L χ - ^ln ∣Σ ® IM ∣

-{-1 (y - (Im ® X)β)'(∑-1 ® Im )(y - (Im ® X)β)

I 2

(3)


(4)


(5)


(6)


Define X :=IM ® X and β = (X'(Σ-1 ® IM )X)-1X'(Σ-1 ® IM ) y so that the loglikelihood (6) can be written as [5]

ln L = - - ln ΣH 2(β - β)X'(Σ-1 ® Im )X(β - β) ∣

2tr[( y -X β)(y -X β)'(Σ"1 ® I m)]

.                    . λ.

X ln( N(β∖β, Σ,X, y)IW(Σ | β, X, y).

Now the normal-inverse-Wishart prior has the form

p(β, Σ ∖ X, y) χ p(y ∖ X, β, Σ)p(β, Σ).

Incorporating prior in (8) yields the following conditional posterior distribution:

p (β∖Σ,X, y) = N (β, Σ),

p (Σ∖β, X, Y) = IW (Ξσ+ (Y - Xβ)'( Y - Xβ), T + γ).

Markov chain Monte Carlo algorithm such as Gibbs

sampling can be used to make inference based on (9). Detail derivation of (9) can be seen in [5] and [6].

  • III.    Results and Discussion

This research uses foreign tourist visits from four major countries: Asia Pacific, ASEAN, America (including Canada and Latin America), and Europe during the period from January 2010 to June 2016. The plot of the visits for the four countries can be seen in Figure 1.

Fig. 1. Foreign tourist visits to Bali during January 2010—June 2016.

Figure 1 shows an increasing stochastic trend and pronounces seasonal effect on the four foreign tourist visits. To model and to forecast the four-time series simultaneously Bayesian vector autoregression with normal-inverse-Wishart prior is applied. The lag selection, in this case, 12, follows from an empirical research study carried out by [7].

The forecasts for foreign tourist visits can be seen in Table 1. The plot of the forecasts and the credible interval can be seen in Figure 2.

TABLE I

Forecasts of foreign tourist visits for the next 12 months

Month

Asia Pacific

America

ASEAN

Europe

Jul. 2016

254,906

23,369

31,105

92,101

Aug. 2016

230,891

21,279

26,208

89,716

Sep. 2016

276,676

22,577

42,292

93,010

Oct. 2016

227,706

22,011

37,748

90,136

Nov. 2016

184,553

21,820

36,001

72,947

Des. 2016

250,688

27,660

57,285

90,310

Jan. 2017

252,709

23,998

30,620

87,931

Feb. 2017

252,574

23,509

33,713

77,574

Mar. 2017

228,883

27,894

47,204

87,810

Apr. 2017

264,296

27,415

42,981

100,601

May 2017

272,261

26,697

48,591

98,512

Jun. 2017

276,951

24,984

47,542

86,176

Fig. 1. Plot of forecasts for period June 2016 to July 2017.

  • IV.    Conclusion

Forecasting foreign tourist visits to Bali using Bayesian vector autoregression shows an increasing stochastic trend albeit considerable fluctuation due to seasonal effect.

Acknowledgment

The authors would like to thank Udayana University for the financial support under the research scheme Hibah Unggulan Program Studi contract number 2731/UN14.1.28/LT/2016 dated 27 June 2016.

References

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S. Karlsson, “Forecasting with Bayesian vector autoregression” in Handbook of Economic Forecasting, vol. 2B, E. Graham and T. Allan, Ed.

New York: Elsevier, 20

Figure 2 shows a plot of forecasts for the next 12 months. As can be seen from the figure there is an increasing (stochastic) trend for all the visits albeit some fluctuation in a certain month (see also Table 1).