ARIMA + GARCH + Bootstrap forecasting method applied to the airline industry

https://doi.org/10.1016/j.jairtraman.2018.05.007Get rights and content

Highlights

  • The ARIMA + GARCH + Bootstrap method is proposed to forecast pax in the air transport industry.

  • Pax is analyzed and represented by conditional mean, variance, and the distribution of the error.

  • The performance of the proposed method is compared to 4 of the most used forecasting methods in the air transport industry.

  • The performance of the ARIMA + GARCH + Bootstrap method is validated by using two tests of superior ability.

Introduction

The aims of forecasting are to make decisions. Hence, the selected forecasting method must consider the characteristics of the data under study. Unfortunately, the existent methods proposed in related air transport literature only consider certain characteristics of data. Normally, they consider trend and seasonality, but volatility and distribution are not. However, both factors also contribute to determine the path behavior of data. Clemen (1989) concludes that forecast accuracy can be substantially improved through the combination of multiple individual forecasts. These are the reasons why, in this paper, the hybrid ARIMA + GARCH + Bootstrap time series method is applied for the first time to forecast air transportation passenger demand (pax). This method can combine the trend, variations, and historical distribution of data to eliminate the detrimental effects on forecasting.

Air transportation executives run different forecasting methods to make decisions. Predictions of pax are very important to them, because estimating pax helps deal with different problems such as coping with seasonality, sudden changes in demand levels, or even for making decisions on planning and investment. The problem of forecasting pax is the accuracy of the prediction delivered by the applied forecasting method.

In literature, many forecasting methods are available; each one has been developed to cope with different data characteristics; and therefore, the selection of the appropriate method depends on many factors, such as the available data and its relevance, the desired degree of accuracy, the time period, and the availability of time to forecast. In this paper, we mainly focus on increasing the degree of accuracy by developing a method that considers trend and seasonality, data variability, and pax distribution over time.

In the air passenger transportation industry, different methods have been developed for forecasting pax depending on the demand components they must capture. These methods are mainly panel data, multiple regressions, and time series. Carson, Cenesizoglu, and Parker (2011) propose an aggregating individual market approach (AIM) to forecast the United States (US) air pax aggregated demand using a multiple regression at the individual market level. The model uses exogenous variables that are specific to each region, and then it adds the forecasts to produce an aggregate forecast at an upper level. Guo and Zhong (2017) develop five econometric models to forecast Singapore air pax demand. They analyze many explanatory variables, such as Singapore GDP, China GDP, exchange rate, tourism, etc. to forecast air pax demand. From the analysis of the relation between variables, five regressions are built. Grosche, Routhlauf and Heinzl (2007) develop two gravity models to forecast air pax demand between city-pairs. The models include economic and geographic variables. They also included a good literature review about developed models for forecasting air pax demand using explanatory variables. Finally, Wang and Song (2010) present an extensive literature review with emphasis on research development, publication sources, geographic focus, and drivers of air pax demand, modeling and forecasting methods. They provide a good overview about models that have been applied to forecast air pax demand. A critical issue when it comes to predicting the behavior of pax is the problem of causality between pax and other economic factors, such as Gross Domestic Product (GDP). For example, Hakim and Merkert (2016) analyze causal relationship between pax and air freight volumes with economic growth for countries with low income. Hakim and Merkert (2016) conclude that causality exists from GDP to pax and from GDP to air freight but not the other way around. Hence, if the interest is on forecasting pax through explanatory variables, economic growth should be considered as an important factor.

In this paper, we analyze the pax behavior as a time series. We assume that pax data can be represented by three components: conditional mean (μt), conditional variance (σt), and the distribution of the error term (εt). The conditional mean is estimated using the AutoRegressive Integrated Moving Average (ARIMA) model (Box et al., 2016), the conditional variance is estimated with the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model (Bollerslev, 1986), and the distribution of the error term is approximated using the Bootstrap model (Pascual et al., 2006). The forecasting times series method that captures these features is the ARIMA + GARCH + Bootstrap.

The ARIMA model is an AutoRegressive Moving Average (ARMA) model applied to a non-stationary time series data. These models are normally applied to analyze risk measures in the financial sector such as the value at risk (VaR). In literature, many studies introduce bootstrap method for ARMA or GARCH models separately. Despite these studies, there are few studies researching the validity of the bootstraps techniques for ARMA-GARCH models (Shimizu, 2009). In 2004, Francq and Zakoian applied the bootstrap method for the stationary ARMA + GARCH model. Gamba-Santamaria et al. (2016) compare Shimizu (2009) subsample bootstrap ARMA + GARCH model with the Asymptotic normality model proposed by Moraux (2001), with the Asymptotic Hill estimator and with the Data tilting proposed by Chan et al. (2007). Gamba-Santamaria et al. (2016) conclude that Shimizu (2009) subsample bootstrap ARMA + GARCH model shows the best performance of the evaluated methods. According to our knowledge, the ARIMA + GARCH + Bootstrap time series method has never been applied in the air transportation industry to forecast pax. However, the method can be applied to forecast data in any other industry without restrictions.

This paper shows that forecasting the size of pax, at national level, using a method that considers μt, σt and εt forecasts much better than traditional forecasting methods commonly used in literature; especially in the air passenger transportation industry, which is why this paper is written for and the model is applied to. We compare its performance to one autoregressive method (ARIMA), two exponential smoothing methods (Holt-Winters Additive and Holt-Winters Multiplicative) (Winters, 1960; Chatfield, 1975; Montgomery et al., 1990), and one artificial intelligent method (Damp Trend Grey Model, DTGM) (Carmona-Benítez et al., 2013). We choose these methods because they are commonly used to estimate pax in related literature. These forecasting methods consider one of three pax components, the conditional mean (trend and seasonality). They have been proven to be superior to other forecasting methods in literature so far (De Gooijer and Hyndman, 2006). However, these methods do not consider conditional variance, and they assume a distribution function for data rather than simulating its real distribution. Therefore, the literature gap to full fill by the proposed method is to forecast the three pax components: conditional mean (trend and seasonality), conditional variance (variability), and distribution of data. The most accurate forecasting method is determined by the Diebold-Mariano test (Diebold and Mariano, 1995), the Superior Predictive Ability (SPA) test (Hansen, 2005), and the mean absolute percentage error (MAPE). We found that the ARIMA + GARCH + Bootstrap is more accurate than these methods. The reason is that the ARIMA + GARCH + Bootstrap has the advantage of considering trend and seasonality, data variability, and pax distribution over time, all at once.

This paper has the following structure: Section 2 presents a literature review of pax forecasting methods. Section 3 explains the ARIMA + GARCH + Bootstrap time series forecasting method. In Section 4, simulation experiments for forecasting US pax using the ARIMA + GARCH + Bootstrap, the ARIMA, the Exponential Smoothing, the Holt-Winters Additive, the Holt-Winters Multiplicative, and the DRGM are shown together with a statistical analysis that determines which method is the best for forecasting pax. In Section 5, conclusions are drawn, and future research topics are provided.

Section snippets

Time series forecasting method for estimating pax

The most important thing when forecasting is the level of accuracy and the quality of the estimation. Therefore, the main problem with forecasting pax is selecting the most accurate technique.

Time series methods are among the most common methods for forecasting. They are commonly used to forecast trend and seasonality components. Exponential smoothing techniques (Williams et al., 1998) and ARIMA are amongst the most generally used to forecast transport demand (Bermudez et al., 2007; Milenkovic

Demand forecasting model

We assume that the structure of the pax demand data can be described by the following model:Paxt=μt+σtεt

Where μtis the conditional mean, σt the conditional standard deviation and εt is the error term with zero mean and unit variance.

For constructing a forecast of pax using equation (1), the application of a time series method is needed (explained in the following subsections) for estimating each component.

Estimation methods

For applying some estimation methods, the ARIMA method, we must analyze the pax behavior

Experimental data

The Bureau of Transportation Statistics publishes the United States (US) monthly aggregated pax at national level from January 1990 to April 2016 (Bureau of Transportation Statistics, 1991–2016). The database is divided into two sets: the first data set is from January 1990 to April 2013, and the second data set is from May 2013 to April 2016. The first data set is used for the estimation of the four-time series methods: Holt-Winters, ARIMA, DTGM and the ARIMA + GARCH + Bootstrap. In total, we

Conclusions

In this study, we prove the hypothesis that the ARIMA + GARCH + Bootstrap is the best time series forecasting method to estimate pax in the US air transport industry among those that are chosen for comparison purposes. The advantage is that the method considers three components of the pax demand: conditional mean (trend and seasonality), conditional variance (variability), and pax distribution over time. Therefore, the method includes the ability to forecast trend, variance and pax distribution

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References (41)

  • C.I. Hsu et al.

    Application of Grey theory and multi objective programming towards airline network design

    European journal of operation research

    (2000)
  • C.I. Hsu et al.

    Reliability evolution for airline network design in response to fluctuation in passenger demand

    Omega – The international Journal of Management Science

    (2002)
  • C.I. Hsu et al.

    Determining flight frequencies on an airline network with demand-supply interactions

    Transport. Res. E Logist. Transport. Rev.

    (2003)
  • Y. Liang

    Forecasting models for Taiwanese tourism demand after allowance for Mainland China tourists visiting Taiwan

    Comput. Ind. Eng.

    (2014)
  • C. Lim et al.

    Time series forecasts of international travel demand for Australia

    Tourism Manag.

    (2002)
  • L. Pascual et al.

    Bootstrap prediction for returns and volatilities in GARCH models

    Comput. Stat. Data Anal.

    (2006)
  • J.D. Bermudez et al.

    Holt-winters forecasting: an alternative formulation applied to UK air passenger data

    J. Appl. Stat.

    (2007)
  • G. Box et al.

    Time Series Analysis: Forecasting and Control

    (2016)
  • Bureau of Transportation Statistics (1991-2016) at https://www.rita.dot.gov/bts/data_and_statistics/index.html,...
  • R.B. Carmona-Benítez et al.

    Comparison of bootstrap estimation intervals to forecast arithmetic mean and median air passenger demand

    J. Appl. Stat.

    (2017)
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