A Comparison of Time Series Models for Forecasting Outbound Air Travel Demand *

Transcription

1 Journal of Aeronautics, Astronautics and Aviation, Series A, Vol.42, No.2 pp (200) 73 A Comparison of Time Series Models for Forecasting Outbound Air Travel Demand * Yu-Wei Chang ** and Meng-Yuan Liao Department of Air Transportation Management, Aletheia University 70- Pei-Shi-Lia, Matou, Tainan, 72, Taiwan, R.O.C. ABSTRACT This paper uses Naïve, moving average, exponential smoothing, seasonal ARIMA (SARIMA) as well as GM (,) models to model outbound air travel demand of Taiwan to three major destinations - Hong Kong, Japan and the USA and compare their forecasting performances. Monthly time series data from Jan 996 to Dec 2006 are used in this research. According to the indicators of forecasting performance, MAPE (mean absolute percentage error) and MAE (mean absolute error), the SARIMA method appears to be the superior model for forecasting seasonal air travel demand both in short- and long-term forecasting. The results also reveal that the forecasting capability of short-term forecasting appears to be more reliable than that of long-term forecasting. Keywords: Air travel, Forecast, Seasonal ARIMA (SARIMA), GM (,) I. INTRODUCTION Accurate forecasting of air travel demand is an essential determinant for successful investments in the transportation industry for both the public and private sectors. For the public sectors, estimation of air travel flows is important for destinations to make efficient use of transportation infrastructure and resources []. For the private sector, such as airlines, it is beneficial information for aircrafts, facilities and manpower planning. As air travel demand forecasting is one of the most important policy tools for decision makers all over the world, selecting a valid method for forecasting future air travel flows is therefore important if there are a number of forecasting methods available. To forecast air travel demand, econometric models and time series models are the two major methods within the quantitative approach [2]. Econometric models make use of economic interrelationships between specific explanatory variables and the independent variable which is of interest, such as regression analysis. Time series models such as the naïve model, moving average method, exponential smoothing techniques and the ARIMA model use only historical data of the dependent variable to forecast future values [3]. Previous studies revealed that time series models often performed better than econometric models in forecasting [4]. Furthermore, time series models are also suitable when only historical data of variables are available or related variables are difficult to explain [5]. The comparison of forecasting performance of competing models is frequently highlighted in many previous studies on air travel and tourism demand forecasting. For instance, Chu [6], Reid [7] and Coshall [] found that ARIMA models are more suitable for tourism demand forecasting. Sheldon [8], Yeung and Law [9] suggest that exponential smoothing methods appear to be outperformed among the competing time series models. Martin and Witt [2], and Turner and Witt [0] concluded that Naïve methods outperform other formal methods in many situations. Recently, although the GM (,) model has been successfully applied in engineering, economics and finance, the use of GM (,) for air travel forecasting is still not widespread. Past research about time series forecasts often have different results for the same problems, it is therefore interesting to examine the forecasting ability of the above competing time series models. The main purpose of this study is to examine the accuracy of various alternative time series models in * Manuscript received, Aug. 6, 2009, final revision, Jan. 30, 200 ** To whom correspondence should be addressed, uwchang@seed.net.tw

2 74 Yu-Wei Chang Meng-Yuan Liao forecasting international outbound air travel demand. In this paper, Naïve, moving average, exponential smoothing, SARIMA as well as GM (,) models are used for outbound travel prediction. Empirical data from three principal outbound air travel destinations- Hong Kong, Japan, and the USA, from Taiwan, are used to compare the forecasting ability of the competing models from 996:-2004:2. Out-of-sample forecasting is used to forecast demand six months (short-term) and twenty four years (long-term) from now, based on the period of 2005:-2006:2. The forecasting ability of short- and long- forecasting will also be compared in this paper. II. LITERATURE REVIEW Over the past two decades, the study of air travel demand forecasting has attracted considerable attention from researchers. The comparison of forecasting performance of competing models is frequently highlighted in reviewing the main empirical data published on passenger demand forecasting (for example, Burger et al. []; Chu [6]; Coshall []; Kulendran and Wong [2]). Sheldon [8] examines the accuracy of six different forecasting models from six countries to the United States, the results of which show that the no change model and exponential models, overall, outperform the other models. Chu [6] examines the accuracy of six time series forecasting techniques and finds that the ARIMA model is, overall, the most accuracy method for forecasting international tourist arrivals. Burger et al. [] compares a variety of time-series forecasting methods to forecast the US demand for travel to Durban, South Africa. A variety of techniques employed in the survey include Naïve, moving average, single exponential smoothing, decomposition, ARIMA, multiple regression, neural networks and genetic regression. The study shows that the neural network method performs the best. Cho [3] investigates three time series forecasting techniques- exponential smoothing, ARIMA and Elman s Model for ANN from six countries to Hong Kong, the results shows that Neural Networks seems to be the best method for forecasting tourist arrivals. Yeung and Law [9] forecasts US outbound travelers to Europe, the Caribbean and Asia using the moving average, exponential smoothing and Naïve II, the results of which indicate that the Winter s exponential smoothing method performs best. Coshall [] applies Holt-Winters, Naïve I, Naïve II and ARIMA models to quarterly United Kingdom air travel flows to 20 principle international destinations, the result shows that the ARIMA model outperforms the others in terms of accuracy. Despite all the studies and empirical evidence on the forecasting performance of different methods, the results still remain inconsistent. III. METHODOLOGY Given that the monthly outbound air travel departures of Taiwan exhibit trend and seasonal pattern. The selection of six types of mode specific models to be used to forecast air travel demand is considered in this study. The models are Naïve I, Naïve ii, moving average, exponential smoothing, SARIMA and GM(,) model. The following paragraphs briefly describe each of the models. 3. Naïve Models There are two Naïve models, they are Naïve I and II. The Naïve I forecasting model in the period t+ is equal to the number of t: F t+ = F t () Where F t is the number of outbound departures in period t and F t+ is the forecasting values. In the Naïve I model, all future forecasting values are equal to the actual value of the most recently observed value. The Naïve II forecasting technique has the following form: F t+ = F t [+ (F t -F t- ) / F t- ] (2) The value of F t+ at time period t+ is calculated by multiplying the growth rate over the previous period. 3.2 Moving Average Model The moving average model is one of the easiest forecasting techniques to use. The value Y at time period t+ is the average of the terms in the moving average. The model is given in equation (3) Y t+ = (Y t +Y t- +Y t-2 + +Y t-n- )/n (3) Where Y t+ is moving average in t; n is the period in the moving average and Y t is the actual value in period t. The moving average model in this research is Four-Quarter moving average, which can be expressed as MA(4) = (Y t +Y t- +Y t-2 +Y t-3 )/4 (4) 3.3 Exponential Smoothing Model Exponential smoothing model is a simple adaptive forecasting method. Most air travel demand is affected by economic and seasonal factors such as holidays and seasons and therefore includes cycle, trend and seasonality. To accommodate these patterns, the exponential smoothing model incorporates them into an exponential smoothing process by permitting estimates of these parameters. Different smoothing constants are used to smooth the levels of intercepts, slopes and seasonal factors in a series. The estimate of the intercept and slope of the trend line at time t are shown in equations (5) and (6) respectively (Bails and Peppers [3]): A t = α( Y t / S t-n ) + (-α) (A t- +B t- ) (5) B t = β(a t -A t- )+(-β)b t- (6) and the seasonal factors are revised based on S t =γ( Y t / A t ) + (-α) S t-n (7)

3 A Comparison of Time Series Models for Forecasting Outbound Air Travel Demand 75 Where A t is the estimated intercept of the trend line at time t, and where α, β and γ are exponential smoothing constants, while B t is the estimated slope of trend, and S t is the seasonality estimate. The forecasting value of T periods in the future is: Y t+t = (A t + B t T) S t+t (8) 3.4 SARIMA Model The SARIMA model proposed by Box and Jenkins [4] can be used when the time series is stationary and there is no missing data in the time series. The SARIMA model involves three stages: (a) Model identification- to estimate the p, q and d parameters by using auto-correlation (ACF) and partial auto-correlation functions (PACF); (b) Parameter estimation- estimation of the p and q components to see if they are significant to the model, or if they should be dropped and include techniques such as the max. likelihood estimation or least squares; (c) Diagnostic checking- diagnosis of the residuals to see if an estimated model is statistically adequate (Box and Jenkins [4]). Suppose n consecutive observations Y,Y 2,,Y n from a series, the goal is to find a good model that suitably represents the process-generating mechanism. The basic SARIMA model has the following form (Vamdaele [5]), φ(b)φ(b S )W t = Θ(B)θ(B s ) ε t (9) where W t and ε t are random terms at time t, respectively. W t = s D d Y t, d =(-B)d and s D =(-B S ) D. B is the backshift operator defined by BW t =W t-. φ(b), Φ(B S ), Θ(B) and θ(b s ) are the polynomials in B while the seasonal-nonseasonal AR, MA operator can be denoted as series to form a new series x 2 (0) ={ x 0 (2),,x 0 (i), x 0 (n+) }. A new GM (,) model will be formed for forecasting in the next step. The process repeats, and then ends when the required forecasting time period is obtained. Following the Wang [8] and Tseng et al.[9] definition, the GM (,) can be calculated as follows: Step - assume the original series is x 0 x 0 = {x 0 (),x 0 (2),,x 0 (i), x 0 (n) } (4) Step 2 - a new sequence x is generated by the accumulated generating operation (AGO) x = {x (),x (2),,x (n) }, (5) where k ( k) ( i) x0 i x (6) Step 3 - establish the following first-order differential equation dx 0 ax u dt (7) Step 4 - from Step 3, we get ( k) () u ak u xˆ ( x0 ) e (8) a a xˆ xˆ xˆ (9) ( k) ( k) ( k) 0 T aˆ a,u ( b b) b r n, where T T φ(b)=-φ B- φ 2 B 2 - -φ p B p, AR operator (0) Φ(B S )=-Φ B S - Φ 2 B S - -ΦpBp S, seasonal AR operator () Θ(B)=-Θ B- Θ 2 B 2 - -Θ q B q, MA operator (2) () () 0.5( x x2 ) () () 0.5( x2 x3 ) b () () 0.5( x n xn ) (20) θ(b s )=-θ B s - θ 2 B 2s - -θ Q B Qs, seasonal MA operator (3) The SARIMA model can also be written as SARIMA (p, d, q) (P,D,Q) S (The details are available in Pankratz [6]). 3.5 GM (,) Model Grey forecasting is a single variable first-order model presented by Deng [7], namely GM (,). Four observations are enough for forecasting the series. In general, the GM (,) includes accumulated generation, inverse accumulated generation and grey modeling. The basic idea is to assume the original series is x 0 = {x () 0,x (2) 0,,x (i) 0, x (n) 0 }, then a new sequence x (n+) 0 is generated by a preliminary transformation called the accumulated generating operation (AGO). Then a new series x (0), which is equal to {x () 0,x (2) 0,,x (n+) 0 }, is () generated. In the next step, x 0 is removed from the r ( x, x,, x ) (2) n ( ) xˆ k (0) (0) (0) T 2 3 n is the predicted value of ( ) ˆ k x at time k+ Step 5 - inverse the accumulated generation operation (0) (IAGO) and we have the predicted value xˆ o ( k ) at time k. xˆ ( k) xˆ xˆ (22) (0) ( k) ( k ) o 3.6 Forecasting Accuracy Measurements For each of the series, out-of-sample forecasting values are calculated and compared with the actual values of the series. The accuracy of the forecasts can be measured by several criteria. The mean absolute percentage error (MAPE) and mean absolute error (MAE) are two most commonly used measures of forecasting

4 76 Yu-Wei Chang Meng-Yuan Liao performance. and commonly used statistical tool. MAPE = tn yt yˆ t (23) n yt t t tn MAE = yt yˆ (24) n t t where y t (t =, 2,, n) is the actual value, and y ˆ t ( t =, 2,, n) represents the forecasted value. The lower the value of MAPE and MAE, the better the forecast is. According to Lewis [20], any scale with MAPE smaller than 0% is highly accuracy, 0%-20% is good, 20% -50% is reasonable, greater than 50% is inaccurate. IV. RESULT 4. Data Outbound air travel data have been taken from various issues of the Annual Report on Tourism published by the Tourism Bureau, Republic of China. Monthly outbound air travel data were obtained for three principal international destinations- Hong Kong, Japan and the USA, within the chosen period of 996:-2006:2. Within-sample data from 996: to 2004:2 which contains 08 observations are used for model building phase. The remaining data, i.e. 2005: -2006:2 are retained for short- term (predicting next 6 months) and long-term (predicting next 24 months) ex-post validation purposes. Figure reports the time plot of outbound air travel from Taiwan to Hong Kong, Japan and the USA, respectively SARIMA Model Log transformations of all the series are applied to the series before fitting SARIMA models to the outbound travel series. In order to estimate the parameter of p, q and P, Q, initially, various SARIMA models are fitted to the logarithms of outbound air travel from Taiwan to Hong Kong, Japan, the USA and the total outbound air travel, respectively. The best fitted models are selected based on various criteria. The AR and MA must be significant at the 5% level. The selected models must have the smallest AIC or SBC values. The diagnostic test of serial correlation, normality and heteroscdasticity are checked for the random residuals of the fitted models. After empirical examination, the most appropriate models are SARIMA (0,,) (,0,) 2 for Hong Kong, SARIMA (,,0) (0,0,) 2 for Japan, SARIMA (,,0) (0,,) 2 for the USA and SARIMA (,,) (,0,0) 2 for the total outbound air travel in Taiwan. 4.3 GM (,) Model The GM (,) model needs the series data to consist of at least four. Here, we take the total outbound departures of Taiwan for the calculation example. Step - the original total outbound air travel data are. x ,49826,, Step 2 - from step, a new sequence series is generated by the AGO. x , ,, Step 3 - using the least-squares method, the parameters. aˆ , Step 4 - the forecast equation of GM (,) is HONGKONG JAPAN USA OUTBOUND Figure Time Plot of Outbound Air Travel from Taiwan to Hong Kong, Japan and the USA Source: Taiwan Tourism Bureau (2006) 4.2 Proposed Models 4.2. The Naïve, Exponential Smoothing and Moving Average Models The results of Naïve and the moving average models were calculated by EXCEL following equations () to (4). The exponential smoothing model was carried out by using computer software, SPSS 2.0, which is a powerful ( k ) xˆ ( / )exp( k) / Step 5 - inverse the accumulated generation. When k=08 (2004:2), the xˆ (k) = ; when k=09 (2005: ), the xˆ (k) = The forecasting value of 2005: is therefore equal to xˆ (09) - xˆ (08) = From the calculation process, we can obtain the forecasting values from 2005: to 2006: Forecasting Results The model forecasting performance is conducted by using the mean absolute percentage error (MAPE) and mean absolute error (MAE) on the basis of out-of-sample fit. The model with the least forecast error is identified as the best model. Table summarizes the MAPE and MAE

5 A Comparison of Time Series Models for Forecasting Outbound Air Travel Demand 77 of short-term forecasts. It is clear that the SARIMA model is the best model because the mean MAPE is 8.8% and the mean MAE is 20,830, which are smaller than the rest of other competing models. All forecasts of the SARIMA models fall in the highly accurate range. The next best model is exponential smoothing model because the mean MAPE is 9.56% and the mean MAE is 24,2, which have the second smallest forecast error, second only to the SARIMA model. The Naïve II and Naïve I methods appear to have the highest MAPE and MAE. To gain a better understanding of the forecasting abilities of these models, the forecasting performance was also evaluated in terms of long-term horizon. Table 2 shows the MASE and MAE of the SARIMA forecasts are uniformly smaller than the others followed by exponential smoothing, MA (4), GM (,), Naïve II and Naïve I. Overall, the widespread use of SARIMA model indicates it s superiority in forecasting both short- and long-term air travel demand. Finally, as far as the forecast period is concerned, the mean MAPE of short- and long-term forecasts in the Naïve I model is 2.20% and 24.86%, respectively; the Naïve II model is 22.6% and 22.50%, respectively; the MA (4) is 2.87% and 4.20%, respectively; the exponential smoothing model is 9.56% and 2.86%, respectively; the SARIMA model is 8.8% and 9.09%, respectively and the GM (,) model is 8.% and 9.4%, respectively. Overall, the mean MAPE and MAE of short-term forecasts are smaller than that of long-term forecasts for the same forecast method. The analyses reveal that the forecasting capability of short-term forecasts appears to be more reliable than that of longterm forecasts. V. CONCLUSIONS A demand forecasting model with high accuracy is of great importance. One of the primary challenges facing air transportation management is to generate accurate forecasts of air travel passengers. In this paper, various models have been proposed to generate forecasts of outbound departures of Taiwan. Results show that the SARIMA model is the best for predicting air travel demand, followed by exponential smoothing, moving average, GM (.) and Naive model both in short- and long-term forecasts. The SARIMA model was therefore chosen to forecast outbound air travel demand in Taiwan. Furthermore, the lower prediction errors of the forecasting models suggest that the forecasting ability performs better with short- than long-term forecasting. REFERENCES [] Coshall, J., Time Series Analysis of UK Outbound Travel by Air, Journal of Travel Research, Vol. 44, No. 3, 2006, pp Table Performance of Forecasting Models (Short-term) Hong Kong Japan USA Total Outbound Mean Rank Forecast MAE MAE MAE MAPE MAE MAE MAPE model MAE Naïve I Naïve II MA (4) Exponential smoothing SARIMA GM (,) Table 2 Performance of Forecasting Models (Long-term) Hong Kong Japan USA Total Outbound Mean Rank Forecast MAE MAE MAE MAPE MAE MAE MAPE model MAE Naïve I Naïve II MA (4) Exponential smoothing SARIMA GM (,)

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