Forecasting air travel demand for a large airline using time series modelling

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Forecasting air travel demand for a large airline using time

series modelling

Simon Luijsterburg

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Preface

For a period of four months, from March 2007 up to mid July 2007, I worked at the Operations Research department of British Airways, located at West-London, UK. Here I was assigned to work on an internship named ‘Project Timebomb’, describing the concern of BA whether their future is secure or eventually low-cost carriers will take over the whole air travel market. Looking back on the project it has been very challenging, requiring me to think beyond the scope of econometrics, focusing a lot on data mining. This paper has been written as a master thesis for the study econometrics, and is based on the econometric part of the research done at BA.

My stay in London allowed me to get a good taste of the British culture, and the friendly people that work at British Airways. Someone once told me that while being in England you must drink tea with milk; I didn’t mind following the British habit of drinking tea in the afternoon, but I refused adding milk to it. Going for drinks after work, having a barbeque on Sunday, even with the dreadful English weather it was all good fun.

This paper could not have been written without the support of many people. First of all some people at BA should be mentioned. Big thanks to Mirjam Maatman for making my internship possible and inviting me to various social activities. Special thanks to Christopher Watt and Susan Robinson for guiding me through the project and to Simon Cumming for sharing his incredible knowledge of data mining and modelling with me and of course all the new-grads at BA who provided a great atmosphere to work in.

Secondly, there are a few remarkable people in London that deserve a special thanks. Paul and Carol Beadsmoore allowed me to stay at their residence and proved to be such great people! The Northfields area was a fabulous place to stay, living in the center of civilization and quite close to BA. Thanks for providing such a nice place to come home to.

Last but not least, there’s my friends and family who helped me through the process of writing this paper. Special thanks go out to my father and Wynfrith Meijwes for reading and commenting on this paper and Kees Praagman for supervising the whole project.

With this paper a period of just over 5 years of study comes to an end. Simon Luijsterburg

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Abstract

Forecasting air travel demand for a large aircraft carrier using time series modelling Author: Simon Luijsterburg

British Airways is a big player in the air travel industry that has a large customer base. To ensure that BA is still a big player in the near future, market research is continuously being done. Because the market can change at any time due to changing travel needs, it is important to know what is going on in the market and what the demand for BA will look like in the near future.

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Introduction

The demographics of the UK population are changing. Many companies want to know what the key segments of the population will be over the next 10 years or so, what their flying behaviour is likely to be and how they should act towards these segments to get as much cus-tomers as they can. As the demographics are ever changing, so is the UK air travel market. Years ago flying used to be a sport only for the higher class; nowadays it is a common mode of transportation. Flying is accessible for everyone at a wide variety of rates; the amount of air travel has increased multi fold over the years. Spread across the world, many new carriers have started operations in a various number of markets. This has caused the markets to change and the travel behaviour of the UK population is adjusting to these changes.

Initiated by British Airways (BA), research was carried out to investigate the UK flight market and identifying the key segments for BA’s success. This paper investigates a segment of the UK flight market, namely the short haul leisure market. Using econometric models, in particular time series modelling, we will try to model the demand for flights on BA. Under-lying issues that drive this research are: How will this segment develop in the next 10 years based on the trends of recent years? What kind of demographic changes might affect the amount of flights flown? Which market segments should BA focus its marketing campaigns on at this moment to maintain a stable customer base in the near future. Is this segments changing in any way and might this compromise BA’s revenues? Is there a risk of BA not engaging with the younger population segment?

This paper will not answer any of the underlying issues, but will try to provide a tool, to make answering these questions easier. If it is possible to accurately model the demand of a segment of UK air travel market, then further research based on this model can be done. Attempts can then be made to answer the questions raised before.

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Chapter 2

Problem setting

In short the issue BA is facing can be described and structured by means of the following diagram, to be read from left to right:

Figure 2.1: Structuring the problem

The main question for BA is whether the changing UK demographics will affect the demand for BA in the next 10 years (box 1). This problem is very complex, and to look into it a specific approach is needed. There are market research papers available that predict how the air travel market will change in the near future (see for example Civil Aviation Authority (2006) and Lovegrove (2004)). However these papers do not provide enough details for BA to understand how the changing market will affect their business. As we will see, the short haul market causes the most reason for concern, and this paper will therefore focus only on the short haul market.

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First we will look into what data is available and pick an appropriate variable to measure demand for BA. Then we will discuss a few models that have been built by other researchers to tackle similar problems. After this, we will show look at how demand has changed for BA in recent years, and several modelling techniques will be used to construct our own models, which will be compared and evaluated.

This paper tries to answer the following question:

Can we build a meaningful model to describe and forecast the demand for short haul air travel on BA using econometric techniques?

Because of the complexity involved in this question, it would never be possible to produce a definitive answer. However, by using and evaluating a few econometric techniques we can get an impression of how good econometric modelling is in the context of modelling air travel demand for BA. Instead of using the Box-Jenkins airline data to test the models on ( Box and Jenkins (1976)), which is often used in theoretical papers, real BA data was used. The following list describes what the next part of the paper will look into.

• Look at models that have been used before for similar problems and their performance. • Build an ARIMA model to describe demand for flights on BA.

• Build a multivariate model, combining demand for multiple airlines, to describe demand for flights on BA.

• Construct a different variable to describe the demand for BA and build a model to describe this variable.

By looking at the results from these models, we will evaluate the following questions. • Are the above models useful? Are the forecasts obtained useful?

• Can any of the models be used to determine how underlying (demographic) factors influence demand?

In total, we will fit three econometric models to air travel series. We will elaborate on this. The first model, the ARIMA model looks at past demand for BA only. The model will be used to forecast the UK short haul market. This model will show to what extent historical data influence the future demand and how much insight this gives in the changing air travel market. Because no external effects are taken into account in this model, we will also elabo-rate on the utility of such a model.

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model transforms a large amount of data into a single meaningful variable and will estimate the influence of various external factors on this variable. The model will also take into account time series components, but will mainly measure the impact of the external factors.

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Chapter 3

Measuring demand and selecting an

appropriate variable

In this chapter we will discuss ways to measure the demand for BA and which variable was chosen to use in this paper. Air travel is a complicated industry and thousands of people fly with BA every day. There are a few ways to put the demand for BA into a number. One way would be to count the number of passengers that fly with BA per month. But there is a problem here, how is one flight defined? If a person is flying from one city to another via an intermediate airport, would that count as one flight or as two flights.

To give a short example on this. Suppose there are three different people flying. The first person wants to go from New York to Amsterdam (landing in between at a London Airport). The 2nd person wants to go from New York to London only, and the 3rd from London to Amsterdam. It is clear that all these people make only one flight, but from a data mining point of view it is not easy to distinguish whether the trip from New York to Amsterdam was one or two flights. It is also not clear whether a flight from New York to London should be weighted equal to a flight from London to Amsterdam, which is many times shorter in duration.

A measure for the amount of people flying with BA is the Passenger Sector Journey (PSJ). PSJ is a measure commonly used in airlines, and can be seen as an approximation for the amount of passengers that flew with a particular carrier. For example when a person flies from London to Amsterdam, it is one PSJ, while a person flying from London to Sydney via Hong Kong generally is 2 PSJ’s.

An obvious disadvantage of using PSJ’s as a measure is that it doesn’t tell you anything about the revenues generated by a PSJ. Perhaps 50 PSJ’s, all corresponding to people flying economy class, would generate the same amount of revenues as 1 PSJ corresponding to a per-son flying first class. In other words, even if you know the PSJ counts, you cannot say much about the revenue corresponding to these counts. Since revenue information is not available for every booking, and cannot be linked to PSJ counts, we will not consider revenues in this paper.

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briefly discuss the available data sources to obtain PSJ counts from.

All data about PSJ counts used in this paper have been gathered from the International Passenger Survey (IPS) database. The IPS is a survey conducted by an external organisation on all London airports on a selection of passengers each month and contains among other things booking-, passenger- and route information. Interviewing is carried out throughout the year and over a quarter of a million face-to-face interviews are conducted each year, rep-resenting about 1 in every 500 passengers. Even though the data in this database are not an exact representation of the PSJ counts, it is generally seen as a good approximation. One of the advantages of using ISP survey data is that competitor information is available too, and that data about all competitors is obtained in the same way. This means that different carri-ers can be compared to each other without any manipulation of the data. The IPS database stores age information so that PSJ counts be split up by age. The survey counts are scaled up using weights to be representative for the population and the data is corrected for outlying values. Because this is done by the IPS survey team, details of this data manipulation are not available. The IPS contains surveys with missing data, and probably surveys that contain wrong information. It is assumed that these errors in the data are found in every category of the database equally and we will therefore not bother with this.

Another source of data that is more specific for BA is the Global Performance Monitor (GPM), which is an on-board survey performed on most BA flights. This survey contains more demographic information about the passengers, but does not have competitor informa-tion. Within BA databases, it is possible to obtain the actual numbers of PSJ’s made with BA, but those data were too complex to obtain in the form needed for this paper, and doesn’t date back more than 1 year.

The reason only the IPS is used as a source is to have consistent data sets. The data about BA from IPS and GPM do not necessarily match up, there are discrepancies. The monthly counts used in this paper are not corrected for the number of days in each month, as this has already been done when the data was put into the database.

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Chapter 4

Models used in literature to

describe air travel demand

The problem of trying to describe the air travel market is not new. Many researchers have been trying to find analytical ways to describe the air travel market and find patterns in it. This chapter will briefly discuss two earlier attempts that use econometric modelling.

4.1 Airline model

Box and Jenkins Box and Jenkins (1976) looked into a series called the airline data, which is a series originating from 1962, describing the number of passengers per month on all car-riers. These data show an upward trend, high seasonality and a variance that increases as the number of passengers increases. Intuitively, taking the log of the data to mitigate the increasing variance seemed a logical thing to do, but Box and Jenkins found that a seasonal autoregressive model1_{fits the data best, a model often referred to as the airline model. Using}

this model, they were capable of getting a good fit of the data, and produce forecasts up to 12 months ahead. This model proved to be a good way of explaining the data, and based on the usual criteria is often found superior to alternative ways of modelling, at least for the airline data set. But this model choice is fairly simple, the only explaining variables are historical data and no step changes, or outside influences affect the outcome of the model. Perhaps this model is too simple to give enough insight on how the air travel market develops. An obvious advantage of this model is that it is easy to use and to interpret. In Findley et al. (2002) a variation on this model was used in an attempt to create a model better than the airline model.

4.2 Neural networks approach

In an attempt to use a more involved model, Faraway and Chatfield Faraway and Chatfield (1998) fitted several neural networks (NN) models to the airline data set. Neural networks are generally more involved than regression or ARIMA models, and it turned out that in order to set up the model a lot of knowledge is needed about modelling. The output of the model depends greatly on the choice of input parameters. It was found that none of the

1_{A (0, 1, 1) × (0, 1, 1)}

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NN models were able to create a better fit than the airline model for the airline data. The best model used parameters for the 1, 12 and 13th lag. It was also found that for some NN model choices, the fit of data was reasonable, while the forecasting performance was very poor. Another comment was made about how NN models are black-box models. Results can be obtained, but understanding the outcome of the model is difficult. This is a major downside; it requires a lot of knowledge of NN models to both apply the model and interpret the results. It would be very hard to build a generalized NN model that would be suitable for forecasting demand.

As much as these papers try to build models to describe patterns in air travel, they fail to describe which factors influence the demand, other than autoregressive or hidden factors. The approach in both papers is from a modelling point of view; aiming to get a good model fit. No exogenous variables are used, and no attempts are made to explain what is causing the changes in the series to occur.

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Chapter 5

An ARIMA model to forecast the

demand for a single carrier

In this section we will try to model the demand for short haul flights on BA by looking at PSJ’s on BA, using only historical data. We will fit an Autoregressive Integrated Moving Average (ARIMA)1 model to the amount of PSJ’s, for UK residents only, using data from January 1996 up to September 2005. The data from October 2005 to September 2006 will be used to evaluate the model, and based on the model, we will forecast 10 years ahead. The forecasts will be then be evaluated.

5.1 Analysis of data

The data shows a seasonal pattern and possibly a trend. In time-series modelling it is a nice property for the series to be analysed to be stationary in order for forecasts based on the model to make sense. Stationarity of a series implies that the autocorrelation, mean and variance do not change over time.

5.1.1 Autocorrelations

Before we fit an ARIMA model to the data, we first look at the autocorrelations of the series. The k-lag autocovariance describes the relation between an observed value and its value k observations before. A high absolute value for lag k implies a possibly significant lag k to include in the ARIMA model. A graphical plot of the autocorrelation function (ACF) and the partial autocorrelation function (PACF) can be found in figure C.1. The graph shows a seasonal 12-month effect. The ACF shows high peaks at lags 2, 3, 10, 11, 12, 13, 14, 22, 23, 24. We can also see that the ACF does not damp out within 24 lags. This might mean that the series are non-stationary, which we will test for in the next section.

5.1.2 Stationarity

Using the Augmented Dickey-Fuller (ADF) test we can test whether a series, for a given model choice, are likely to contain a unit root, implying the series is non-stationary. Because the data is likely to contain a seasonal effect, in the test we allow for a maximum lag of 24.

1

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The null hypothesis is that the model contains a unit root, thus is non-stationary, against the alternative hypothesis of the model being stationary. For a model with an intercept this hy-pothesis is tested at a 99% significance level. The results of this test can be found in figure C.2. At a 99% significance level we cannot reject the hypothesis that a model with 11 lags and an intercept contains a unit root. For alternative setups including a linear trend and intercept or no intercept does the ADF test fail to reject the assumption of a unit root too. Because of this, we will look at the first difference of the series. The ADF test performed on the differenced series rejects the hypothesis of a unit root; the first difference series can be assumed stationary. Continuing from here, it might be useful to examine the first differenced series further and see what the correlogram looks like. The correlogram of the series ∆Ytshows high peaks at lag 12

and 24 and 36 that do not damp out, and this indicates a possible seasonal trend. Because we already suspected the series to have a seasonal trend, we will look at the 12th differences of the first differences series, filtering the seasonal effect out (a model was estimated using only the first differences, but this lead to parameter estimations indicating an unstable model). The ADF test indicates that with very high probability (> 99%) the hypothesis of having a unit root is rejected, and we will therefore assume that the resulting series are stationary. The correlogram shows that the ACF cuts off after lag 13, which could indicate that MA terms up to an order of 13 are present in the series.

5.1.3 Model estimation

Having seen the correlogram and the results of the unit root test, we can decide on our exact model choice. As we have seen, it is appropriate to look at the twelfth differences of the first differences of the series. The next step is to determine which AR and MA terms to include. Let us assume that the obtained differenced series are generated by an unknown underly-ing ARMA model of orders p and q. Ideally we would be able to estimate the true model orders and parameters, but there is no criterion for model selection that guarantees selecting the true model. In Luijsterburg (2006) it was shown that when estimating ARMA models, where the Akaike Information Corrected Criterion (AICC) is used as an order-selection cri-terion, the AICC tends to pick up the correct model in about 63 % of the models that were used in the paper. It was also shown that models selected by the AICC provide forecasts that are nearly as good as forecasts done based on the true model, if the true model was in fact an ARMA model. For this reason we will use the AICC as the order-selection criterion and select the model specification that has the lowest AICC.

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Parameter Estimate Standard Error t-value Pr > |t| MA(1) -0.9163 0.038 -24.33 0.0000 MA(12) -0.8198 0.038 -21.36 0.0000 MA(13) 0.7495 0.049 15.23 0.0000 Table 5.1: Parameters estimates for the MA(13) model

against the hypothesis that this is not true. Results show that the null hypothesis cannot be rejected, therefore instead using a MA(12)xSMA(12) model would result in a model with one parameter less. Parameter estimations of this model can be seen in table 5.2

Parameter Estimate Standard Error t-value Pr > |t| MA(1) -0.8783 0.047 -18.86 0.0000 SMA(12) -0.6637 0.071 -9.38 0.0000 Table 5.2: Parameters estimates for the MA×SMA(12) model

The R-squared is 0.54 . Mathematically this model can be written as formula 5.1:

(1 − L)(1 − L12)yt= (1 − βL)(1 − φL12)et (5.1)

Here L is the lag operator. This type of model looks (coincidently) very much like the airline model (see for example Findley et al. (2002) ). Before proceeding we need to make sure that the true values of β and φ are not equal to one, because then the model would be redundant. Again a Wald test is used to test the hypotheses β = 1 and φ = 1. Both hypotheses are rejected at a 99% significance level.

Another way to verify how well the model fits the data is by looking at the residuals of the estimation. It is straightforward to obtain estimated values of Yt, ˆYt, ∀t ∈ [0, T ]. If the

model fits the data well, then, from the structure of the regression, the relation shown in formula 5.2 should hold.

Yt= ˆYt+ et, et∼ N (0, σ2) (5.2)

The series et, the residuals, are white noise with some variance σ2, identically and

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5.2 Forecasting

In this section, we will use the fitted ARIMA model to forecast the number of PSJ’s up to ten years ahead. We will discuss in detail the results of the model estimated.

In figure C.6 one year of forecasting is shown, where the differenced series were converted back to the undifferenced series. The forecasts for the evaluation period October 2005 -September 2006 seem to fit the actual data quite well. In order to get a measure of fit for the forecasts we use the Mean Absolute Percentage Error (MAPE), which is defined as in formula 5.3. MAPE = 1 T t∗+T X t=t∗ |At− Ft At | (5.3)

Atis the actual observation at time t and Ft is the forecasted value for time t. t∗ is the start

of the evaluation period and T is the number of months observed. For the evaluation year, the MAPE for each individual observation can be found in table C.1. The MAPE over the whole evaluation period is 15.3%. The month where the prediction is most off is January 2006 where the MAPE was 30.0 %.

Considering we are interested in long term forecasts up to 10 years, the individual discrep-ancies of the forecasts are not important, as long as the overall performance of the forecasts is acceptable. Forecasting for a long time period is very tricky when the series involved has a high variance. To get an idea of how reliable the longer term forecasts are, we will look at the 50% and 95% confidence intervals of the forecasts. We will do this by using simulation.

5.2.1 Simulating confidence intervals for the forecasts

Because we want to forecast ten years ahead, or 120 months, we need a non-standard method to forecast and determine how reliable the forecasts are. To create forecasts and confidence intervals, a simulation method is used that simulates residuals. This method assumes that the distribution of the residuals remains the same during the whole forecasting period. In literature a few methods to simulate the confidence intervals for forecasts can be found, see for example Prescott and Stengos (1987) where a bootstrap method is described.

In short this bootstrap method works as follows. Let us assume a series Yt, t = 1, . . . , n

is being looked at. Following the usual procedure, autoregressive parameters are estimated and denoted by θ, and the residuals denoted by e := {e1. . . en}. If N denotes the number of

simulations desired, then N vectors ei := {ei1. . . ein}, i = 1, . . . , N are created by sampling

with replacement N × n values from the vector of residuals e. For every i = 1, . . . , N , artificial series Yit, t = 1, . . . , n are created. For every artificial series autoregressive parameters can be

estimated, denoted by θi. To forecast beyond time n, more sampling is done from the residual

vector e. For every simulation i, iteratively values for Yit, t > n can be obtained, using the

actual values of Ytfor t ≤ n and Yitfor t > n for lagged values, by using the sampled residuals

and the parameters estimated in θi. This yields N forecasted series. By looking at the N

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The method used in this paper is similar to the method described in Prescott and Sten-gos (1987), albeit more simplified; we assume all scenarios have the same underlying model and therefore the model parameters are not re-estimated for every simulation. The reason this assumption was made is because programming this method into EViews proved to be a too difficult task for the multivariate model described later in this paper; to keep the meth-ods used consistent it was decided to use the simplified simulation method throughout the paper. We do not expect this adjustment to have a major influence on the results. First the coefficients for the MA(1)xSMA(12) model are estimated. Then for the data from January 1996 to September 2005, using these model results, the model will be fitted and the residuals will be calculated. From these residuals an empirical distribution can be found. By drawing random values from this distribution we can simulate the forecasts for the residual series up to 120 months. Given these residuals, iteratively the forecasts of the differenced series can be found, which can be converted back into PSJ counts. Repeating this process for 500 simulated residual series leads to 500 different scenarios.

These scenarios represent different possibilities of how the PSJ series can behave in the next 10 years. For every month past September 2005 we can evaluate the 500 different scenarios and look at the range of values. That way we can look at the middle 99 % scenarios to get a 99 % confidence interval for the forecast of that month. Similarly we can look at any amount of % scenarios to get other confidence intervals. The median value for each point in time will be calculated too. If we put all median values in one series, we get a series covering the centre of all distributions. This ’median’ series is probably the best estimation of the ten year ahead forecasts, even though any of the scenarios could function as forecasts.

To verify if the simulations have been done correctly, we look at how for a given time t, the simulated forecasts are spread. According to the model, the simulated values Yit, t >

n, i = 1, . . . , N should be normally distributed, as the only source of uncertainty are the simulated residuals, which we have found to be normally distributed. In figure C.7 QQ-plots can be found for a few values of time, and it clearly shows that for these t > n the forecasted values are normally distributed. From this it is reasonable to assume to for any t > n the Yit, i = 1, . . . , N are normally distributed, where obviously the parameters of this distribution

can change over time.

For a period from October to September every year, we will show the upper- and lower confidence limits as a percentage of the median forecasted value, averaged over all months. The results can be found in Table C.3. The table shows the standard error, and the value of the 50% and 99% upper- and lower confidence intervals as a percentage of the forecasted value. The last column displays the total width of the 99% confidence interval as a percentage of the forecasted value. The bigger the width, the less reliable the forecasts are.

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Figure 5.1: Simulated 50% and 95% confidence limits of the forecasts

5.3 Evaluation of the model and its forecasts

We have seen that the ARIMA model provides a suitable modelling method to fit the data. The ARIMA model recognizes a trend and seasonality in the data, which is very plausible for airline data. In the short term, the model provides accurate forecasts. Forecasts in the long term become increasingly unreliable, and from the 95% confidence intervals observed it seems that the ARIMA model is not a good model to forecast 10 years ahead without losing a lot of accuracy. However, the 50% confidence intervals are quite narrow. It is questionable whether we care about the very broad 99% confidence intervals. Air travel is (implicitly) restricted by a lot of factors: airport capacity, number of aircrafts available, size of population, CO2 pollution to name a few. Experts say that because of these restrictions it would be near impossible for any of the ’extreme scenarios’ predicted in the model to happen. In reality, we should expect one of the more conservative scenarios to be likely. From this point of view, it seems reasonable to use the 50% confidence intervals as a measure to where the demand for BA will most likely be in the near future.

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5.3.1 About seasonal unit roots

In this model, we assumed that after taking first differences, a 12th order seasonal unit root was apparent. A detailed look at the data and correllogram gave evidence that this step was plausible. However, as shown in appendix B it can be tested whether seasonal unit roots are present at different frequencies. For example, it could be that taking 6th differences suffices to make the series stationary. Following the test procedure described in appendix B for the first differenced data, regression is done to determine πi, i = 1, . . . , 12 and the t-values

corre-sponding to the appropriate tests.

In table C.4 the test results can be seen. The critical values shown have been taken from Beaulieu and Miron (1993) for a model with no trend, no seasonal dummies and no intercept. Only for π6 is the null hypothesis of π6 = 0 rejected, but the hypothesis that π5 = 0 is not rejected,

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Chapter 6

A multivariate model for modelling

the short haul market

In this section we construct a multivariate model to simultaneously model the number of PSJ’s for a number of carriers. Data from British Airways, Ryanair (RY), EasyJet (EA) and British Midlands (BMI) are available from January 1999 up to September 2006, for UK residents of various ages. The reason the data starts at January 1999 (instead of January 1996 as in the previous chapter) is because Ryanair and EasyJet only appear in the IPS database from this data onwards. We will look at cointegration and compare a model that takes cointegration into account with a model that does not. For the purpose of evaluation, data from October 2005 to September 2006 will be used. Then, using the best performing model we will construct forecasts up to 10 years ahead and construct confidence intervals, and its performance will be evaluated and compared to the results from the airline model. To build a multivariate model we will use PSJ data not only of British Airways, but also data of three competiting airlines. Considering these carriers are all very active in the UK air travel market it could be that similar underlying factors drive the movement of the series involved. Ideally we would fit a model that involves all carriers that operate within the UK on the short haul market, but this would cause estimation issues as we need to estimate too many parameters compared to the amount of data available, resulting in a low degree of freedom.

The model types that will be used are a Vector Autoregression (VAR) and a Vector Er-ror Correction Model (VECM). Both model types can model multiple series at the same time, but in a different way.

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6.1 Cointegration

Before we fit a multivariate model, we will check for cointegration. According to Christof-fersen and Diebold (1998) correctly recognizing a cointegrating relationship between series can greatly increase the forecasting performance of a model. Cointegrating relationships can occur when one or more of the individual series are nonstationary, which we have seen to be the case in the previous chapter. A cointegrating relationship may be apparent when two of the series wander around in similar motion, never wandering too much apart from each other. In technical terms the existence of a cointegrating relationship implies that when two series are integrated of order k1 and k2, where the value of ki, i = 1, 2 indicates the times a series

needs to be differenced to obtain a stationary series, there exists a linear relation between the two series that results in a series that is integrated of order 0, hence is stationary (See Engle and Granger (1987)).

We know that differencing all series once results in stationary series; this was shown in the previous chapter. In the previous chapter the 12th differences of the differenced series was used because the model was unstable otherwise; for the multivariate model the parameter es-timations caused no reason to believe the model is unstable. A test to check whether variables influence each other is the Granger causality test. The test pair wise compares series and tests the null hypothesis that a series does not influence another series. Because the Granger Cause test is better performed on stationary series we compare all first differenced series. In figure D.1 the results for this test (allowing for 3 lags) can be found. As can be seen, the test does reject several of the null hypothesises, which could imply the existence of cointegrating relationships.

In order to test more formally we can use the Johansen Trace test on the undifferenced series. The reason the undifferenced series are used here is because taking differences leads into stationary series, which will no longer contain cointegrating relationships. In figure D.2 the results of the Johansen Trace test are shown. The test assumes an intercept in the coin-tegrating equation and no trend in the differenced form of the VAR. The results show that the existence of one cointegrating equation is likely.

Based on these results, which model choice should be used? In general, a VAR model does not take into account cointegrating relationships. If a cointegrating equation would exist, es-timating a VAR model would leave out valuable information about this equation, which may affect the quality of the model. A VECM is a more appropriate way to model cointegrating relationships; it models the first differences of the series to obtain (more) stationary series and allows for cointegrating relationships when fitting the model. A disadvantage of using VECM models is that including exogenous variables in the model can result in strange behaviour of the critical values of limit distributions. This means that if we were to fit such a model, we could not say much about the significance of the estimated parameters.

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6.2 Estimating a VAR model without cointegrating

relation-ships

For now, we will ignore the existence of any cointegrating relation. Estimating a VAR model using the undifferenced series would certainly result in unstable model estimates; we will take the first difference of each of the series to estimate a VAR model on. This gives a set of four stationary series to work with.

Estimating which AR order for the model to use is not straightforward, as visual indica-tors like a correlogram only provide information about a single series and are not sufficient for detecting relationships between the series. We manually look at the AICC for different orders of the AR-components and select the model with the lowest AICC. In table D.1 the AICC can be found for various orders chosen for the AR-component. We see that a model with an AR(1)- and AR(12)-component is preferred if our goal is to build a model that describes all four carriers (a constant is not included in the model as it turned out to be insignificant). Using the software package EViews we estimate the parameters. A summary of the model can be found in Figure D.3. The R2for the BA series is 0.745 which is really good.

From this we can see that only a few of the estimated parameters are significantly differ-ent from zero. Almost every variable has a significant parameter for its twelfth lag. BA depends on the number of PSJ’s on Ryanair last month and on BMI 12 months ago, as does BMI depend on the number of PSJ’s on BA 12 months ago. The parameter estimations do partly coincide with the results from the Granger causality test; some of the parameters that, according to the Granger causality test, are deemed important, are found significantly differ-ent from zero. The Granger causality test however is sensitive to the amount of lags specified, according to Johnston and Dinardo (1997)), and we will not put too much weight to this result. In order to check the validity of the model, we look at the properties of the residuals for the BA series. In figure D.6 the results of a BDS test can be found, showing that the residu-als show no significant autocorrelations for orders up to 12 (the test shows similar results for setting a lower maximum order). The residuals can be considered independent. In figure D.7 a QQ-plot of the residuals for the BA series can be found. Even though the higher and lower values are not on the straight line, the middle values seem to match the straight line fairly well. A test for normality of the residuals (see figure D.9) confirms this.

6.3 Estimating a VECM model with cointegrating

relation-ships

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The R2 of this model for the BA series is 0.766 which is higher than the R2 of the VAR model. It appears to be that including a cointegrating relation into the model leads to better performance.

Again we look at the residuals of the estimation. In figure D.10 the results of the test for autocorrelation of the residuals can be found, showing no significant autocorrelations. Ac-cording to figure D.11 and a Jarque-Bera test the residuals are independently and identically normally distributed, even though again the tails are skewed. We conclude from this that the VECM model provides a good fit for the BA data.

Based on the R2 and the AICC the VECM model seems to provide a better fit than the VAR model. Apparently, incorporating the cointegrating relation allows for a better model.

6.4 Forecasts

Because the VECM model appears to be a better model to describe demand, we will only use that model from now on. Based on the VECM model estimated, we can forecast 10 years ahead. As we are only interested in modelling the demand for PSJ’s on BA, we will focus on the forecasts generated for the BA series. Because EViews doesn’t provide a satisfactory way of forecasting a VECM model far into the future, we will use a simulation method again, identical to the method used for the ARIMA model. This simulation method is used for both the forecasting and determining confidence intervals for the forecasts.

The simulation method works as follows. A VECM model gets fitted to the model from January 1999 to September 2005, and the residuals for the four series get calculated. An empirical distribution is estimated for all series. Then the residuals are being forecasted by sampling residuals from the empirical distributions. Using these residuals we can iteratively forecast the four series month by month.

By ordering the values of different scenarios for each month we can find the median fore-casted value and its confidence limits. In figure 6.1 three arbitrarily chosen scenarios can be seen.

In order to measure how good the forecasts fit the data we look at the MAPE for each month of the evaluation period. The values for the VECM model can be found in table D.3. The MAPE is low for most months, with its highest value being 8.63% in May 2006. Over the whole period the MAPE is 4.90%.

6.4.1 Reliability of the forecasts

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Figure 6.1: 3 scenarios for the BA PSJ’s

both directions as time progresses. This means that, even though the predicted values might look reasonable, there is no way of telling how accurate they are. The true value could be nowhere near the predicted value, based on the confidence limits. The 50% confidence limits are by far less extreme than the 99% limits and are probably more realistic than the 99% confidence limits, based on similar arguments used to analyse the confidence limits for the ARIMA model.

We could argue whether the forecasts produced have any meaning, as when looking at just the confidence intervals, there is no guarantee that the forecasted values are correct. But the MAPE for the evaluation period shows that the model with taking into account cointegrating relationships does fit the data accurately. From that point of view, we would expect the forecasts further ahead to be sensible. We could even argue whether confidence limits have any meaning in this research just as we did with the forecasts made with the ARIMA model.

6.5 Evaluation of the multivariate model and comparison with

the ARIMA model

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Figure 6.2: Simulated confidence limits for the VECM model

The average MAPE over the evaluation period is 4.90% which is, compared to an average MAPE of 15.3% for the ARIMA model, a lot better than the ARIMA model.

The forecasts up to 2015 should therefore be more reliable than the ARIMA forecasts. How-ever, the simulated confidence intervals for the VECM forecasts are much wider than those of the ARIMA model. As is mentioned, it is questionable how important this result is.

6.5.1 Adding exogenous variables

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Chapter 7

A different look at the data: market

share

In this section, instead of modelling the amount of PSJ’s, we will model a different variable. As was mentioned before, the more data about various carriers you can implement in your model, the more accurate your model can be. We have got time series data on the amount of PSJ’s on a lot of carriers from January 1996 up to September 2006. Potentially, a model using all this data could be very powerful and describe most of what is going on in the aircraft market. But as the amount of parameters that need to be estimated grows increasingly with each variable added, there is no way of putting all this data into one big VAR- or VECM model. In this section we will build a regression model that uses less variables, but still implicitly contains the original data. We will also include exogenous variables.

7.1 Transforming the observed variables

To build our model, we have got data from 95 carriers flying in and out of the UK using any of the airports around London. One way of reducing the amount of variables used, is to simply leave out data. This way, model estimates might be more reliable, but possibly valuable information is not used to create the model. Another way, is to transform the variables into a new variable by applying a formula on it. In the case of PSJ data, we could measure the fraction of BA PSJ’s per month compared to PSJ’s on any other carrier. For example by applying formula 7.1 (we will not go into great detail on the symbols used as it is straightforward):

PMS = PSJBA

PSJOTHERS + PSJBA (7.1)

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Because the series for each carrier inhibit high seasonality, the PMS will vary highly per month. This leads to a series with a very high variance compared to the observation values. Because market share is usually something that is measured over a longer period of time, we will average out the market share to get quarterly averages (this also makes it easier to fit the exogenous variables, of which some are quarterly data).

In figure E.3 the PMS series are shown. A seasonal effect can still be seen, as well as a downward trend. Even though it is clear that this series is non-stationary, we will not take first differences. We also won’t adjust the series to prevent the model from predicting negative market share, because we want the exogenous variables to have direct and linear effect on the PMS. We will get back on this later.

7.2 Adding exogenous variables

Instead of modelling only the PMS, we can think of factors that are likely to influence the market share. In table 7.1 a list of exogenous variables (all variables are for the UK) that was available is given. The data for these exogenous variables has been gathered from various sources, and have been adjusted and rescaled to match the data we’re trying to fit it to. It should be noted that not all data is of high quality, but it is the best that was publicly available. There are more exogenous variables that can be thought of, but due to a lack of data they have not been included in the list. For each regressor the correlation between the PMS and the regressor is calculated to see how strong the relation between the two variables is.

Variable Correlation with PMS Oil prices per tonne -0.66

Disposable income -0.51 Population size -0.73 No. of carriers operating -0.62 Step change sept. 2001 0.53

Table 7.1: Correlation of PMS with exogenous factors

We would have expected disposable income to have a positive correlation with BA’s market share. BA is a carrier that has passengers mainly from people with higher incomes, from that point of view a higher level of income should lead to more BA passengers. Because all the correlations are different from zero, we will try to incorporate all factors into the model. The variable oil prices is meant as a replacement for flight fares. The pricing structure of air travel is too complex to model as one variable, so a variable that is (assumed to be) correlated with the air travel prices is used instead.

7.3 Model selection

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As a measure of fit, we use the AICC, which allows us to compare models with different amount of parameters used. In table E.1 we see the AICC for several model choices. In-cluding a constant in the regression increases the fit significantly. The best model includes a constant and the population size and oil prices as exogenous variables. The model estimates can be seen in table 7.2.

Parameter Estimate Standard Error t-value Pr > |t| Constant -1.7019 0.478 -18.86 0.0010 Oil prices -0.0437 0.053 -0.82 0.4170 Pop. size -2.9708 0.938 -3.17 0.0031 Table 7.2: Parameters estimates for the regression model

The R-squared for this model is 0.478 which is not very high, but is not surprising given that we are fitting a linear model to highly fluctuating data. The variable oil prices is found to not have a significant influence on the PMS. However, leaving the variable out of the regression does not improve the fit. It is surprising that the number of carriers operating is not found important for the model.

Because we believe that the model fit is too poor with just these variables, we will investigate if adding autoregressive components improve the fit by explaining more of the seasonality. In table E.2 the AICC values of some models can be found. Adding both an AR(4) and MA(4) component to the model results in the lowest AICC.

Parameter Estimate Standard Error t-value Pr > |t| Constant 1.8328 0.687 2.67 0.0121 Oil prices -0.0527 0.024 -2.18 0.0368 Pop. size -3.1997 1.289 -2.48 0.0186 AR(4) 0.8647 0.028 31.23 0.0000 MA(4) -0.9551 0.028 -34.63 0.0000

Table 7.3: Parameters estimates for the regression model with AR and MA components

The R-squared is 0.919 which is almost double the value of the model without AR and MA components. This seems like a good result, however this high R-squared can indicate an in-correctly estimated model, we will get back on this. It can be seen that on a 95% significance level all the parameters are found different from zero.

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So what would the parameter estimates mean if this model were true? The level of the PMS depends negatively on the population size; perhaps BA is not attracting enough new customers, or the capacity of BA does not expand fast enough to keep up with the growing UK population. The higher the oil price, the lower BA’s market share. This makes sense, oil price influences the flight fares, and while a carrier like BA would forward oil price increases to its customers, low-cost carriers would most likely keep their prices low, which would cause customers that care about flight prices to shift carrier.

We should keep in mind that a different set of exogenous variables could have lead to different parameter estimates. But considering this estimation gives the best fit, we will not bother with this.

7.4 Scenario analysis

Continuing with the PMS model, we can look at what the PMS would look like under changes of the exogenous variables. This section will give one example. Because the population size grows at a predictable rate, the effect of the growing population on the PMS can be seen directly from the parameter estimations. However, oil prices are more vulnerable to changes. Because it would be interesting to see what effect changes in the oil prices would have on the PMS, we will look at some scenarios and construct a response function for the PMS on changes in the oil prices at the 4th quarter in 2015.

First we forecast the population size variable up to 2015. Because the population size shows a near linear growth, a model with only a trend is estimated and used to obtain the forecasts. Oil prices up to the 4th quarter of 2005 are known. To keep things simple we will assume that the oil prices from 2006 onwards grow by a fixed amount c every quarter (given the volatile nature of oil prices this is not very realistic, but this analysis can easily be extended to model the oil price series more accurately). Using the PMS model and the forecasts for both exoge-nous variables, forecasts for the PMS can be obtained for different values of c. The oil prices value at the 4th quarter of 2005 is approximately 0.60 (ignoring the unit size). In table 7.4 the scenarios for the oil prices are shown. The first column shows what the oil price at the 4th quarter of 2015 will be (values of the oil price for the period inbetween 2005 and 2015 can be intrapolated), and the second column shows the corresponding c needed to obtain that growth.

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Using EViews to forecast the series, different forecasts for the value of the PMS at the 4th quarter of 2015 can be compared. In figure 7.1 the relation between the value of c and the PMS can be seen. To no surprise this relation is linear and negative, but the added value of such a graph is that the forecasted value of the PMS conditional on a possible behaviour of the oil prices can be seen.

To illustrate how this graph should be read, we present an example. If we assume the oil prices will linearly grow to twice its current value in the next ten years of time (corresponding to c = 0.015), from the graph it can be read that the PMS will obtain a value of almost −0.5, while if the oil prices were to remain the same, the PMS would be around 1.2. (The values on the y-axis are adjusted). This shows that there is a large influence of the oil prices on the value of the PMS.

Figure 7.1: Relation between c and the PMS value at the 4th quarter of 2015

This scenario analysis is just an example to illustrate how the PMS can be used to determine the influence of an exogenous variable on the demand for BA. We could perform all kinds of scenario analysis on the PMS model, but it is clear how the PMS can be used to measure the effect of exogenous variables on the demand for BA.

7.5 Evaluation of the PMS model

The PMS model provides a good fit of the market share data. Using real world variables combined with an autoregressive component it describes the movement of BA’s market share. The MAPE of the forecasts is good, but the model tends to overestimate the PMS for dates beyond 2005. By means of scenario analysis forecasts can be obtained for the PMS conditional on behaviour of the exogenous variables. If the PMS model is anywhere near representative for BA’s market share, then this scenario analysis can be a great way to measure the impact of factors on the demand for BA.

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is logical that BA’s market share is declining; BA is limited to the airport capacity of the two airports it operates from (and cannot expand to other airports within a short time frame), while some other carriers on other airports can expand much more because those airports are still expanding. It could be that BA’s market share, looking at for example Heathrow only, is increasing, while BA’s market share on all London airports (of which some BA doesn’t operate from) is declining. For BA it would be more alarming if their market share on Heathrow was declining than their total market share, as the latter doesn’t necessarely mean BA is loosing ground to other carriers.

Based on this, an alternative way of setting up the PMS model is to look at data from London Heathrow Airport only. Unfortunately data was lacking to look into this. But if such a model could be estimated, that may prove to be quite powerful.

7.5.1 Model stability

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Chapter 8

Conclusions

In this paper we have tried to shed light on a major issue in the air travel industry. As the flying behaviour of the population is changing, BA wants to know if the changing needs of the population affects their customer base, now and in the near future. This paper made an attempt to model passenger demand for BA, and (to a lesser extend) to determine important factors that influence this demand. The actual issue of what consequences there are for BA was not tackled in this paper; the goal was merely to provide ways of looking into the complex problem.

The majority of this paper concerns using econometrics to describe the demand for British Airways. We have seen that in the past several econometric models have been fitted to air travel data. The Box-Jenkins airline model is the best performing model for the dataset used in these papers: the airline data. A complicated neural networks model was not able to improve the model fit, and proved complicated when trying to interpret the outcomes of the model.

In this paper passenger sector journey counts are used to represent the demand for BA per month. Ideally a measure taking into account revenues would have been used, but determin-ing monthly revenue is a project by itself and proved too complicated for this paper. Because it is easy to obtain data on PSJ’s, for practical reasons this is a good way to describe the demand for BA, and also its competitors.

Using three different econometric models, it was attempted to fit the PSJ data. The first model discussed in this paper is an ARIMA approach to fitting only the BA series. Following the appropriate ARIMA modelling procedure and using the AICC as a selection criterion, it turned out a (0, 1, 1) × (0, 1, 1)12model gives the best fit of the data; a model almost identical

to the well-known airline model. This result is remarkable; the same model is best for fitting two series of air travel data that have completely different sources.

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mean value for each month, making the forecasts appear to be very unreliable. However, 50% confidence intervals showed that the majority of the forecasts lies within a much smaller range of values. It was argued that 50% confidence intervals can be used to describe the reliability of the forecasts.

The second model used data from BA and three competitive airlines, Ryanair, EasyJet and BMI. By estimating two multivariate models, a VAR and a VECM model, it was found that there is a significant influence of the carriers on each other. Apparently competitor infor-mation is useful when describing the demand for BA. Because the VECM model includes cointegrating equations, it provided a better fit of the data and was used for forecasting. Some exogenous variables were added to the VECM model, but the influence on the model was too minor to be useful.

By means of simulation forecasts and its confidence intervals were produced. The MAPE of this model for the evaluation was found to be only 4.90% which is very low compared to the ARIMA model. The confidence intervals are even more wide than those of the ARIMA model, and the 99% intervals are not useful. The range of the 50% intervals is a lot smaller, but still make the forecasts seem unreliable. However, given the very good model fit, it would be plausible to ignore the confidence intervals and assume that the median forecasted series is a fair estimate of what BA’s demand will be in the future.

The third model is different from the previous two. Instead of using PSJ counts directly, it transforms PSJ counts for many carriers into a ‘pseudo market share’ series. This has the advantage of having an intuitively useful variable to work with and requires only a few param-eters to be estimated. A regression model with a selection of exogenous variables and ARMA components was fitted to the PMS data. By combining exogenous and ARMA components, not all exogenous variables were found significant.

The best model was a (4, 0, 4) model with the UK population size and the price of oil per tonne as exogenous variables. The MAPE for the evaluation period was 12.64%. Forecasts up to 2015 have not been produced for this model in the same way as for the previous two models, as data was lacking for the regressor variables to be projected into the future. In-stead, as an example, we looked at the effect of changing oil prices on the demand for BA in 2015 by means of scenario analysis. The analysis showed that by looking at various scenarios, more insight could be obtained on what will happen to BA’s demand in the future.

All three the models managed to reproduce the fitted series well, according to statistical measures. Because the PMS model is differently setup, no comparison can be made between the three series based on model fit. The VECM model proved to be superior to the ARIMA in terms of model fit. Based on this, we expect the VECM forecasts up to 2015 to be the most useful, but there is no way to confirm this.

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to explain why the results are as they are. In other words, from an econometrical point of view, the model is a good way to fit the BA series, but from a practical point of view, the model doesn’t give insight in which factors are influencing the demand for BA. The PMS model is simpler to work with and exogenous variables have a significant role in this model, allowing for some understanding of how the exogenous variables influence the demand for BA. To get back to the questions posed in the second chapter: all three models constructed are useful in some way. If the goal is only to obtain forecasts for the demand on BA, the ARIMA and VECM models can be used; the PMS model is harder to use for forecasting because of the inclusion of exogenous variables. Based on the model fit, the VECM model produces the most useful forecasts. However, part of the aim of this paper was to measure the effect of changing (demographic) factors on the demand for BA. From the outcomes of the ARIMA and VECM model, this effect cannot be seen. The way those models explain the data, by using time series components only, makes it impossible to increase our understanding of how and why the air travel market is changing.

To answer the question of ‘Can any of the models be used to determine how underlying (de-mographic) factors influence demand?’, the answer would be no for the ARIMA and VECM model. Only the PMS model provides a way of relating some underlying (demographic) fac-tors to the demand for BA. The PMS model can be used to measure the impact of changing factors on the pseudo market share, and can therefore partly determine how underlying fac-tors influence demand, but this paper shows that its utility is limited as only a select number of factors have been found significant, leaving some obviously important factors unused. To conclude we get back to the main question of this paper: Can we build a meaningful model to describe and forecast the demand for short haul air travel on BA using economet-rical techniques? This paper has shown that there are ways to accurately model the demand for BA and even to obtain good (short-term) forecasts. These forecasts are meaningful, but in terms of really describing the short haul market they are not that useful, as none of the models tested have really been able to explain which factors influence the demand for BA, and therefore fail to describe the demand properly.

8.1 Recommendations for future research

Getting back on the results shown in chapter 3, looking at more specific data can be useful. Using the IPS database, monthly PSJ counts can be obtained for specific age groups. For each age group, one of the models discussed in this paper could be fitted, to obtain the fore-casted demand up to 2015. It is likely that people from different age groups have different flying behaviour. By splitting up the data into smaller groups by for example age and level of affluence, perhaps more interesting models can be fitted.

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in chapter 7), more care can be taken into fitting models to the exogenous variables for the purpose of forecasting.

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Forecasting air travel demand for a large airline using time series modelling