Modified panel data regression model and its ...

Accepted Manuscript Modified panel data regression model and its applications to the airline industry: modelling the load factor of Europe North and Europe Mid Atlantic flights Yohannes Yebabe Tesfay PII:

S2095-7564(15)30649-8

DOI:

10.1016/j.jtte.2016.01.006

Reference:

JTTE 80

To appear in:

Journal of Traffic and Transportation Engineering (English Edition)

Received Date: 2 October 2015 Revised Date:

14 January 2016

Accepted Date: 28 January 2016

Please cite this article as: Tesfay, Y.Y., Modified panel data regression model and its applications to the airline industry: modelling the load factor of Europe North and Europe Mid Atlantic flights, Journal of Traffic and Transportation Engineering (English Edition) (2016), doi: 10.1016/j.jtte.2016.01.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Original research paper

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Modified panel data regression model and its applications to the airline industry: modelling the load factor of Europe North and Europe Mid Atlantic flights

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Yohannes Yebabe Tesfay

Faculty of Economics, Informatics and Social Change, Molde University College, Molde 6402, Norway

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Abstract

This article conducts an econometric analysis on the passenger load factor of the airline industry. Used to measure competence and performance of the airline, load factor is the percentage of seats filled by revenue passengers. It is considered a complex metric in the airline industry. Thus, it is affected by several dynamic factors. This paper applies advanced stochastic models to obtain the best fitted trend of load factor for Europe’s North Atlantic (NA) and Mid Atlantic (MA) flights in the Association of European

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Airlines. The stochastic model’s fit helps to forecast the load factor of flights within these geographical regions and evaluate the airline’s demand and capacity management. The paper applies spectral density estimation and dynamic time effects panel data regression models on the monthly load factor flights of NA and MA from 1991 to 2013. The results show that the load factor has both periodic and serial correlations. Consequently, the author acknowledges that the use of an ordinal panel data model is inappropriate for a

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realistic econometric model of load factor. Therefore, to control the periodic correlation structure, the author modified the existing model was modified by introducing dynamic time effects. Moreover, to eradicate serial correlation, the author applied the Prais–Winsten methodology was applied to fit the

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model. In this econometric analysis, the study finds that AEA airlines have greater demand and capacity management for both NA and MA flights. In conclusion, this study prosperous in finding an effective and efficient dynamic time effects panel data regression model fit, which empowers engineers to forecast the load factor off AEA airlines.

Keywords: Airlines, Load factor, Spectral density estimation, Dynamic time effects panel data model Tel.: +47 45085680 E-mail: [email protected].

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Introduction

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The yield, which measures return per unit of output sold, is an immensely significant metric in the airline industry. By definition, it is the mathematical outcome of two additional fundamental metrics: output sold and revenue earned. In recent periods, the overall yield across the airline industry has been in decline due to several dynamic factors. The price incentive from the decline accounts for a significant portion of

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the traffic growth during this period (Netessine and Shumsky, 2002). Very broadly, yields will soften under the following conditions.

used to sustain higher load factors.

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·Traffic growth is relatively stable or insufficient to captivate output growth. Low and stable prices are

·Intensified competition, lower prices, and yields will toughen when

(1) load factors are already high and output is growing no faster than traffic. (2) traffic growth is outstripping output growth.

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(3) lower competition keeps prices unchanged.

The traffic load factor, and thus revenue and yield is exaggerated by these modification types, exemplifying how intimately associated the variables are within the context of a vacant output (Talluri and Ryzin, 2001).

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The main purpose of this article is that the airline industry’s passenger load factor. The load factor quantifies the percentage of an airline’s sold output, which, in effect, is a measure of the extent to which

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supply and demand are balanced at prevalent price points. The industry’s load factors hide marked variations between diverse types of airlines, with regional carriers at the lower end of the spectrum and charter airlines at higher load factors than scheduled carriers (Cross, 1997). The average load factor for any single airline masks variations between different markets and cabins, with economy/coach achieving higher load factor. This is because customers tend to book economy seats in advance and expect lower levels of seat convenience than premium cabins. It also conceals noticeable daily, weekly, and, in particular, seasonal variations.

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Load factors are mainly driven by six aspects. The first driver is the industry’s production decisions compared to demand growth. The production growth must be carried into closer configuration with

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demand growth. The second driver is pricing. Contingent upon what decisions are made with respect to output, fare discounts usually simulate demand and, generate higher load factor. The third driver is traffic mix. Precisely, the higher the percentage of business travelers carried by an airline, the lower the average

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seat factor. That is, the random element in demand for business travels (unstable) suggests a lower average load factor in business and first class cabins (McGill and van Ryzin, 1999). The fourth factor is payment policies. A carrier accepting non-refundable payment at the time of reservation is likely to have

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relatively fewer no-shows and a comparatively higher seat factor than on selling a greater portion of tickets on a fully flexible basis. The fifth driver is commercial success. A success in product designs, promotions, marketing communications, distributions, and service conveyance will undoubtedly influence current load factor (U.S. General Accounting Office, 1990). The sixth driver is revenue management. The efficiency of a revenue management systems (RMS) will influence load factor. RMS capabilities,

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specifically, the enhancement of demand forecasting tools, contribute significantly (Marriott and Cross, 2000; Tesfay and Solibakke, 2015).

The first driver of the load factor reflects the effectiveness and efficiency of the airline’s management

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efforts. In addition, the other drivers (i.e., two to six) of load factor reflect the effectiveness and efficiency of the airline’s demand management efforts. The author of this paper evaluates the management capacity

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of airlines by investigating the relationship between the load factor (LF) and available seat kilometer (ASK). Similarly, by analyzing the relationship between the load factor (LF) and revenue passenger kilometer (RPK), this paper evaluates the demand management of airlines (Barnhart et al., 2012; Tesfay and Solibakke, 2015).

In the airline industry, yield management describes practices and techniques used to allocate and assign limited resources to a variety of customers in order to optimize the total revenue of the investment. The limited resources are the available seat kilometer (ASK) of a forthcoming flight, and the varieties of customers comprise first class, business, and economy travelers. In short, yield management is

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concerned with the airline’s capacity and demand management (Netessine and Shumsky, 2002). Therefore, evaluations of capacity and demand management offer momentous contributions to the airline

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industry’s yield management.

Load factor is a metric that measures the success of an airline´s capacity and demand management efforts. These efforts are hindered by the fact that at the same time as demand varies in units of single

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seat-departures in different origins and destination markets and is volatile, supply can only be produced in units equivalent to the capacity of whichever aircraft type is available to operate the flight-legs and routes designed to serve targeted origin and destination markets and is broadly fixed in the short run. Moreover,

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the requirements to uphold both the high flight completion rate and integrity of network connections and aircraft and crew assignments might impede a scheduled passenger carrier from cancelling a striking number of its lightly loaded flights (Bruckner and Whalen, 2000).

Depending on the prevailing market, conditions, load factor, and yield trade off against each other, except that demand is particularly strong and output growth is under firm control, it is likely that

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intensifying yield will be related with sliding pressure on load factor. On the other hand, deteriorating yields are often related to higher load factor. Hence, airline carriers normally want to arrive at a capacity plan with target load factor that strikes a balance between the costs of turning passengers away and

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costs of meeting all peak demand coming forward and overwhelming the market at other times (“double-edged sword”). From an operational viewpoint, it is much easier to manage an airline when load

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factors are at 64% than when they are at 84% (Cross et al., 2010). A moderate average load factor may be acceptable if the break-even load factor is sufficiently low, as when, for example, a high yield product is being offered. A high average load factor is not necessarily enough to ensure an acceptable operational performance if the break-even load factor is high, as when, for example, the unit cost is high or the yield is low. If the average load factor rises while the yield and unit cost, and therefore the break-even load factor, remain constant, then operating performance will improve (and vice versa). In the airline line industry, it is crucially important to have quantitative information about the future values of load factor as input in the decision making of setting airfare. However, the econometric

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modelling and forecasting of load factors is a challenging task in the industry. Hence, several complicated and dynamic factors directly or indirectly affect the airline industry’s load factor. One factor is that the

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airline industry is an emblematic example of a cyclical industry, which leads to load factor wavering (Ľubomír and Jakub, 2013). Secondly, airlines apply a dynamic aircraft assignment for any given flight to absorb demand, providing that the airlines capacity management strategy and policy affect the load factor

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(Bertsimas and Popescu, 2003; Li et al., 2007). Third, the industry is dynamic, complex, and extremely subjected to external factors. For example, the dynamics of oil prices directly affect the cost of the airline, and consequently the airfare, which in turn affects the airline’s load factor (Babikian et al., 2002;

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Borenstein 1992; Chua et al., 2005; Doganis, 2010; Flew, 2008). Finally, the direct outcome of deregulation in the industry is intense competition among airlines, and the Computer Reservation System (CRS) puts power in the customer’s hands. This leads to dynamic decisions by the airline yield manages, which affects the load factor (Doganis, 2010; Cento, 2009; Kahn, 1988).

Classic panel data econometric models, such as fixed or random effect panel data regression models,

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are sufficient in producing an effective and efficient forecast of the airline’s load factor. Hence, the load factor is a multifarious variable that must be, advanced to modify current modelling techniques. This study modifies the existing panel data model by defining dynamic time effects, which are neither a linear nor

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nonlinear fuction of time t. Such a model modification helps to capture the important and hidden variation of an airline’s load factor for its geographical flights.

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The major focus of this paper is to apply stochastic models capable of controlling the variability of the load factors for Association European Airlines (AEA) flights in Europe’s North Atlantic (NA) and Mid Atlantic (MA). The stochastic model’s fit helps to forecast the load factor of the flights in these geographical regions and evaluate the airline’s demand and capacity management. Nevertheless, several issues are encountered.

First, appropriate models for the analysis of load factor must be identified. Two most important stochastic quantitative models, which are helpful in forecasting, are regression and time series analyses. Regression analysis is a type of multivariate analysis that is applied to the cross-sectional (spatial) data to

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stochastically measure the impact of the exogenous variables (predictors) on the endogenous variables (response variables). Time series analysis studies the dynamic aspects of the endogenous variables

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(Granger, 1991; Pearl, 2000). However, a time series analysis does not appropriately measure the spatial effect and regression analysis does not measure the dynamic effect of the exogenous variables on the endogenous variables. Thus, these two fundamental econometric models are complementary. Therefore,

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in this study’s stochastic modelling of the load factor, the author uses a panel data regression model to combine the advantages of both regression and time series analyses (Pearl, 2000). Second, the panel data model’s time effect can be treated as either fixed or random, according to the

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results of the Hausman specification test statistic (Fitzmaurice, 2004). However, for this airline’s passenger load factor analysis, the Hausman test statistic is in sufficient to completely control the cyclical dynamics of time effects on the load factor. Therefore, the major challenge in this stochastic analysis is identifying an autocorrelation configuration for the load factor. Normally, the serial-correlation diminishes with more distant lags. However, the autocorrelation configuration of the load factor has a multifarious

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structure and contains both serial and periodic autocorrelations (Tesfay and Solibakke, 2015). To control such a multifaceted autocorrelation arrangement, the classical panel data analysis is extended to express the load factor’s time effect as a dynamic linear or nonlinear function of the parameters integrated to

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geographical NA and MA flights. The modification helps to control the periodic autocorrelation configuration. Moreover, the author applies the Prais–Winsten recursive autoregression estimation (Prais

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and Winsten, 1954) is applied to control the classical serial correlation. The best-fitted dynamic time effects panel data regression model brings new and improved information to AEA European airlines.

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Literature review

The global airline industry is responsible for effectively providing transportation to passengers, cargo, and postal services to every country in the world. The industry has played an essential role in establishing of today’s global economy. By itself, the airline industry is an important economic entity, and it also supports econormic integration worldwide. When one includes the expansion of tourism and recreation, the

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industry’s role has a significant impact on facilitating the world economy (Belobaba et al., 2009). In general, the global airline industry is complex, dynamic, and profitable (Doganis, 2010). It is also

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subject to prompt changes and innovation, making it both cyclical and subjective to external dynamics. An important trend is the reliably righteous growth in demand, which is happening at a diminishing rate. Within the industry, pricing refers to the various service amenities and capabilities of airline products,

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bearing in mind different tariffs in an origin‐destination market. Revenue management is the process of determining the number of seats available at each tariff level. The revenue management of the airline is therefore a function of its tariff strategy and the resulting load factor.

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The success of the airline is determined by its ability to make unit revenues (Yield.LF) that exceed its unit costs (Total Cost/ ASK). Consequently, in addition to minimizing the unit cost, the airline manager’s central is to quickly take advantage of the yield and load factor. An alternative evaluation of an airline’s profitability is its operating ratio. The operating ratio signifies the efficiency of an airline by comparing its

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operating expense to its net sales (Brueckner, 2004; ICAO, 2013; Kellner, 2000).

Load factor-measure of performance of airlines

Yield management is variety of systems, strategies, and tactics that airlines use to scientifically manage

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the demand for their services and products. Airlines are one of the most prominent users of yield management systems. From its roots in the airline industry, yield management rehearsal has developed

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to prominence today as an emblematic business tool in other industries, such as fashion retail, hospitality, energy, and manufacturing (Link, 2004). A bid price is the highest price that a passenger is willing to pay for air transport service. Consequently, this price is contingent on the type of customer at a time. The bid price reflects the dynamic network models and optimal network solution. The success of dynamic network models involves the ability to control the optimal revenues in the market (Kaul, 2009). Passenger load factor, often called “load factor”, measures the degree that an airline reaches its passenger carrying capacity. In other words, the load factor is a measure of an airline’s efficiency and

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performance. The success of a high load is indispensable and a primary indicator of the airline’s profitability. Literature regarding the airline industry can be divided into the demand structures, fleet,

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network, and revenue modelling, as well as its market structures and operating performance (Dender, 2007).

As discussed in the introduction, this paper’s main objective is operating performance focusing on the

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unit revenue, which is inversely proportional to the load factor. This shows the load factor is the degree to which supply and demand are balanced at predominant price points (Distexhe and Perelman, 1994).The percentage of seats an airline has in service and must sell at a given yield, or price level, to cover its

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costs is called the break-even load factor. In order to avoid a negative profit, every single airline has a break-even load factor. The cost of the airline is always positive, and thus, the price of the airline is negatively correlated with the break-even load factor (Bamber, 2011; Flores-Fillol and Moner-Colonques, 2007).

The load factor’s magnitude for a given airline directly reflects the competency of that airline. As a result,

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it is supposed to be infuriating to scrutinize factors that are potentially affecting the load factor of the airline. Normally, operational factors play a noteworthy role in affecting the load factor of airlines. Precisely, an airline’s capacity, the journey measured in distance, tourist, code-share agreement (aviation

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business configuration, where more than one airline shares the same flight), and market concentration HHI index (universally recognized measure of market concentration) are the most important factors. Each

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has a positive and significant effect on the load factor (Cho et al., 2007). The GINI index, which measures the price dispersion or price discrimination in the airline of the same flight, is the factor that negatively affects the airline’s load factor. Other important factors affecting the load factor include airport features, performance limitations, flight conditions, seasonal demand, time of traveler schedule, frequency of flights and dynamic route networks (Cho et al., 2007; Karagiannis and Kovacevic, 2000). The data includes important information about available seat-kilometres (ASK), revenue passenger-kilometres (RPK) and load factor (LF). The data is organized to be suitable for the objectives set by the newly modified panel data regression model (see next section).

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Knowing and identifying these potential factors will benefit the airline, helping to make it more effective strategic and tactical decisions. These strategic and tactical decisions include the staff training, adapting

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the airline staff’s mind-set, determining the optimal number of travel agencies and advertisement, adapting airline management practices, optimizing human resources, and many other related activities

Data and methodology

4.1

Data

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(Talluri and Ryzin, 2004).

The dataset, downloaded from Research & Statistics (http://www.aea.be/research/traffic/index.html), is obtained from Association of European Airlines (AEA) and AEA Traffic and Capacity Data from January 1991 to December 2013.

The AEA is a trustworthy contributor with the following key objectives: raise aviation’s role in Europe’s

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future, increase customer benefit, contribute to move cost-effective regulation, speed up the aviation progress towards a single European Sky, decarbonize aviation to protect the global environment, safeguard circumstances for the fair competition of airlines, and title holder a global security framework of

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airlines.

The AEA brings together more than 30 major European airlines and is comprised of Adria Airways

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(Slovenia), Aegean Airlines (Greece), Air Baltic (Latvia), Air Berlin (Germany), Air France (France), Air Malta (Malta), Air Serbia (Serbia), Alitalia (Italy), Austrian Airlines (Austria), British Airways (United Kingdom), Belgium Brussels Airlines (Belgium), Cargolux (Luxembourg), Croatia-Airlines (Croatia), Cyprus Airways (Cyprus), DNAtsche Lufthansa (Germany), DHL (Germany), Airlines (Spain),

Icelandair (Iceland), KLM (The Netherlands),

Finnair (Finland), Iberia

LOT Polish Airlines (Poland), Luxair

(Luxembourg), Meridiana (Italy), Scandinavian Airlines System (Sweden, Norway, Denmark), Swiss (Switzerland),

TAP Portugal (Portugal), Tarom (Romania), TNT Airways (Belgium),

(Turkey ), and Ukraine International Airlines (Ukraine).

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Turkish Airlines

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Moreover, the North Atlantic-Europe (NA) was defined as any scheduled flights between Europe and North, Central or South America via gateways in the continental USA, including Alaska and Hawaii, and

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Canada. Mid Atlantic-Europe (MA) is defined as any scheduled flights between Europe and North, Central and South America via gateways in the Caribbean, Central America, or South American mainland north of Brazil (i.e. Bolivia, Colombia including the San Andres Islands, Ecuador, French Guinea, and Guyan).

Methodology

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4.2

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4.2.1 Signal processing

Signal processing is a stochastic process of the time series data that is formulated as the series of harmonic functions (Hamilton, 1994). Specifically, the spectrum decomposes the stochastic process’s component into different frequencies helping to identify periodicities. The signal processing stochastic

2015)

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model for a discrete variable is stated as follow (Kammler, 2000; Priestley, 1991; Tesfay and Solibakke,

yt = µt* + ∑ [ ak cos ( 2πvk t ) + bk sin ( 2πvk t ) ]

(1)

k

µt*

is the expected value of the spectrum at time t, a k and

bk are coefficients of cosine and

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where

sine waves respectively, where the independent zero mean normal random variables, v k is the distinct

k is the summation index.

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frequency, and

The mean and variance of the spectrum are

µt∗

2 and E  [ ∑ (ak cos(2πvk t ) + bk sin(2πvk t ))]   k 

respectively. Moreover, the covariance of the spectrum is expressed as follow

  Cov(yt , yt −τ ) = E [∑ (ak cos(2πvk t ) + bk sin(2πvk t ))][∑ (ak cos(2πvk t −τ ) + bk sin(2πvk t −τ ))] k  k 

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(2)

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To simplify the computation, one assumes time is a continuous variable. Further, one assumes that the

function is expressed as below

Auto(yt , yt −τ ) =

∫

T

−T

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spectrum extends infinitely [T ∈ ( −∞ , ∞ ) ] in time towards both directions. Then, the autocorrelation

[ak cos(2πvk t ) + bk sin(2πvk t )][ak cos(2πvk t − τ ) + bk sin(2πvk t − τ )]dt

∫

T

−T

[ak cos(2πvk t ) + bk sin(2πvk t )] dt 2

(3)

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Spectral density analysis illustrates the autocorrelation function’s natural configuration on the observed time series data. This creates a good opportunity to observe the exact nature of autocorrelation in the

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time series data in the Fourier space (Boashash, 2003). The most important advantage of spectral density analysis is its ability to show as hidden periodicities of the time series data (Engelberg, 2008). Estimation techniques for spectral density involve parametric or non-parametric approaches based on the time domain or frequency domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model. A common non-parametric technique is the

4.2.2

Ljung–Box test

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periodogram (Hajivassiliou, 2008; Tesfay, 2015).

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The Ljung–Box test simultaneously tests the existence and order of autocorrelation on the time series data. The null hypothesis (H0) of the Ljung–Box test procedure is defined as the serial correlation equals zero up to order h versus an alternative hypothesis (H1). At least one of the serial correlations up to lag h

4.2.3

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is nonzero (Davidson, 2000; Tesfay and Solibakke, 2015).

Dynamic time effects panel (T-panel) data regression model

The dynamic time effect panel data regression model specifically models the load factor’s variations in the airline industry (see section 1). Using different theoretical and practical quantitative models, the dynamic time effects panel data regression model is expressed below (Barreto and Howland, 2005; Chan et al.,

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2012; Davidson and Mackinnon, 1993; Davies and Lahiri, 2000; Fahrmeir et al., 2009; Hayashi, 2000;

yit = ηi + λt + β1 x1it + β 2 x2it + β3 x3it + L + β k xkit + ε it

λt = f (t ;ϕi )

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Hsiao, 2003; Luc et al., 2000; Kalli and Griffin, 2014; Schittkowski, 2002) (4)

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ε it = U (ε i ,t −1 , ε i ,t − 2 ,L, ε t ,t − h ; ρi1 , ρi 2 ,L, ρih ) + vit , vit ~ iiDN (0, σ v2 ) i = 1, 2, L, n, t = 1, 2,L, T , l = 1, 2,L, k ηi is

specific time effect, xlit represents the exogenous imputes of coefficients function

of

time

t

and

a

vector

of

λt

the ith specific spatial effect,

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where yit is the response from cross section i at time t,

parameter,

U(·)

is

βl , a

is the t

th

f(·) is any real valued linear

function

of

ϕ = [(ϕ11 , ϕ12 , L , ϕ1m ), (ϕ 21 , ϕ 22 , L , ϕ 2 m ), L, (ϕ n1 , ϕ n 2 , L , ϕ nm )] , ε i ,t − j and ρij ,

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j=1,2, ,h.

The newly introduced specification, i.e.,

λt = f (t; ϕi ) ,

causes the existing panel data regression

model. If

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model to have a dynamic time effect. Thus, the modified model is named the T-panel data regression

λt = f (t ;ϕi ) is

a linear function, then, under the complete fulfilment of the Gauss-Markov

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assumption, results from the Least Squares Dummy Variable (LSDV) estimation are the model parameters’ best linear unbiased estimators (BLUE) (Baltagi, 2008). Otherwise, under the complete fulfilment of the Gauss-Markov-Aitkin assumption, results from the Generalized Least Squares (GLS) estimation are the model parameters’ best linear unbiased estimators (BLUE) (Amemiya, 1985; Hassibi et al., 1999; Hsiao 2003; Kailath et al., 2000; Voinov and Nikulin, 1993).

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If

λt = f (t; ϕi ) is a nonlinear function, then the following estimation procedure is applied to estimate

Wooldridge, 2013).

yit = F (t ; X ; θ ) + ε it

Let

where F (t ; X ; θ ) = ηi + f (t ; ϕi ) + β1 x1it + β 2 x2 it + β 3 x3it + L + β k xkit ,

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the model parameters (Billings, 2013; Kelley, 1999; Meade and Islam, 1995; Seber and wild, 1989;

(5)

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Now minimizing the total sum of the errors, one achieves

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θ = [θ1 , θ 2 , L , θ n ( m +1) + k ] = [η i , ϕ , β ] is the vector of model parameters.

n T  min ∑∑ [ yit − F (t ; X ; θ )]2   i =1 t =1 

(6)

∂  n T  [yit − F (t ; X ; θ )]2  = 0 , s = 1, 2,L, n(m + 1) + k ∑∑  ∂θ s  i =1 t =1  n T  ∂F (t ; X ; θ )  ⇒ ∑ ∑ [ yit − F (t ; X ; θ ) ]  =0 ∂θ s i =1 t =1  

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⇒

n T   ∂F (t; X ; θ )     ∂F (t; X ; θ )  ⇔ ∑∑  yit   − F (t; X ; θ )   = 0 ∂θs ∂θs i =1 t =1       

(7)

as

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In order to solve Eq. (7), the Newton-Raphson recursive algorithm is applied by defining a new function

  ∂F (t; X ; θ )   ∂F (t; X ; θ )   Gs (t; X ; θ ) = ∑∑  yit   − F (t; X ; θ )   ∂θs ∂θs i =1 t =1        T

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n

G = [Gs (t; X ; θ )]n(m+1)+k

Afterwards derive the Jacobean matrix (Hazewinkel, 2001) from Eq.(8)

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(8)

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(9)

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∂G1 (t ; X ; θ1 ) ∂G1 (t ; X ; θ1 )   L   ˆ ˆ ∂θ1 ∂θ n (m +1) + k     ∂G2 (t ; X ; θ 2 ) ∂G2 (t ; X ; θ 2 ) L   ˆ ˆ ∂θ1 ∂θ n (m +1) + k JG =     M M   ∂Gn (m +1) + k (t ; X ; θ n (m +1) + k )   ∂Gn (m +1) + k (t ; X ; θ n (m +1) + k ) L   ∂θˆn (m +1) + k ∂θˆ1  [n (m +1) + k ]×[n (m +1) + k ]

To get the numerical solution of the model parameters’, apply the Newton-Raphson recursive as an

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algorithm (Bonnans et al., 2006; Ortega and Rheinboldt, 2000).

[θˆs ]r +1 = [θˆs ]r − [ J G−1 ][G] where

J G−1

4.2.4

Model adequacy and diagnostics

r = 1, 2, L

(10)

is the inverse of the Jacobian matrix.

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This section involves removing the serial correlation from the time series data to obtain efficient model parameters. In order to remove the serial correlation, the following algorithm is used. Step 1: Compute residuals as below (Cook and Weisberg, 1985; Weisberg, 1985).

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εˆit = yit − F (t; X ; θˆ )

(11)

Step 2: Test the presence of a serial correlation on the estimated residuals. Here, one applies the

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Ljung–Box test of serial autocorrelation.

Step 3: If autocorrelation is identified, then the Prais–Winsten methodology is applied to remove the serial correlation (Amemiya, 1985; Davies and Lahiri, 1995; Frees, 2004; Prais and Winsten, 1954; Verbeek, 2004; Wooldridge, 2008, 2013). Step 4: Repeat step 1 to step 3 unless the Ljung–Box test of autocorrelation approves that there is no serial correlation on the estimated residuals.

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Results and discussion

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5.1

Evaluation of load factor’s regional characteristics

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Before fitting the panel data model, it is indispensible to analyze the relationship between the load factor (LF) of both the Europe-North Atlantic (NA) and Europe-Mid Atlantic (MA) airlines with reverence of the available seat-kilometres (ASK) and revenue passenger-kilometers (RPK).

Bootstrap result estimates of the RPK and ASK of NA and MA flights are given in Table 1, which shows

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that the mean RPK estimates (millions) of NA and MA flights are 13,514.0 (with bias of -18.82) and 3265.5 (with bias of -1.11), respectively. The mean ASK estimates (millions) of NA and MA are 17,018.90

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(with bias of +0.92) and 4090.49 (with bias of +4.68), respectively. These results confirm that both the RPK and ASK of NA flights are higher than those of MA flights. Furthermore, an average of 3504.90 ASK and 824.99 ASK (millions) is out of use every month for NA and MA flights, respectively. Table 1

Mean estimates of RPK (millions) and ASK (millions) of NA and MA flights. Estimates of RPK (millions)

Statistic Est.

Std. error

95% confidence interval

Lower

Upper

Mean

13514.0

-18.82

246.31

13,003.8

14,003.0

Std. dev

4115.4

-7.89

143.22

3819.9

Mean

3265.5

-1.11

72.47

Std. dev

1214.3

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NA

MA

Bias

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Flight

-1.81

Bias

95% confidence interval

Std. error

Lower 17,018.90

Upper

0.92

239.33

16,560.50

17,469.18

4384.3

3940.39 -8.91

132.62

3674.34

4203.10

3128.6

3404.1

4090.49

4.68

80.90

3922.51

4259.37

1138.7

1286.9

1336.16 -3.33

47.25

1236.82

1419.24

Bootstrap results are based on 1000 bootstrap samples

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Estimation method

38.46

Est.

Estimates of ASK (millions)

Referring to the results in Table 2, average NA flights have 10,248.52 RPK (millions) and 12,928.45 ASK (millions) more than MA flights. Moreover, the results confirm that the load factor of NA and MA flights are statistically equal. Table 2

Variable

Comparison of ASK, RPK and load factor of NA and MA flights.

Comparison of flights

Mean difference

Std. error

15

t-cal

Sig. (2-tailed)

95% confidence interval Lower

Upper

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NA vs. MA flights

12,928.45

254.24372

50.85

0.001

12,444.25

13,440.09

RPK (millions)

NA vs. MA flights

10,248.52

267.22149

38.49

0.001

9697.97

10,753.41

LF (percent)

NA vs. MA flights

-0.281

0.57551

-0.49

Estimation method

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ASK (millions)

0.642

-1.42

0.83545

Bootstrap results are based on 1000 bootstrap samples

The estimation results from Table 3 and Table 4, along with Fig.1 and Fig.2 show that there is a strong

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and positive linear relationship between RPK and ASK, with determination coefficients of 90.7% and 97.5%, respectively, for both NA and MA flights. Results also show that, generally, operational decisions

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taken by the airlines to balance the supply and demand of these geographical flights are good. The linear regression fit using the Prais-Winsten recursive parameter estimation indicates that from a one million increases in ASK, the RPK of the NA and MA flights is increased by 1.067 and 0.907 millions,

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respectively.

Fig. 1 Time series plot of ASK and RPK (millions) of NA flights.

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Fig.2

Table 3

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Time series plot of ASK and RPK (millions) of MA flights.

Prais-Winsten recursive parameter estimation of an RPK (millions) linear regression in response to ASK (millions) of NA flights. Std. error

t

App. sig

R square

ASK-NA

1.067

0.021

51.451

0.000

0.907

Constant

-4634.960

-12.561

0.000

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B

368.984

Note: Prais-Winsten autoregression estimation method is used; dependent variable: RPK.

Prais-Winsten recursive parameter estimation of an RPK (millions) linear regression in

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Table 4

response to ASK (millions) of MA flights.

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B

Std. error

t

App. sig

R square 0.975

ASK-MA

0.907

0.009

102.809

0.000

Constant

-446.588

38.132

-11.712

0.000

Note: Prais-Winsten estimation method is used; dependent variable: RPK. To evaluate the managerial performance of the airlines, one must analyze the results reported in Fig.3 and Fig.4. The estimation results from Table 5, together with Fig.3, show that there is a moderate and positive linear relationship between LF and PRK for both NA and MA flights with determination

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coefficients of 67.4% and 60.1%, respectively. The fit of the linear regression using the Prais-Winsten recursive parameter estimation indicates that, for a one million increase in RPK, LF is improved by 0.002

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and 0.003 percent for NA and MA flights, respectively. Moreover, Table 6’s estimation results, together with Fig.4, show that there exists a considerable and positive linear relationship between LF and ASK for both NA and MA flights with determination coefficients of 41.5% and 48.9%, respectively. The fit of the

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linear regression using the Prais-Winsten recursive parameter estimation indicates that, for a one-million increase in ASK, LF is improved by 0.002 and 0.004 percent for NA and MA flights, respectively. The

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management of both NA and MA flights.

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overall analysis of Table 5 and Table 6 confirms that airlines have improved the demand and capacity

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Fig.3 Scatter plot of the load factor versus RPK.

Fig. 4 Scatter plot of the load factor versus ASK.

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Table 5

Prais-Winsten recursive parameter estimation of a linear regression of load factor

Flight

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(percent) in response to RPK (millions).

Predictor

Estimator

Std. error

t-cal

Sig.

RPK-NA

0.002

0.0001

23.738

0.000

Constant

45.975

1.6000

28.729

0.000

RPK-MA

0.004

0.0002

20.260

0.000

Constant

63.571

0.7800

81.527

MA

Model std. error

0.674

2.890

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NA

R square

0.601

Prais-Winsten recursive parameter estimation of a linear regression of load factor

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Table 6

2.678

0.000

(percent) in response to ASK (millions).

Flight

Predictor

Estimator

Std. error

t-cal

Sig.

ASK-NA

0.002

0.0001

13.902

0.000

Constant

47.067

2.3140

13.902

0.000

ASK-MA

0.004

0.0002

16.161

0.000

Constant

63.507

0.9640

65.892

0.000

MA

Model std. error

0.415

3.967

0.489

3.126

Evaluation of load factor’s autocorrelation configuration

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5.2

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NA

R square

In the time series econometric analysis, it is necessary to obtain the exact autocorrelation configuration of

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the time series. The spectral density analysis is a powerful method of ascertaining the alignment of the autocorrelation function. The non-parametric spectral density and analysis provide graphical information about how the autocorrelation function behaves in Fourier space. One of the graphical methods is the response of periodogram of the autocorrelation function of the frequency of the time series observation. This method is tremendously sensitive to the series’ optimum autocorrelation. Another method is the response of density of the autocorrelation function of the frequency of the time series observation. This technique is sensitive to the weighted autocorrelation of the

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series. Thus, both plots provide imperative information about the autocorrelation structure of the load factor. The results of the spectral density estimation and the Ljung–Box test for the load factor of NA and

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MA flights are given in Table 7.

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Monthly distributions of load factor over years

Spectral density estimation using periodogram

Spectral density estimation using density

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Flight

Structure of autocorrelation of load factor of NA and MA flights.

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Table 7

DF

Sig.

224.41

14

0.000

324.36

16

0.000

Key

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MA

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NA

Ljung-Box test Chi-Sq.

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5.2.1 Load factor’s autocorrelation configuration of North Atlantic-Europe flights

The non-parametric plot of the periodogram and spectral density of the LF of the NA flights indicate that there exists a strong periodic autocorrelation, which is observed after jumping a certain period of months. The LF recurrent yearly plot of NA flights over several months show that, from year to year, there are

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strong periodic pattern with small variance.

In the recurrent yearly plot of LF over several months, one noticed an important pattern. The smallest LF is observed in January, then it underway grows until July before deteriorating until December.

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The periodogram plot and spectral density suggest that the LF distribution of NA flights is serially correlated up to a certain month lags. Moreover, the Ljung-Box parametric test proposes that there exists an important serial correlation of order 17 months, which is unrestrained after the 18th month.

5.2.2

Load factor’s autocorrelation configuration of Mid Atlantic-Europe flights

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The non-parametric plot of the periodogram and spectral density of the LF of flights of MA indicate that there exists a strong periodic autocorrelation which is observed after jumping a certain period of months. The repeated yearly LF plot of MA flights over several months shows that there is a strong periodic

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pattern.

In the recurrent yearly LF plot over several months, one observes that there is an important pattern.

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The smallest LF is observed in November, December and January. Then LF starts to grow during July, August and September before deteriorating until November. The periodogram plot and spectral density suggest that the LF distribution of MA flights is serially correlated up to a certain month lags. Furthermore, Ljung-Box parametric test proposes that there is an important serial correlation of order 15 months, which is dissipated after the 16th month.

5.3

Fitting load factor’s dynamic time effects panel (T-panel) data regression model

Analysis from section 5.1 identifies that both the RPK and ASK of NA flights are higher than those of MA

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flights. Moreover, the average LF of MA flights is higher than the average of NA flights. This confirms that the panel data model’s spatial effects exert control for such important variability.

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The analysis of section 5.1 describes how LF that the characteristics behave differently for MA and NA flights. The load factor significantly correlates with both the RPK and ASK for the NA and MA flights in different magnitude. Using these variables (i.e., both RPK and ASK) as common exogenous imputes to

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predict the load factor is inadequate.

The analysis of the autocorrelation’s configuration in section 5.2 identifies that both periodic and serial correlations exist on the LF. The autocorrelation structure is different for both MA and NA flights. This

associated with the regional flights.

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shows that the time effect on LF is not simply fixed or random. Rather, it is dynamic ally and uniquely

Therefore, the appropriate model that is fit to analyze the LF of airline flights is the dynamic time effects two way panel data regression model. The dynamic time effect two way panel data regression model’s fit for the load factor of MA and NA flights is given in Table 8 and Fig.5.

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The significance of the time function harmonic component suggests that the LF is seasonal by its nature. Generally, the fit of the model suggests that the load factor is improving (i.e., growing) with time for both MA and NA flights. Specifically, the load factor’s spatial effect MA flights (62.18%) is smaller than

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that of the NA flights (64.48%).

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Rho (AR1)

Forecasting of load factor of 2014 (in %)

Estimates

t-cal

Std. error

Approx. sig.

0.25622

0.05931

4.31968

0.00002

0.03602

0.00748

4.81731

0.00000

1.83815

0.61281

2.99952

0.00296

sin(ω2t)

-0.69541

0.29154

-2.38533

sin(ω6t) cos(ω2t) cos(ω6t) cos(ω5t) Spatial effect of NA flights

-4.05655 0.96537 -6.34351 -0.70343 64.48071

0.38286 0.29145 0.38093 0.37299 2.01055

-10.59536 3.31234 -16.65287 -1.88594 32.07111

cos(ω6t)

-1.35234 62.17773

0.01776

0.00000 0.00105 0.00000 0.06039 0.00000

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0.00605 0.49259 0.20627 0.24770 0.20620

95% prediction interval UP

69.84714

83.74768

Feb

76.97245

70.02218

83.92272

Mar

80.95242

74.00215

87.90269

Apr

84.84587

77.89560

91.79614

May Jun Jul Aug Sep Oct Nov Dec

87.16869 90.17115 93.48762 93.31471 89.20248 85.12541 82.67023 79.66971

80.21842 83.22088 86.53735 86.36444 82.25221 78.17514 75.71996 72.71944

94.11896 97.12142 100.00000 100.00000 96.15275 92.07568 89.6205 86.61998

6.01170

0.00000

Jan

84.58222

79.54898

89.61545

0.00000 0.00000 0.00000 0.00000 0.00894

Feb Mar Apr May Jun

87.05071 87.95908 84.27398 80.79658 83.20649

82.01748 82.92585 79.24074 75.76335 78.17326

92.08394 92.99231 89.30721 85.82982 88.23972

0.24720

-5.47060

0.00000

Jul

88.05483

83.0216

93.08807

1.61428

38.51721

0.00000

Aug

88.36816

83.33493

93.40139

Sep

84.94909

79.91586

89.98233

π

Where t=12 (Current year-1991)+Current month and

3.53325

76.79741

LB

4.84263 5.27510 -7.85658 9.18377 2.63374

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Spatial effect of MA flights

0.05685

0.02931 2.59844 -1.62055 2.27482 0.54307

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Time function coefficients of MA flights

0.34176 t ln(t) sin(ω2t) sin(ω6t) cos(ω2t)

Jan

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t

Rho (AR1)

Month

Expected

ln(t) Time function coefficients of NA flights

Model S.E

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Parameter estimates

Spatially integrated dynamic time effects

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Table 8 Fit of dynamic time effects panel data regression model of load factor of NA and MA flights.

ωi = , i = 1, 2, L

are the periods

i

24

2.5587

Oct

83.41876

78.38553

88.45199

Nov

84.26842

79.23519

89.30165

Dec

84.52316

79.48993

89.55639

Comparison of dynamic time effects panel data regression model’s fit with actual values.

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Fig.5

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Conclusions and recommendations

6.1

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6

Conclusions

This paper applies advanced econometric analysis to the load factor (LF) of AEA airline’s MA and NA flights. The econometric analysis provides the following conclusions. The bootstrap estimation results suggest that the mean RPK estimates (millions) of NA and MA flights are 13,514.0 (with a bias of -18.82) and 3265.5 (with a bias of -1.11), respectively. The mean ASK estimates (millions) of NA and MA flights are 17,018.90 (with a bias of +0.92) and 4090.49 (with a bias of

25

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+4.68), respectively. The results confirm that both the RPK and ASK of NA flights are higher than those of MA flights. Furthermore, an average of 3504.90 and 824.99 ASK (millions) is out of use for every month in

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the NA and MA flights, respectively.

LF for both NA and MA flights are significantly and positively correlated with RPK and ASK. The results suggest that airlines have a better demand and capacity management for both NA and MA flights.

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LF of both MA and NA flights have periodic (i.e., season to season) correlations. The smallest LF is observed in November, December, and January then starts to grow during July, August and September before declining until November. The smallest LF of NA flights is observed in January, then grows until

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July and then declines until December. Furthermore, the LF of both MA and NA flights have serial (i.e., month to month) correlations. The LF of MA and NA flights have order of 15 and 17 months respectively. The overall autocorrelation structure suggests that the dynamic time effect and two way panel data regression model is an appropriate and realistic forecasting model for the load factor of MA and NA flights. By using the fitted model forecast monthly values for LF with upper and lower 95% prediction intervals for

prediction intervals, for 2014.

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2014. Furthermore, Fig. 5 provides the monthly forecasted values of the LF. with upper and lower 95%

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6.2 Recommendations and policy implications

This paper applies a two-way panel data econometric model for the LF AEA airline’s MA and NA flights.

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Results from the study have important managerial implications, helping to form the following policy recommendations.

Using the dynamic time effect panel data model, the load factor’s fit is more robust and realistic. AEA may, therefore, use the model to predict the LF for the distribution of MA and NA flights. In this regard, airlines should apply the model to their regional flights across the globe to more precisely forecast the load factor. In addition to reducing costs in the airline industry, the profitability of any given airline relies on the joint maximization of yield and LF. In order to push up the LF and yield simultaneously and produce strategic

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decisions regarding profitability, AEA may extend the LF analysis by considering individual airlines (i.e., spatial effects) into the dynamic time effects panel data model. Such analysis can give rigorous

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information about the LF of each airline. Consequently, AEA will have an improve quantitative input regarding how to restructure yield management, network design, etc., with respect to their specific flights over the time periods.

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The econometric analysis indicates that the response of the airlines adopt the demand is found generally good for both the MA and NA flights. Furthermore, the paper found that airlines have a good demand and capacity management of both regional flights. It is recommended that one keeps up with the

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existing demand and capacity management strategy.

Finally, as suggested by many scholars, the airline industry is seasonal. This paper found that the LF of MA and NA flights are both seasonal and different. Implying that LF is far from stable. Any of the airline’s stabilizing policies have so far failed. AEA may, therefore, continuously focus on LF stabilization and

7

Limitations

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improvement.

Since, it is crucial to modify the existing panel data regression model to analyze the load factor of the

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airline industry, it is important to recognize that the modified model specifications have certain limitations. The main limitations of this econometric analysis are detailed. First, this study does hot include all

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important variables in the airline industry with potential impact to the load factor. The author focused the analysis on the aggregated load factor of all member airlines of the Association of European Airlines (AEA). Therefore, it is recommended that other important variables are included when future researchers apply a similar analysis to airlines. Secondly, the introduction of dynamic time effects in the existing panel data regression model demands specific formulation. Therefore, future researchers must recognize and model the load factors of particular airline with dynamic time effects panel data regression model that requires advanced acquaintance of curve estimation and numerical mathematical methods.

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Acknowledgments First and most, I give ultimate thanks to my creator God, Jesus Christ, the only son of St. Mary, who gave

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me space to live and time to think. Secondly, I would like to thank St. Gabriel and St. Michael, who have protected me since childhood. Special thanks go to professor Per Bjarte Solibakke for his continuous support and valuable comments on this paper. I would also like to thank the scholars referenced in the

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brother, Abebe E. Berhe, and my aunt, Shashitu E. Berhe.

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article. Finally, special thanks go to my mother Birhan E. Berhe, my father, Yebabe T. Gebremariam, my

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Kluwer Academic Publishers, Amsterdam. Weisberg, S., 1985. Applied Linear Regression, second ed. John Wiley & Sons, New York.

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Wooldridge, J.M., 2013. Introductory Stochastics: A Modern Approach, fifth ed. South Western, Australia.

Yohannes Yebabe Tesfay is a Senior Quantitative Scientist, Econometrician and Industrial Supply Chain and Process Engineer. He is a contributor of: 1) new econometric models: the Hecksher-Ohlin-Tesfay effect, the Tesfay-generalised extensive margin effect and the Tesfay-eigenvalue technique of detecting structural changes on the country’s participation in the international trade; 2) the hyper-hybrid

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coordination and the Kraljic-Tesfay portfolio matrix; 3) T-panel data regression model. Furthermore, he is the author of many books, academic articles, and is a manuscript reviewer in well-known academic

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journals published by Springier, Taylor & Francis, Elsevier and Open Sage.

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