The data
The dataset is obtained from the Association of European Airlines (AEA) and is downloaded from http://www.aea.be/research/traffic/index.html. The data is collected for the period 1991 to 2013 and contains information about Available Seat-Kilometres (ASK), Revenue Passenger-Kilometres (RPK) and Load factor (LF).
Moreover, Europe-Far East (EF) is defined as any scheduled flights between Europe and points east of the Middle East region, including Trans-Polar and Trans-Siberian flights. Europe-Middle East (EM) is defined as any scheduled Terminating flights between Europe and Bahrain, Iran, Iraq, Israel, Jordan, Kuwait, Lebanon, Oman, Saudi Arabia, Syria, United Arab Emirates, Yemen and the Democratic Republic of Yemen (Available at www.aea.be).
Methodology
One way analysis of variance (ANOVA)
One way analysis of variance (ANOVA) is used to see the existences of the main differences of a certain random variables with a single treatment over its levels. The linear statistical model for ANOVA is given as([6, 43]:
$$ {y}_{ij}=\mu +{\alpha}_i+{\varepsilon}_{ij},\kern0.5em i=1,2,3,\dots, a\kern0.5em and\kern0.5em j=1,2,3,\dots, n $$
(1)
where: μ the grand mean of y
ij
, α
i
the i
th
level effect on y
ij
and ε
ij
∼ iidN(0, σ
2). The bootstrapping estimation method is applied to estimate the model parameters. Usually the method of estimation of the model parameters is either using ordinal least square (OLS) or generalized least square (GLS) estimators according to the parameters are fixed or random, respectively [17, 11]. Nevertheless, modern econometric methods used bootstrapping to acquire thorough information about the estimated parameters. In this particular case we apply the Bias-Estimation Bootstrap technique. The estimation method gives information about bias of the estimates due to resampling in addition to the estimates of OLS or GLS [13].
Signal processing
Signal processing represents a time series as a stochastic sum of harmonic functions of time [20]. Signal processing helps to identify the autocorrelation structure of the time series data. The signal processing stochastic model for stochastic process in discrete time is given as [23, 35]:
$$ {y}_t={\mu}_t^{\ast }+{\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {\upsilon}_kt\right)+{b}_k sin\left(2\pi {\upsilon}_kt\right)\right]} $$
(2)
Where: μ
∗
t
is the mean of the series at time t, a
k
, b
k
(Fourier transformation coefficients of cosine and sine waves) are independent zero mean normal random variables, v
k
are distinct frequencies.
The mean, variance and covariance of the spectrum of the time series data are derived as follows:
$$ \begin{array}{l}E\left[{y}_t\right]=E\left[{\mu}_t^{\ast}\right]+E\left\{{\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {\upsilon}_kt\right)+{b}_k sin\left(2\pi {\upsilon}_kt\right)\right]}\right\}\hfill \\ {}E\left[{y}_t\right]={\mu}_t^{\ast }+{\displaystyle \sum_kE\left[{a}_k \cos \left(2\pi {\upsilon}_kt\right)+{b}_k sin\left(2\pi {\upsilon}_kt\right)\right]}\hfill \\ {}\therefore E\left[{y}_t\right]={\mu}_t^{\ast}\hfill \end{array} $$
(3)
$$ \begin{array}{l}Var\left[{y}_t\right]=E{\left\{\left[{y}_t\right]-E\left[{y}_t\right]\right\}}^2\hfill \\ {}Var\left[{y}_t\right]=E{\left\{{\mu}_t^{\ast }+{\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {\upsilon}_kt\right)+{b}_k sin\left(2\pi {\upsilon}_kt\right)\right]}-{\mu}_t^{\ast}\right\}}^2\hfill \\ {}\therefore Var\left[{y}_t\right]=E{\left\{{\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {\upsilon}_kt\right)+{b}_k sin\left(2\pi {\upsilon}_kt\right)\right]}\right\}}^2\hfill \end{array} $$
(4)
$$ Cov\left[{y}_t,{y}_{t-\tau}\right]=E\left\{\left({y}_t-E\left[{y}_t\right]\right)\left({y}_{t-\tau }-E\left[{y}_{t-\tau}\right]\right)\right\} $$
(5)
Since \( {y}_t-E\left[{y}_t\right]={\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {v}_kt\right)+{b}_k \sin \left(2\pi {v}_kt\right)\right]} \) and \( {y}_{t-\tau }-E\left[{y}_{t-\tau}\right]={\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {v}_kt-\tau \right)+{b}_k \sin \left(2\pi {v}_kt-\tau \right)\right]} \)
Therefore, Eq. 5 can be expressed as:
$$ Cov\left[{y}_t,{y}_{t-\tau}\right]=E\left\{\left({\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {v}_kt\right)+{b}_k \sin \left(2\pi {v}_kt\right)\right]}\right)\left({\displaystyle \sum_k\left[{a}_k \cos \left(2\pi {v}_kt-\tau \right)+{b}_k \sin \left(2\pi {v}_kt-\tau \right)\right]}\right)\right\} $$
(6)
Spectral density is a powerful tool to analyse the nature of the autocorrelation of the time series data in the Fourier space that contains infinite sum of sine and cosine waves of different amplitudes [16, 3]. This creates good prospect to remove the problem of autocorrelation and to choose appropriate an econometric model that capture the possible variations of the time series data. Estimation techniques of spectral density can involve parametric or non-parametric approaches based on time domain or frequency domain analysis. A common parametric technique involves fitting the observations to an autoregressive model. A common non-parametric technique is the periodogram. The important advantage of applying the periodogram spectral estimator is determining possible hidden “periodicities” in the time series [36].
Ljung–Box test
There are a number of parametric methods that detect autocorrelation. However, the Ljung–Box test is preferable in this case because it simultaneously detects the existence and the order of autocorrelation on the time series data. The Ljung–Box test procedure is given as [10]: the Null Hypothesis H
0
: serial correlation equals zero up to order h versus H
1
: at least one of the serial correlations up to lag h is nonzero. The test statistic of Ljung–Box is given as:
$$ Q=n\left(n+2\right){\displaystyle \sum_{l-1}^h\frac{{\widehat{\rho}}_l^2}{n-l}} $$
(7)
where: n is the sample size, \( {\widehat{\rho}}_l \) is the sample autocorrelation at lag l, and h is the number of lags being tested. The null hypothesis is rejected for α level of significance if Q > χ
2
l − α,h
.
Multivariate trend analysis
The aim of the trend analysis is to get the best fitted model to be applied for forecasting the long run behaviour of the series as a function of time [28]. The general form of multivariate trend analysis is given as [5, 41].
$$ {y}_{it}={f}_i\left(t;{\beta}_i\right)+{\varepsilon}_{it} $$
(8)
\( {\varepsilon}_{it}=U\left({\varepsilon}_{it-1},{\varepsilon}_{it-2},\dots {\varepsilon}_{it-h};{\rho}_{i1},{\rho}_{i2},\dots, {\rho}_{i{h}_i}\right)+{v}_{it} \) and v
it
~ iiDN(0, σ
2
iv
) where: f
i
(t; β
i
) is any real valued function of time “t” and a vector of parameters \( {\beta}_i=\left({\beta}_{i0},{\beta}_{i1},{\beta}_{i2},\dots, {\beta}_{i{k}_i}\right) \), \( U\left({\varepsilon}_{it-1},{\varepsilon}_{it-2},\dots {\varepsilon}_{it-h};{\rho}_{i1},{\rho}_{i2},\dots, {\rho}_{i{h}_i}\right) \) is a linear function of ε
it − ij
, ρ
ij
and j = 1, 2, 3, …. h
i
, ε
it
and v
it
are random error terms.
To find suitable estimation method of the model parameters, it is necessary to have acquaintance about the mathematical structure of f
i
(t; β
i
). In this case we have two major categories of f
i
(t; β
i
), linear and nonlinear models [41]. If the model is linear then we simply apply the ordinary least square (OLS) estimation method to estimate the model parameters [39, 20].
Steps of controlling serial autocorrelation
After controlling for periodic autocorrelation by setting the time effects as a function of time, we need to remove the serial correlation. Therefore, the following steps (algorithm) are used to remove serial correlation:
-
Step 1:
First estimate the model fit residuals as [43, 7]:
$$ {\widehat{\varepsilon}}_{it}={y}_{it}-{f}_i\left(t;{\widehat{\beta}}_i\right) $$
(9)
-
Step 2:
Determine the structure of autocorrelation. At this step we use the Ljung–Box test of autocorrelation.
-
Step 3:
If we do not reject our null-hypothesis we take the model fit is free from the problem of autocorrelation. Otherwise, we apply the Prais–Winsten estimation recursive estimation to remove serial correlation [44, 19, 42, 12, 1, 34]. The estimated variance covariance matrix is given as:
$$ {\widehat{\varOmega}}_i=\frac{1}{1-{\widehat{\rho}}_i^2}\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill {\widehat{\rho}}_i\hfill & \hfill {\widehat{\rho}}_i^2\hfill & \hfill \cdots \hfill & \hfill {\widehat{\rho}}_i^{n-1}\hfill \\ {}\hfill {\widehat{\rho}}_i\hfill & \hfill 1\hfill & \hfill {\widehat{\rho}}_i\hfill & \hfill \cdots \hfill & \hfill {\widehat{\rho}}_i^{n-2}\hfill \\ {}\hfill {\widehat{\rho}}^2\hfill & \hfill \widehat{\rho}\hfill & \hfill \ddots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \hfill & \hfill 1\hfill & \hfill {\widehat{\rho}}_i\hfill \\ {}\hfill {\widehat{\rho}}_i^{n-1}\hfill & \hfill {\widehat{\rho}}_i^{n-2}\hfill & \hfill \cdots \hfill & \hfill {\widehat{\rho}}_i\hfill & \hfill 1\hfill \end{array}\right] $$
(10)
The inverse of the variance covariance matrix can be expressed as:
$$ \begin{array}{lllll}{\widehat{\varOmega}}_i^{-1}={\widehat{\psi}}_i^{\prime },{\widehat{\psi}}_i\hfill & where\hfill & {\widehat{\rho}}_i=\frac{{\displaystyle \sum_{t=2}^n{\widehat{\varepsilon}}_{it}{\widehat{\varepsilon}}_{it-1}}}{{\displaystyle \sum_{t=1}^n{\widehat{\varepsilon}}_{it-1}^2}}\hfill & and\hfill & {\widehat{\psi}}_i=\left[\begin{array}{ccccc}\hfill \sqrt{1-{\widehat{\rho}}_i^2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ {}\hfill -{\widehat{\rho}}_i\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -{\widehat{\rho}}_i\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array} $$
(11)
-
Step 4:
Transform the original trend equation as [2]:
$$ {\widehat{\psi}}_i\left[{y}_{it}\right]={\widehat{\psi}}_i\left[{f}_i\left(t;{\beta}_i\right)\right]+{\widehat{\psi}}_i\left[{\varepsilon}_{it}\right] $$
(12)
where: [y
it
] denotes the vector of stacked output variables [y
it
] for t = 1, 2, 3, …, T, [ε
it
] is similarly constructed from the error terms and [f
i
(t; β
i
)] denotes the stacked Regressors vector.
-
Step 5:
Re-estimate model parameters using the data transformed according to Eq. 12.
-
Step 6:
Repeat from Step 1 to Step 5 unless the Ljung–Box test of autocorrelation confirms that there is no serial correlation on the random error terms.