Post-Recession Recovery in US Air Transport: Forecasting with a Temporal Causal Model

Abstract

This study examines the causal relationship between air transport demand and socio-economic indicators, with a focus on post-recession recovery dynamics. Using monthly data from 1990 to 2022, the research explores causality between air transport indicators and socio-economic indicators, based on which a temporal causal model for forecasting is created. A temporal causal model, integrating air transport and socio-economic metrics, is introduced to improve passenger air transport forecasting. Forecasted values from the temporal causal model and an ARIMA model were compared with actual realized traffic to assess the quality and accuracy of the models. Based on the comparison, the temporal causal model enables instant analysis of circumstances and causes in dynamic environments, as well as reliable forecasting of upcoming intermediate periods. This research contributes to airlines and other air transport stakeholders by delivering a reliable short-term forecasting tool for informed decision-making and sustainable growth.

1. Introduction

Quantifying the future demand for air transport is a prerequisite for airlines to plan all activities and technological processes on a tactical and strategic level: flight schedule planning, crew schedules, maintenance, fleet and network planning, revenue management, and management and marketing [1]. The importance of forecasting demand in air transport takes on an even more significant role during recessions. The characteristics of air transport, such as volatility, rigidity, and seasonality, are an extremely important reason for predicting the future air transport demand. The air transport demand is closely related to socio-economic trends, as the industry promotes international trade and enables businesses worldwide to access various markets quickly and efficiently. In addition to socio-economic indicators, air transport demand can also be influenced by special events, for example, military conflicts, terrorist attacks, sporting events, public health crises, and geological activities, such as volcanic eruptions or earthquakes.

The COVID-19 pandemic is one of the events with a significant impact on the air transport industry, stronger than previous recessions (9/11, the SARS epidemic in 2003. or the global financial crisis in 2008). In the first wave of the pandemic, between March and June 2020, around 17,000 aircraft were grounded globally. In 2020, a decrease of 69.7% in revenue passenger kilometers was recorded globally compared to that in 2019 [2]. After previous recessions that shocked the airline industry, air transport growth (in the long term) has remained stable, but the downturn caused by the COVID-19 pandemic was unprecedented. The recovery of RPKs after the COVID-19 pandemic has been unprecedented compared to previous global shocks—as shown in Figure 1—with the industry experiencing a 93% drop in April 2020 but fully rebounding by early 2024. Domestic air travel returned to pre-pandemic levels in early 2023, while international routes reached 2019 levels by February 2024, despite regional variations due to geopolitical factors and slower recovery in certain markets, like China [3].

The subject of the research in this paper is forecasting the air transport passenger demand after recessions. The primary motivation for this research stemmed from existing studies on the recovery of the air transport industry following recessions that were published after the COVID-19 pandemic [4,5,6,7].

This paper analyzes a causal relationship between the air transport demand (and capacity) and the relevant socio-economic indicators to quantify the future trend of the air transport demand. Traditional ARIMA (AutoRegressive Integrated Moving Average) models and their variations, widely used in this field, require prior forecasting of input variables, such as gross domestic product (GDP) and oil prices, before projecting the dependent variable—such as the number of revenue passengers. This approach might fail to adequately capture structural shifts, like the COVID-19 pandemic, leading to a reliance on assumptions and adjustments within the model to achieve relatively realistic forecasts. These adjustments often result in artificially constructed projections that do not fully reflect the data’s inherent complexity. The proposed approach, based on temporal causal modeling (TCM), offers significant advantages over the conventional ARIMA methodology. This novel method introduces a framework that analyzes causal relationships between target variables and potential input variables without manual adjustments or assumptions, as the model independently detects and accounts for all relevant interdependencies. Consequently, forecasts are entirely data-driven, allowing for a more robust and unbiased model. An additional advantage of this approach is the use of monthly time series data, which facilitates more detailed and timely forecasting. Forecasts based on annual time series provide limited business relevance as they do not adequately respond to real-time shifts in the industry. Finally, TCM enables fully automated modeling without the need to introduce dummy variables, ensuring swift and straightforward application whenever new data become available. This ensures timely and reliable forecasts that always reflect the latest available data.

The structure of this paper is as follows: Section 2 presents a literature review. Section 3 outlines the data sources and the rationale for using monthly data. Section 4 details the TCM methodology applied. Section 5 provides an analysis of the data and the results of causality tests within the TCM. Section 6 discusses the short-term TCM forecasting results, as well as the direct and indirect causality test outcomes from the TCM applied to the selected system. Section 7 focuses on result validation, comparing the TCM- and ARIMA-forecasted trends of recovery against the actual observed traffic during the forecast period (12 months in advance). Finally, Section 8 offers a discussion and conclusion for the research.

2. Literature Review

The inspiration for this research primarily stems from existing studies that analyze the recovery of the air transport industry following economic recessions, with particular focus on the recovery trends post COVID-19. One significant study in this area [4] examines the global air transport recovery timeline after the pandemic, emphasizing that a return to pre-pandemic passenger and freight demand levels is projected to occur within 2 to 2.4 years, although regional variations are significant. The authors utilize ARIMAX models to explore the relationship between air transport demand and key economic indicators such as GDP and oil prices, highlighting the sensitivity of passenger demand to recovery patterns following economic crises. In contrast, freight demand recovers more quickly, reflecting differences in the market’s response to external shocks. The study further suggests that while the ARIMAX (AutoRegressive Integrated Moving Average with eXogenous variables) model effectively captures the macroeconomic factors influencing demand, it is limited by its reliance on annual data inputs. This data frequency, though useful for long-term forecasting, can undermine the precision of short-term predictions, which are critical for the aviation industry as it navigates rapidly changing market conditions in the aftermath of a crisis. The integration of variables like GDP, fuel prices, and macroeconomic forecasts across various regions provides valuable insights; however, the study’s focus on annual data limits the ability to generate timely forecasts that can aid in operational planning and resource allocation. Consequently, the findings suggest the need for more granular, short-term forecasting approaches that can offer higher accuracy and responsiveness to immediate shifts in the market.

Furthermore, analyses focused on specific regional studies during and after the pandemic provide deeper insights into recovery at both the global and regional levels. Reference [5] analyzes the impact of the COVID-19 pandemic on air transport markets in nine African countries, forecasting recovery timelines for domestic and international demand. Using SARIMAX (Seasonal AutoRegressive Integrated Moving Average with eXogenous variables) with data from August 2003 to December 2021, it predicts a recovery of approximately 28 months for domestic flights and 34 months for international flights. The findings suggest that the fluctuations in the aviation market are cyclical rather than indicative of a structural change. In [6], the authors provide information on the impact of the COVID-19 pandemic on capacity and demand in passenger air transport and emphasize the need for causality research to identify the key indicators that drive demand for air transport during and after the pandemic.

The authors in [7] investigate the causal relationship between personal income and passenger air transport demand to provide insight into the future trends of the air transport industry, with particular emphasis on the post-crisis recovery period. The authors use gross domestic product as a proxy for the analysis of personal income and crude oil prices. The results indicate that the recovery of passenger air traffic will be slower than the recovery of the economy. The study leverages annual data on the number of passengers carried, real GDP, and oil prices from 1970 to 2018, specifically from sources like the World Bank, and finds that these dynamic elasticity adjustments enhance the accuracy of air travel forecasts, particularly in post-crisis scenarios. The authors of [8] explore the impact of the COVID-19 pandemic on air transport demand, distinguishing between supply restrictions and demand depression. The authors propose a method to separate these factors by segmenting passengers, simulating scenarios, and comparing predictions with actual data. Applying this to Air France–KLM data, they found a 40.3% demand drop, with 57.4% due to demand depression and 42.6% due to supply restrictions. The study highlights the varying impact on different passenger segments and offers insights for airline recovery and broader industry applications. Another study [9] examines the recovery of the global aviation system following the COVID-19 pandemic, with a focus on the period since May 2020. By analyzing both spatial and temporal aspects of recovery, the authors explore market entry patterns and identify drivers behind the heterogeneous recovery processes observed across different regions. The study offers valuable insights into the varying trajectories of recovery in the aviation sector, contributing to the broader understanding of the pandemic’s impact on air transport.

While studies on post-pandemic recovery are crucial, research conducted before COVID-19 is equally important, as it helps to better understand the fundamental factors shaping air transport demand. Prior to COVID-19, there were several studies that examined forecasting air travel demand. Reference [10] investigates the relationship between air traffic volume and macroeconomic development in Taiwan from 2001 to 2014 using data mining techniques such as K-means clustering and Decision Tree C5.0 classification. The study identifies four critical macroeconomic factors influencing air traffic: the IE Index (Information and Electronics Industrial Production Index), National Income Per Capita, Employed Population, and the Japanese Nikkei 225 Stock Average. The findings were validated by comparing forecasted results with actual data from 2015 to 2016, confirming the relevance of these factors for predictive modeling. This methodology offers valuable insights for policymakers and industry stakeholders to optimize resource allocation and make informed decisions regarding air transport strategy and operations. In a study that analyzed 146 regional airports across 21 EU countries [11], the determinants of air traffic volumes and carrier types at smaller airports serving under one million passengers annually were investigated. The study focused on airport choice factors for airline customers and examined the characteristics of small airports and their catchment areas to explain variations in traffic volumes in 2016. Using multiple linear regression and correlation analysis, the research identified population size, airport charges, and capacity coordination as key factors influencing passenger numbers. It also explored the relationship between the share of low-cost, full-service, and charter carriers, extending existing findings on small regional airports. The study concluded that while certain relationships were found, their statistical significance was moderate, indicating the need for further research on smaller airport samples.

While analyzing air passenger transport determinants in Turkey [12], researchers examined provincial-level data from 2004 to 2014, focusing on factors influencing traffic in an emerging economy. The findings revealed that GDP per capita, population, proximity to alternative airports, tourism activity, leading cities, and international migrations positively affect air traffic. Conversely, higher market concentration correlates with reduced traffic, while the presence of academies appears to boost it. Residual mapping suggested additional influences, such as catchment areas, surface transport availability, domestic migrations, and geopolitical considerations. The study concludes that Turkey’s air transport determinants align closely with those of developed economies and emphasizes the importance of these factors in guiding the development of new airports.

In a study [13] focusing on air passenger demand in the Baltimore–Washington metropolitan area, researchers developed a model as part of an ACRP project investigating the role of disaggregated socio-economic data. The model incorporated traditional causal variables such as population, employment, average household income, and airfares, alongside a variable reflecting change in household income distribution. Using annual data from 1990 to 2010, the model yielded statistically significant coefficients for all variables. Household income distribution significantly altered the estimated elasticity of demand. This finding highlights the importance of considering income distribution in forecasting, as future changes in income distribution and average income could influence projected air passenger travel differently. Reference [14] introduces a novel MIV-based nonlinear vector auto-regression neural network (NVARNN) approach to improve the forecasting of volatile and irregular air passenger flows. Testing with data from Beijing International Airport, the model demonstrated superior accuracy and robustness compared to single and hybrid models, highlighting its effectiveness in predicting both the direction and level of passenger demand. In a study [15] that examines the effects of real income, population, exchange rates, and exchange rate volatility on South Korean outbound tourism demand to five destinations, long-run results reveal that income growth drives tourism demand, while exchange rate volatility, particularly from third-country risks, significantly influences destination choice, with passengers favoring less-volatile destinations. In the short run, Korean tourists are less responsive to third-country exchange rate risks.

In a study comparing the performance of various forecasting models for predicting passenger demand in the transport sector [16], the authors analyzed models ranging from basic techniques, such as Naïve and Drift, to more sophisticated methods, including ARIMA and Holt–Winters. The results demonstrated that ARIMA and Holt–Winters models performed significantly better, particularly in capturing seasonality, whereas simpler models provided linear forecasts and failed to account for seasonal variations. The research highlights the critical role of accurate demand forecasting in a competitive transport market, where understanding future passenger trends is essential for sustaining operations and addressing market challenges effectively.

A study that develops an econometric model to analyze passenger air travel demand in Jordan, [17], focuses on data from 2006 to 2017. Using stepwise regression and multiple linear regression analysis, the research identifies gross domestic product (GDP) in USD as the most significant determinant of air travel demand.

Using data from Beijing, Guangzhou, and Pudong airports, reference [18] proposes a hybrid VMD-ARMA/KELM-KELM approach to address the challenges of forecasting air passenger demand, which exhibits nonlinearity and non-stationarity. The method combines Variational Mode Decomposition (VMD) to reduce data complexity, the ARMA model for stationary components, and Kernel Extreme Learning Machine (KELM) for non-stationary components, with a final integration using another KELM model. This method was presented as a reliable tool for short-term air passenger demand forecasting.

Most studies that were reviewed in [19] have greatly overlooked causal relationships between air transport demand and socio-economic indicators, focusing mainly on GDP as the traditional indicator for demand forecasting. Among the factors recognized by the reviewed studies, socio-economic factors such as GDP and income, population, and price indicators are the most mentioned factors causing the variation in air transport demand. It can be concluded that, on the macroeconomic level, regional air transport demand is determined by economic activities, market size, and air travel affordability. Recognizing these gaps, this research aims to provide a tool that could assist airlines in operational planning and management, particularly by generating more accurate short-term demand forecasts. By utilizing monthly data and incorporating causality, the approach presented in this paper introduces a more responsive and precise forecasting framework. This framework is intended to improve decision-making processes and optimize resource allocation within the air transport industry.

Various scientific papers [20,21,22,23,24,25,26,27,28,29,30,31] research the causality between air transport demand and socio-economic indicators. Most of the data used in those studies were annual data, and the most common proxy for economic growth was GDP [20,21,22,23,24,25,26,27,28]. The most common variable to determine air transport demand in previous studies was revenue passengers (number of domestic and/or international passengers according to the IATA) [20,21,22,23,24,25,26,27,28,29,30]. The applied methodology in these papers [20,21,22,23,25,26,27,28,29,30,31] is mainly based on the Granger causality analysis of time series. Granger causality [32] is a statistical method used to determine cause-and-effect relationships between two variables using time series analysis. This method has been widely applied in research in economics, finance, marketing, meteorology, transport, and other areas based on the analysis of time series.

Global passenger traffic in 2024 is projected to exceed 2019 levels for the first time since the onset of COVID-19, reaching 9.7 billion passengers, or 106% of the 2019 figures, with a year-on-year growth rate of 12%. This growth rate is anticipated to gradually slow in the following years as more markets complete their recovery from the impacts of the pandemic [33]. Macroeconomic factors, including elevated global inflation, a slowdown in global GDP growth, diminished business confidence, and ongoing geopolitical conflicts, introduce significant risks and uncertainties to long-term air traffic projections. Considering these challenges, the importance of short-term forecasts becomes evident, as they allow for more precise demand predictions, enabling the air transport industry to adapt to rapidly changing market conditions and mitigate potential disruptions.

Short-term forecasts play a pivotal role in navigating the uncertainties introduced by these macroeconomic factors. For airlines, they enable the optimization of capacity through the adjustment of flight schedules, aircraft allocation, and pricing strategies tailored to anticipated demand. Airports leverage these forecasts to manage terminal operations, allocate aircraft parking positions, and coordinate ground handling services effectively. Air traffic service providers also benefit by using short-term forecasts to optimize airspace capacity and maintain efficient flight operations.

Moreover, short-term forecasts are instrumental in operational planning, including workforce scheduling, fuel procurement, and supply chain management, ensuring cost-efficiency and resilience. By providing a clearer picture of immediate demand trends, these forecasts empower industry stakeholders to respond swiftly to economic fluctuations and market disruptions. In a period marked by volatility, short-term forecasting stands as a critical tool for ensuring the aviation industry remains agile, efficient, and capable of delivering a high level of service to passengers worldwide.

In recent years, leading organizations in the aviation industry, including the IATA, ICAO, Boeing, Airbus, and ACI, have increasingly emphasized the importance of short-term forecasts alongside their traditional long-term projections. For example, ACI’s latest air travel outlook highlights the value of integrating short-term demand forecasts to navigate rapidly changing market conditions, particularly in the wake of the COVID-19 pandemic [33]. Similarly, the IATA includes short-term forecasting in its 20-Year Passenger Forecast, addressing critical factors such as consumer confidence, exchange rates, and unemployment levels to provide alternative scenarios for mitigating market uncertainties [34]. In addition, the ICAO has developed forecasting tools that allow users to generate customized predictions for passenger traffic, RPKs, and passenger load factors, which are crucial for short-term planning. The ICAO also regularly publishes monthly air traffic reports, providing updated data and analysis useful for short-term operational planning and adjustment [35].

Boeing’s World Air Cargo Forecast also demonstrates how detailed analysis of regional and global air cargo markets can aid stakeholders in understanding and responding to short-term industry trends [36]. Airbus, while traditionally focused on long-term projections, has recognized short-term market recovery dynamics post COVID. In its 2024–2043 Global Market Forecast, Airbus highlights an 8.4% average annual growth rate until 2027, reflecting the short-term recovery of lost traffic during the pandemic [37].

These examples underscore the growing recognition that short-term forecasts are indispensable for managing immediate challenges and making data-driven decisions in an increasingly volatile and dynamic aviation landscape. By combining these forecasts with long-term projections, industry stakeholders can achieve greater operational resilience and adaptability, ensuring sustainable growth of the sector.

3. Data

For this research, predictor variables were selected based on previous research, such as gross domestic product, crude oil and jet fuel prices, unemployment rates, and the industrial production index. In addition to previously researched predictor variables, data that are indirectly significant for air transport demand, which were not considered in earlier research, were also collected. Some of them are homeownership rate, probability of recession, uncertainty index, and effective exchange rates. The data set used for the study covers the period from April 1990 to February 2022, monthly. The selected period includes five significant crises in the air transport industry of the United States of America (illustrated in Figure 2): The Gulf War (1990–1991), which caused the price of fuel to rise and thereby forced as many as seven major airlines (including PanAm) into bankruptcy, from which they did not recover; the terrorist attack on the United States of America (11 September 2001); the SARS epidemic (2003); the global financial crisis (2008); and the latest crisis caused by the COVID-19 pandemic (started in the first quarter of 2020).

The research is based on domestic and international data representing the air transport demand: the number of revenue passengers (RPs) and the revenue passenger kilometers (RPKs). In addition to these data, the research also used data on revenue tonne kilometers (RTKs), revenue freight tonne kilometers (FTKs), and available seat kilometers (ASKs). All these data were collected from publicly available data from the Bureau of Transportation Statistics (BTS) of the US Department of Transport, i.e., the Air Carrier Summary table: T1: U.S. Air Carrier Traffic and Capacity Summary by Service Class [38]. The table contains traffic data reported by United States airlines. The monthly summary is compiled according to carrier entities (geographic regions in which the carrier operates) and service classes and includes available seat miles (ASMs), available tonne miles (ATMs), revenue passengers (RPs), revenue passenger miles (RPMs), revenue tonne miles (RTMs), and revenue freight tonne miles (FTMs). The data were collected for scheduled and non-scheduled transport for passengers and cargo. The table offers individual data points for each carrier for each month, for different classes of transport, and for different destinations (domestic or international transport). After summing up the individual values for each month, sums of data are used at the monthly level. The data used represent aggregated data on air transport demand and supply in passenger and cargo air transport of US airlines in domestic and international transport for the selected time period.

In addition to data representing air transport indicators, various data points representing socio-economic indicators were also used. These data were taken from various sources and are listed in

Table 1. Socio-economic indicator labels, frequencies, and sources.

Socio-Economic Indicators	Label	Freq.	Source
Population	S01	M	US Census Bureau
Employment rate	S02	M	Bureau of Labour Statistics—BLS
Unemployment rate	S03	M	International monetary fund—IMF
Employees in air transportation	S04	M	Bureau of Labour Statistics—BLS
Gross domestic product (GDP)	S05	Q	International monetary fund—IMF
Crude oil price	S06	M	U.S. Energy Information Administration
Jet fuel oil	S07	M	U.S. Energy Information Administration
Employment cost index	S08	Q	Bureau of Labour Statistics—BLS
Product price index (PPI)	S09	M	International monetary fund—IMF
Consumer price index (CPI)	S10	M	International monetary fund—IMF
Industrial production index	S11	M	International monetary fund—IMF
Import price index	S12	M	Bureau of Labour Statistics—BLS
Export price index	S13	M	Bureau of Labour Statistics—BLS
National currency per SDR	S14	M	International monetary fund—IMF
Real effective exchange rate based on CPI	S15	M	International monetary fund—IMF
Real effective exchange rate based on ULC	S16	M	International monetary fund—IMF
Personal income	S17	M	Bureau of Economic Analysis—BEA
Disposable personal income	S18	M	Bureau of Economic Analysis—BEA
Personal saving rate	S19	M	Bureau of Economic Analysis—BEA
Homeownership rate	S20	Q	US Census Bureau
Smoothed US recession probabilities	S21	M	Federal Reserve Bank of St. Louis
Economic policy uncertainty index	S22	M	Federal Reserve Bank of St. Louis
Coincident economic activity index	S23	M	Federal Reserve Bank of St. Louis
Airline ticket price as a component of CPI	S25	M	Bureau of Labour Statistics—BLS
Source: [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]

, along with the labels used for those indicators in the research and frequencies of publication.

Since some data were available on a monthly basis and others on a quarterly basis, the quarterly data were converted to a monthly format using the cubic spline interpolation method to create new data points within the range of known values [58]. Cubic spline interpolation was chosen due to its ability to provide a smooth transition between data points while preserving the continuity of the first and second derivatives. This ensures that the interpolation aligns closely with the inherent trends and seasonality of air transport demand data, avoiding the oscillations.

By analyzing the time series, using the ARIMA model, the justification for using monthly data for short-term forecasts was established and is presented in

Table 2. ARIMA analysis of air transport variables.

Model ID		Model Type
RPd	Model_1	ARIMA (1, 1, 0) (0, 1, 1)
RPi	Model_2	ARIMA (0, 1, 0) (0, 1, 1)
RPKd	Model_3	ARIMA (0, 1, 0) (0, 1, 1)
RPKi	Model_4	ARIMA (0, 1, 0) (0, 1, 1)
RTKd	Model_5	ARIMA (1, 1, 10) (0, 1, 1)
RTKi	Model_6	ARIMA (0, 1, 4) (0, 1, 1)
FTKd	Model_7	ARIMA (2, 1, 9) (0, 1, 1)
FTKi	Model_8	ARIMA (1, 1, 0) (0, 1, 1)
ASKd	Model_9	ARIMA (0, 1, 2) (0, 1, 1)
ASKi	Model_10	ARIMA (2, 1, 0) (0, 1, 1)

.

Based on the ARIMA time series model, it was determined that all observed time series reveal the same seasonal component (time series with one differentiation and seasonal moving average), and it is more appropriate to analyze and forecast time series on a monthly basis, rather than annually.

4. TCM Methodology

In this chapter, a procedure is explained for detecting temporal relationships that combines Granger causality and multiple linear regression analysis for selecting input variables into the model. Otherwise, the TCM procedure is based on the greedy orthogonal matching pursuit (GOMP) algorithm. For more information about the given GOMP algorithm, see [59,60].

Thus, TCM starts from Granger’s causality, which is defined as follows: time series aa causally affects time series bb if the current value of series bb is better explained based on the past values of series bb and aa, rather than only based on its own past values, i.e., the past values of series bb.

Or, in the form of an expression, we have

$${bb}_t\approx {\sum }_{j=1}^L{\beta }_j·{bb}_{t-j}$$

(1)

$${bb}_t\approx {\sum }_{j=1}^L{\alpha }_j·{aa}_{t-j}+{\sum }_{j=1}^L{\beta }_j·{bb}_{t-j}$$

(2)

where L represents the number of delays in the time series. So, if the current value of time series bb is better explained by expression (2) compared to expression (1), we say that time series aa causally affects time series bb.

From the above, Granger causality applies to only one pair of time series. However, in modern business projects there are rarely only two time series, but multiple ones—so the interest is in detecting which input time series influences the target time series. For this reason, the group greedy regression procedure is used. Therefore, it is an important feature to form natural groups of input time series from the entire set of input time series. For example, it is natural to determine whether the input time series aa is statistically significant when it is modeled as a whole, referring to the entire set of time series $\left\{{aa}_{t-1},{aa}_{t-2},\dots ,{aa}_{t-L}\right\}$ and not just to a single time series, say ${aa}_{t-2}$.

The natural formation of groups of input time series that affect the target time series is achieved by using the complex GOMP algorithm. In this algorithm, the input variables are X, y G, M, S, L, ε, and K^*, where $y\in {\mathbb{R}}^{\left(m-L\right)\times 1}$ represents the target vector of the time series, where it is needed to establish Granger causality. Note that the first L values of the vector y are excluded. Next, $X\in {\mathbb{R}}^{m\times n}$ represents the matrix of input time series, where the first L values are not excluded. Next, L represents the number of delays for each target time series, while K* represents the maximum number of input time series in the model, and finally ε represents the value based on which it is decided whether a certain input time series will be included in the model.

Next, G, M, and S represent grouping, centering, and scaling operators. More details about these operators can be found in [61]. Note that GOMP is an iterative algorithm, where $J_{sel}^0$ represents the set of indices for the input time series at the initial, i.e., zero, iteration. Based on these input variables, the task is to greedily find the input time series, which will solve the system $X\beta =y$.

So, the essence of the GOMP iterative algorithm is to select the best input time series in each iteration. To select the best input time series, the argmin function is used, which is represented by a special algorithm. See more about the algorithm for this function at IBM [2017]. As the output of the GOMP algorithm, we have J_sel and $\widehat{\beta }$ for all target time series, where J_sel represents the indices of the input time series and $\widehat{\beta }$ represents the estimated coefficients from the multiple linear regression.

Based on these output values of the GOMP algorithm, a forecast can be formed. For example, if data are available up to time interval m, then the forecast for y for one step ahead will be given by the expression

$${\widehat{y}}_m\left(1\right)={\widehat{\beta }}_0+{\sum }_{j\in J_{sel}}{\sum }_{l=1}^L{\widehat{\beta }}_{j,l}·X_{j,m+1-l}$$

(3)

5. Analysis

Using the temporal causal model, key relationships in the time series data are identified, which enables the prediction of the time series based on these relationships. The TCM creates forecasts based on established temporal causality, without creating dummy variables or subjective assumptions by the researcher. The model is based on statistical and mathematical methods applied to time series. This approach includes the application of autoregressive models, vector autoregressive (VAR) models, vector autoregressive error correction models (VECMs), and Granger causality test. Statistical methods include hypothesis testing, parameter estimation, stationarity testing, and series cointegration testing, which are applied to time series to identify causal relationships between variables in a temporal context. TCM builds an autoregressive time series model for each target series and includes only those time series that have a causal relationship with the target variable. This approach differs from traditional time series modeling in which predictors for the target time series must be explicitly specified (for example, ARIMA models).

Since temporal causal modeling usually involves building a model for several connected time series, it is called a model system. Time series representing air transport indicators that are usually used to forecast demand—revenue passengers and revenue passenger kilometers (domestic and international)—were taken as target variables. In addition to them, the transport indicators revenue tonne kilometers, revenue freight tonne kilometers, and available seat kilometers were used. So, ten time series are marked as target variables. Socio-economic indicators (S01–S25) are defined as predictor variables, with the assumption that these time series cause some of the mentioned transport indicators. It is assumed that air transport indicators are interconnected, considering the characteristics of air transport, and for further research it is assumed that these time series are both target and predictor variables at the same time—which is why they are assigned the role of being “both” variables.

IBM SPSS Statistics is widely recognized as a suitable tool for applying statistical methods and models to time series data, enabling the in-depth analysis and modeling of temporal dependencies and dynamic relationships over time. It provides tools and functionality for time series analysis, including the ability to create VAR models, VECMs, Granger causality tests, and other techniques through the Temporal Causal Model (TCM). The model enables the selection of several predictor variables that the program will print, based on the best fit of the model. This means that the predictors that have the best ability to predict the target variables are selected for display in the results. This is often achieved by comparing different models based on the value of the coefficient of determination R² (R square). Figure 3 shows a histogram of overall model quality. On the histogram shown, the value of R² is very high (higher than 0.88), which indicates a very good model fit.

Figure 4 shows the complexity of simultaneous modeling of several time series using the temporal causal model. It shows the causal connections between the series in the system. Causal connections between time series are indicated by lines of different colors and thicknesses, with arrows indicating the direction of causality.

In Figure 5, output results from IBM SPSS Statistics for the conducted TCM system are displayed, showing the most significant causal relationships for the variable domestic and international revenue passengers (RPd and RPi). These represent the layers of Figure 4, which are provided to enhance the understanding of Figure 4. Additional layers are included in Figure 6 and in Appendix A (Figure A1, Figure A2 and Figure A3) for a more comprehensive understanding.

As seen in Figure 5a, the variable domestic revenue passengers (RPd) is both a target and predictor variable: it is a predictor since it has causal effect on ASKi (available seat kilometers in international traffic) and RPi (revenue passengers in international traffic), and it is a target since it is effected by the indicators industrial production index (S11), homeownership rate (S20), and revenue freight tonne kilometers (FTKd). International revenue passengers (RPi) is also a target and predictor variable. It causes FTKd (which has an effect back on RPi), and RTKi, and is caused by the economic policy uncertainty index (S22), revenue passengers in domestic traffic (RPd), homeownership rate (S20), industrial production index (S11), and crude oil price (S06).

Figure 6 displays the output results of IBM SPSS Statistics for the conducted TCM of the most significant causal relationships for the variable domestic and international revenue passengers’ kilometers (RPKd and RPKi). Both Figure 6a,b show that these variables are only target variables—there is no reverse connection between target variables and predictor variables in the diagrams.

6. Forecasting

The established TCM system enables the creation of short-term forecasts based on causal relationships between target and predictor time series. When the model identifies significant causal relationships between time series, it independently predicts the behavior of these predictor series and generates a forecast of the target variable based on these forecasts. Thus, the model considers significant causal relationships and dependencies among series to generate a forecast of the target variable based on learned patterns and knowledge of causal relationships. Figure 7 illustrates demand forecasts in passenger air transport, i.e., forecasts of time series’ domestic revenue passengers (RPd), and Appendix A contains Figure A2, Figure A3 and Figure A4 of forecasts for international revenue passengers (RPi) and domestic and international revenue passenger kilometers (RPKd and RPKi) for US airlines. The forecasts presented in Appendix A were also generated using the same TCM and follow the same methodology as those for domestic revenue passengers. However, these additional forecasts offer a broader perspective by focusing on the international segment of air travel (RPi) and the corresponding passenger kilometers (RPKd and RPKi) for both domestic and international flights. These projections are important as they allow for a more comprehensive understanding of demand dynamics in both domestic and international markets.

The forecast horizon extends 12 months into the future, with predictions based on historical data from 383 previous periods, spanning monthly data from April 1990 to February 2022. The graphical representations of these forecasts, which were created in Excel based on data obtained from IBM SPSS Statistics, provide visual clarity and enable easier comparison of different forecasted variables (RPd, RPi, RPKd, and RPKi).

As outlined earlier in this paper, these forecasts are crucial for understanding the evolving demand patterns for US airlines, informing strategic decisions related to fleet planning, route development, and capacity management. By analyzing these forecasts, airlines can better anticipate market shifts and adjust their strategies to maximize efficiency and profitability in both domestic and international markets.

Causality diagrams for the previously forecasted target variables are presented for two levels: direct and indirect causal relationships. Figure 6 demonstrates the direct causal relationships affecting RPd, as shown in Figure 8. The most significant causal relationship affecting RPd is represented by S11, which is shown as the thickest line of causality, indicating a very strong influence. Following S11, the next significant causal factors are FTKd (revenue freight tonne kilometers), S20, S22, S05, S06, S10, S14, and another instance of S22, listed in descending order of significance.

In addition to these direct causal relationships, the diagram also highlights indirect causal influences. These include a combination of indicators, such as FTKd, RPd, S01, S05, S10, S14, S25, and RPi, which indirectly affect S14.

Regarding the significance of these relationships, the arrows in the diagram are color-coded and weighted to indicate the level of statistical significance. If the arrow is black and bold, the significance is less than 1%, suggesting a very strong causal relationship. If the arrow is black, the significance lies between 1% and 5%, while a gray arrow indicates a significance between 5% and 10%, denoting a weaker but still noteworthy relationship.

Also, Appendix A contains figures that represent both direct and indirect causes based on the proposed Temporal Causal Model (TCM) system for several key variables, including international revenue passengers (RPi) and domestic and international revenue passenger kilometers (RPKd and RPKi).

7. Validation

The results, i.e., the performance of the proposed temporal causal model used for forecasting demand in passenger air transport, were compared with the results of the implemented ARIMA model on the same dependent variables that were applied in the TCM, for the same period: from April 1990 to February 2022. The ARIMA model was developed specifically for time series forecasting and uses historical values of a time series to predict future values of that same series. ARIMA models describe the mechanical, i.e., statistical, properties of time series. Figure 9, Figure 10, Figure 11 and Figure 12 represent the revenue passengers and revenue passenger kilometers of US airlines (in domestic and international traffic) in the period from March 2022 to February 2023 [3]. Additionally, forecasts for the same period, generated using both the Temporal Causal Model and the ARIMA model, are presented for comparison in those figures.

Figure 9 shows how, for the first quarter, the TCM-forecasted values (line RPd_TCM) correspond to the actual realized revenue passengers carried by US airlines in domestic traffic (RPd). After that period, the lines begin to separate, but the TCM forecast is still more suitable than the ARIMA forecast (line RPd_ARIMA).

Figure 10 shows trend of the revenue passengers in international traffic. The trend of actual realized revenue passengers (RPis) and TCM-forecasted revenue passengers (RPi_TCMs) are quite close, while the line representing revenue passengers forecasted by the ARIMA model (RPi_ARIMA) still diverges from the actual realized data.

As in Figure 9 and Figure 11, in the first quarter, the actual and TCM-forecasted trend of revenue passenger kilometers in domestic traffic largely coincide; however, after six months the line begins to diverge from the actual data. Forecasts using the ARIMA model continue to provide more accurate results in the longer term for this observed indicator (RPKd).

Observing the movement of actual data (shown in Figure 12) on revenue passenger kilometers in international traffic (RPKi), none of the forecasts overlap with the actual data. The reason for this may lie in the fact that the forecasts show international traffic of US airlines, which are mostly oriented towards European and Asian destinations, which were still under the influence of COVID-19 restrictions and affected by the war between Ukraine and Russia.

8. Discussion and Conclusions

In a dynamic environment in which the air transport industry operates, it is important to assess the passenger and cargo air transport demand as accurately as possible to adjust the marker supply, especially during recovery after a recession. Airlines, airports, and other air transport stakeholders rely on demand forecasts to optimize their operations. This includes planning flight schedules, capacity adjustments, crew scheduling, and ground handling. Accurate forecasts help optimize resources and manage operations more efficiently. Short-term demand forecasts allow stakeholders to make informed decisions about resource allocation, operational planning, and, crucially, financial implications. In the short term, demand forecasts enable strategic adjustments, such as cost reduction measures and investment realignment, to mitigate the impact of economic downturns.

Although this research deals with the forecasting of air transport indicators, it requires more than a simple understanding of statistics. A comprehensive grasp of the underlying causal factors affecting air transport demand is necessary, as well as insights into causality. In this context, it is possible to argue that ARIMA models, although effective in capturing time series patterns, are not based on the theory of air transport demand, which is one of the disadvantages of this approach. Temporal causal models are primarily used to analyze causal relationships between time series and provide information on how changes in one series affect changes in another series and provide insight into the causal relationships between variables. They are built based on the selection of relevant variables while testing the statistical significance of causal relationships. They provide understanding of the causality between variables. Forecasts are achieved by applying a model that is derived from the causality and uses data from past values of predictor and target variables to predict future values. Both models are suitable for short-term forecasting, and forecasting in a temporal causal model has limitations and relies on assumptions about the constancy of causal relationships, which can be challenging in dynamic environments. However, once a system is created it works independently, requiring no subjective intervention from the researcher, and enables semi-automatic updates as new data become available. This makes TCM especially useful in dynamic environments, where they can generate real-time forecasts that respond promptly to the latest data.

One of the challenges in implementing a TCM system lies in the initial data collection process. In this study, 10 air transport indicators and 24 socio-economic indicators were processed over a span of 383 months. Each data point had to be adapted to fit the model’s requirements, a process that required substantial initial investment. Nevertheless, once this data set is established, integrating new information becomes relatively straightforward, especially with knowledge of data sources and the model’s requirements. With each new data set, the model can be “self-updated”, seamlessly incorporating recent data as soon as they become available.

In stable periods without recessions or extraordinary disruptions, ARIMA models remain valuable for predicting passenger air traffic demand. These models are adept at handling seasonality, trends, and fluctuations, making them suitable for consistent, trend-based forecasting.

However, to enhance forecast accuracy, it is crucial to continually assess causative factors, as TCM’s adaptability enables it to recognize and incorporate shifts in predictor–target relationships over time. The reliability of forecasts is generally highest for the immediate 3–4 future values (typically covering 3–4 months), with accuracy diminishing over extended periods. The advantage of temporal causal models lies in their simplicity of recalibration: once new data from the ongoing time series are available, forecasts can be refreshed, producing reliable short-term predictions in real time. This results in a live system that adapts to the latest available data at any moment, and therefore provides quite reliable forecasts.

While this study highlights the potential of temporal causal models (TCMs) for improving short-term demand forecasting in air transport, further studies could explore the long-term impacts of global disruptions, such as the COVID-19 pandemic, geopolitical tensions, and economic shocks, on international air transport demand. It would be particularly valuable to examine how such events alter travel patterns and recovery dynamics over time, which could enhance model accuracy. Additionally, a more focused analysis of the unique factors influencing demand for long-haul flights, particularly those connecting Europe and Asia, is needed. This could involve examining the role of specific socio-economic drivers, such as cross-border trade relations, political stability, and regional economic growth, in shaping demand for international routes.

Future research could investigate which variables are most reliable in predicting air transport demand in the short and medium term. For instance, exploring the influence of demographic shifts, income distribution, and regional mobility trends could improve the precision of TCM forecasts.

Although TCMs provide semi-automatic updates as new data become available, further studies could focus on improving their real-time forecasting capabilities, especially in response to sudden shifts in demand due to external events. Investigating the integration of machine learning techniques could improve the dynamic recalibration of these models, allowing them to better capture unpredictable fluctuations in demand. By focusing on these specific areas, future research can contribute to the development of more robust and actionable forecasting models that better support decision-making and strategic planning in the air transport industry.