Nonlinear Time Series Models Research Articles

The Huber estimation for UMAR model with imprecise observations

A nonlinear uncertain time series model was proposed which was defined as the uncertain max-autoregressive (UMAR) model. After the model is established, we pay more attention to its corresponding prediction and analysis. Before that, the estimation of unknown parameters is the top priority of a series of work. So, this paper presents the Huber estimation to estimate the unknown parameters. Then, the residual analysis and relevant prediction are discussed. Furthermore, the data of China's unemployed population, from 2002 to 2021 via the State Statistical Bureau, are used to verify the feasibility of the model and the Huber estimation. So, the relevant comparison with uncertain autoregressive (UAR) model are given to verify the practicability of the UMAR model under the Huber estimation. In addition, the relevant comparative analysis are given to present that the Huber estimation has better robustness than the least square (LS) estimation when the outliers appear.

Unveiling an asymmetric relationship between global crude oil and local food prices in an oil-importing economy

Recent swift comovements of local food and global crude oil prices have attracted the attention of policymakers and researchers. To evaluate this relationship, many studies have used time series models to explore global crude oil and local food prices. However, robust research based on advanced nonlinear time series models that incorporate control variables for their formation is lacking. In this paper, nonlinear techniques are applied to assess the asymmetric nexus between Brent oil prices and local retail food prices in Slovakia. To estimate this value, we extend the single-threshold NARDL approach to the MTNARDL model. The nominal exchange rate and industrial production index are used as the control variables. Compared with conventional NARDL models, the MTNARDL model provides a more detailed representation of global oil‒local food price linkages and detects the asymmetric effect of global oil prices on food prices from both long- and short-term perspectives. Interestingly, with respect to long- and short-term food price volatility, changes in response to oil price fluctuations are greatest under a regime with rather a small number of positive and moderate changes.

Modal volatility function

We in this article propose a novel non‐parametric estimator for the volatility function within a broad context that encompasses nonlinear time series models as a special case. The new estimator, built on the mode value, is designed to complement existing mean volatility measures to reveal distinct data features. We demonstrate that the suggested modal volatility estimator can be obtained asymptotically as well as if the conditional mean regression function were known, assuming observations are from a strictly stationary and absolutely regular process. Under mild regularity conditions, we establish that the asymptotic distributions of the resulting estimator align with those derived from independent observations, albeit with a slower convergence rate compared to non‐parametric mean regression. The theory and practice of bandwidth selection are discussed. Moreover, we put forward a variance reduction technique for the modal volatility estimator to attain asymptotic relative efficiency while maintaining the asymptotic bias unchanged. We numerically solve the modal regression model with the use of a modified modal‐expectation‐maximization algorithm. Monte Carlo simulations are conducted to assess the finite sample performance of the developed estimation procedure. Two real data analyses are presented to further illustrate the newly proposed model in practical applications. To potentially enhance the accuracy of the bias term, we in the end discuss the extension of the method to local exponential modal estimation. We showcase that the suggested exponential modal volatility estimator shares the same asymptotic variance as the non‐parametric modal volatility estimator but may exhibit a smaller bias.

Drone motion prediction from flight data: a nonlinear time series approach

In this paper, we explore the application of data-driven predictive systems in enhancing unmanned aerial vehicle (UAV) control capabilities. We introduce a new model for predicting the motion of individual drones by utilizing fundamental flight control data. The model aims to improve the autonomy of individual drones and circumvent the complexity of traditional flight control systems, thus eliminating intricate nested controls. The proposed model lays the foundation for studying collective behaviours within a cluster of drones, thereby advancing the research into swarm behaviour exhibited by drones. The research findings demonstrate the potential of data-driven methods in the construction of UAV control systems. In particular, we here show a comparison of the prediction performances between two neural network architectures using real drone flight data involved in various kinds of motions. We explore the utility of using long short term memory (LSTM) and nonlinear autoregressive with exogenous inputs (NARX) family of nonlinear time series models in developing a virtual drone model using real experimental data.

Uncertain statistics models for characterizing annual total energy production

In the field of annual total energy production, it is hard to determine whether the experimental data is linear or nonlinear growth characteristics. Therefore, different regression models are used to test the properties of the observed data. This article uses actual data to perform parameter estimation and residual analysis of linear and nonlinear regression models. In addition, uncertain nonlinear time series model is used to fit the experimental data in order to find a more suitable statistics model after obtaining the growth properties of the observed data. A test is introduced to prove that the disturbance terms presented in the regression and time series models are actually uncertain variables, not random variables. Moreover, the uncertain hypothesis tests show that the uncertain time series model is in better agreement with the real data of the annual total energy production.

Maximum likelihood inference for a class of discrete-time Markov switching time series models with multiple delays

Autoregressive Markov switching (ARMS) time series models are used to represent real-world signals whose dynamics may change over time. They have found application in many areas of the natural and social sciences, as well as in engineering. In general, inference in this kind of systems involves two problems: (a) detecting the number of distinct dynamical models that the signal may adopt and (b) estimating any unknown parameters in these models. In this paper, we introduce a new class of nonlinear ARMS time series models with delays that includes, among others, many systems resulting from the discretisation of stochastic delay differential equations (DDEs). Remarkably, this class includes cases in which the discretisation time grid is not necessarily aligned with the delays of the DDE, resulting in discrete-time ARMS models with real (non-integer) delays. The incorporation of real, possibly long, delays is a key departure compared to typical ARMS models in the literature. We describe methods for the maximum likelihood detection of the number of dynamical modes and the estimation of unknown parameters (including the possibly non-integer delays) and illustrate their application with a nonlinear ARMS model of El Niño–southern oscillation (ENSO) phenomenon.

Development of out-of-sample forecast formulae for the FIGARCH model

Volatility is a matter of concern for time series modeling. It provides valuable insights into the fluctuation and stability of concerning variables over time. Volatility patterns in historical data can provide valuable information for predicting future behaviour. Nonlinear time series models such as the autoregressive conditional heteroscedastic (ARCH) and the generalized version of the ARCH model, i.e. generalized ARCH (GARCH) models are popularly used for capturing the volatility of a time series. The realization of any time series may have significant statistical dependencies on its distant counterpart. This phenomenon is known as the long memory process. Long memory structure can also be present in volatility. Fractionally integrated volatility models such as the fractionally integrated GARCH (FIGARCH) model can be used to capture the long memory in volatility. In this paper, we derived the out-of-sample forecast formulae along with the forecast error variances for the AR (1) -FIGARCH (1, d, 1) model by recursive use of conditional expectations and conditional variances. For empirical illustration, the modal spot prices of onion for Delhi, Lasalgaon and Bengaluru markets, India and S&P 500 index (close) data are used.

Assessing Asymmetry in Okun's Law: A Nonlinear Approach in the Turkish Economy

In this paper, the presence of asymmetric behavior in Okun’s Law for Turkey during the period 2005-2023 is analysed. The relationship between economic growth and the unemployment rate has been assessed in the context of nonlinear time series models. Nonlinear methods are typically favoured in macroeconomic series analyses as they generate more robust findings. In this paper, nonlienear symmetric KSS cointegration and nonlinear asymmetric Hepsağ cointegration tests were used to test for the existence of a long-run relationship between the variables, and the Hatemi-j test was used to test for nonlinear causality. Our findings demonstrate no evidence of co-integration between the variables. Hatemi-j test results, on the other hand, indicate the presence of a causality relationship from negative growth components to positive unemployment components. Specifically, a decrease in the growth was found to be the only one of all the components that was a statistically significant cause of an increase in the unemployment.

Nonlinear hydrological time series modeling to forecast river level dynamics in the Rio Negro Uruguay basin.

Floods significantly impact the well-being and development of communities. Hence, understanding their causes and establishing methodologies for risk prevention is a critical challenge for effective warning systems. Complex systems such as hydrological basins are modeled through hydrological models that have been utilized to understand water recharge of aquifers, available volume of dams, and floods in diverse regions. Acquiring real-time hydrometeorological data from basins and rivers is vital for establishing data-driven-based models as tools for the prediction of river-level dynamics and for understanding its nonlinear behavior. This paper introduces a hydrological model based on a multilayer perceptron neural network as a useful tool for time series modeling and forecasting river levels in three stations of the Rio Negro basin in Uruguay. Daily time series of river levels and rainfall serve as the input data for the model. The assessment of the models is based on metrics such as the Nash-Sutcliffe coefficient, the root mean square error, percent bias, and volumetric efficiency. The outputs exhibit varying model performance and accuracy during the prediction period across different sub-basin scales, revealing the neural network's ability to learn river dynamics. Lagged time series analysis demonstrates the potential for chaos in river-level time series over extended time periods, mainly when predicting dam-related scenarios, which shows physical connections between the dynamical system and the data-based model such as the evolution of the system over time.

Time Series Analysis of Nonlinear Head Dynamics Using Synthetic Data Generated with a Variably Saturated Model.

The performance of time series models is assessed using synthetic head series simulated with a numerical model that solves Richards' equation for variably saturated flow. Heads were simulated in a homogeneous unconfined aquifer between two parallel canals; measured daily precipitation and potential evaporation are specified at the land surface and root water uptake is simulated. The head response to a precipitation event is nonlinear and depends on the saturation degree and rainfall before and after the precipitation event while evaporation reduction occurs during summers. Synthetic series were generated for 27 years and three different soil types; the unsaturated zone thickness varies between 0 and >5 m. The synthetic head series were simulated with a linear and nonlinear time series model. Performance of a linear time series model with four parameters, using a scaled Gamma response, gave R2 values ranging from 0.67 to 0.96. The nonlinear time series model with five parameters simulates recharge using a root zone reservoir after which the head response to recharge is simulated with a scaled Gamma response function. The nonlinear time series model was able to simulate all synthetic head series very well with R2 values above 0.9 for almost all models. The head response of the nonlinear model to a single precipitation event compares well to the response of the variably saturated groundwater model. The provided scripts may be used to simulate synthetic head series for other climates or for systems with additional complexity to assess the performance of other data-driven models.

Modelling the GDP of KSA using linear and non-linear NNAR and hybrid stochastic time series models.

Gross domestic product (GDP) serves as a crucial economic indicator for measuring a country's economic growth, exhibiting both linear and non-linear trends. This study aims to analyze and propose an efficient and accurate time series approach for modeling and forecasting the GDP annual growth rate (%) of Saudi Arabia, a key financial indicator of the country. Stochastic linear and non-linear time series modeling, along with hybrid approaches, are employed and their results are compared. Initially, conventional linear and nonlinear methods such as ARIMA, Exponential smoothing, TBATS, and NNAR are applied. Subsequently, hybrid models combining these individual time series approaches are utilized. Model diagnostics, including mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE), are employed as criteria for model selection to identify the best-performing model. The findings demonstrated that the neural network autoregressive (NNAR) model, as a non-linear approach, outperformed all other models, exhibiting the lowest values of MAE, RMSE and MAPE. The NNAR(5,3) projected the GDP of 1.3% which is close to the projection of IMF benchmark (1.9) for the year 2023. The selected model can be employed by economists and policymakers to formulate appropriate policies and plans. This quantitative study provides policymakers with a basis for monitoring fluctuations in GDP growth from 2022 to 2029 and ensuring the sustained progression of GDP beyond 2029. Additionally, this study serves as a guide for researchers to test these approaches in different economic dynamics.

Stability Analysis of Deep Belief Network: Based SD-AR Model for Nonlinear Time Series

As for nonlinear time series prediction, many different kinds of varying-coefficient models have been proposed and analysised in recent years. A kind of varying functional-coefficient autoregressive model, called the deep belief network-based state-dependent autoregressive (DBN-AR) model is considered in this paper. The stability conditions and existing conditions of limit cycle of the DBN-AR model are also studied. An especial designed parameter estimation method is used to identify the DBN-AR model. The DBN-AR model is used to predict the famous Canadian lynx data and Henon chaotic series, the prediction capability of the DBN-AR model is compared with other prediction models, the experimental results show that the DBN-AR model obtains better prediction accuracy.

A novel comparative study of NNAR approach with linear stochastic time series models in predicting tennis player's performance

BackgroundPrediction models have gained immense importance in various fields for decision-making purposes. In the context of tennis, relying solely on the probability of winning a single match may not be sufficient for predicting a player's future performance or ranking. The performance of a tennis player is influenced by the timing of their matches throughout the year, necessitating the incorporation of time as a crucial factor. This study aims to focus on prediction models for performance indicators that can assist both tennis players and sports analysts in forecasting player standings in future matches.MethodologyTo predict player performance, this study employs a dynamic technique that analyzes the structure of performance using both linear and nonlinear time series models. A novel approach has been taken, comparing the performance of the non-linear Neural Network Auto-Regressive (NNAR) model with conventional stochastic linear and nonlinear models such as Auto-Regressive Integrated Moving Average (ARIMA), Exponential Smoothing (ETS), and TBATS (Trigonometric Seasonal Decomposition Time Series).ResultsThe study finds that the NNAR model outperforms all other competing models based on lower values of Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). This superiority in performance metrics suggests that the NNAR model is the most appropriate approach for predicting player performance in tennis. Additionally, the prediction results obtained from the NNAR model demonstrate narrow 95% Confidence Intervals, indicating higher accuracy and reliability in the forecasts.ConclusionIn conclusion, this study highlights the significance of incorporating time as a factor when predicting player performance in tennis. It emphasizes the potential benefits of using the NNAR model for forecasting future player standings in matches. The findings suggest that the NNAR model is a recommended approach compared to conventional models like ARIMA, ETS, and TBATS. By considering time as a crucial factor and employing the NNAR model, both tennis players and sports analysts can make more accurate predictions about player performance.

Comparison of non-linear time series models (Beta-t-EGARCH and NARMAX models) with Radial Basis Function Neural Network usingReal Data

This paper presents a comparison of three different non-linear time series modelling approaches: NARMAX (Non-linear Autoregressive Moving Average with Exogenous Inputs), Beta-t-EGARCH (Beta t Exponential Generalized Autoregressive Conditional Heteroscedasticity), and Radial Basis Function Neural Networks (RBFNN) applied to weekly stock market index data. We will explain three types of models and compare their compositions and structures. Then, we will show which model gives better predictions. To study series data, the comparison involved analysing the structure of the model and its errors in various time series models and summarising their findings. We divide the data into two parts: training data to structure the time series and testing. The training data tests the model's predictions. Then, we can analyse the model with the errors and the best deterrence predictions. After selecting the NARMAX and Beta-t-EGARCH models, we test them with specific criteria. The best choice is finding the model with the lowest average errors. For this study, we analysed the weekly average closing of the Aramco 2222 index from 15 December 2019 to 16 July 2023 and made 187 observations.

Nonlinear Modeling of Mortality Data and Its Implications for Longevity Bond Pricing

Human mortality has been improving faster than expected over the past few decades. This unprecedented improvement has caused significant financial stress to pension plan sponsors and annuity providers. The widely recognized Lee–Carter model often assumes linearity in its period effect as an integral part of the model. Nevertheless, deviation from linearity has been observed in historical mortality data. In this paper, we investigate the applicability of four nonlinear time-series models: threshold autoregressive model, Markov switching model, structural change model, and generalized autoregressive conditional heteroskedasticity model for mortality data. By analyzing the mortality data from England and Wales and Italy spanning the years 1900 to 2019, we compare the goodness of fit and forecasting performance of the four nonlinear models. We then demonstrate the implications of nonlinearity in mortality modeling on the pricing of longevity bonds as a practical illustration of our findings.

Oil price shocks in the age of surging inflation

This study examines the nonlinear impact of oil price shocks on inflation in the euro area. We apply nonlinear vector time series models to identify regime shifts in recent times of geopolitical uncertainty. For the period from January 1999 to October 2022, we reveal the influence of geopolitical risks on exacerbating the inflationary impact of oil price shocks. Our impulse response analysis indicates that oil passthrough is more pronounced and persistent when the risks associated with adverse geopolitical events surpasses a given threshold. Eurozone inflation is found to be more vulnerable to geopolitical conflicts compared with the US economy. Our results underscore the nonlinear dynamic of oil price passthrough and the amplifying effect of geopolitical events that European policymakers should consider when addressing growing inflationary pressure.

Nonlinear GARCH-type models for ordinal time series

AbstractDespite their relevance in various areas of application, only few stochastic models for ordinal time series are discussed in the literature. To allow for a flexible serial dependence structure, different ordinal GARCH-type models are proposed, which can handle nonlinear dependence as well as kinds of an intensified memory. The (logistic) ordinal GARCH model accounts for the natural order among the categories by relying on the conditional cumulative distributions. As an alternative, a conditionally multinomial model is developed which uses the softmax response function. The resulting softmax GARCH model incorporates the ordinal information by considering the past (expected) categories. It is shown that this latter model is easily combined with an artificial neural network response function. This introduces great flexibility into the resulting neural softmax GARCH model, which turns out to be beneficial in three real-world time series applications (air quality levels, fear states, cloud coverage).

The Role of Public Debt in Economic Growth: An Empirical Analysis Evidence for Western Balkans and European Countries

Public debt was often considered as a supplementary source of public finances. In six Western Balkan states, a fixed regression model was employed to gauge the effect of public debt on economic growing using panel data. Findings demonstrate that public debt significantly affects economic growth. It was found too that direct foreign investments have a substantial influence into the economic growth. As an option for financing, state governments use external and internal public debt, which is often part of the debate approximately how much public debt should be used. Use of public debt varies from country to country and depends on economic growth and budgetary factors and economic and social demands. Reviewing the literature, it can be observed impact of public debt on overall economy and its growth and this was too used by states as an opportunity for economic growth. This is also reflected in the case of our study where we analyzed the data for EU member countries and Great Britain compared to the Western Balkans countries. During the research it is observed influence of public debt in economic growth on these countries and as such many states have exceeded the limits of the use of public debt based on legal frameworks, with the sole purpose of financing public demands and influence of economic growth. Public debt in this paper identifies the affiliation with economic growth through the specification of linear and non-linear time series models using the panel model for the years 2012-2021 for 35 countries in total. Also, the regression analysis shows us a support and correlation among public debt versus economic growth of countries part of this study. This is observed in developed countries, which have a higher potential and possibility of financing economic activity through public debt, while the Western Balkans countries are often challenged with the possibilities of financing from public debt. As variables we have GDP, Foreign Direct Investments and Inflation.

Estimation of prediction error in time series

Summary The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impossible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), up to now there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, a method for accurate estimation of prediction errors in both univariate and multivariate stationary time series is developed in this article. In particular, several estimates are derived for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, with causal invertible autoregressive moving average and nonparametric autoregressive processes discussed as lead examples. Simulations demonstrate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modelling is prediction.

Forecasting Day-Ahead Brent Crude Oil Prices Using Hybrid Combinations of Time Series Models

Crude oil price forecasting is an important research area in the international bulk commodity market. However, as risk factors diversify, price movements exhibit more complex nonlinear behavior. Hence, this study provides a comprehensive analysis of forecasting Brent crude oil prices by comparing various hybrid combinations of linear and nonlinear time series models. To this end, first, the logarithmic transformation is used to stabilize the variance of the crude oil prices time series; second, the original time series of log crude oil prices is decomposed into two new subseries, such as a long-run trend series and a stochastic series, using the Hodrick–Prescott filter; and third, two linear and two nonlinear time series models are considered to forecast the decomposed subseries. Finally, the forecast results for each subseries are combined to obtain the final day-ahead forecast result. The proposed modeling framework is applied to daily Brent spot prices from 1 January 2013 to 27 December 2022. Six different accuracy metrics, pictorial analysis, and a statistical test are performed to verify the proposed methodology’s performance. The experimental results (accuracy measures, pictorial analysis, and statistical test) show the efficiency and accuracy of the proposed hybrid forecasting methodology. Additionally, our forecasting results are comparatively better than the benchmark models. Finally, we believe that the proposed forecasting method can be used for other complex financial time data to obtain highly efficient and accurate forecasts.