Moment estimation of uncertain threshold autoregressive model

Pakistan is considered among the top five countries with the highest CO2 emissions globally. This calls for pragmatic policy implementation by all stakeholders to bring finality to this alarming situation since it contributes greatly to global warming, thereby leading to climate change. This study is an attempt to make a comparative analysis of the linear time series models with nonlinear time series models to study CO2 emission data in Pakistan. These linear and nonlinear time series models were used to model and forecast future values of CO2 emissions for a short period. To assess and select the best model among these linear and nonlinear time series models, we used the root mean square error (RMSE) and the mean absolute error (MAE) as performance indicators. The outputs showed that the nonlinear machine learning models are the best among all other models, having the lowest RMSE and MAE values. Based on the forecasted value of the nonlinear machine learning neural network autoregressive model, Pakistan's CO2 emissions will be 1.048 metric tons per capita by 2028. The increasing trend in emissions is a frightening and clear warning, suggesting that innovative policies must be initiated to reduce the trend. We encourage the Pakistan government to price CO2 emissions by companies and entities per ton, adapt electricity production from hydro, wind, and different sources with no emissions of CO2, initiate rigorous planting of more trees in the populated areas of Pakistan as forest covers, provide incentives to companies, organisations, institutions, and households to come out with clean technologies or use technologies with no CO2 emissions or those with lower ones, and fund more studies to develop clean and innovative technologies with less or no CO2 emissions.

This PhD dissertation deals with the world of multivariate time series models where the behaviour of the observed process is described by using a time-varying parameter. In particular, this thesis explore three different dynamic multivariate nonlinear models which are able to deal with multivariate time series gathered from heavy-tailed phenomena. Although the popularity of linear and univariate time series models, empirical evidences have shown that variables generated from complex phenomena are typically inter-related both contemporaneously and across time. This is the case for several fields of science such as economics, finance, biology or physics, where it is widely accepted that with a univariate approach it is difficult to obtain a satisfactory representation of the reality or to make good predictions about the future. For these reasons, the literature of linear multivariate Gaussian time series models has received increasing attention. However, these models are known for their unsatisfactory performances when the collected data are contaminated by outliers, yielding biased estimates and unreliable forecasts. In fact, when departure from the hypothesis of normality is confirmed by the observed data, it is reasonable to switch into the realm of nonlinear or non-Gaussian time series models. Unfortunately, despite the development of recent technologies, the estimation of nonlinear time series models might be really challenging, since they require simulation-based and computer-intensive methods. In addition, statistical properties of such estimators are not always easy to be derived. This thesis contributes to the literature by defining dynamic multivariate and heavy-tailed models that are relatively simple. The emphasis is models which are analytically tractable and can be easily estimated by means of maximum likelihood. For each of the models, a very detailed statistical and asymptotic analysis it is provided. Their practical usefulness is highlighted with several simulation studies and empirical applications.

In this article, we are concerned with least absolute deviation (LAD) estimation applied to linear regression and linear time series models. LAD estimation is widely used in statistical modeling and analysis. Compared to classical least squares (LS), LAD is a robust estimation procedure and is typically more efficient when handling heavy‐tailed data. We provide a selective overview on its developments in linear regression and time series models. This overview includes a description of a basic method for establishing the asymptotic properties of the LAD estimator for finite variance autoregressive moving average (ARMA) models. Efficiency of LAD estimation relative to LS estimation as well as computational issues are also discussed.

Introduction: Time series models are generally categorized as a data-driven method or mathematically-based method. These models are known as one of the most important tools in modeling and forecasting of hydrological processes, which are used to design and scientific management of water resources projects. On the other hand, a better understanding of the river flow process is vital for appropriate streamflow modeling and forecasting. One of the main concerns of hydrological time series modeling is whether the hydrologic variable is governed by the linear or nonlinear models through time. Although the linear time series models have been widely applied in hydrology research, there has been some recent increasing interest in the application of nonlinear time series approaches. The threshold autoregressive (TAR) method is frequently applied in modeling the mean (first order moment) of financial and economic time series. Thise type of the model has not received considerable attention yet from the hydrological community. The main purposes of this paper are to analyze and to discuss stochastic modeling of daily river flow time series of the study area using linear (such as ARMA: autoregressive integrated moving average) and non-linear (such as two- and three- regime TAR) models. Material and Methods: The study area has constituted itself of four sub-basins namely, Saghez Chai, Jighato Chai, Khorkhoreh Chai and Sarogh Chai from west to east, respectively, which discharge water into the Zarrineh Roud dam reservoir. River flow time series of 6 hydro-gauge stations located on upstream basin rivers of Zarrineh Roud dam (located in the southern part of Urmia Lake basin) were considered to model purposes. All the data series used here to start from January 1, 1997, and ends until December 31, 2011. In this study, the daily river flow data from January 01 1997 to December 31 2009 (13 years) were chosen for calibration and data for January 01 2010 to December 31 2011 (2 years) were chosen for validation, subjectively. As data have seasonal cycles, statistical indices (such as mean and standard deviation) of daily discharge were estimated using Fourier series. Then ARMA and two- and three-regime SETAR models applied to the standardized daily river flow time series. Some performance criteria were used to evaluate the models accuracy. In other words, in this paper, linear and non-linear models such as ARMA and two- and three-regime SETAR models were fitted to observed river flows. The parameters associated to the models, e.g. the threshold value for the SETAR model was estimated. Finally, the fitted linear and non-linear models were selected using the Akaike Information Criterion (AIC), Root Mean Square (RMSE) and Sum of Squared Residuals (SSR) criteria. In order to check the adequacy of the fitted models the Ljung-Box test was used. Results and Discussion: To a certain degree the result of the river flow data of study area indicates that the threshold models may be appropriate for modeling and forecasting the streamflows of rivers located in the upstream part of Zarrineh Roud dam. According to the obtained evaluation criteria of fitted models, it can be concluded the performance of two- and three- regime SETAR models are slightly better than the ARMA model in all selected stations. As well as, modeling and comparison of SETAR models showed that the three-regime SETAR model have evaluation criteria better than two-regime SETAR model in all stations except Ghabghablou station. Conclusion: In the present study, we attempted to model daily streamflows of Zarrineh Rood Basin Rivers located in the south of Urmia Lake by applying ARMA and two- and three-regime SETAR models. This is mainly because very few efforts and rather less attention have been paid to this non-linear approach in hydrology and water resources engineering generally. Therefore, two types of data-driven models were used for modeling and forecasting daily streamflow: (i) deseasonalized ARMA-type model, and (ii) Threshold Autoregressive model, including Self-Existing TAR (SETAR) model. Each ARMA and SETAR models were fitted to daily streamflow time series of the rivers located in the study area. In general, it can be concluded that the overall performance of SETAR model is slightly better than ARMA model. Furthermore, SETAR model is very similar AR model, therefor, it can be easily used in water resources engineering field. On the other hand, due to apply these non-linear models, the number of estimated parameters in comparison with linear models has decreased.

It is known that the occurrence of outliers in linear or non-linear time series models may have adverse effects on the modelling and statistical inference of the data. Consequently, extensive research has been conducted on developing outlier detection procedures so that outliers may be properly managed. However, no work has been done on the problem of outliers in circular time series data. This problem is the focus of this paper. The main objective is to develop novel numerical and graphical procedures for detecting these outliers in circular time series data.A number of circular time series models have been proposed including the circular autoregressive model. We extend the iterative outlier detection procedure which has been successfully used in linear time series models to the circular autoregressive model. The proposed procedure shows a good performance when investigated via simulation for the circular autoregressive model of order one. At the same time, several statistical techniques have been used to detect the change of preferred trend in time series data using SLIME and CUSUM plots. While the methods fail to indicate directly the outliers in circular time series data, we use the ideas employed to develop three novel graphical procedures for identifying the outliers. For illustration, we apply the procedures to a particular set of wind direction data. An agreement between the results of the graphical and iterative detection procedures is observed. These procedures could be very useful in improving the modelling and inferential processes for circular time series data.

Chapter 3 - Introduction of multiple/multivariate linear and nonlinear time series models in forecasting streamflow process

We propose a new diagnostic test for linear and nonlinear time series models, using a generalized spectral approach. Under a wide class of time series models that includes autoregressive conditional heteroskedasticity (ARCH) and autoregressive conditional duration (ACD) models, the proposed test enjoys the appealing "nuisance-parameter-free" property in the sense that model parameter estimation uncertainty has no impact on the limit distribution of the test statistic. It is consistent against any type of pairwise serial dependence in the model standardized residuals and allows the choice of a proper lag order via data-driven methods. Moreover, the new test is asymptotically more efficient than the correlation integral–based test of Brock, Hsieh, and LeBaron (1991, Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence) and Brock, Dechert, Scheinkman, and LeBaron (1996, Econometric Reviews 15, 197–235), the well-known BDS test, against a class of plausible local alternatives (not including ARCH). A simulation study compares the finite-sample performance of the proposed test and the tests of BDS, Box and Pierce (1970, Journal of the American Statistical Association 65, 1509–1527), Ljung and Box (1978, Biometrika 65, 297–303), McLeod and Li (1983, Journal of Time Series Analysis 4, 269–273), and Li and Mak (1994, Journal of Time Series Analysis 15, 627–636). The new test has good power against a wide variety of stochastic and chaotic alternatives to the null models for conditional mean and conditional variance. It can play a valuable role in evaluating adequacy of linear and nonlinear time series models. An empirical application to the daily S&P 500 price index highlights the merits of our approach.We thank the co-editor (Don Andrews) and two referees for careful and constructive comments that have lead to significant improvement over an earlier version. We also thank C.W.J. Granger, D. Tjøstheim, and Z. Xiao for helpful comments. Hong's participation is supported by the National Science Foundation via NSF grant SES–0111769. Lee thanks the UCR Academic Senate for research support.

Uncertain regression analysis is a powerful analytical tool to model the relationships between explanatory variables and the response variable by uncertainty theory. One of the core problems in uncertain regression analysis is to estimate the unknown parameters of an uncertain regression model and the uncertain disturbance term. In this paper, the moment estimation of uncertain regression model is proposed, which can determine both the uncertain regression model and the disturbance term at one time. After that, the uncertain hypothesis test is used to test whether the estimated uncertain regression model is appropriate. Furthermore, a real-world example of factors analysis of grain yield is provided to illustrate the moment estimation. Finally, as a byproduct, this paper also indicates that the stochastic regression model cannot model the agriculture data.

Rapeseed and mustard are vital oilseed crops in Uttar Pradesh, India, contributing to agricultural livelihoods and food security. This study analyzes long-term trends and variability in area, production, and yield in Lucknow district from 1999–2000 to 2022–23 using linear and nonlinear time series models. Data on area (hectares), production (tonnes), and yield (tonnes/hectare) were modeled with Linear, Power, Mechanistic Growth, Logistic 3-Parameter (3P), and Gompertz 3P models. The Linear model best described area and production, while the Logistic 3P model outperformed for yield, capturing its sigmoidal growth. Challenges include managing high data variability due to weather and policy fluctuations and ensuring convergence of nonlinear models. Results show modest growth in area (+57.8 ha/year) and production (+133.5 t/year), with yield rising (+0.015 t/ha/year). High variability (coefficient of variation: 25.7% for area, 46.2% for production, 31.4% for yield) and instability indices (19.5%–34.3%) suggest external influences like weather or policy changes. Decomposition analysis revealed that yield improvements drove 60.2% of production growth, particularly post-2012 (73.6%). Sensitivity analysis confirmed model robustness, and residual diagnostics validated fit. Forecasts predict stable yields (0.97 t/ha by 2027) and modest increases in area and production. Compared to Uttar Pradesh’s higher yields (1.0–1.2 t/ha), Lucknow’s lag suggests policy needs for hybrid seeds and irrigation. These findings, supported by transparent data access, inform sustainable agricultural planning. This study contributes reliable forecasting tools for regional agricultural planning, a reproducible methodology via transparent data access and insights into the efficacy of Linear versus Nonlinear models for oilseed crops, advancing sustainable agriculture in resource-constrained regions.

In the last five decades, Box Jenkins methodology has been in existence to model univariate time series data but fails or has limitations on modeling volatility. Most financial time series data do exhibit heavy tail and thick distribution, to this effect various parametric and semi-parametric non –linear time series models have been proposed two or three decades ago to capture volatility. However, this research entails measuring volatility and its forecasting using time series exchange rate annual data over the period from 1981 to 2020 (wide periodicity). The exchange rate was transformed to return, and parametric non –linear time series was modeled on it. It was found out that GARCH (1,2) reveals continuous volatility for short while and was the best model to predict the exchange rate volatility based on the evidence from measurement volatility tool; RMSE, MAE, MAPE among other extensions of GARCH models; EGARCH and TGARCH. EGARCH (1, 4) captures the asymmetry effect revealing that negative shocks will persistently have an effect on the volatility of the naira/dollar exchange rate.

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.

In this paper we assess the short-term forecasting power of different time series models in the electricity spot market. In particular we calibrate AR/ARX (X stands for exogenous/fundamental variable—system load in our study), AR/ARX-GARCH, TAR/TARX and Markov regime-switching models to California Power Exchange (CalPX) system spot prices. We then use them for out-ofsample point and interval forecasting in normal and extremely volatile periods preceding the market crash in winter 2000/2001. We find evidence that (i) non-linear, threshold regime-switching (TAR/TARX) models outperform their linear counterparts, both in point and interval forecasting, and that (ii) an additional GARCH component generally decreases point forecasting efficiency. Interestingly, the former result challenges a number of previously published studies on the failure of non-linear regime-switching models in forecasting.

This study investigates the effects of outliers on the estimates of ARIMA model parameters with particular attention given to the performance of two outlier detection and modeling methods targeted at achieving more accurate estimates of the parameters. The two methods considered are: an iterative outlier detection aimed at obtaining the joint estimates of model parameters and outlier effects, and an iterative outlier detection with the effects of outliers removed to obtain an outlier free series, after which a successful ARIMA model is entertained. We explored the daily closing share price returns of Fidelity bank, Union bank of Nigeria, and Unity bank from 03/01/2006 to 24/11/2016, with each series consisting of 2690 observations from the Nigerian Stock Exchange. ARIMA (1, 1, 0) models were selected based on the minimum values of Akaike information criteria which fitted well to the outlier contaminated series of the respective banks. Our findings revealed that ARIMA (1, 1, 0) models which fitted adequately to the outlier free series outperformed those of the parameter-outlier effects joint- estimated model. Furthermore, we discovered that outliers biased the estimates of the model parameters by reducing the estimated values of the parameters. The implication is that, in order to achieve more accurate estimates of ARIMA model parameters, it is needful to account for the presence of significant outliers and preference should be given to the approach of cleaning the series of outliers before subsequent entertainment of adequate linear time series models.

Uncertain autoregressive model is a powerful analytical tool that uses uncertainty theory to predict future values based on previously observed values. In the study of uncertain autoregressive model, one of the core problems is how to estimate the unknown parameters and uncertain disturbance term in the model. In this paper, a moment estimation method for uncertain autoregressive model is proposed to determine these unknown parameters and uncertain disturbance term. Following that, the uncertain hypothesis test is used to verify the suitability of the estimated uncertain autoregressive model. In addition, we also provide a case study of Disney stock prices to illustrate the advantages of moment estimation method over other statistical inference methods.

This paper presents a simple test for dependence in the residuals of a linear parametric time series model fitted to non gaussian data. The test statistic is a third order extension of the standard correlation test for whiteness. but the number of lags used in this test is a function of the sample size. The power of this test goes to one as the sample size goes to infinity for any alternative which has non zero bicovariances c e3(r,s)= E[e(t)e(t + r)e(t + s)] for a zero mean stationary random time series. The asymptotic properties of the test statistic are rigorously determined. This test is important for the validation of the sampling properties of the parameter estimates for standard finite parameter linear models when the unobserved input (innovations) process is white but not gaussian. The sizes and power derived from the asymptotic results are checked using artificial data for a number of sample sizes. Theoretical and simulation results presented in this paper support the proposition that the test wi...