Search Results (17)

There is now a vast body of literature on ARMA and VARMA models with time-dependent or time-varying coefficients. A large part of it is based on local stationary processes using time rescaling and assumptions of regularity with respect to time. A recent paper has presented an alternative asymptotic theory for the parameter estimators based on several distinct assumptions that seem difficult to verify at first look, especially for time-dependent VARMA or tdVARMA models. The purpose of the present paper is to detail several examples that illustrate the verification of the assumptions in that theory. These assumptions bear on the moments of the errors, the existence of the information matrix, but also how the coefficients of the pure moving average representation of the derivatives of the residuals (with respect to the parameters and evaluated at their true value) behave. We will do that analytically for two bivariate first-order models, an autoregressive model, and a moving average model, before sketching a generalization to higher-order models. We also show simulation results for these two models illustrating the analytical results. As a consequence, not only the assumptions can be checked but the simulations show how well the small sample behavior of the estimators agrees with the theory. Full article

Several authors have attempted to compute the asymptotic Fisher information matrix for a univariate or multivariate time-series model to check for its identifiability. This has the form of a contour integral of a matrix of rational functions. A recent paper has proposed a short Wolfram Mathematica notebook for VARMAX models that makes use of symbolic integration. It cannot be used in open-source symbolic computation software like GNU Octave and GNU Maxima. It was based on symbolic integration but the integrand lacked symmetry characteristics in the appearance of polynomial roots smaller or greater than 1 in modulus. A more symmetric form of the integrand is proposed for VARMA models that first allows a simpler approach to symbolic integration. Second, the computation of the integral through Cauchy residues is also possible. Third, an old numerical algorithm by Söderström is used symbolically. These three approaches are investigated and compared on a pair of examples, not only for the Wolfram Language in Mathematica but also for GNU Octave and GNU Maxima. As a consequence, there are now sufficient conditions for exact model identifiability with fast procedures. Full article

The global outbreak of coronavirus disease 2019 (COVID-19) has posed a severe threat to public health and caused widespread socioeconomic disruptions in the past several years. While the pandemic has subsided, it is essential to explore effective disease surveillance tools to aid in controlling future pandemics. Several studies have proposed methods to capture the epidemic trend and forecast new daily confirmed cases. In this study, we propose the use of exponentially weighted moving average (EWMA) control charts integrated with time series models to monitor the number of daily new confirmed cases of COVID-19. The conventional EWMA control chart directly monitors the number of daily new confirmed cases. The proposed methods, however, monitor the residuals of time series models fitted to these data. In this study, two time series models—the auto-regressive integrated moving average (ARIMA) model and the vector auto-regressive moving average (VARMA) model—are considered. The results are compared with those of the conventional EWMA control chart using three datasets from India, Malaysia, and Thailand. The findings demonstrate that the proposed method can detect disease outbreak signals earlier than conventional control charts. Full article

Detecting anomalies in large, complex systems is a critical and challenging task, and this is especially true for high-dimensional anomaly detection due to the underlying dependency structures among sensors. To incorporate the interrelationships among various sensors, a novel sparsity-constrained vector autoregressive moving average (scVARMA) model is proposed for anomaly detection in complex systems with multisensory signals. This model aims to leverage the inherent relationships and dynamics among various sensor readings, providing a more comprehensive and accurate analysis suitable for complex systems’ complex behavior. This research uses convex optimization to search for a parameterization that is sparse based on the principal of parsimony. This sparse model will not only contribute to meeting the real-time requirements of online monitoring strategies but also keeps the correlations among different sensory signals. The performance of the proposed scVARMA model is validated using real-world data from complex systems. The results affirm the superiority of the proposed scheme, demonstrating its enhanced performance and potential in practical applications. Full article

Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations. Full article

This paper deals with the identifiability of VARMA models with VAR order greater than or equal to the MA order, in the context of mixed-frequency data (MFD) using extended Yule–Walker equations. The main contribution is that necessary and sufficient conditions for identifiability in the single-frequency data case are expressed in an original way and yield new results in the MFD case. We also provide two counterexamples that answer an open question in this topic about whether certain sufficient conditions are necessary for identifiability. Therefore, this paper expands the set of models that can be identified with MFD using extended Yule–Walker equations. The main idea is that with MFD, some autocovariance blocks are not available from observed variables and, in some cases, the new conditions in this paper can be used to reconstruct all the non-available covariance blocks from available covariance blocks. Full article

When using vector autoregressive (VAR) models for approximating time series, a key step is the selection of the lag length. Often this is performed using information criteria, even if a theoretical justification is lacking in some cases. For stationary processes, the asymptotic properties of the corresponding estimators are well documented in great generality in the book Hannan and Deistler (1988). If the data-generating process is not a finite-order VAR, the selected lag length typically tends to infinity as a function of the sample size. For invertible vector autoregressive moving average (VARMA) processes, this typically happens roughly proportional to $logT$. The same approach for lag length selection is also followed in practice for more general processes, for example, unit root processes. In the I(1) case, the literature suggests that the behavior is analogous to the stationary case. For I(2) processes, no such results are currently known. This note closes this gap, concluding that information-criteria-based lag length selection for I(2) processes indeed shows similar properties to in the stationary case. Full article

The allocation of pension funds has important theoretical value and practical significance, which improves the level of pension investment income, achieves the maintenance and appreciation of pension funds, and resolves the pension payment risk caused by population aging. The asset allocation of pension funds is a long-term asset allocation problem. Thus, the long-term risk and return of the assets need to be estimated. The covariance matrix is usually adopted to measure the risk of the assets, while calculating the long-term covariance matrix is extremely difficult. Direct calculations suffer from the insufficiency of historical data, and indirect calculations accumulate short-term covariance, which suffers from the dynamic changes of the covariance matrix. Since the returns of main assets are highly autocorrelated, the covariance matrix of main asset returns is time-varying with dramatic dynamic changes, and the errors of indirect calculation cannot be ignored. In this paper, we propose a novel Black–Litterman model with time-varying covariance (TVC-BL) for the optimal asset allocation of pension funds to address the time-varying nature of asset returns and risks. Firstly, the return on assets (ROA) and the covariance of ROA are modeled by VARMA and GARCH, respectively. Secondly, the time-varying covariance estimation of ROA is obtained by introducing an effective transformation of the covariance matrix from short-term to long-term. Finally, the asset allocation decision of pension funds is achieved by the TVC-BL model. The results indicate that the proposed TVC-BL pension asset allocation model outperforms the traditional BL model. When the risk aversion coefficient is 1, 1.5, and 3, the Sharp ratio of pension asset allocation through the TVC-BL pension asset allocation model is 13.0%, 10.5%, and 12.8% higher than that of the traditional BL model. It helps to improve the long-term investment returns of pension funds, realize the preservation and appreciation of pension funds, and resolve the pension payment risks caused by the aging of the population. Full article

This study adopted the exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model to examine the 10 ozone precursors of the highest concentrations among the 54 that were assessed over a number of years at the four photochemical assessment monitoring stations (PAMSs) in the Kaohsiung–Pingtung Area in Taiwan. First, the 10 ozone precursors, which were all volatile organic compounds (VOCs), were analyzed using the factor analyses in multiple statistical analyses that had the most significant impact on the area’s ozone formation: mobile pollution factor, which included 1,2,4-Trimethylbenzene (C₉H₁₂), toluene (C₇H₈), and Isopropyl benzene (C₉H₁₂). Then, taking into consideration that the number sequences might be affected by standardized residuals, this study applied the vector autoregressive moving average-EGARCH (VARMA-EGARCH) model to analyze the correlation between the three VOCs under different polluting activities. The VARMA-EGARCH model in this research included dummy variables representing changing points of variance structures in the variance formula to predict the conditional variance. This process proved able to effectively estimate the relevant coefficients of the three VOCs’ dynamic conditions that changed with time. The model also helped to prevent errors from occurring when estimating the conditional variance. Based on the testing results, this study determined the VARMA(2,1)-EGARCH(1,0) as the most suitable model for exploring the correlation between the three VOCs and meteorological phenomena, as well as the interplay between them in regard to interaction and formation. With the most representative of the three, toluene (TU), as the dependent variable and 1,2,4-Trimethylbenzene (TB) and Isopropyl benzene (IB) as the independent variables, this study found it impossible to calculate the TU concentration with TB and IB concentrations in the same period; estimations of TB and IB concentrations with a period of lag time were required because TU was mainly contributed by automobiles and motorcycles in Kaohsiung. TB and IB resulted from other stationary pollution sources in the region besides cars and motorbikes. When TU was evenly distributed and stayed longer in the atmosphere, the TB and IB concentrations were lower, so distribution conditions and concentrations could not be used to effectively estimate the concentration of toluene. This study had to wait until the next period, or when stationary pollution sources started producing TB and IB of higher concentrations during the daytime, in order to estimate the TU concentrations in a better photochemical situation. Full article

This paper investigates the asymptotic properties of estimators obtained from the so called CVA (canonical variate analysis) subspace algorithm proposed by Larimore (1983) in the case when the data is generated using a minimal state space system containing unit roots at the seasonal frequencies such that the yearly difference is a stationary vector autoregressive moving average (VARMA) process. The empirically most important special cases of such data generating processes are the I(1) case as well as the case of seasonally integrated quarterly or monthly data. However, increasingly also datasets with a higher sampling rate such as hourly, daily or weekly observations are available, for example for electricity consumption. In these cases the vector error correction representation (VECM) of the vector autoregressive (VAR) model is not very helpful as it demands the parameterization of one matrix per seasonal unit root. Even for weekly series this amounts to 52 matrices using yearly periodicity, for hourly data this is prohibitive. For such processes estimation using quasi-maximum likelihood maximization is extremely hard since the Gaussian likelihood typically has many local maxima while the parameter space often is high-dimensional. Additionally estimating a large number of models to test hypotheses on the cointegrating rank at the various unit roots becomes practically impossible for weekly data, for example. This paper shows that in this setting CVA provides consistent estimators of the transfer function generating the data, making it a valuable initial estimator for subsequent quasi-likelihood maximization. Furthermore, the paper proposes new tests for the cointegrating rank at the seasonal frequencies, which are easy to compute and numerically robust, making the method suitable for automatic modeling. A simulation study demonstrates by example that for processes of moderate to large dimension the new tests may outperform traditional tests based on long VAR approximations in sample sizes typically found in quarterly macroeconomic data. Further simulations show that the unit root tests are robust with respect to different distributions for the innovations as well as with respect to GARCH-type conditional heteroskedasticity. Moreover, an application to Kaggle data on hourly electricity consumption by different American providers demonstrates the usefulness of the method for applications. Therefore the CVA algorithm provides a very useful initial guess for subsequent quasi maximum likelihood estimation and also delivers relevant information on the cointegrating ranks at the different unit root frequencies. It is thus a useful tool for example in (but not limited to) automatic modeling applications where a large number of time series involving a substantial number of variables need to be modelled in parallel. Full article

We consider a generalization of Awad–Shannon entropy, namely Awad–Varma entropy, introduce a stochastic order on Awad–Varma residual entropy and study some properties of this order, like closure, reversed closure and preservation in some stochastic models (the proportional hazard rate model, the proportional reversed hazard rate model, the proportional odds model and the record values model). Full article

The presence of risk premium is an issue that weakens the rational expectation hypothesis. This paper investigates changing behavior of time varying risk premium for holding 10 year maturity bond using a bivariate VARMA-DBEKK-AGARCH-M model. The model allows for asymmetric risk premia, causality and co-volatility spillovers jointly in the global bond markets. Empirical results show significant asymmetric partial co-volatility spillovers and risk premium exist in the bond markets. The estimates of the bivariate risk premia show bi-directional causality exist between the Australia and France Bond markets. Overall results suggest nonexistence of pure rational expectation theory in the risk premium model. This information is useful for the agents’ strategic policy decision making in global bond markets. Full article

Benzene is a pollutant which is very harmful to our health, so models are necessary to predict its concentration and relationship with other air pollutants. The data collected by eight stations in Madrid (Spain) over nine years were analyzed using the following regression-based machine learning models: multivariate linear regression (MLR), multivariate adaptive regression splines (MARS), multilayer perceptron neural network (MLP), support vector machines (SVM), autoregressive integrated moving-average (ARIMA) and vector autoregressive moving-average (VARMA) models. Benzene concentration predictions were made from the concentration of four environmental pollutants: nitrogen dioxide (NO₂), nitrogen oxides (NO_x), particulate matter (PM₁₀) and toluene (C₇H₈), and the performance measures of the model were studied from the proposed models. In general, regression-based machine learning models are more effective at predicting than time series models. Full article

This study adopted the Exponential Generalized Autoregressive Conditional Heteroscedasticity (EGARCH) model to analyze seven air pollutants (or the seven variables in this study) from ten air quality monitoring stations in the Kaohsiung–Pingtung Air Pollutant Control Area located in southern Taiwan. Before the verification analysis of the EGARCH model is conducted, the air quality data collected at the ten air quality monitoring stations in the Kaohsiung–Pingtung area are classified into three major factors using the factor analyses in multiple statistical analyses. The factors with the most significance are then selected as the targets for conducting investigations; they are termed “photochemical pollution factors”, or factors related to pollution caused by air pollutants, including particulate matter with particles below 10 microns (PM₁₀), ozone (O₃) and nitrogen dioxide (NO₂). Then, we applied the Vector Autoregressive Moving Average-EGARCH (VARMA-EGARCH) model under the condition where the standardized residual existed in order to study the relationships among three air pollutants and how their concentration changed in the time series. By simulating the optimal model, namely VARMA (1,1)-EGARCH (1,1), we found that when O₃ was the dependent variable, the concentration of O₃ was not affected by the concentration of PM₁₀ and NO₂ in the same term. In terms of the impact response analysis on the predictive power of the three air pollutants in the time series, we found that the asymmetry effect of NO₂ was the most significant, meaning that NO₂ influenced the GARCH effect the least when the change of seasons caused the NO₂ concentration to fluctuate; it also suggested that the concentration of NO₂ produced in this area and the degree of change are lower than those of the other two air pollutants. This research is the first of its kind in the world to adopt a VARMA-EGARCH model to explore the interplay among various air pollutants and reactions triggered by it over time. The results of this study can be referenced by authorities for planning air quality total quantity control, applying and examining various air quality models, simulating the allowable increase in air quality limits, and evaluating the benefit of air quality improvement. Full article

In this paper the theory on the estimation of vector autoregressive (VAR) models for I(2) processes is extended to the case of long VAR approximation of more general processes. Hereby the order of the autoregression is allowed to tend to infinity at a certain rate depending on the sample size. We deal with unrestricted OLS estimators (in the model formulated in levels as well as in vector error correction form) as well as with two stage estimation (2SI2) in the vector error correction model (VECM) formulation. Our main results are analogous to the I(1) case: We show that the long VAR approximation leads to consistent estimates of the long and short run dynamics. Furthermore, tests on the autoregressive coefficients follow standard asymptotics. The pseudo likelihood ratio tests on the cointegrating ranks (using the Gaussian likelihood) used in the 2SI2 algorithm show under the null hypothesis the same distributions as in the case of data generating processes following finite order VARs. The same holds true for the asymptotic distribution of the long run dynamics both in the unrestricted VECM estimation and the reduced rank regression in the 2SI2 algorithm. Building on these results we show that if the data is generated by an invertible VARMA process, the VAR approximation can be used in order to derive a consistent initial estimator for subsequent pseudo likelihood optimization in the VARMA model. Full article