No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Multiple Time Series
... Günter (2000), in his work, "Asymptotic confidence bands for the estimated autocovariance and autocorrelation functions of vector autoregressive models" provided formulae for computing the asymptotic standard errors of the estimated autocovariance and autocorrelation functions of stable VAR models; these he stipulated can be used to construct asymptotic confidence bands, where the sample autocovariance and the sample autocorrelations are asymptotically normal. Hannan (1970) and Anderson (1971) indicated that the test relying on sample autocorrelations is easily conducted, with standard errors approximately equal to 1/√T. However, Dufour and Roy (1985) demonstrated that these tests may exhibit a lower frequency of rejecting the null hypothesis than expected. ...
The primary aim of this study was to conduct a comparative analysis of the performance of parsimonious models, specifically the Diagonal Vector Autoregressive (VAR) and Multivariate Autoregressive Distributed Lag (MARDL) Models, using their respective Autocovariance and Autocorrelation properties. This comparison was driven by the imposition of restrictions on parameters within the coefficient matrices, specifically limiting them to diagonal elements. To assess the efficacy of these novel multivariate lag models, we utilised data derived from key macroeconomic variables, including Nigeria's Gross Domestic Product (GDP), Crude Oil Petroleum (C/PET), Agriculture (AGRIC), and Telecommunication (TELECOM). The data was subjected to first-order differencing of the logarithm of the series to ensure stationarity. Subsequently, the models were estimated, and autocovariances and autocorrelations of the processes were derived for the analysis. The empirical findings revealed notable patterns, particularly the direct converse autocorrelation observed in both VAR and MARDL models. The negative autocorrelation identified in the macroeconomic variables suggests that periods of economic expansion were succeeded by contractions and vice versa. This implies a complementary relationship between the two models in effectively capturing the dynamics of multivariate lag variables. In conclusion, our study underscores the significance of considering the Diagonal Vector Autoregressive and Multivariate Autoregressive Distributed Lag Models with restricted parameters in the diagonal elements when modelling multivariate lag variables. These findings contribute to a nuanced understanding of the interplay between economic variables and provide valuable insights for researchers and practitioners in the field.
... As a rule, such functions are sufficiently smoothed to have second and even third partial derivatives in the components of the vector v . Assumption B is related to the definition of regularity [1] of additive noise, which always holds for stationary Gaussian processes. Estimate (14) belongs to the class of M-estimates. ...
This article is devoted to the synthesis and analysis of the quality of the statistical estimate of parameters of a multidimensional linear system (MLS) with one input and m outputs. A nontrivial case is investigated when the one-dimensional input signal of MLS is a deterministic process, the values of which are unknown nuisance parameters. The estimate is based only on observations of MLS output signals distorted by random Gaussian stationary -dimensional noise with a known spectrum. It is assumed that the likelihood function of observations of the output signals of MLS satisfies the conditions of local asymptotic normality. The -consistency of the estimate is established. Under the assumption of asymptotic normality of an objective function, the limiting covariance matrix of the estimate is calculated for case where the number of observations tends to infinity.
... Observe that this representation is similar to the spectral representation of stationary processes (see Anderson, 1971;Brillinger, 1975;Hannan, 1970;Priestley, 1981) for introductory concepts]. The main difference is that Aðt=T, xÞ and lðt=TÞ are not constant in t. ...
We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroscedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the tradeoff between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews, we establish how nonstationarity affects the bias-variance tradeoff. We use the plug-in approach to construct data-dependent bandwidths for the DK-HAC estimators and compare them with the DK-HAC estimators from Casini that use data-dependent bandwidths obtained from a sequential MSE criterion. The former performs better in terms of size control, especially with stationary and close to stationary data. Finally, we consider long-run variance (LRV) estimation under the assumption that the series is a function of a nonparametric estimator rather than of a semiparametric estimator that enjoys the usual T rate of convergence. Thus, we also establish the validity of consistent LRV estimation in nonparametric parameter estimation settings.
... This representation can accommodate SAR models in the error term (so-called spatial error models (SEMs)) as a special case, as well as variants like SARMA and MESS, whence its generality is apparent. The linear-process structure shares some similarities with that familiar from the time series literature (see, e.g., Hannan, 1970). Indeed, time series versions may be regarded as very special cases, but, as stressed before, the features of spatial dependence must be taken into account in the general formulation. ...
We propose a series-based nonparametric specification test for a regression function when data are spatially dependent, the “space” being of a general economic or social nature. Dependence can be parametric, parametric with increasing dimension, semiparametric or any combination thereof, thus covering a vast variety of settings. These include spatial error models of varying types and levels of complexity. Under a new smooth spatial dependence condition, our test statistic is asymptotically standard normal. To prove the latter property, we establish a central limit theorem for quadratic forms in linear processes in an increasing dimension setting. Finite sample performance is investigated in a simulation study, with a bootstrap method also justified and illustrated. Empirical examples illustrate the test with real-world data.
The boosted HP (bHP) trend filter iterates the standard HP filter until the resulting trend deviation is free of any stochastic trend with the latter determined by suitable stopping rules. Here the performance and properties of the bHP trend filter for growth cycle analysis are considered based on the time-invariant moving-average representation of the bHP filter in the body of the series. We propose alternative trend selection criteria based on a constant cut-off frequency and maximising sharpness. We find there is a strong case for selecting a bHP trend filter with an 8-year rather than a 10-year cut-off period, and that using a bHP filter with 2 iterations (twicing) or a sharpened HP filter is preferable to using the standard HP filter.
Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).
The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a periodically correlated stochastic sequence from observations of the sequence with missings is considered. Formulas for calculation the mean-square error and the spectral characteristic of the optimal estimate of the functionals are proposed in the case where spectral densities of the sequences are exactly known. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed in the case of spectral uncertainty, when spectral densities of sequences are not exactly known but the class of admissible spectral densities is given.
Для стационарного гауссовского процесса выводится расстояние Хеллингера $T(f,g)$ между спектральными плотностями $f$ и $g$. Оценивая $T(f_\theta,f_{\theta+h})$ как $O(h^\alpha)$, мы выводим $1/\alpha$-состоятельную асимптотику оценки максимального правдоподобия для $\theta$ в случае нерегулярных спектральных плотностей. В случае же регулярных спектральных плотностей мы вводим оценку, основанную на минимизации расстояния Хеллингера: $\widehat{\theta}=\operatorname{arg}\min_\theta T(f_\theta,\widehat{g}_n)$, где $\widehat{g}_n$ - непараметрическая оценка спектральной плотности. Мы показываем, что оценка $\widehat\theta$ является асимптотически эффективной и более робастной, чем оценка Уиттла. Представлены также некоторые численные исследования.
The chapter considers singularities of characterization and application of linear AR processes in the problems of development and construction of diagnostic systems of power equipment and power systems. The characteristic functions of such processes are presented. Some methods for development of decision rules when using such processes as mathematical models of the information signals of power equipment and power systems are considered. An example of linear AR process application for wind generator vibration diagnosis are presented. The advantages of using such models as mathematical models of information signals of power equipment and power systems are discussed.
Predicting the value of stock prices is separate from external investment exposure to the stock market. The aim of the article is to evaluate selected time series forecasting models in forecasting stock data – exactly Apple Inc. (AAPL) stock price. There are not many articles in the literature on the possibility of using ML models in forecasting stock prices. It is important to ask the question whether ML models give better results than traditional models. To answer this question, in this article following regression models ARIMA, Logistic Regression were analyzed but also were used VEC, LSTM, XGBoost, Prophet.
Linear ARMA model fitting requires to select the order of the model as accurately as possible. Many past studies are based on portmanteau tests, originally derived for the white noise hypothesis, but applied to the autocorrelations of the estimated residuals, and employ sums of the scaled squares of the first d residual autocorrelations. An automatic choice of d was proposed by Escanciano and Lobato (J Econom 151:140–149, 2009) based on the largest (in absolute value) residual autocorrelation. Such maximal autocorrelation may be itself employed as a test statistic, and Baragona et al. (Test 31:675–698, 2022) proposed white noise tests based on a bivariate statistic consisting of both the portmanteau and the largest autocorrelation. However, the derivation of the asymptotic null distribution requires asymptotic independence of the autocorrelation estimates and this is not true when they are computed on the residuals of a fitted ARMA model. Therefore, we propose to use a linear transformation of the estimated residual autocorrelation in order to achieve independence, in the same spirit as recursive residuals in regression. Monte Carlo experiments are performed for comparing the effectiveness of our new method to the Escanciano Lobato and a more classical portmanteau test. We find that the two data-driven tests are approximately equivalent if non negligible residual autocorrelation is found only at the first lags, while our test is more powerful if autocorrelations at larger lags arises.
In psychology, the use of portable technology and wearable devices to ease participant burden in data collection is on the rise. This creates increased interest in collecting real-time or near real-time data from individuals within their natural environments. As a result, vast amounts of observational time series data are generated. Often, motivation for collecting this data hinges on understanding within-person processes that underlie psychological phenomena. Motivated by the body of Dr. Peter Molenaar's life work calling for analytical approaches that consider potential heterogeneity and non-ergodicity, the focus of this paper is on using idiographic analyses to generate population inferences for within-person processes. Meta-analysis techniques using one-stage and two-stage random effects meta-analysis as implemented in single-case experimental designs are presented. The case for preferring a two-stage approach for meta-analysis of single-subject observational time series data is made and demonstrated using an empirical example. This provides a novel implementation of the methodology as prior implementations focus on applications to short time series with experimental designs. Inspired by Dr. Molenaar's work, we describe how an approach, two-stage random effects meta-analysis (2SRE-MA), aligns with recent calls to consider idiographic approaches when making population-level inferences regarding within-person processes.
While ANOVA for independent errors has been well-tuned, ANOVA for time-dependent errors is in its infancy. In this chapter, we extend the one-way fixed model with independent errors and groups to a one-way fixed model with time-dependent errors and independent groups.
We derive the finite sample bias of the sample cross‐covariance estimator based on a stationary vector‐valued time series with an unknown mean. This result leads to a bias‐corrected estimator of cross‐covariances constructed from linear combinations of sample cross‐covariances, which can in theory correct for the bias introduced by the first lags of cross‐covariance with any not larger than the sample size minus two. Based on the bias‐corrected cross‐covariance estimator, we propose a bias‐adjusted long run covariance (LRCOV) estimator. We derive the asymptotic relations between the bias‐corrected estimators and their conventional Counterparts in both the small‐ and the fixed‐ limit. Simulation results show that: (i) our bias‐corrected cross‐covariance estimators are very effective in eliminating the finite sample bias of their conventional counterparts, with potential mean squared error reduction when the data generating process is highly persistent; and (ii) the bias‐adjusted LRCOV estimators can have superior performance to their conventional counterparts with a smaller bias and mean squared error.
The fluctuation of stationary time series often shows a certain periodic behavior and this pattern is usually summarized via a spectral density. Since spectral density is a periodic function, it can be modeled by using a circular distribution function. In this paper, several time series models are studied in relation to a circular distribution. As an introduction, we illustrate how to model bivariate time series data using complex-valued time series in the context of circular distribution functions. These models are extended to have a skewed spectrum by incorporating a sine-skewing transformation. Two parameter estimation methods are considered and their asymptotic properties are investigated. These theoretical results are verified via a Monte Carlo simulation. Real data analyses illustrate the applicability of the proposed model.KeywordsCircular statisticsMaximum likelihood estimationSine-skewed modelSpectral densityTime series analysis
We discuss the evolution in the presentation of statistical science in English‐language textbooks, focusing on the period 1900–1970 as the field became increasingly influenced by research contributions of R. A. Fisher and Jerzy Neyman. George Udny Yule authored an early popular book that had 14 editions. Methods books authored by Fisher and George Snedecor guided scientists in implementing modern statistical methods. In the World War 2 era, textbooks authored by Maurice Kendall, Samuel Wilks, and Harald Cramér presented a dramatically different “mathematical statistics” portrayal that centered on theoretical foundations. The textbook emergence of the Bayesian approach occurred later, influenced by books by Harold Jeffreys and Leonard J. Savage. The quarter century after World War 2 saw an explosion of books in mathematical statistics and in particular topic areas. In addition to his highly cited research contributions, Sir David Cox was a prolific author of books on a great variety of topics. Most were published after the 1900–1970 period considered in this article, but we also summarize them as part of this special issue to honor his memory. We conclude by discussing the future of textbooks on the foundations of statistical science in the emerging, ever‐broader, era of data science.
We consider the estimation and uncertainty quantification of the tail spectral density, which provide a foundation for tail spectral analysis of tail dependent time series. The tail spectral density has a particular focus on serial dependence in the tail, and can reveal dependence information that is otherwise not discoverable by the traditional spectral analysis. Understanding the convergence rate of tail spectral density estimators and finding rigorous ways to quantify their statistical uncertainty, however, still stand as a somewhat open problem. The current article aims to fill this gap by providing a novel asymptotic theory on quadratic forms of tail statistics in the double asymptotic setting, based on which we develop the consistency and the long desired central limit theorem for tail spectral density estimators. The results are then used to devise a clean and effective method for constructing confidence intervals to gauge the statistical uncertainty of tail spectral density estimators, and it can be turned into a visualization tool to aid practitioners in examining the tail dependence for their data of interest. Numerical examples including data applications are presented to illustrate the developed results.
Though globalization, industrialization, and urbanization have escalated the economic growth of nations, these activities have played foul on the environment. Better understanding of ill effects of these activities on environment and human health and taking appropriate control measures in advance are the need of the hour. Time series analysis can be a great tool in this direction. ARIMA model is the most popular accepted time series model. It has numerous applications in various domains due its high mathematical precision, flexible nature, and greater reliable results. ARIMA and environment are highly correlated. Though there are many research papers on application of ARIMA in various fields including environment, there is no substantial work that reviews the building stages of ARIMA. In this regard, the present work attempts to present three different stages through which ARIMA was evolved. More than 100 papers are reviewed in this study to discuss the application part based on pure ARIMA and its hybrid modeling with special focus in the field of environment/health/air quality. Forecasting in this field can be a great contributor to governments and public at large in taking all the required precautionary steps in advance. After such a massive review of ARIMA and hybrid modeling involving ARIMA in the fields including or excluding environment/health/atmosphere, it can be concluded that the combined models are more robust and have higher ability to capture all the patterns of the series uniformly. Thus, combining several models or using hybrid model has emerged as a routinized custom.
The aim of this paper is to make inference about a general class of time series models including fractional Brownian motion. The spectral of these processes is supported on lines not parallel to the diagonal [Formula: see text], [Formula: see text], [Formula: see text], in spectral square [Formula: see text], and this class includes stationary, cyclostationary, almost cyclostationary time series and specially fractional Brownian motions. First, the periodogram of these processes is defined and auxiliary operator is applied to explore the distribution of the periodogram. Then the asymptotical estimation for the spectral density function is proposed and asymptotical Wishart function is found. Finally, the validity of the theoretical results is studied using simulated data sets.
This paper constructs a class of isotropic vector random fields on the probability simplex via infinite series expansions involving the ultraspherical polynomials, whose covariance matrix functions are functions of the metric (distance function) on the probability simplex, and introduces the scalar and vector fractional, bifractional, and trifractional Brownian motions over the probability simplex, while the metric is shown to be conditionally negative definite.
In this paper, we propose tests for the existence of random effects and interactions for two-way models with dependent errors. We prove that the proposed tests are asymptotically distribution-free which have asymptotically size \({{\tau }}\) and are consistent. We elucidate the nontrivial power under the local alternative when a sample size tends to infinity and the number of groups is fixed. A simulation study is performed to investigate the finite-sample performance of the proposed tests. In the real data analysis, we apply our tests to the daily log-returns of 24 stock prices from six countries and four sectors. We find that there is no strong evidence to support the existence of substantial differences in the log-return across countries, nor to the existence of interactions between countries and sectors. However, there exists random effect differences in the daily log-return series across different sectors.
Bringing together a collection of previously published work, this 2004 book provides a discussion of major considerations relating to the construction of econometric models that work well to explain economic phenomena, predict future outcomes and be useful for policy-making. Analytical relations between dynamic econometric structural models and empirical time series MVARMA, VAR, transfer function, and univariate ARIMA models are established with important application for model-checking and model construction. The theory and applications of these procedures to a variety of econometric modeling and forecasting problems as well as Bayesian and non-Bayesian testing, shrinkage estimation and forecasting procedures are also presented and applied. Finally, attention is focused on the effects of disaggregation on forecasting precision and the Marshallian Macroeconomic Model that features demand, supply and entry equations for major sectors of economies is analysed and described. This volume will prove invaluable to professionals, academics and students alike.
In this chapter, we discuss so-called autoregressive processes, i.e. stationary solutions \((x_t)\) of difference equations of the form $$ x_{t}=a_{1}x_{t-1}+\cdots +a_{p}x_{t-p}+\epsilon _{t},\,\,\forall t\in \mathbb {Z}$$where \((\epsilon _{t})\) is white noise. AR models are probably the most widely used class of models for practical applications of time series analysis. Autoregressive models enable us to model processes with an “infinite” memory (i.e. where the present values of the process correlate with values that lie arbitrarily far back in time) and, in contrast to general MA\((\infty )\) processes, with a finite number of parameters. AR models are, for instance, well suited for describing processes with pronounced peaks in the spectral density. These are processes with dominating, “almost periodic” components, which can be found in many applications, for example, in electrocardiogram (ECG) signals. Moreover, any regular process can be approximated with arbitrary accuracy by an AR process if the order p is chosen large enough. Another important advantage of autoregressive processes is the simplicity of their prediction. Under the stability condition, the one-step-ahead prediction from the infinite past is simply \(\hat{x}_{t,1}=a_{1}x_{t-1}+\cdots +a_{p}x_{t-p} \). This means that the least squares prediction depends only on the last p past values and the corresponding coefficients are exactly the coefficients of the AR model. Therefore, the AR model is an explicit description of the intertemporal dependence structure. The model decomposes \(x_t\) into the part determined by the past and the innovation. Last but not least, the AR model can also be estimated in a very simple way, e.g., using the so-called Yule-Walker equations. The model can be interpreted as a regression model. This explains the name “autoregressive” and shows that the model can also be estimated with the ordinary least squares method. In the first section, we shall briefly discuss the stationary solution of the AR system under the stability condition. We already did essential preliminary work on this in the previous chapter. Next, we shall cover the prediction of AR processes from finite or infinite past and discuss the main characteristics of the spectral density of AR processes. The penultimate section focuses on the Yule Walker equations, which establish the interrelation between the parameters of the AR system and the covariance function. As stated above, these equations also form the basis for one of the most important methods used to estimate AR systems. In the last section, we will omit the stability condition and briefly discuss the stationary solutions of AR systems in general. This section will also consider special non-stationary solutions, so-called integrated and co-integrated processes, which occur in the case of a so-called unit root. One of the most important results in this context is Granger’s representation theorem.
The computation of “good” predictions and a quantitative analysis of the prediction quality are among the most important applications of time series analysis. Prediction in general is concerned with approximating a future process variable by a function of the observed values up to the present time. Here, we discuss a special prediction problem, namely the so-called linear least-squares prediction, which means that we restrict ourselves to linear (to be more precise affine) prediction functions and the quality of the predictions is measured by the expectation of the squared prediction errors. We consider an idealized problem for which we assume that we know the exact properties of the underlying process (expectation and covariance function). This idealized prediction problem (which is an important step for the development of “feasible” predictions) can be handled simply and elegantly by using the projection theorem. Despite the fact that the restriction to linear predictions and the quadratic performance criterion is a limitation, this prediction scheme is the one that is used most often. The first section of this chapter uses the last k observed values for prediction. Hence, one also speaks of prediction based on a finite past. The transition for k going to infinity, which is accordingly called prediction based on the infinite past, subsequently leads to the so-called Wold decomposition of stationary processes. This Wold decomposition is important for prediction and, moreover, crucial for the understanding of the structure of stationary processes.
Estimating the spectral density function f(w) for some w∈[−π,π] has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, i.e., approximating f(w) by a constant over a window of small width. Although f(w) is uniformly continuous and periodic with period 2π, in this paper we recognize the fact that w = 0 effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for w=±π. It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when w is at (or near) the points 0 or ±π. The case w = 0 is of particular importance since f(0) is the large-sample variance of the sample mean; hence, estimating f(0) is crucial in order to conduct any sort of inference on the mean.
Autocovariances are a fundamental quantity of interest in Markov chain Monte Carlo (MCMC) simulations with autocorrelation function (ACF) plots being an integral visualization tool for performance assessment. Unfortunately, for slow-mixing Markov chains, the empirical autocovariance can highly underestimate the truth. For multiple-chain MCMC sampling, we propose a globally centered estimator of the autocovariance function (G-ACvF) that exhibits significant theoretical and empirical improvements. We show that the bias of the G-ACvF estimator is smaller than the bias of the current state-of-the-art. The impact of this improved estimator is evident in three critical output analysis applications: (a) ACF plots, (b) estimates of the Monte Carlo asymptotic covariance matrix, and (c) estimates of the effective sample size. Under weak conditions, we establish strong consistency of our improved asymptotic covariance estimator, and obtain its large-sample bias and variance. The performance of the new estimators is demonstrated through various examples. Supplementary materials for this article are available online.
The problem of the optimal linear estimation of functionals that depend on the unknown values of the stochastic stationary sequence of observations of a sequence with missing values is considered. Formulas for calculating the root-mean-square error and the spectral characteristic of the optimal linear estimate of the functionals are derived under the spectral determinacy, where the spectral density of the sequence is known exactly. The minimax (robust) method of estimation is applied in the case where the spectral density of the sequence is not known exactly while some classes of feasible spectral densities are given. Formulas that determine the least favourable spectral densities and minimax spectral characteristics are derived for optimal linear estimation of functionals for some special classes of spectral densities.
Since economic trends have a great influence on corporate activities, predicting whether the economy is in an expansion period or in retreat is important. Business condition indexes used in Japan that quantify the economy include the diffusion index (DI) and the composite index (CI). A method for predicting economic judgement is presented in this study. An economic trend is taken as an objective function and the DI and CI values are explanatory variables. The prediction model is defined as a Bayesian network. In Bayesian networks, random variables are used as nodes, and the dependency between variables is represented by a directed graph. Japan’s economic trends and DI and CI values from 1985 to 2020 are taken as experimental data. The forecast model is determined using the data from 1985 to 2017 as learning data, and the economic trend from 2018 to 2020 is predicted. The proposed algorithm is compared with a linear model for time-series data. The proposed algorithm shows better accuracy than the linear models.
he problem of the mean-square optimal estimation of the linear functionals which depend onthe unknown values of a stochastic stationary process from observations of the process with missingsis considered. Formulas for calculating the mean-square error and the spectral characteristic of theoptimal linear estimate of the functionals are derived under the condition of spectral certainty, wherethe spectral density of the process is exactly known. The minimax (robust) method of estimation isapplied in the case where the spectral density of the process is not known exactly while some sets ofadmissible spectral densities are given. Formulas that determine the least favourable spectral densitiesand the minimax spectral characteristics are derived for some special sets of admissible densities.Key Words: stationary stochastic proce
We propose MultiRocket, a fast time series classification (TSC) algorithm that achieves state-of-the-art accuracy with a tiny fraction of the time and without the complex ensembling structure of many state-of-the-art methods. MultiRocket improves on MiniRocket, one of the fastest TSC algorithms to date, by adding multiple pooling operators and transformations to improve the diversity of the features generated. In addition to processing the raw input series, MultiRocket also applies first order differences to transform the original series. Convolutions are applied to both representations, and four pooling operators are applied to the convolution outputs. When benchmarked using the University of California Riverside TSC benchmark datasets, MultiRocket is significantly more accurate than MiniRocket, and competitive with the best ranked current method in terms of accuracy, HIVE-COTE 2.0, while being orders of magnitude faster.
Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields Z ( x ) Z(x) and construct approximations of solutions U ( x ) U(x) of ordinary or partial differential equations whose random coefficients depend on Z ( x ) Z(x) . FD models of Z ( x ) Z(x) and U ( x ) U(x) constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of Z ( x ) Z(x) and U ( x ) U(x) for two types of random fields Z ( x ) Z(x) and a simple stochastic equation. Some of these conditions are illustrated by numerical examples.
We use information from higher order moments to achieve identification of non-Gaussian structural vector autoregressive moving average (SVARMA) models, possibly nonfundamental or noncausal, through a frequency domain criterion based on higher order spectral densities. This allows us to identify the location of the roots of the determinantal lag matrix polynomials and to identify the rotation of the model errors leading to the structural shocks up to sign and permutation. We describe sufficient conditions for global and local parameter identification that rely on simple rank assumptions on the linear dynamics and on finite order serial and component independence conditions for the non-Gaussian structural innovations. We generalize previous univariate analysis to develop asymptotically normal and efficient estimates exploiting second and higher order cumulant dynamics given a particular structural shocks ordering without assumptions on causality or invertibility. Finite sample properties of estimates are explored with real and simulated data.
This paper employs the local Bayesian likelihood methodology to estimate a medium-scale dynamic stochastic general equilibrium (DSGE) model on different frequencies and uses frequency-domain tools to evaluate the time-varying parameter model and the fixed-parameter model. These techniques yield fresh insights into theoretical and empirical implications conveyed by alternative models beyond what conventional time-domain approaches can offer. We show that parameter estimates are sensitive to frequencies, and goodness-of-fit varies substantially with the frequency bands. Overall, the estimated time-varying parameter model captures the properties of the U.S. data better in the business cycle frequency band, and beyond this band, the fixed-parameter model performs better. Additionally, our study also reveals the importance of structural shocks in improving the fit between models and data. Finally, we utilize the spectral representation of generalized forecast error variance decomposition to investigate the frequency dynamics of volatility connectedness. We find shocks to economic activity have an impact on variables at different frequencies with different strengths, and markets become more connected during crisis periods. Furthermore, this study provides insights into a question policymakers are much concerned with: which shocks are major sources of economic volatilities and which sectors serve as major recipients of shocks?
Herein, we propose a novel nonparametric frequency Granger causality test. We apply a filtering process in the time domain to remove possible spurious causality, thereby eliminating potential interference. Thereafter, in the frequency domain, we perform a local kernel regression for each frequency and test the noncausality hypothesis from the distance between each estimate to zero. We provide asymptotic results for strict stationary series concerning α‐mixing conditions. Our method can also perform group causality tests, a feature that is absent in most alternative methods. Monte Carlo experiments illustrate that our method is comparable, and in some cases, performs better than alternative methods in the literature. Finally, we test the causality between monetary policy variables and stock prices.
By locating the running maxima and minima of a time series, and measuring the current deviation from them, it is possible to generate processes that are relevant for the analysis of the business cycle and for characterizing bull and bear phases in financial markets. First, the measurement of the time distance from the running peak originates a first order Markov chain, whose characteristics can be used for testing time reversibility of economic dynamics and specific types of asymmetries in financial markets. Secondly, the gap processes can be combined to provide a nonparametric measure of the growth cycle. The paper derives the time series properties of the gap process and other related processes that arise from the same measurement context, and proposes new nonparametric tests of time reversibility and new measures of the output gap.
We survey the literature on spectral regression estimation. We present a cohesive framework designed to model dependence on frequency in the response of economic time series to changes in the explanatory variables. Our emphasis is on the statistical structure and on the economic interpretation of time-domain specifications needed to obtain horizon effects over frequencies , over scales , or upon aggregation . To this end, we articulate our discussion around the role played by lead-lag effects in the explanatory variables as drivers of differential information across horizons. We provide perspectives for future work throughout.
Time series occur naturally in a wide range of practices. For example, the opening price of a certain stock at the New York Stock Exchange, the monthly rainfall total of a certain region, and the CD4+ cell count over time of an individual infected with the HIV virus may all be viewed as time series. A time series is usually denoted by X
t, t ∈ T, where T is a set of times, or X(t), t ∈ T, although in this book the latter notation is reserved for (continuous-time) stochastic processes (see the next chapter). An observed time series is a sequence of numbers, one at each observational time.
We propose a randomized approach to the consistent statistical analysis of random processes and fields on \({\mathbb {R}}^m\) and \({\mathbb {Z}}^m, m=1,2,...\), which is valid in the case of strong dependence: the parameter of interest \(\theta \) only has to possesses a consistent sequence of estimators \({\hat{\theta }}_n\). The limit theorem is related to consistent sequences of randomized estimators \({\hat{\theta }}_n^*\); it is used to construct consistent asymptotically efficient sequences of confidence intervals and tests of hypotheses related to the parameter \(\theta \). Upper bounds for “admissible” sequences of normalizing coefficients in the limit theorem are established for some statistical models in Part 2.
Nonlinear wavelet-based estimators for spectral densities of non-Gaussian linear processes are considered. The convergence rates of mean integrated squared error (MISE) for those estimators over a large range of Besov function classes are derived, and it is shown that those rates are identical to minimax lower bounds in standard nonparametric regression model within a logarithmic term. Thus, those rates could be considered as nearly optimal. Therefore, the resulting wavelet-based estimators outperform traditional linear methods if the degree of smoothness of spectral densities varies considerably over the interval of interest, such as sharp spike, cusp, bump, etc, since linear estimators are not able to attain these rates. Unlike in classical nonparametric regression with Gaussian noise errors where thresholds are determined by normal distribution, we determine the thresholds based on a Bartlett type approximation of a quadratic form with dependent variables by its corresponding quadratic form with independent identically distributed (i.i.d.) random variables and Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. The theory is illustrated with some numerical examples, and our simulation studies show that our proposed estimators are comparable to the current ones.
ResearchGate has not been able to resolve any references for this publication.