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This book provides a solution to the ecological inference problem, which has plagued users of statistical methods for over seventy-five years: How can researchers reliably infer individual-level behavior from aggregate (ecological) data? In political science, this question arises when individual-level surveys are unavailable (for instance, local or comparative electoral politics), unreliable (racial politics), insufficient (political geography), or infeasible (political history). This ecological inference problem also confronts researchers in numerous areas of major significance in public policy, and other academic disciplines, ranging from epidemiology and marketing to sociology and quantitative history. Although many have attempted to make such cross-level inferences, scholars agree that all existing methods yield very inaccurate conclusions about the world. In this volume, Gary King lays out a unique--and reliable--solution to this venerable problem. King begins with a qualitative overview, readable even by those without a statistical background. He then unifies the apparently diverse findings in the methodological literature, so that only one aggregation problem remains to be solved. He then presents his solution, as well as empirical evaluations of the solution that include over 16,000 comparisons of his estimates from real aggregate data to the known individual-level answer. The method works in practice. King's solution to the ecological inference problem will enable empirical researchers to investigate substantive questions that have heretofore proved unanswerable, and move forward fields of inquiry in which progress has been stifled by this problem.

We investigate a test for the conditional variance of stationary and ergodic integer-valued time series. This hypothesis testing problem is motivated by the fact that the form of the conditional variance of the process is determined by the conditional distribution and the conditional mean. First, we estimate the unknown parameters of the intensity function using an M-estimator and prove strong consistency and asymptotic normality. Second, we show that the proposed test has asymptotic size α and is consistent. Finally, we discuss the nontrivial power of the proposed test for the local alternative. The proposed test statistic can be applied to various problems, such as specification tests for intensity functions, tests for overdispersion and underdispersion, and goodness-of-fit tests for ergodic and stationary integer-valued time series. A simulation study illustrates the finite-sample performance of the proposed test. Lastly, in a real-data application, we analyze the number of patients with Escherichia coli in Germany.

With the exponential growth of data, solving classification or regression tasks by mining time series data has become a research hotspot. Commonly used methods include machine learning, artificial neural networks, and so on. However, these methods only extract the continuous or discrete features of sequences, which have the drawbacks of low information utilization, poor robustness, and computational complexity. To solve these problems, this paper innovatively uses Wasserstein distance instead of Kullback–Leibler divergence and uses it to construct an autoencoder to learn discrete features of time series. Then, a hidden Markov model is used to learn the continuous features of the sequence. Finally, stacking is used to ensemble the two models to obtain the final model. This paper experimentally verifies that the ensemble model has lower computational complexity and is close to state-of-the-art classification accuracy.

One of the most widely used properties of the multivariate Gaussian distribution, besides its tail behavior, is the fact that conditional means are linear and that conditional variances are constant. We here show that this property is also shared, in an approximate sense, by a large class of non-Gaussian distributions. We allow for several conditioning variables and we provide explicit non-asymptotic results, whereby we extend earlier findings of Hall and Li (Ann. Statist. 21 (1993) 867–889) and Leeb (Ann. Statist. 41 (2013) 464–483).

The class of α -stable distributions is widely used in various applications, especially for modeling heavy-tailed data. Although the α -stable distributions have been used in practice for many years, new methods for identification, testing, and estimation are still being refined and new approaches are being proposed. The constant development of new statistical methods is related to the low efficiency of existing algorithms, especially when the underlying sample is small or the distribution is close to Gaussian. In this paper, we propose a new estimation algorithm for the stability index, for samples from the symmetric α -stable distribution. The proposed approach is based on a quantile conditional variance ratio. We study the statistical properties of the proposed estimation procedure and show empirically that our methodology often outperforms other commonly used estimation algorithms. Moreover, we show that our statistic extracts unique sample characteristics that can be combined with other methods to refine existing methodologies via ensemble methods. Although our focus is set on the symmetric α -stable case, we demonstrate that the considered statistic is insensitive to the skewness parameter change, so our method could be also used in a more generic framework. For completeness, we also show how to apply our method to real data linked to financial market and plasma physics.

Accurately forecasting Bitcoin’s conditional variance is essential for reliable Value-at-Risk (VaR) estimation yet remains challenging due to nonlinear dynamics, volatility clustering, and heavy-tailed return distributions. This study developed a novel stacking ensemble that integrates econometric and machine-learning models through XGBoost meta-learning to produce improved variance forecasts. Hybrid ML–GARCH specifications are incorporated separately to enrich the comparative analysis. All estimators are trained with time-aware cross-validation to ensure temporal coherence and prevent look-ahead bias. Using Bitcoin data from 2014 to 2020, the empirical results show that the stacking ensemble consistently outperforms both standalone and hybrid alternatives in conditional variance forecasting and VaR accuracy, including during periods of severe market stress such as the COVID-19 episode. Residual diagnostics confirm that the ensemble effectively captures persistent temporal dependencies in volatility dynamics. Overall, the proposed methodology offers an innovative and interpretable risk-management tool for financial institutions, combining statistical rigor with the adaptability of machine-learning techniques in digital asset markets.

Multivariate time series modeling approaches are known as valuable methods for simulating and forecasting the temporal evolution of hydroclimatic variables. These approaches are also useful for modeling the temporal dependence and cross-dependence between variables and sites. Although multiple linear time series approaches, such as vector autoregressive (VAR) and multiple generalized autoregressive conditional heteroscedasticity (MGARCH) approaches are ordinarily applied in finance and econometrics, these methods have not been broadly applied in hydrology science. The present research employs the VAR and VAR–MGARCH methods to model the mean and conditional variance (heteroscedasticity) of daily streamflow data in the Zarrineh Rood dam watershed, in northwestern Iran. The bivariate diagonal vectorization heteroscedasticity (DVECH) model, as one of the key MGARCH models, demonstrates how the conditional variance, covariance, and correlation structures change in time between the residual time series from VAR model. In this regards, in the present study, five experiments which present different combinations of twofold streamflows (including both upstream and downstream stations) are conducted. The VAR approach is fitted to the twofold daily time series in each of the experiments with different orders. The Portmanteau test, as a formal test for demonstrating time-varying variance (or so-called ARCH effect), indicates the existence of conditional heteroscedastic behavior in the twofold residual time series obtained from the VAR models fitted to the twofold streamflows. Thus, the VAR–DVECH approach is suggested to capture the inherent heteroscedasticity in daily streamflow series. The bivariate DVECH approach indicates short-term and long-term persistency in the conditional variance–covariance structure of the twofold residuals of streamflows. Results show also that the use of the nonlinear bivariate DVECH model improves streamflow modeling efficiency by capturing the heteroscedasticity in the twofold residuals obtained from the VAR model for all experiments. The assessment criteria indicate also that the VAR–DVECH approach leads to a better performance than the VAR model.

This book shows how current and recent market prices convey information about the probability distributions that govern future prices. Moving beyond purely theoretical models, Stephen Taylor applies methods supported by empirical research of equity and foreign exchange markets to show how daily and more frequent asset prices, and the prices of option contracts, can be used to construct and assess predictions about future prices, their volatility, and their probability distributions. Stephen Taylor provides a comprehensive introduction to the dynamic behavior of asset prices, relying on finance theory and statistical evidence. He uses stochastic processes to define mathematical models for price dynamics, but with less mathematics than in alternative texts. The key topics covered include random walk tests, trading rules, ARCH models, stochastic volatility models, high-frequency datasets, and the information that option prices imply about volatility and distributions. Asset Price Dynamics, Volatility, and Predictionis ideal for students of economics, finance, and mathematics who are studying financial econometrics, and will enable researchers to identify and apply appropriate models and methods. It will likewise be a valuable resource for quantitative analysts, fund managers, risk managers, and investors who seek realistic expectations about future asset prices and the risks to which they are exposed.

The predominant approach for studying volatility is through various GARCH specifications, which are widely utilized in model-based analyses. This study focuses on assessing the predictive performance of specific GARCH models, particularly the Markov-Switching GARCH (MS-GARCH). The primary objective is to determine the optimal number of regimes within the MS-GARCH framework that effectively captures the conditional variance of the Muscat Securities Market Index (MSMI). To achieve this, we employ the Akaike Information Criterion (AIC) to compare different MS-GARCH models, estimated via Maximum Likelihood Estimation (MLE). Our findings indicate that the chosen models consistently exhibit at least two regimes across various GARCH specifications. Furthermore, a validation using the Value at Risk (VaR) confirms the accuracy of volatility forecasts generated by the selected models.

Since the electricity market liberalisation of the mid-1990s, forecasting energy demand and prices in competitive markets has become of primary importance for energy suppliers, market regulators and policy makers. In this paper, we propose a non-parametric model to obtain point and interval predictions of price and demand. It does not require any parametric assumption on the distribution of the error term or on the functional relationships linking the response variable to covariates. The assumed location–scale model provides a non-parametric estimation of the conditional mean and of the conditional variance by means of a Generalised Additive Model. Interval forecasts, at any given confidence level, are then obtained using a further non-parametric estimation of the innovation’s quantile. Since both the conditional mean and the conditional variance of the response variable are non-linear functions of covariates depending on calendar factors, renewable energy productions and other market variables, the resulting model is very flexible. It easily adapts to market conditions as well as to the non-linear characteristics of demand, supply and prices. An application to hourly data for the Italian electricity market, over the period 2015–2019 period, shows the one-day-ahead forecasting performance of the model for zonal electricity prices and level of demand.