Many financial time series not only have varying structures at different quantile levels and exhibit the phenomenon of conditional heteroscedasticity at the same time but also arrive in the stream. Quantile double‐autoregression is very useful for time series analysis but faces challenges with model fitting of streaming data sets when estimating other quantiles in subsequent batches. This article proposes a renewable estimation method for quantile double‐autoregression analysis of streaming time series data due to its ability to break with storage barrier and computational barrier. Moreover, the proposed flexible parametric structure of the quantile function enables us to predict any interested quantile value without quantile curve crossing problem or keeping the desirable monotone property of the conditional quantile function. The proposed methods are illustrated using current data and the summary statistics of historical data. Theoretically, the proposed statistic is shown to have the same asymptotic distribution as the standard version computed on an entire data stream with the data batches pooled into one data set, without additional condition. Simulation studies and an empirical example are presented to illustrate the finite sample performance of the proposed methods.

In the context of functional time series, we propose a significance test to distinguish between short memory with a change point and long range dependence. The test is based on coefficients of projections onto an optimal direction that captures the dependence structure of the latent stationary functions that are not observable due to a potential change point. The optimal direction must be estimated as well. The test statistic is constructed using the local Whittle estimator applied to these coefficients. It has standard normal distribution under the null hypothesis (change point) and diverges to infinity under the alternative (long range dependence). The article includes asymptotic theory, a simulation study and an application to curve‐valued time series derived from intraday asset prices.

We propose a methodology for detecting multiple change points in the mean of an otherwise stationary, autocorrelated, linear time series. It combines solution path generation based on the wild contrast maximisation principle, and an information criterion‐based model selection strategy termed gappy Schwarz algorithm. The former is well‐suited to separating shifts in the mean from fluctuations due to serial correlations, while the latter simultaneously estimates the dependence structure and the number of change points without performing the difficult task of estimating the level of the noise as quantified e.g. by the long‐run variance. We provide modular investigation into their theoretical properties and show that the combined methodology, named WCM.gSa, achieves consistency in estimating both the total number and the locations of the change points. The good performance of WCM.gSa is demonstrated via extensive simulation studies, and we further illustrate its usefulness by applying the methodology to London air quality data.

The article introduces a new subclass of nonlinear moving average model, called the smooth transition moving average (STMA) model, and studies its probabilistic properties. It is shown that, under some mild conditions, the least squares estimation (LSE) is strongly consistent and asymptotically normal. A powerful score‐based goodness‐of‐fit test for the STMA model is presented. A different parametrization from the classical one is applied to numerically improve the identification and estimation of this model. Simulation studies are conducted to assess the performance of the LSE and the score‐based test in finite samples. The results are illustrated with an application to the weekly exchange rate of the USA Dollar to the British Pound.

In the general setting of long‐memory multivariate time series, the long‐memory characteristics are defined by two components. The long‐memory parameters describe the autocorrelation of each time series. And the long‐run covariance measures the coupling between time series, with general phase parameters. It is of interest to estimate the long‐memory, long‐run covariance and general phase parameters of time series generated by this wide class of models although they are not necessarily Gaussian nor stationary. This estimation is thus not directly possible using real wavelets decomposition or Fourier analysis. Our purpose is to define an inference approach based on a representation using quasi‐analytic wavelets. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure including the phase term. Consistent estimators based on a local Whittle approximation are then proposed. Simulations highlight a satisfactory behavior of the estimation on finite samples on multivariate fractional Brownian motions. An application on a real neuroscience dataset is presented, where long‐memory and brain connectivity are inferred.

Variance estimation is an important aspect in statistical inference, especially in the dependent data situations. Resampling methods are ideal for solving this problem since these do not require restrictive distributional assumptions. In this paper, we develop a novel resampling method in the Jackknife family called the stationary jackknife. It can be used to estimate the variance of a statistic in the cases where observations are from a general stationary sequence. Unlike the moving block jackknife, the stationary jackknife computes the jackknife replication by deleting a variable length block and the length has a truncated geometric distribution. Under appropriate assumptions, we can show the stationary jackknife variance estimator is a consistent estimator for the case of the sample mean and, more generally, for a class of nonlinear statistics. Further, the stationary jackknife is shown to provide reasonable variance estimation for a wider range of expected block lengths when compared with the moving block jackknife by simulation.

In this article, we develop additive autoregressive models (Add‐ARM) for the time series data with matrix valued predictors. The proposed models assume separable row, column and lag effects of the matrix variables, attaining stronger interpretability when compared with existing bilinear matrix autoregressive models. We utilize the Gershgorin's circle theorem to impose some certain conditions on the parameter matrices, which make the underlying process strictly stationary. We also introduce the alternating least squares estimation method to solve the involved equality constrained optimization problems. Asymptotic distributions of the parameter estimators are derived. In addition, we employ hypothesis tests to run diagnostics on the parameter matrices. The performance of the proposed models and methods is further demonstrated through simulations and real data analysis.

The autocovariance and cross‐covariance functions naturally appear in many time series procedures (e.g. autoregression or prediction). Under assumptions, empirical versions of the autocovariance and cross‐covariance are asymptotically normal with covariance structure depending on the second‐ and fourth‐order spectra. Under non‐restrictive assumptions, we derive a bound for the Wasserstein distance of the finite‐sample distribution of the estimator of the autocovariance and cross‐covariance to the Gaussian limit. An error of approximation to the second‐order moments of the estimator and an ‐dependent approximation are the key ingredients to obtain the bound. As a worked example, we discuss how to compute the bound for causal autoregressive processes of order 1 with different distributions for the innovations. To assess our result, we compare our bound to Wasserstein distances obtained via simulation.