We propose a Cramer-von Mises type test for diagnostic checking of weak vector autoregressive models, in which the errors are assumed to be uncorrelated but not necessarily independent. The test statistic is constructed from the integrated squared distance between the sample periodogram of the residuals and a constant spectrum. Unlike time domain portmanteau tests, which rely on the first residual autocorrelations, the proposed spectral test is sensitive to correlations at all lags. We study the asymptotic behavior of the test statistic and show that, due to the dependent structure of the errors and the effect of parameter estimation, the limiting distribution is not asymptotically pivotal. To address this issue, we employ a blockwise random weighting bootstrap to approximate critical values and establish its asymptotic validity. Finite sample performance is assessed through extensive Monte Carlo simulations and illustrated with a real data application.
We consider the problem of nonparametric testing for rough volatility, using high-frequency data with a fixed time span, in a setting where the price is purely discontinuous. More specifically, we analyze the asymptotic properties of a test we developed in previous work in a pure-jump setting. We show that the asymptotic behavior of the test remains unchanged, regardless of whether the underlying price contains a diffusion or is of purely discontinuous type. This adaptive nature of the test is very convenient from an empirical point of view as it implies that no pretest for the underlying features of the price process, that is, whether it contains a diffusion, is needed when testing for rough volatility. A Monte Carlo study shows good finite sample behavior of the test in purely discontinuous price settings.
This paper studies -penalized estimation for location models , where is defined by a possibly non-Markovian recursion and is a martingale difference sequence with possibly time-varying conditional variance. In such settings, standard LS/QML criteria are typically non-convex. A two-step plug-in scheme is considered: a first-step estimator (e.g., WLS or QMLE) is assumed to be strongly consistent, its fitted recursions are frozen, and a weighted least-squares criterion with an penalty is minimized in a second step. The resulting objective is convex and compatible with standard LASSO algorithms. Under mild regularity conditions, the second-step estimator is strongly consistent as soon as the penalties attached to the nonzero coordinates of the true parameter vanish. For penalties of order , its asymptotic distribution is derived, and adaptive penalties yield selection consistency and an oracle property. The second-step estimator is unconstrained and, when combined with an unconstrained first-step estimator, it yields a standard Gaussian limit with a tractable covariance matrix, in contrast with the non-standard limits that typically arise for QMLE when some components of the true parameter are zero. The general results are specialized to several time-series models and illustrated by Monte Carlo experiments and a real-data application to interest rates.
This note develops a rigorous analytical framework for computing exact MA() coefficients of mixed causal-noncausal autoregressive MAR processes. While analytical solutions exist only for the MAR specification in the existing literature (Gouri & eacute;roux and Jasiak, 2016), general MAR processes are typically handled through recursive approximation algorithms that suffer from numerical approximations. Using complex contour integration and the residue theorem, we derive explicit closed-form expressions valid for arbitrary orders . The derived expressions enable direct simulation algorithms, eliminating the recursive approximation bias while retaining only the standard truncation error inherent to any finite-order approximation, and facilitate implementation of forecasting methodologies. Numerical comparisons with existing recursive methods demonstrate improvements in accuracy, and an experiment with -stable innovations illustrate the empirical relevance of our results.
We develop a sparse causal dynamic regression framework for long multivariate time series. With very long time series, the potentially large number of lags and leads in a dynamic regression model often makes time-domain estimation numerically unstable or intractable. Frequency-domain estimation offers a stable and computationally efficient alternative, but it tends to generate noncausal, non-sparse solutions. Noncausality prevents real-time prediction by relying on future predictor values, and insufficient sparsity hinders interpretability and practical application. Our method applies thresholding operators to the frequency-domain estimates to obtain causal models that retain only a small, relevant set of variables and lags. The procedure is supported by theory showing that, under mild conditions, it achieves the optimal sparsity rate with only a small increase in mean squared prediction error. A frequency-domain cross-validation scheme, with an optional one-standard-error rule, selects tuning parameters and promotes parsimony. Simulation studies and a stock index return application demonstrate accurate lag identification, competitive predictive performance, and clear interpretability.
We develop a new test procedure for detecting changes in the distribution of the errors in (dynamic) linear models. Our framework accommodates misspecification of the dynamic linear model, thereby allowing for the inclusion of lagged dependent variables as regressors and autocorrelated errors. Under the null hypothesis, the distribution of the errors remains the same throughout the sample period, while there are multiple changes in the distribution of the errors under the alternative. Our procedure is based on the cumulative sum (CUSUM) process that compares the empirical distribution functions of the residuals in the first part of the observations and the whole sample. We derive the asymptotic properties of the proposed test statistics. Monte Carlo simulations show that the proposed test has good size control and high power. We provide empirical applications to Phillips curves and capital asset pricing models.
In this paper, we study the statistical inference methods for the periodic asymmetric power GARCH model (PAPGARCH), which can simultaneously capture the periodicity of volatility, the clustering effect, and the asymmetry of market responses. We proposed a data-driven weighted composite quantile regression (WCQR) estimator based on adaptive weights. This estimator improves efficiency by minimizing the asymptotic variance and its consistency and asymptotic normality are proved. To test for the asymmetry of conditional heteroscedasticity, a Wald-type test statistic is proposed. We derived the limiting distributions under the null hypothesis, alternative hypothesis, and local alternative hypothesis for the proposed Wald-type asymmetric test. Monte Carlo experiments showed that WCQR performed better than quasi-maximum likelihood estimation (QMLE) in small samples, and the test results of the proposed Wald-type asymmetric test were satisfactory. The modeling experiment of the USD/DZD exchange rate demonstrated the effectiveness of this framework.
Given a sample from a -dimensional stationary time series , the most commonly used estimator for the spectral density matrix at a given frequency is the Daniell smoothed periodogram which is an average over many periodograms at slightly perturbed frequencies. We prove that the Marchenko-Pastur law holds for the eigenvalues of uniformly in , when and grow with such that and for some . This demonstrates that high-dimensional effects can cause to become inconsistent, even when the dimension is much smaller than the sample size . Notably, we do not assume independence of the components of the time series. The Marchenko-Pastur law thus holds for Daniell smoothed periodograms, even when it does not necessarily hold for sample auto-covariance matrices of the same processes.
Both the U.S. stock and bond returns exhibit distinct Markovian regimes. However, because these regimes display limited coherence, conventional models typically require highly parameterized systems to adequately capture their joint distribution. This paper proposes a novel framework for the bivariate MS-VAR model that more effectively characterizes the regime dynamics of U.S. stock and bond returns. The specification leverages six latent Markov chains to govern the evolution of the model's parameters. Empirical evidence shows that this approach delivers more interpretable estimates of conditional moments and yields more informative decoding of the latent states than the standard MS-VAR model. The maximum likelihood estimator is implemented via an expectation-conditional maximization algorithm with closed-form conditional maximization steps. Moreover, the large-sample properties of the estimator are formally established.
This paper considers the trajectory of a time series with time-varying coefficients and proposes to test the adequacy of these parameters at a finite and fixed number of instants of the trajectory. For this purpose, a Wald test is constructed from point estimates of the parameters obtained by minimization of a kernel contrast. This can take the form of a localized near-maximum likelihood estimator for ARMA or GARCH processes, or a localized least squares estimator for a GLARCH process, but many other time-varying time series such as AR, ARCH, ARMA-GARCH, APARCH,& mldr;, could be considered. Above all, this allows the introduction of a new stationarity test for these processes, whose very good numerical performance has been demonstrated by numerical experiments.
We develop an online framework for forecast combination that integrates distributional robustness with randomized weights in dependent time-series settings. Unlike deterministic approaches that update a single weight vector, our approach models the weights as random draws from parametric families, with parameters updated sequentially to minimize a worst-case expected loss over a Wasserstein ambiguity set centered at the empirical joint distribution of forecasts and realizations. Randomization facilitates adaptive exploration of alternative weight configurations and improves calibration by accounting for combination uncertainty, while distributional robustness provides protection against heavy tails and model misspecification. We establish finite-sample concentration bounds for -mixing processes, derive oracle-type excess risk and online regret guarantees, and characterize the bias-variance trade-off induced by randomization. In addition, we show that a batch version of the estimator is asymptotically normal under standard identification and mixing conditions. Monte Carlo experiments across a variety of data-generating processes demonstrate that the proposed method achieves lower worst-case error, faster post-break adjustment, and improved predictive coverage relative to existing forecast combination methods. An empirical application to macroeconomic forecasting further illustrates its robustness and effectiveness under dependence and evolving dynamics.
We develop a novel nonparametric test for Granger causality in distribution based on entropic optimal transport. Unlike classical mean-based approaches, the proposed method directly compares the full conditional distributions of a response variable with and without the history of a candidate predictor. This allows for the detection of causal effects that arise from changes in dispersion, skewness, tail behavior, or multimodality. The test statistic is constructed as a temporal average of entropically regularized Wasserstein distances between kernel-smoothed conditional distributions. We establish uniform consistency of the estimators, derive the non-asymptotic exponential concentration bound, and prove a central limit theorem under weak temporal dependence. The test achieves the minimax optimal detection boundary over H & ouml;lder classes and remains robust to mild misspecification of the conditioning set. For valid inference in finite samples, we develop a stationary bootstrap procedure that preserves temporal dependence and consistently approximates the sampling distribution of the test statistic. Simulation studies demonstrate reliable size control and high power across a broad range of nonlinear and distributional data generating processes. An empirical application to financial volatility forecasting reveals dynamic causal patterns that are not detected by existing Granger causality tests.
The paper deals with the construction of a synthetic indicator of economic growth, obtained by projecting a quarterly measure of aggregate economic activity, namely gross domestic product (GDP), into the space spanned by a finite number of smooth principal components, representative of the medium-to-long-run component of economic growth of a high-dimensional time series, available at the monthly frequency. The smooth principal components result from applying a cross-sectional filter distilling the low-pass component of growth in real time. The outcome of the projection is a monthly nowcast of the medium-to-long-run component of GDP growth. After discussing the theoretical properties of the indicator, we deal with the assessment of its reliability and predictive validity with reference to a panel of macroeconomic US time series.
The present article introduces new methods for quantifying how intensively the distribution changes over time in a quite general locally stationary framework. Concretely, two characteristic function-based measures for quantifying the intensity of distribution changes are introduced and estimated. By combining these estimators with suitable dependent wild bootstrap procedures, confidence bands for both measures are estimated. The coverage ratios of the estimated confidence intervals are investigated for finite sample sizes by using simulation studies. As applications, the distribution change intensity of the log returns corresponding to several stocks in chosen time periods is investigated.
Forecasting presents a complex estimation challenge, as it involves balancing multiple, often conflicting, priorities and objectives. Conventional forecast optimization methods typically emphasize a single metric, such as minimizing the mean squared error (MSE), which may neglect other crucial aspects of predictive performance. To address this limitation, the recently developed Smooth Sign Accuracy (SSA) framework extends the traditional MSE approach by simultaneously accounting for sign accuracy, MSE, and the frequency of sign changes in the predictor. This addresses a fundamental trade-off, the so-called accuracy-smoothness (AS) dilemma, in prediction. We extend this approach to the multivariate M-SSA, leveraging the original criterion to incorporate cross-sectional information across multiple time series. As a result, the M-SSA criterion enables the integration of various design objectives related to AS forecasting performance, effectively generalizing conventional MSE-based metrics. To demonstrate its practical applicability and versatility, we explore the application of the M-SSA in three primary domains: Forecasting, real-time signal extraction (nowcasting), and smoothing. These case studies illustrate the framework's capacity to adapt to different contexts while effectively managing inherent trade-offs in predictive modeling.
We propose a new fiducial Markov Chain Monte Carlo (MCMC) method for fitting parametric Gaussian models. We utilize the Cayley transform to decompose the parametric covariance matrix, which in turn allows us to formulate a general data generating algorithm for Gaussian data. Leveraging constrained generalized fiducial inference, we are able to create the basis of an MCMC algorithm, which can be specified to parametric models with minimal effort. The appeal of this novel approach is the wide class of models which it permits, ease of implementation and the posterior-like fiducial distribution without the need for a prior. We provide background information for the derivation of the relevant fiducial quantities, and a proof that the proposed MCMC algorithm targets the correct fiducial distribution. We need not assume independence nor identical distribution of the data, which makes the method attractive for application to time series and spatial data. Well-performing simulation results of the MA(1) and Mat & eacute;rn models are presented.
We study the estimation of the intercept parameter in an integrated Galton-Watson process, an important building block for many count-valued time series models. In this unit root setting, the ordinary least squares estimator is known to be inconsistent, whereas the existing weighted least squares (WLS) estimator is consistent only in the case where the process is transient, a condition that depends on the unknown intercept parameter. We propose an alternative WLS estimator based on the new weight function of , and show that it is consistent regardless of whether the process is transient or null recurrent, with a common convergence rate of .
This article examines the filtering and approximation-theoretic properties of score-driven time series models. Under specific Lipschitz-type and tail conditions, new results are derived, leading to maximal and deviation inequalities for the filtering approximation error using empirical process theory. This approach allows the study of the asymptotic behavior of the empirical distribution function and empirical process of the approximated noise, extending the results of Francq and Zako & iuml;an (2022) for generalized autoregressive conditional heteroskedasticity models. For general score-driven models, however, it is proven that the asymptotic distribution of the empirical process of the approximated noise is model-dependent and influenced by the estimation of model parameters. This contrasts with well-known results for linear and some nonlinear time series models, and is mainly due to the fact that the finite-dimensional static model parameters affect both the noise density and the score-driven process. The goodness-of-fit problem is then considered, and an application of these results is demonstrated with the Beta--GARCH(1,1) model, a popular score-driven time series model.
