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on Econometric Time Series |
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Issue of 2026–11–02
fifteen papers chosen by Simon Sosvilla-Rivero, Instituto Complutense de Análisis Económico |
| By: | Muyi Li; Yajia Chen; Yuqing Xu; Feiyu Jiang |
| Abstract: | We propose a semiparametric real-time GARCH (SRT-GARCH) model that combines a smooth deterministic long-run volatility component with a stationary short-run process driven by contemporaneous return innovations. The model accommodates gradual changes in unconditional variance while allowing short-run volatility to respond directly to current information rather than relying solely on lagged observations. We derive the variance normalization, conditional moments, and closed-form conditional density of the model. For estimation, we develop a two-step procedure that combines kernel smoothing with conditional Gaussian maximum likelihood. Under Gaussian innovations and mild regularity conditions, we establish consistency and asymptotic normality of the short-run parameter estimator. The analysis exploits geometric moment contraction to control serial dependence and shows that estimating the long-run component adds a positive semidefinite rank-one adjustment to the oracle covariance matrix obtained when the long-run component is known. The resulting asymptotic covariance matrix admits a plug-in estimator that requires neither lag truncation nor additional tuning parameters. Monte Carlo simulations show good finite-sample performance of the estimator and the associated inference procedure. Applications to FTSE~100 returns and U.S. natural gas futures show that allowing for a time-varying long-run component improves forecasting relative to the parametric RT-GARCH benchmark, while the additional real-time feedback mechanism provides more pronounced gains in the natural gas application than in the FTSE application. The proposed model also delivers competitive one-step-ahead Value-at-Risk forecasts. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.36739 |
| By: | Peter Fuleky (University of Hawai‘i at MÄ noa) |
| Abstract: | This chapter surveys a century of time series econometrics, from its statistical foundations to modern, data-rich practice. It traces the field’s major turning points: univariate ARIMA modeling, the rise of vector autoregressions and the identification debates that followed, and the treatment of nonstationarity through unit roots, cointegration, and error correction. Later sections take up volatility and nonlinear modeling, state-space signal extraction, and Bayesian estimation. The chapter then follows the shift to high-dimensional forecasting and real-time nowcasting, and the spread of machine learning into time series workflows. It compares the major approaches on forecasting performance, structural interpretability, and the measurement of uncertainty. The methods that endured answered a concrete empirical need, whether for better out-of-sample forecasts, credible policy analysis, or an honest measure of risk. The chapter closes with lessons from the field’s evolution and an agenda for future research as data and computation expand. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:hae:wpaper:2026-2 |
| By: | MD Nazmul Ahsan; Jean-Marie Dufour |
| Abstract: | We propose computationally inexpensive and efficient estimators for multivariate stochastic volatility (MSV) mod- els with cross-dependence, Granger causality, and higher-order persistence in latent volatilities. The proposed class of estimators is based on a small set of moment equations derived from the weak VARMA representations of MSV models. Except for the cross-dependence parameters, closed-form expressions are obtained for the remaining pa- rameters, so that no numerical optimization is required. To improve the stability and efficiency of the volatility- persistence parameter estimates, we develop shrinkage-type VARMA estimators based on averaging different simple estimates (winsorization) and matrix-variate regressions (MVR). We also derive the asymptotic distribution of these estimators. Due to their computational simplicity, VARMA estimators make reliable — possibly exact — simulation- based inference tractable through Monte Carlo test techniques. We formulate volatility commonality as a testable common-C-space restriction on the MSV transition matrices, and develop finite-sample Monte Carlo tests for this hypothesis. Simulation results show that the proposed shrinkage-type VARMA-MVR estimators compare favorably with Bayesian and QML benchmarks in terms of bias, RMSE, and computational cost. They also remain feasible in very high dimensions, and, using auxiliary particle filtering, deliver accurate latent-state recovery and covariance forecasts even under model misspecification. In empirical applications to stock returns, we find that unrestricted higher-order MSV models perform well for covariance forecasting and portfolio building, while the data provide little support for an exact low-dimensional common dynamic subspace. Nous proposons des estimateurs efficaces et peu coûteux en ressources de calcul pour les modèles de volatilité stochastique multivariée (MSV) intégrant des dépendances croisées, une causalité au sens de Granger et une persistance d’ordre supérieur des volatilités latentes. La classe d’estimateurs proposée repose sur un nombre restreint d’équations de moments dérivées des représentations VARMA faibles des modèles MSV. À l’exception des paramètres de dépendance croisée, des expressions analytiques explicites sont obtenues pour tous les autres paramètres, ce qui élimine la nécessité de recourir à une optimisation numérique. Afin d’améliorer la stabilité et l’efficacité des estimations des paramètres de persistance de la volatilité, nous développons des estimateurs VARMA de type shrinkage (rétrécissement), fondés sur la combinaison de différentes estimations simples (winsorisation) et sur des régressions matricielles (matrix-variate regressions, MVR). Nous établissons également la distribution asymptotique de ces estimateurs. Grâce à leur simplicité de calcul, les estimateurs VARMA permettent de réaliser des inférences statistiques fiables, voire exactes, fondées sur la simulation au moyen de techniques de tests de Monte-Carlo. Nous formulons l’existence d’une composante commune aux volatilités sous la forme d’une restriction d’espace C commun (common-C-space) sur les matrices de transition des modèles MSV, et élaborons des tests de Monte-Carlo en échantillon fini pour vérifier cette hypothèse. Les résultats des simulations montrent que les estimateurs VARMA-MVR de type shrinkage proposés se comparent favorablement aux méthodes de référence bayésiennes et à celles fondées sur le quasi-maximum de vraisemblance (QML), tant sur le plan du biais et de la racine de l’erreur quadratique moyenne (RMSE) que du coût de calcul. Ils demeurent également applicables à des modèles de très grande dimension et permettent, grâce au filtrage particulaire auxiliaire, de reconstituer avec précision les états latents et de produire des prévisions fiables des covariances, même lorsque le modèle est mal spécifié. Dans les applications empiriques aux rendements boursiers, nous constatons que les modèles MSV non restreints d’ordre supérieur offrent de bonnes performances pour la prévision des covariances et la construction de portefeuilles, tandis que les données fournissent peu d’éléments à l’appui de l’existence d’un sous-espace dynamique commun exact de faible dimension. |
| Keywords: | Simple methods, matrix-variate regression, Markov Chain Monte Carlo, Monte Carlo tests, multivariate stochastic volatility, asymptotic distribution, Méthodes simples, régression matricielle, méthodes de Monte-Carlo par chaînes de Markov, tests de Monte-Carlo, volatilité stochastique multivariée, distribution asymptotique |
| JEL: | C15 C32 C53 C55 C58 G12 |
| Date: | 2026–10–21 |
| URL: | https://d.repec.org/n?u=RePEc:cir:cirwor:2026s-20 |
| By: | Marco Brianti; Mario Forni; Luca Gambetti; Antonio Granese |
| Abstract: | The factors of a stationary Dynamic Factor Model (DFM) are the first differences of singular I(1) processes. Being singular, such processes are always cointegrated. This however does not prevent the existence of a VAR for their first differences, so that the standard estimation procedure of the stationary DFM remains valid. Problems arise only when there is a linear combination of the shocks that has transitory effects on all the variables of the panel. We argue that this case is implausible for macroeconomic data and show that in fact it does not occur for two widely used US data sets, the FRED-QD and the FRED-MD. A Monte Carlo exercise shows that the standard procedure outperforms alternative methods based on the factors in levels, especially at long horizons. |
| JEL: | C32 E32 |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:bol:bodewp:wp1234 |
| By: | Joel M. David; Raffaella Giacomini; Xiyu Jiao; Weining Wang |
| Abstract: | State-dependent local projections (LPs) are widely used to study how causal effects vary as a function of economic states, but shock exogeneity alone does not identify this response function. We show that identification follows when the underlying conditional mean is linear in the shock with a state-dependent coefficient, a condition satisfied in canonical micro-macro environments, including first-order perturbation solutions of heterogeneous-agent and macro-finance models. Even then, standard linear-interaction LPs generally recover only a projection of the response function, motivating LPs with nonparametric state dependence. We develop a sieve estimator and establish pointwise and uniform inference for micro-macro panels, where a distinctive challenge is that the estimator can converge at different rates across the state space. Applied to firm investment, the method uncovers a hump-shaped response to monetary policy shocks and shows that standard linear-interaction LPs substantially understate the aggregate role of financial heterogeneity. |
| Keywords: | Impulse response functions; Identification; uniform inference; heterogeneous agent |
| JEL: | C12 C14 C23 E32 |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:fip:fedhwp:103968 |
| By: | Ioannis Kasparis |
| Keywords: | Domains of attraction, fractional process; heavy-tailed process, kernel regression, IVX, mildly integrated process, nearly integrated process, stable distribution, weakly nonstationary process |
| JEL: | C22 C12 C13 C26 C14 |
| Date: | 2026–10–21 |
| URL: | https://d.repec.org/n?u=RePEc:ucy:cypeua:03-2026 |
| By: | Joshua C. C. Chan; Wei Zhang |
| Abstract: | Bayesian comparison of nonlinear state space models requires integrating over a growing latent state path and a low-dimensional parameter vector. We propose a marginal likelihood approximation that integrates out the states with a numerical or simulation-based evaluator and applies a second-order corrected Laplace approximation to the integral over the static parameters. The correction is expressed through smoothing cumulants of the complete-data log density and, under derivative-growth and posterior-concentration conditions, reduces the log marginal likelihood error from $O(T^{-1})$ to $O(T^{-2})$. In stochastic volatility experiments, it removes 64\% to 99\% of the Laplace error at far lower cost than simulation-based estimators. Applications to bivariate exchange-rate volatility models and to a sample of $T = 19{, }266$ five-minute stock returns find the strongest support for heavy-tailed observations and for a transitory volatility factor. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.34629 |
| By: | Schmitz, Luca |
| Abstract: | This paper develops a High-Dimensional Distributional Vector Autoregression (HiDVAR), a real-time, data-rich framework for forecasting the joint predictive distribution of GDP growth, inflation, and unemployment without imposing a parametric density shape. The motivation is policy risk monitoring: standard quantile-to-density pipelines can compress one-sided inflation risks in turbulent periods, especially when they rely on restrictive distributional modeling or forecast anchors. HiDVAR combines distribution regression (many binary threshold models), local projections, elastic-net shrinkage, and factor-based information extraction from an augmented real-time FRED-QD panel, and then constructs coherent multivariate, path-consistent scenarios using a triangular factorization and copula-based simulation. In real-time out-of-sample evaluation (February 2013-June 2025), HiDVAR improves probabilistic forecast accuracy relative to the FRBNY Outlook-at-Risk framework and standard benchmarks, with the largest gains concentrated in inflation tail-risk assessment. Relative to the FRBNY model, HiDVAR reduces CPI inflation CRPS by 28% and the multivariate Energy Score by 22%, with a weakly significant Energy Score improvement in Diebold-Mariano tests. During the post-pandemic inflation surge, HiDVAR assigns substantially more probability mass to persistent high-inflation outcomes and produces earlier, better-calibrated upper-tail risk signals. A policy application maps the predictive distributions into risk-scenario probabilities and rule-implied policy-rate distributions, illustrating the value of flexible density forecasting for macro risk monitoring. |
| Keywords: | Inflation, Forecasting and Simulation: Models and Applications, Central Banks and their policies, Nonparametric methods, Time-Series Models, Forecasting, Large Data Sets |
| JEL: | E31 E37 E58 C14 C32 C53 C55 |
| Date: | 2026 |
| URL: | https://d.repec.org/n?u=RePEc:zbw:imfswp:344237 |
| By: | Alessandro Saccal |
| Abstract: | The present work provides the first structural decomposition of business cycle fluctuations in Uzbekistan using a Real Business Cycle (RBC) Structural Vector Auto-Regression (SVAR). Employing a recursive identification scheme on a core vector of macroeconomic variables one is to recover the inaugural set of Orthogonalised Impulse Response Functions (OIRFs) and Forecast Error Variance Decompositions (FEVDs) for the Uzbek economy. The findings furnish a critical empirical benchmark for Uzbekistan, scrutinise the theoretical relevance of the RBC framework for transition economies and lay the essential empirical foundation for the subsequent development and parametrisation of a structural Dynamic Stochastic General Equilibrium (DSGE) model. The analysis thereby bridges a significant gap in the scientific literature, moving the study of Central Asian business cycles from empirical causality to structural explanation. |
| Keywords: | capital, consumption, labour, RBC, TFP, VARs, Uzbekistan. |
| JEL: | C22 C60 E13 E32 E37 |
| Date: | 2026–01–10 |
| URL: | https://d.repec.org/n?u=RePEc:eei:rpaper:eeri_rp_2026_10 |
| By: | Sascha A. Keweloh; Shu Wang |
| Abstract: | Proxy SVARs often rely on a single credible proxy, leaving proxy exogeneity untestable. Narrative arguments used to justify proxy validity imply a stronger form of exogeneity. We formalize the strong exogeneity assumption and propose a corresponding test based on internal synthetic proxies. The approach uses the external proxy to locate the target-shock direction in the reduced-form innovations and synthesize the candidate shock into an additional internal synthetic proxies, which allows to test the joint exogeneity of the original and internal synthetic proxy. We derive the large-sample theory for the ISP test and show that the identification strength is governed by a direct measure of the target shock's non-Gaussianity. We study weak identification under asymptotic Gaussianity and develop an identification-robust version of the test. Monte Carlo experiments demonstrate the favorable power of the test when the proxy has a low signal-to-noise ratio. Applications to monetary policy and oil supply news shocks find no evidence against the exogeneity of the benchmark proxies. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.36093 |
| By: | Linton, O. B.; Michalopoulos, C. |
| Abstract: | We develop bootstrap-based inference for the threshold parameter in a threshold quantile regression (TQR) model. Replacing the least-squares loss with the quantile loss, changes the bootstrap empirical process significantly since it removes the maximal inequality’s dependence on the magnitude of the regression error, and creates a bootstrap nuisance-scale problem with no least-squares counterpart. Building on the grid-bootstrap confidence interval of Hansen (1999) and the least-squares validity theory of Hidalgo, Lee and Seo (2019), we prove that the correctly rescaled bootstrap statistic converges to the correct limiting distribution under the null. The key theoretical ingredient is a conditional maximal inequality for the quantile bootstrap score process. Since the bootstrap error enters the quantile score only through its sign, the score is, conditional on the data, an i.i.d., hence a martingale argument delivers the required tightness directly, without importing the magnitude of the regression error into the bootstrap calculation. This reduces the required moment condition from the eighth-order requirement of Hidalgo et al. (2019), with no additional moment constraints contributed by the bootstrap step itself. We further show that the bootstrap has its own nuisance scale factor, generally distinct from the one governing the original-sample likelihood-ratio statistic of Kuan et al. (2017) and derive a closed-form studentizer for it directly from the known bootstrap weight density, avoiding a separate nonparametric density estimation step inside the bootstrap calculation. An extensive Monte Carlo study com-pares four nonparametric estimators of the original-sample nuisance parameter, against the asymptotic approximation across two data-generating processes, three error distributions, three sample sizes, three threshold magnitudes, and five quantile levels. The results show that under weak threshold identification, both procedures are oversized and stay that way even at larger samples. Under adequate identification the asymptotic interval is preferred at small samples, but it is conservative and does not improve with more data. The grid bootstrap’s size falls smoothly toward nominal with increasing sample size, especially at central and tail quantiles. Our implementation pairs the quantile wild bootstrap of Feng, He and Hu (2011) with a smoothed, floored conditional-scale construction that keeps both the bootstrap error density and its closed-form studentizer well behaved. An application to real data concludes. |
| Keywords: | Threshold Regression, Quantile Regression, Grid Bootstrap, Change-Point Estimation, Bootstrap Inference, Nonparametric Density Estimation |
| JEL: | C12 C14 C15 C22 |
| Date: | 2026–10–09 |
| URL: | https://d.repec.org/n?u=RePEc:cam:camdae:2684 |
| By: | Dante Amengual (Centro de Estudios Monetarios y Financieros); Lluc Puig-Codina (Banco de España and University of Alicante) |
| Abstract: | The Cox-Ingersoll-Ross model (CIR) is a diffusion process with widespread applications in economics, finance and population biology. In this paper we first obtain a full Gibbs Sampler for the exact CIR process observed at equidistant times. We do not require the imposition of the Feller condition. We show how to carry out smoothing and handling of missing data. Additionally, the introduction of a tailored single Metropolis-Hastings step allows us to handle irregular and random time intervals. Secondly, we obtain several asymptotic properties which include Local Asymptotic Normality, consistency and asymptotic normality of conditional maximum likelihood estimation and a Bernstein-von Mises theorem. Lastly, we provide an empirical illustration using U.S. Treasury yields under a variety of priors and a comparison with other models such as the Vasicek and the 3/2 models. |
| Keywords: | Cox-Ingersoll-Ross process, square-root diffusion, auto-regressive gamma process, latent variables, data augmentation, Gibbs sampling, Bernstein-von Mises theorem |
| JEL: | C11 C15 C22 C58 G12 |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:bde:wpaper:2640e |
| By: | Dimitri Sotnikov; Adeline Viot |
| Abstract: | We establish a martingality criterion for the Exponentially Fading Memory (EFM) signature volatility model, extending the recent result of Abi Jaber, Gassiat, and Sotnikov (2026). In the proposed framework, the volatility process is given by a linear functional of the EFM signature of a time-augmented Brownian motion, which yields a globally well-behaved, fully non-Markovian volatility model admitting a stationary distribution. The stationary EFM signature volatility model is also covered by allowing for a random initial value of the signature. Moreover, the volatility process can be adjusted to the initial forward variance curve. For this model, excluding trivial cases, we prove that the price process is a true martingale if and only if the truncation order is odd and a sign condition involving the correlation parameter holds. |
| Date: | 2026–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2610.35022 |
| By: | Sara Boughanou (UM5 - Université Mohammed V de Rabat [Agdal]); Adil El Marhoum (UM5 - Université Mohammed V de Rabat [Agdal]) |
| Abstract: | Abstract Realized-volatility forecasting supports risk management and allocation. GARCH and HAR-RV mostly exploit numerical memory, whereas LSTM-GARCH and FinBERT-LSTM hybrids are often monolingual and rarely receive out-of-sample economic evaluation. We propose a bilingual FinBERT-CamemBERT framework that constructs the daily textual stress index Sₜ and the event vector Eₜ, which enter HAR-X for the S&P 500 and the MASI. The corpus contains 12.3 million French and English items over 2015-2024. Expanding-window validation reserves 2023-2024 for final testing. Against HAR-RV, HAR-X reduces the S&P 500 RMSE by 21.1%; the Diebold-Mariano test yields DM = -2.44, p = .015, and qBH = .033. For the MASI, DM = -2.62 and qBH = .033. Benjamini-Hochberg correction covers six comparisons. The net-of-cost Sharpe ratio rises from .64 to .81, but this gain remains descriptive without a direct Ledoit-Wolf test. Twenty-three of twenty-four monthly differentials are positive, with p < .001 under an exact sign test. The framework connects multilingual textual information, heterogeneous memory, and allocation. SHAP assigns a major predictive role to Sₜ and Eₜ without establishing a causal relationship. Limitations concern annotation reliability, foreign exchange, and concept drift. |
| Abstract: | Résumé Prévoir la volatilité réalisée soutient la gestion du risque et l'allocation. GARCH et HAR-RV exploitent la mémoire numérique ; les hybrides LSTM-GARCH et FinBERT-LSTM sont souvent monolingues et rarement évalués hors échantillon. Nous proposons un cadre bilingue FinBERT-CamemBERT qui construit l'indice de stress Sₜ et le vecteur événementiel Eₜ, intégrés à HAR-X pour le S&P 500 et le MASI. Le corpus rassemble 12, 3 millions d'items sur 2015-2024. Une validation par fenêtre croissante réserve 2023-2024 au test final. Face à HAR-RV, HAR-X réduit le RMSE du S&P 500 de 21, 1 % ; le test de Diebold-Mariano donne DM = -2, 44, p = 0, 015 et qBH = 0, 033. Pour le MASI, DM = -2, 62 et qBH = 0, 033. Les six comparaisons sont corrigées par Benjamini-Hochberg. Le ratio de Sharpe net de frais passe de 0, 64 à 0, 81, mais ce gain reste descriptif faute de test direct de Ledoit-Wolf. Vingt-trois écarts mensuels sur vingt-quatre sont positifs, avec p < 0, 001 au test exact des signes. Le cadre relie information textuelle multilingue, mémoire hétérogène et allocation. SHAP attribue un rôle prédictif à Sₜ et Eₜ sans établir de relation causale. Les limites concernent l'annotation, le change et la dérive conceptuelle. |
| Keywords: | portfolio optimization, MASI, HAR-X, sentiment analysis, Transformer, Financial volatility, Volatilité financière, analyse de sentiment, optimisation de portefeuille |
| Date: | 2026–08–20 |
| URL: | https://d.repec.org/n?u=RePEc:hal:journl:hal-05723648 |
| By: | Marie-Claude Beaulieu; Jean-Marie Dufour; Lynda Khalaf; Olena Melin |
| Abstract: | This paper presents finite-sample tests and confidence procedures for a multivariate linear regression system with cross-dependent possibly non-Gaussian disturbances, allowing for potentially missing explanatory variables. We consider a wide class of multivariate statis- tics based on the roots of a determinantal equation involving restricted and unrestricted least squares residuals. We characterize the exact distribution of these roots and identify important pivotal cases. General conditions for asymptotic validity are also given. To deal with missing explanatory variables, we introduce an identifiable ex-post parameter which embeds the impact of left-out variables, and show that usual tests remain valid from a conservative perspective. We derive simultaneous confidence bands for a vector (as well as linear combinations) of regression coefficients. The results are applied to asset pricing, with focus on risk-return analysis. We revisit the distinction between the alpha (risk-adjusted outperformance) and beta (risk component) of returns, from our ex-post parametrization perspective. Empirically, we analyze the risk-return profile of Catastrophe bond mutual funds. Cet article présente des tests et des procédures de construction d’intervalles de confiance en échantillon fini pour un système de régressions linéaires multivariées dont les termes d’erreur peuvent présenter des dépendances croisées et ne pas suivre une distribution gaussienne, tout en permettant l’omission éventuelle de variables explicatives. Nous considérons une vaste classe de statistiques multivariées fondées sur les racines d’une équation déterminantielle faisant intervenir les résidus des moindres carrés des modèles restreints et non restreints. Nous caractérisons la distribution exacte de ces racines et identifions d’importants cas où les statistiques sont pivots. Nous établissons également des conditions générales de validité asymptotique. Pour traiter le problème des variables explicatives omises, nous introduisons un paramètre ex post identifiable qui intègre l’effet de ces variables et montrons que les tests usuels demeurent valides dans une perspective conservatrice. Nous établissons des bandes de confiance simultanées pour un vecteur de coefficients de régression, ainsi que pour leurs combinaisons linéaires. Les résultats sont appliqués à l’évaluation des actifs financiers, en mettant l’accent sur l’analyse du risque et du rendement. Nous réexaminons la distinction entre l’alpha (surperformance ajustée au risque) et le bêta (composante du risque) des rendements, à la lumière de notre paramétrisation ex post. Sur le plan empirique, nous analysons le profil risque-rendement des fonds communs de placement investissant dans des obligations catastrophes. |
| Keywords: | Multivariate regression, uniform mixed linear hypothesis, exact test, incomplete regression, risk-return decomposition, simultaneous inference, risk-return profiling, Régression multivariée, hypothèse linéaire mixte uniforme, test exact, régression incomplète, décomposition risque-rendement, inférence simultanée, profil risque-rendement |
| JEL: | C12 C30 C58 |
| Date: | 2026–10–28 |
| URL: | https://d.repec.org/n?u=RePEc:cir:cirwor:2026s-22 |