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on Econometric Time Series |
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Issue of 2026–09–28
twelve papers chosen by Simon Sosvilla-Rivero, Instituto Complutense de Análisis Económico |
| By: | Huan Gong; Feiyu Jiang |
| Abstract: | Modern economic and financial data are increasingly organized as multiway arrays, with observations indexed simultaneously by geographic regions, industrial sectors, asset categories, and other economic characteristics. Representing such data as tensor-valued time series preserves their intrinsic multiway structure. Although substantial effort has been devoted to modeling the conditional mean of tensor-valued time series, comparatively less attention has been paid to their conditional covariance dynamics. The latter remains challenging because unrestricted multivariate covariance models involve many parameters and substantial computational cost. To address these challenges, we propose the Tensor-BEKK (T-BEKK) model, a tensor-structured BEKK specification that retains the positive definite covariance recursion for the vectorized process while imposing Kronecker structures on the intercept and the ARCH and GARCH coefficient matrices. The model reduces the parameter dimension and provides mode-specific interpretations of the covariance intercept, ARCH effects, and GARCH persistence. We establish stationarity, identification, and the asymptotic properties of the Gaussian quasi-maximum likelihood estimator. We further develop mode-specific restricted score tests tailored to the tensor structure, inference procedures for nonzero spillover intensities within each mode, and a portmanteau diagnostic test based on quadratic form residuals. For higher-dimensional settings, we also introduce the Tensor-Factor-BEKK (TF-BEKK) model. Under a first-step negligibility condition, its feasible second-step QMLE is asymptotically equivalent to the oracle QMLE based on the latent factors. Simulations and two empirical applications, covering currency futures and Chinese equity tensor portfolio allocation, illustrate the finite-sample behavior and empirical usefulness of the proposed methods. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.18157 |
| By: | Hong, Y.; Lin, Z.; Linton, O. B.; Newey, W. K.; Sun, J. |
| Abstract: | We propose affine-equivariant adjusted-range self-normalization for joint inference on time-series parameters. The method uses the projected ranges of a centered influence path to normalize estimation error, yielding pivotal limiting inference without estimating the long-run covariance matrix. The resulting statistic is invariant to nonsingular linear reparameterizations, and its inversion yields affine-equivariant confidence regions. In the univariate case, the proposed method reduces exactly to adjusted-range self-normalization. We derive scalar reference distributions and accommodate proportional variance accumulation through appropriate path centering. Simulations show power gains over quadratic self-normalization and quantify size distortions under persistent dependence. An application to U.S. fiscal multipliers illustrates joint inference across horizons and its sensitivity to concentrated identifying variation. |
| Keywords: | Self-Normalized Inference, Long-Run Covariance, Affine Equivariance, Influence Functions, Local Projections |
| JEL: | C12 C13 C22 C32 |
| Date: | 2026–09–08 |
| URL: | https://d.repec.org/n?u=RePEc:cam:camdae:2678 |
| By: | Blagov, Boris; Krause, Clara |
| Abstract: | This paper introduces a mixed-frequency Gaussian state-space framework that embeds forecast reconciliation into Bayesian VAR modeling for hierarchical macroeconomic data. Using precision-based sampling to generate high-frequency latent estimates, we construct a consistent proxy for the forecast-error covariance matrix, enabling optimal reconciliation with short datasets. We prove the symptotic convergence of this estimator and show that forecast reconciliation can be formally derived as a special case of conditional forecasting, allowing straightforward implementation with standard state-space algorithms. We derive reconciled impulse response functions that ensure bottom-level structural responses aggregate exactly to the top-level impulse response. Applying the framework to UK and German regional economic data, we demonstrate improvements in the forecast accuracy alongside structurally consistent impulse responses. |
| Abstract: | In diesem Beitrag wird ein Modell vorgestellt, das die Prognoseabstimmung in die bayessche VAR-Modellierung für hierarchische makroökonomische Daten mit fehlenden Daten integriert. Durch den Einsatz präzisionsbasierter Sampling-Verfahren zur Erzeugung hochfrequenter latenter Schätzwerte konstruieren wir einen konsistenten Proxy für die Kovarianzmatrix der Prognosefehler, was eine optimale Abstimmung auch bei kurzen Datensätzen ermöglicht. Wir beweisen die asymptotische Konvergenz dieses Schätzers und zeigen, dass sich die Prognoseabstimmung formal als Sonderfall der bedingten Prognose ableiten lässt, was eine unkomplizierte Implementierung mit Standard-Zustandsraumalgorithmen ermöglicht. Wir leiten abgestimmte Impulsantwortfunktionen ab, die gewährleisten, dass sich die strukturellen Antworten der untersten Ebene exakt zur Impulsantwort der obersten Ebene summieren. Durch die Anwendung des Modells auf regionale Wirtschaftsdaten aus Großbritannien und Deutschland zeigen wir Verbesserungen der Prognosegenauigkeit bei gleichzeitig strukturell konsistenten Impulsantworten. |
| Keywords: | forecast reconciliation, regional forecasts, mixed-frequency, hierarchical impulse responses |
| JEL: | C32 C53 R11 |
| Date: | 2026 |
| URL: | https://d.repec.org/n?u=RePEc:zbw:rwirep:343573 |
| By: | Matthew Read |
| Abstract: | Sign restrictions on the slopes of supply and demand curves are often used to identify historical decompositions in structural vector autoregressions. I show that the identifying power of these restrictions depends on both reduced-form parameters and realised forecast errors. Consequently, unlike many other structural objects, the strength of identification cannot be assessed from reduced-form parameters alone. Empirically, identified sets for historical decompositions of US inflation are typically largely uninformative, both in aggregate and in most expenditure categories. Existing inflation decompositions are therefore sensitive to auxiliary assumptions used to select among observationally equivalent models. |
| Keywords: | historical decomposition, set identification, sign restrictions, structural vector autoregression |
| JEL: | C32 E31 E32 |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:een:camaaa:2026-82 |
| By: | Guo, Honfei; Marín Díazaraque, Juan Miguel; Veiga, Helena |
| Abstract: | Rankings of volatility models often change with the market, the evaluation period, and the forecast criterion, so a gain found in one setting may not carry over to another. We test this directly. Eight volatility models, four observation-driven and four stochastic volatility, are estimated for five equity indices by data cloning and compared on one-step-ahead forecasts. We assess joint value-at-risk and expected shortfall accuracy, calibration, predictive density accuracy, and forecast availability. In 2018–2023, nominally significant gains cluster in DAX and NIKKEI, with weaker evidence in FTSE. Neither holdout comparison is corroborated: the DAX estimate changes sign, and the NIKKEI estimate keeps its direction but is imprecise. Different criteria also favour different models. The results argue for validating a model in its intended market and for the criterion it is meant to serve, and they show how a prespecified holdout in a later period can test whether an advantage persists. |
| Keywords: | Forecast evaluation; Holdout validation; Model comparison; Stochastic volatility; Tail risk forecasting; Volatility asymmetry |
| JEL: | C22 C52 C53 G17 |
| Date: | 2026–09–18 |
| URL: | https://d.repec.org/n?u=RePEc:cte:wsrepe:50798 |
| By: | Ayla Jungbluth; Johannes Lederer; Simon Trimborn |
| Abstract: | Modeling the joint distribution of extreme values in high-dimensional financial time series is challenging because extremes are sparse and locally extreme observations are not necessarily extreme relative to their full marginal distribution. To address this, we introduce a time-dependent network H\"usler-Reiss model in which market-informed adjacency matrices determine how strongly observations contribute to the estimation. We propose binary and weighted specifications, including the Joint Extremes Adjacency Matrix (JEAM) which combines information about individual extremeness with historical patterns of joint extreme movements. In the forecasting evaluation part, covering one-minute stock returns from three sectors of the S&P 100, JEAM achieves the best out-of-sample log scores for both tail directions; improving scores by 12.5-13.6% in the lower tail and 11.4-14.9% in the upper tail. The results show that incorporating market-informed network structures in the estimation, improves forecast evaluation of extremes across time series. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.11575 |
| By: | Paritosh Shankarrao Junare |
| Abstract: | Two frequent approaches for identifying structural VARs are external instruments, which carry economic content but are often weak, and non-Gaussianity of the shocks which provides statistical identification but carries no economic meaning. We combine the two strategies in a single generalized method of moments framework that stacks proxy exclusion restrictions with higher-order moment conditions of the structural shocks. This hybrid approach point-identifies the target shocks while also identifying the non-target shocks up to sign and ordering. Under suitable rank conditions, the higher-order moments anchor the identification uniformly over the instrument strength. Consequently, under local-to-zero proxy relevance, estimators of the dynamic causal effects remain consistent, and standard asymptotic inference remains valid. Moreover, the Anderson-Rubin confidence sets are substantially narrower than their instrument-only counterparts. The hybrid estimator is also more efficient than either source of identification used in isolation: at any fixed proxy relevance, even a weak instrument increases efficiency of the estimator through its covariance with the non-Gaussian moment block. Under local deviations from proxy exogeneity, we provide asymptotic bias bounds and show that stronger non-Gaussianity of the shocks compresses the bias. Finally, the over-identified structure yields two mutually orthogonal specification tests, for proxy exogeneity and validity of higher-order moment conditions. We derive their limiting distributions and provide a bootstrap procedure for finite-sample critical values. Monte Carlo simulations and two applications with identification of oil news-shock and a Euro-area MP shock demonstrate the potential of our framework. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.14398 |
| By: | Donia Besher; Rajdeep Pathak; Madhurima Panja; Tanujit Chakraborty |
| Abstract: | Probabilistic forecasting is central to decision-making under uncertainty, yet its methodological landscape has become increasingly fragmented across temporal and spatiotemporal forecasting, statistical modeling, machine learning, and deep generative modeling. This survey develops a unified perspective by organizing probabilistic forecasting methods according to where and how uncertainty is introduced into the forecasting pipeline. Our taxonomy connects model-agnostic approaches including ensembles and distribution-free calibration, with model-intrinsic approaches spanning Bayesian modeling, parametric predictive distributions, distributional regression, and modern generative models, and further examines the emerging role of time series foundation models. Beyond methodological synthesis, we identify the assumptions, computational demands, and forms of uncertainty represented by different paradigms, and translate these distinctions into data-driven and domain-specific guidance for method selection. We complement the survey with a cross-paradigm empirical study on univariate, multivariate, and spatiotemporal forecasting tasks. The results reveal that no single uncertainty-quantification paradigm dominates across settings. Calibration, sharpness, predictive accuracy, and computational efficiency can lead to substantially different model preferences, while expressive generative models and zero-shot foundation models exhibit markedly different accuracy-efficiency trade-offs. Lastly, we identify unresolved challenges surrounding uncertainty in evolving dependency structures, physics-informed predictive distributions, forecasting extreme events, handling count-valued, directional, and continuous-time series, and the development of unified software resources. Our survey provides both a conceptual framework and a practical roadmap for probabilistic forecasting research. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.13345 |
| By: | Xu, Yongdeng (Cardiff University, Cardiff, UK) |
| Abstract: | Realized measures are biased for the moments of returns. We propose a two-step approach to forecasting large correlation matrices: machine-learned forecasts of realized measures enter a dynamic conditional correlation model as drivers, and their weights are estimated by quasi-maximum likelihood on returns. The correlation recursion is run in a matrix-logarithm parametrization, so every forecast is a valid correlation matrix. For Dow Jones stocks, a learned volatility driver and this recursion improve on a dynamic conditional correlation model with realized drivers at horizons of one, five and 22 days. For S&P 500 stocks, the model fits monthly returns better than machine-learned projections of realized correlations, has no invalid forecast in any month, and trades less. The criterion that selects the weight on a realized driver matters as much as the model, and practitioners can obtain valid forecasts calibrated to returns with standard software. |
| Keywords: | DCC-GARCH-X; Realized measures; Covariance forecasting; Forecast evaluation; Positive definiteness; Log-correlation coordinates; Nonlinear shrinkage |
| JEL: | C32 C53 C58 G11 G17 |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:cdf:wpaper:2026/12 |
| By: | Frank, Luis |
| Abstract: | The seasonal adjustment of the series comprising the Monthly Estimator of Economic Activity (EMAE) requires an appropriate selection of the ARIMA models used in the pre-adjustment stage. Since the characteristics of the series may change over time, the specifications automatically selected by X-13ARIMA-SEATS should be reviewed periodically. This paper proposes a model selection procedure based on a restricted search over ARIMA specifications and compares it with the automatic procedure implemented in the seas() function in R. Model selection is initially based on the corrected Akaike information criterion (AICc) and subsequently on a hierarchy of diagnostics that includes residual seasonality (QS), the quality of the seasonal adjustment (Q(M)), the M_i statistics, residual normality, and model parsimony. The results show that the proposed procedure generally selects models with lower AICc values, whereas the automatic procedure tends to select more parsimonious specifications. However, differences in the quality of the seasonal adjustment are small in most cases. The results suggest that combining both procedures makes it possible to exploit their respective advantages and provides a suitable strategy for the systematic review of the models used in the seasonal adjustment of the EMAE activity branches. |
| Keywords: | seasonal adjustment; ARIMA models; model selection; X-13ARIMA-SEATS; EMAE; time series |
| JEL: | C22 |
| Date: | 2026–08–11 |
| URL: | https://d.repec.org/n?u=RePEc:pra:mprapa:131073 |
| By: | Aashish Bohra; Vivek Vijay |
| Abstract: | Stock price prediction remains challenging due to the non-stationary and noisy nature of financial time series. Existing deep learning models often rely on rigid decision-level fusion, ad hoc hyperparameter tuning, and compressed final-layer outputs, causing information loss, overfitting, and limited cross-market generalization. We propose VertiFuseX, a hybrid LSTM architecture using penultimate-layer vertical fusion of multi-scale temporal representations. VertiFuseX stacks and reweights penultimate features from LSTM, Bi-LSTM, and St-LSTM branches, integrates a parallel DNN stream, and jointly optimizes all components via backpropagation under a fixed hyperparameter configuration. This preserves richer intermediate temporal information across scales. Evaluated on 15 years (2010-2024) of closing prices from 10 global equity indices using strict chronological out-of-sample testing with the final 365 trading days held out, VertiFuseX achieves 30-54% MAPE reductions and over 40% improvements in MAE and RMSE versus LSTM-based baselines, and outperforms seven state-of-the-art models across 33 metric-dataset comparisons. Ablation studies confirm penultimate-layer fusion drives these gains over final-layer fusion and decision-level ensembling. Gradient-based saliency analysis shows consistent emphasis on mid-range dependencies at lags 9-15 days. Economic validation via algorithmic trading simulation under extreme market regimes shows reduced maximum drawdowns and superior risk-adjusted returns. With 675k parameters, a 2.6 MB memory footprint, and 1.5 ms/sample inference latency, VertiFuseX offers a lightweight, interpretable, deployment-ready framework for robust financial forecasting. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.12793 |
| By: | Jaehyung Choi |
| Abstract: | We develop Entropic Value-at-Risk (EVaR) parity for tempered stable returns. EVaR-based inverse risk parity (IRP) and equal risk contribution (ERC) portfolios are constructed using multivariate normal tempered stable models and independent component analysis with tempered stable components. We derive the corresponding asset-level EVaR and EVaR-deviation contributions and use the latter to separate the fitted location term from EVaR risk contributions. Under Gaussian returns, EVaR-deviation IRP and ERC recover conventional volatility IRP and ERC weights. We evaluate the resulting portfolios in three investment universes. Empirically, EVaR-based ERC portfolios achieve positive Sharpe differences relative to equal weight across the universes. |
| Date: | 2026–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2609.11905 |