Econometrics

22 pages, 369 KB

Open AccessArticle

Nonlinear Trading-Performance Patterns Among Novice Participants in an Incentivized Trading Simulation

by Alain Finet, Kevin Kristoforidis and Julie Laznicka

Econometrics 2026, 14(2), 30; https://doi.org/10.3390/econometrics14020030 - 22 Jun 2026

Viewed by 196

This article analyses trading-performance patterns in a stock market simulation conducted with 134 second-year students at the University of Mons (Belgium) on 11 December 2025. Participants had a virtual capital of 100,000 euros and were free to trade CAC 40 securities without any [...] Read more.

This article analyses trading-performance patterns in a stock market simulation conducted with 134 second-year students at the University of Mons (Belgium) on 11 December 2025. Participants had a virtual capital of 100,000 euros and were free to trade CAC 40 securities without any restrictions on the number or volume of transactions. An academic incentive scheme, combining a participation bonus and bonuses for the three best portfolios, created a tournament-style environment with continuous ranking feedback. This feature is considered as part of the experimental context rather than as a separately identified causal mechanism. We estimate a quadratic model linking performance to activity, measured by the number of mean-centered transactions to reduce the collinearity between the first-degree term and its square, and control exposure via the average percentage of cash in the portfolio, portfolio variability (measured as the standard deviation of portfolio value) and the average trade size. Breusch–Pagan and White tests indicate heteroscedasticity, justifying a robust inference. The results highlight a convex relationship between activity and performance: the marginal association is initially negative but becomes positive above a model-implied upper-tail level corresponding to approximately 46 transactions. This value should not be interpreted as a behavioral level or as a trading rule. The percentage of cash in the portfolio and the average trade size are negatively associated with performance, while the portfolio variability does not show a statistically significant association with performance. Overall, the results indicate heterogeneous trading patterns rather than a single activity–performance profile. Full article

30 pages, 543 KB

Open AccessArticle

General Data Protection Regulation (GDPR) and Cross-Border M&A by Chinese E-Commerce Firms

by Aining Sun and IKM Mokhtarul Wadud

Econometrics 2026, 14(2), 29; https://doi.org/10.3390/econometrics14020029 - 22 Jun 2026

Viewed by 223

Abstract

The General Data Protection Regulation (GDPR), adopted by the European Union in 2018, aims to enhance consumer trust and market efficiency by strengthening data protection. The concurrent stringent compliance requirements raise operational costs and could reshape competition by favoring larger firms with greater [...] Read more.

The General Data Protection Regulation (GDPR), adopted by the European Union in 2018, aims to enhance consumer trust and market efficiency by strengthening data protection. The concurrent stringent compliance requirements raise operational costs and could reshape competition by favoring larger firms with greater regulatory capacity. While the GDPR reduces data-related risks and promotes global digital trade through its extraterritorial reach, the potential advantage to larger firms could incentivize strategic responses such as mergers and acquisitions (M&A) to consolidate market power. Given the rapid expansion of Chinese digital firms in e-commerce, social media, and cloud services across the EU, this study examines how the GDPR has affected their cross-border M&A activities between 2014 and 2021. Based on difference-in-difference analysis, the study finds that the GDPR did not have a statistically significant impact on the number or value of mergers and acquisitions by Chinese digital firms in the EU in the short term. This suggests that firms may enhance their institutional adaptability by strengthening their compliance capabilities. However, institutional and cultural differences pose long-term entry barriers for the firms. The study contributes by highlighting how firms adjust internationalization strategies under stringent regulatory regimes, offering policy-relevant insights for governments and regulatory authorities. Full article

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30 pages, 532 KB

Open AccessArticle

Threshold-Dependent Dominance in Tail Risk Approximation

by Terence D. Agbeyegbe

Econometrics 2026, 14(2), 28; https://doi.org/10.3390/econometrics14020028 - 17 Jun 2026

Viewed by 262

Abstract

Regulatory risk measurement under Basel III’s Fundamental Review of the Trading Book places Expected Shortfall (ES) at the center of market risk capital, yet the fourth-order Edgeworth expansion, still widely used for Value-at-Risk (VaR) and ES calculations, can produce negative densities in the [...] Read more.

Regulatory risk measurement under Basel III’s Fundamental Review of the Trading Book places Expected Shortfall (ES) at the center of market risk capital, yet the fourth-order Edgeworth expansion, still widely used for Value-at-Risk (VaR) and ES calculations, can produce negative densities in the tail regions where these measures concentrate, while saddlepoint approximations preserve positivity but face their own limits in heavy-tailed and sub-Gaussian settings. Whether either method delivers reliable tail estimates in the rare-disaster regimes documented in the empirical consumption-disaster literature therefore remains an open question. We address it by comparing the two approximations across 648 rare-disaster parameter combinations and five additional distributional families (Student-t, Hansen skewed-t, generalised error distribution (GED), two-sided jump mixture, and generalised hyperbolic), and by deriving a closed-form characterisation of the Edgeworth validity envelope. We establish three core findings. First, the validity envelope is bounded above by a sharp kurtosis ceiling at

γ_{2} = 4

and laterally by a non-monotone skewness boundary peaking at

| γ_{1}^{*, \max} | \approx 0.685

at

γ_{2} \approx 2.533

;

87.5 %

of the rare-disaster grid falls outside it. Second, accuracy is threshold-dependent: Edgeworth dominates at moderate quantiles, saddlepoint at extreme quantiles, with negative-density regions inflating Edgeworth ES error from

6.20 %

inside the envelope to

47.04 %

outside it. Third, these results reconcile only when point probability, density validity, and integrated-tail accuracy are treated as distinct accuracy criteria. The findings have direct implications for ES-based regulatory capital in heavy-tailed regimes and motivate a regime-conditional rather than universal approximation choice. Full article

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17 pages, 2719 KB

Open AccessArticle

A Natural Copula

by Peter B. Lerner

Econometrics 2026, 14(2), 27; https://doi.org/10.3390/econometrics14020027 - 16 Jun 2026

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Abstract

Copulas are widely used in financial economics, as well as in other areas of applied mathematics. Yet, there is much arbitrariness in their choice. The author proposes a “natural copula” concept that minimizes the Wasserstein distance between distributions in a space in which [...] Read more.

Copulas are widely used in financial economics, as well as in other areas of applied mathematics. Yet, there is much arbitrariness in their choice. The author proposes a “natural copula” concept that minimizes the Wasserstein distance between distributions in a space in which both distributions are embedded. Transport properties and hydrodynamic interpretation are discussed with two examples of distributions of financial significance. In 2D, a natural copula can be parsimoniously estimated by linear programming methods. A discussion of the construction of multivariate copulas follows. Finally, the quality of the multivariate copula approximation is investigated using the Kolmogorov–Arnold neural network (KAN). Full article

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22 pages, 3546 KB

Open AccessArticle

India’s Macroeconomic Response to Global Shocks: Evidence from Oil Prices, Financial Crisis and COVID-19

by Nikhil Bhardwaj, Ivana Miklošević and Nalinee Chauhan

Econometrics 2026, 14(2), 26; https://doi.org/10.3390/econometrics14020026 - 12 Jun 2026

Viewed by 317

Abstract

In past decades, the macroeconomic stability of India has been tested repeatedly by major global disruptions, including oil price shocks, the 2008 global financial crisis and the COVID-19 pandemic. Analysing how macroeconomic variables respond to these shocks is essential for evaluating external vulnerability [...] Read more.

In past decades, the macroeconomic stability of India has been tested repeatedly by major global disruptions, including oil price shocks, the 2008 global financial crisis and the COVID-19 pandemic. Analysing how macroeconomic variables respond to these shocks is essential for evaluating external vulnerability and policy resilience in emerging economies. Our study provides a comprehensive empirical investigation of the dynamic responses of wholesale price inflation, industrial output, oil prices and exchange rates in India by employing monthly data from January 1993 to December 2024. To examine long-run equilibrium relationships along with short-run adjustment dynamics, the present study employs co-integration analysis within a Vector Error Correction Model (VECM) framework. Further, we applied impulse response functions and forecast error variance decomposition to track volatility spillover mechanisms. Quantile regression and ARCH–GARCH models were further estimated to account for distributional heterogeneity and time-varying volatility. The findings of our study suggested stable long-run linkages among the selected variables, where oil price shocks emerged as a key external source of macroeconomic fluctuations. Short-run dynamics suggested that shocks in oil prices are transmitted primarily through inflation and exchange rate channels and then affect industrial output. Distributional estimates revealed the effects were stronger during stress periods, indicating tail risks that were not captured by the mean-based models. Lastly, volatility analysis confirmed persistent clustering, especially during phases of crisis. Overall, the findings suggest that India’s macroeconomic system remains externally sensitive, with adjustment mechanisms that operate gradually but come under strain during global disruptions. These results underscore the importance of energy risk management and crisis-responsive macroeconomic stabilisation policies. Full article

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27 pages, 5579 KB

Open AccessArticle

Modeling the Dynamic Relationship Between Stock Market Performance and Key Macroeconomic Indicators in Saudi Arabia: An ARDL-ECM Approach

by Mohamed Sharif Bashir and Sharif Mohd

Econometrics 2026, 14(2), 25; https://doi.org/10.3390/econometrics14020025 - 16 May 2026

Viewed by 958

Abstract

This study investigates the short-term and long-term impacts of gross domestic product (GDP), inflation, foreign capital flows, trade balance and interest rate on stock market performance in Saudi Arabia for the period 1990–2023. The autoregressive distributed lag (ARDL) approach and error correction model [...] Read more.

This study investigates the short-term and long-term impacts of gross domestic product (GDP), inflation, foreign capital flows, trade balance and interest rate on stock market performance in Saudi Arabia for the period 1990–2023. The autoregressive distributed lag (ARDL) approach and error correction model (ECM) are employed to empirically examine the short-run and long-run relationships. The ARDL-ECM technique is effective for analyzing cointegration and assessing adjustment processes. Additionally, impulse response function (IRF) analysis based on the vector autoregression (VAR) model, estimated using these macroeconomic indicators, is applied in this paper. This study provides novel insights and addresses emerging gaps in the literature concerning Saudi Arabia as a developing economy. The long-term relationship in the bounds test results confirms its existence. In the long run, inflation and interest rate exert a statistically significant negative effect on stock market performance, while the trade balance has a significant positive impact. GDP and foreign capital inflows do not exhibit statistically significant long-run effects. Short-run dynamics indicate persistence in stock market performance along with significant effects from inflation and interest rate changes, while GDP and foreign capital inflows remain statistically insignificant in the long-run scenario. Forecast error variance decomposition (FEVD) results show that approximately 68.5% of the variation in market performance is explained by its own shocks, followed by foreign capital flows (16.3%) and inflation (8.4%). While foreign capital flow does not exhibit statistical significance in the ARDL long-run estimates, its contribution in variance decomposition highlights its role as an important source of external shocks. These findings are relevant to various stakeholders, including investors and policymakers. Additionally, policy emphasis should be placed on controlling inflation and maintaining stable interest rates while improving trade balance conditions. Although foreign capital flow does not show a direct long-run effect, its role in influencing market variability suggests the need for a stable and well-regulated investment environment. Full article

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21 pages, 785 KB

Open AccessArticle

Measuring the Return to Online Advertising: Estimation and Inference of Endogenous Treatment Effects

by Shakeeb Khan, Denis Nekipelov and Justin Rao

Econometrics 2026, 14(2), 24; https://doi.org/10.3390/econometrics14020024 - 12 May 2026

Viewed by 373

Abstract

In this paper we aim to conduct inference on the “lift” effect generated by an online advertisement display: specifically we want to analyze if the presence of the brand ad among the advertisements on the page increases the overall number of consumer clicks [...] Read more.

In this paper we aim to conduct inference on the “lift” effect generated by an online advertisement display: specifically we want to analyze if the presence of the brand ad among the advertisements on the page increases the overall number of consumer clicks on that page. A distinctive feature of online advertising is that the ad displays are highly targeted—the advertising platform evaluates the (unconditional) probability of each consumer clicking on a given ad, which leads to a higher probability of displaying the ads that have a higher a priori estimated probability of click. As a result, inferring thecausal effect of the ad display on the page clicks by a given consumer from typical observational data is difficult. To address this we propose a multi-step estimator that focuses on the tails of the consumer distribution to estimate the true causal effect of an ad display. This “identification at infinity” approach alleviates the need for independent experimental randomization but results in nonstandard asymptotic theory, motivating our novel inference method. To validate our results, we use a set of large-scale randomized controlled experiments that Microsoft has run on its advertising platform. Our dataset has a large number of observations and a large number of variables and we employ LASSO to perform variable selection. Providing a basis for comparison with our estimates, we use a study conducted by Microsoft with approximately 9.3 million search sessions focusing on consumer click behavior across search result pages of a major search engine. Randomized experiments indicate that displaying a brand advertisement increases the probability of visiting the advertiser’s website by about 2.27 percentage points relative to a baseline visit rate of roughly 78 percent. Our non-experimental estimates exhibit broadly similar patterns to those obtained from randomized controlled trials, suggesting that the proposed observational estimator can recover qualitatively comparable treatment effects in large-scale advertising data. Full article

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22 pages, 417 KB

Open AccessArticle

Internationalization and Financing Decisions of Chinese Enterprises: Evidence from Hong Kong Listings

by Pujie Lin and Tsz Leung Yip

Econometrics 2026, 14(2), 23; https://doi.org/10.3390/econometrics14020023 - 7 May 2026

Viewed by 653

Abstract

This study explores the impact of internationalization on the financing decisions and finance costs of Chinese enterprises listed in Hong Kong, extending the pecking order theory to an international context. Utilizing data from 785 companies from 2010 to 2020, the research investigates how [...] Read more.

This study explores the impact of internationalization on the financing decisions and finance costs of Chinese enterprises listed in Hong Kong, extending the pecking order theory to an international context. Utilizing data from 785 companies from 2010 to 2020, the research investigates how the degree of internationalization influences corporate finance strategies, with a focus on the mediating role of the pecking order and the moderating effects of international business factors. The findings reveal that while broader internationalization increases finance costs, deeper internationalization reduces them. Legal distance is found to negatively moderate this relationship, whereas the structure of the financial system positively influences it. The results suggest that multinational enterprises with extensive overseas resource allocation demonstrate greater flexibility in financing decisions, particularly in foreign markets characterized by strong investor protection and efficient direct finance mechanisms. Managers should be cautious about pursuing wide geographic expansion without adequate operating depth because a broad but shallow international presence may increase financing frictions. By contrast, deeper resource commitment abroad can strengthen financing flexibility and improve access to lower-cost funds, especially when institutional conditions in the financing market are favorable. Full article

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20 pages, 511 KB

Open AccessArticle

Estimation of Two-States Proportional Hazard Rates Models with Unobserved Heterogeneity

by Emilio Congregado, David Troncoso-Ponce, Nicola Rubino and Alejandro Morales-Kirioukhina

Econometrics 2026, 14(2), 22; https://doi.org/10.3390/econometrics14020022 - 28 Apr 2026

Viewed by 453

Abstract

This article examines two-state proportional hazard rate models with unobserved heterogeneity specific to each state, a framework that is especially relevant for labor market transitions. To make estimation feasible in large longitudinal datasets, we implement hshaz2s, a Stata routine that uses analytical expressions [...] Read more.

This article examines two-state proportional hazard rate models with unobserved heterogeneity specific to each state, a framework that is especially relevant for labor market transitions. To make estimation feasible in large longitudinal datasets, we implement hshaz2s, a Stata routine that uses analytical expressions for the gradient vector and Hessian matrix of the log-likelihood function through the dual second-order moment (d2 ml) method. The empirical application estimates a discrete-time duration model for transitions between employment and unemployment using Spanish labor market microdata for young low-skilled workers over 2000–2019. The results show that apprenticeship contracts are associated with lower exit rates from employment than other temporary contracts, but not with faster transitions from unemployment back into employment. The estimates also reveal substantial state-specific unobserved heterogeneity, with a large latent group characterized by persistent spells in both states. Analytical second-order information also markedly reduces convergence time under richer heterogeneity structures. Overall, the article makes this class of two-state hazard models operational for applied research and provides new evidence on apprenticeship and temporary contracts in Spain. Full article

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24 pages, 405 KB

Open AccessArticle

Edgeworth Expansions When the Parameter Dimension Increases with Sample Size

by Christopher Stroude Withers

Econometrics 2026, 14(2), 21; https://doi.org/10.3390/econometrics14020021 - 27 Apr 2026

Viewed by 398

Abstract

Suppose that we have a statistical model with q unknown parameters w, and an estimate

\hat{w}

, based on a sample of size n. A basic question is: what is the covariance of the estimate? The covariance is needed for [...] Read more.

Suppose that we have a statistical model with q unknown parameters w, and an estimate

\hat{w}

, based on a sample of size n. A basic question is: what is the covariance of the estimate? The covariance is needed for the Central Limit Theorem (CLT). This gives a first approximation for the distribution of

\hat{w}

. But what if

q_{n} = n

increases with n? How fast can it increase and the CLT still hold? An answer has so far only been given for the sample mean. The same is true for the Edgeworth expansions. These are expansions in powers of

n^{- 1 / 2}

for the density and distribution of

\hat{w}

. For fixed q, these expansions are important, as they show how small n can be for the CLT to apply. When it does, they can greatly improve the accuracy of the CLT. I give conditions that allow for the Edgeworth expansions to remain valid when

q_{n} = q

increases with n. Earlier Edgeworth expansions when

q_{n} = q

increases, have only been done for a sample mean, and only for a 2nd order Edgeworth expansion. In contrast, I consider a very large class of estimates, the class of non-lattice standard estimates. An estimate is said to be a standard estimate if its mean converges to its true value as n increases, and for

r \geq 1

, its rth order cumulants have magnitude

n^{1 - r}

and can be expanded in powers of

n^{- 1}

. For this class of estimates, I show that the Edgeworth expansions hold if

q_{n}

grows as a power of n less than

1 / 6 .

That is, I give these expansions in powers of

n^{- 1 / 2} q_{n}^{3}

. This large class of estimates has a huge range of potential applications, as estimates of high dimension are common in nearly all areas of applied statistics. The most important type of standard estimate is when

\hat{w}

is a smooth function of a sample mean, of dimension p say. When either or both

q_{n} = q

and

p_{n} = p

increase with n, I give conditions on their growth for the Edgeworth expansions for

\hat{w}

to remain valid: the eighth power of p times the sixth power of q cannot grow as fast as n. This holds for fixed

q = q_{n}

if

p_{n}

grows less than a power of n less than

1 / 8

. This appears to be the first time when Edgeworth expansions have been given when not one, but two dimensions, are allowed to increase to ∞ with n. This gives two different pathways for allowing an increase in dimensionality. When

q = 1

, I give 5th order Edgeworth-Cornish-Fisher expansions for the standardized distribution and its quantiles of any smooth function of a sample mean of dimension

p_{n}

, when

p_{n}

is a power of n less than

1 / 2

. However for the special case when this function is linear, there is no restriction whatever on how fast

p_{n}

can increase! If also the components of the sample mean are independent, then these expansions are in powers of

{(n p)}^{- 1 / 2}

. I also give a method that greatly reduces the number of terms needed for the 2nd and 3rd order terms in the Edgeworth expansions, that is, for the 1st and 2nd order corrections to the CLTs. I also extend these results to the case where

\hat{w} \in R^{q}

is a function of several independent sample means, each of dimension increasing with n, with total dimension p. Full article

15 pages, 1258 KB

Open AccessArticle

Fuzzy Approach to Analysis of Investment Alternatives

by Tamara Kyrylych and Yuriy Povstenko

Econometrics 2026, 14(2), 20; https://doi.org/10.3390/econometrics14020020 - 13 Apr 2026

Viewed by 840

Abstract

With significant market unsureness, “static” methods fail to account for economic uncertainty, may be less precise and, accordingly, less helpful when selecting investment alternatives. Methods that take into account the current economic situation and allow for adapting the alternative selection to external uncertainty [...] Read more.

With significant market unsureness, “static” methods fail to account for economic uncertainty, may be less precise and, accordingly, less helpful when selecting investment alternatives. Methods that take into account the current economic situation and allow for adapting the alternative selection to external uncertainty are becoming more relevant. One of such methods is the fuzzy set theory. This article addresses the mathematical framework of such an approach for the economic analysis of investment project selection. A step-by-step scheme for implementing the fuzzy set method for investment projects is presented. Studies performed on the example of three investment alternatives give grounds for asserting the compatibility and feasibility of using two methods (the fuzzy set method may be partly based on the results of pairwise comparisons of experts according to the Saaty method) and confirmation or refutation of previous intuitive decisions of investors based on a comprehensive analysis of the criterion composition and the use of mathematical grounded technique. Full article

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16 pages, 1311 KB

Open AccessArticle

When Better Prediction Reduces Overlap: The Predictability Paradox in Propensity Score Matching with Machine Learning

by Foong Soon Cheong

Econometrics 2026, 14(2), 19; https://doi.org/10.3390/econometrics14020019 - 1 Apr 2026

Viewed by 990

Abstract

Evidence from observational studies plays a central role in shaping public policy in health, education, and financial regulation, where randomized experiments are rarely feasible. Propensity score matching (PSM) is a widely used method to approximate fair comparisons between treatment and control groups. Incorporating [...] Read more.

Evidence from observational studies plays a central role in shaping public policy in health, education, and financial regulation, where randomized experiments are rarely feasible. Propensity score matching (PSM) is a widely used method to approximate fair comparisons between treatment and control groups. Incorporating machine learning into the estimation of propensity scores can strengthen prediction and enhance the credibility of findings. However, stronger predictive models create a “predictability paradox”. As predictive accuracy improves, estimated propensity scores for treated and control units become more distinct when treatment assignment is strongly predictable from observed covariates, revealing limited overlap between groups. In the limit, near-perfect prediction produces near-complete separation between groups, rendering traditional matching infeasible and confining inference to a narrow subset of units near the boundary of the propensity score distribution, a setting analogous to a regression discontinuity design (RDD). Researchers thus face perverse incentives to use weaker models for statistically significant but spurious results. These dynamics jeopardize the reliability of evidence for policy. To safeguard decision-making, we propose a simple reform: require that studies using PSM disclose model error rates, including false positive and false negative rates, along with information on overlap and effective sample size. Full article

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16 pages, 2264 KB

Open AccessArticle

Propensity Score and the Double Robust Estimator in the Tails

by Marilena Furno

Econometrics 2026, 14(2), 18; https://doi.org/10.3390/econometrics14020018 - 31 Mar 2026

Viewed by 563

Abstract

This study analyzes the performance of the double robust estimator to compute the treatment effect, not only at the mean but also in the tails in a Monte Carlo experiment. While previous research focused on shifting the regression component of the double robust [...] Read more.

This study analyzes the performance of the double robust estimator to compute the treatment effect, not only at the mean but also in the tails in a Monte Carlo experiment. While previous research focused on shifting the regression component of the double robust estimator toward the tail, here we focus on the behavior of the propensity score away from the mean. Investigating the tails of the regression outcome allows for a closer look at the observations that are either highly or poorly responsive to treatment. Examining the tails of the propensity score distribution scrutinizes the observations with a higher or lower probability of being treated, which can be non-constant and even asymmetric. The goal is to assess the behavior of the double robust estimator when both components are computed away from the sample mean, in the tails of the treatment and control distributions. A case study on Italian education concludes the analysis. We find a positive double robust difference in higher education across regions, larger at the top location, due to the significant internal migration of qualified workers toward the northern regions. Women’s employment is higher for highly educated women, and gender has a significant impact: the analysis of the mismatch between probabilities and outcomes signals that women achieve higher education at rates exceeding their probabilities; they are more likely to exceed their predicted likelihood of attaining higher education. Full article

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26 pages, 4096 KB

Open AccessArticle

Nonparametric Autoregressive Copula Forecasting via Boundary-Reflected Kernel Estimation

by Guilherme Colombo Soares and Márcio Poletti Laurini

Econometrics 2026, 14(2), 17; https://doi.org/10.3390/econometrics14020017 - 28 Mar 2026

Viewed by 909

Abstract

We propose a fully nonparametric empirical autoregressive copula framework for univariate time series, designed to capture nonlinear and asymmetric serial dependence while exactly preserving the empirical marginal distribution. The method decouples marginal behavior from temporal dependence by (i) constructing a shape-preserving empirical marginal [...] Read more.

We propose a fully nonparametric empirical autoregressive copula framework for univariate time series, designed to capture nonlinear and asymmetric serial dependence while exactly preserving the empirical marginal distribution. The method decouples marginal behavior from temporal dependence by (i) constructing a shape-preserving empirical marginal via monotone interpolation and mapping observations to the unit interval, and (ii) estimating the lag–lead dependence through a nonparametric conditional AR(1) copula density on

{(0, 1)}^{2}

. To ensure stable estimation near the boundaries, we employ reflection-based kernel methods that mitigate edge effects and yield well-behaved conditional densities on the unit support. Forecasts are obtained from the implied conditional predictive density: we compute point forecasts either as conditional modes (maximum a posteriori) on the copula scale or as conditional means, and then back-transform exactly using the empirical quantile function, guaranteeing marginal fidelity and support-respecting predictions. Empirically, we evaluate the approach on three CBOE volatility indices (VIX, VXD, and RVX) and benchmark it against linear ARMA models, copula-based parametric competitors, and state-space/heteroskedasticity baselines (Local level, TVP–AR, and ARMA–GARCH). The results highlight that modeling the full conditional transition density nonparametrically can deliver competitive—often best or near-best—forecast accuracy across horizons, particularly in the presence of pronounced volatility regimes and asymmetric adjustments. Full article

(This article belongs to the Special Issue Advancements in Macroeconometric Modeling and Time Series Analysis)

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