arXiv subjects

Peter Reinhard Hansen

Publications and source records attributed to Peter Reinhard Hansen.

At least 19 recordsLinked to original sources

arXiv2026

Bartlett Couplings of the Onion and Vine LKJ Samplers

The extended-onion and C-vine constructions of Lewandowski, Kurowicka and Joe (2009) are standard methods for sampling from the LKJ_n(eta) distribution on correlation matrices. We show that both arise from the simpler row-normalized Bartlett construction associated with the restricted-Wishart representation of Wang, Wu and Chu (2018), which exposes redundancies hidden in standard gamma-based implementations of the classical constructions. Two exact row-wise couplings establish this: the squared norm of the Gaussian vector supplying the onion's direction has exactly the Gamma law required for one component of the Beta radius, and the same vector, with one chi-squared variate, generates the entire row of mutually independent C-vine partial correlations with their required symmetric-Beta laws. We also show that, relative to flat off-diagonal measure, the LKJ family maximizes entropy at fixed expected log-determinant; the dual natural parameter is eta-1. Under gamma-ratio accounting, normalized Bartlett, the onion, and the conventional symmetric-Beta C-vine require n-1, 2(n-1), and n(n-1) Gamma-equivalent calls. Controlled benchmarks confirm a low-dimensional advantage over the onion implementation and a persistent advantage over the C-vine implementations examined; direct Bartlett normalization also avoids subtractive complements, moving the small-eta zero-diagonal threshold from machine-epsilon scale toward the subnormal range. The sampler is valid for every real eta > 0 and requires only standard normal and chi-squared variates.

math.ST↗

arXiv2026

Correlation Matrices in High Dimensions: The Elliptope as a Sample-Correlation Ensemble

The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular: its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. More generally, distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(η)$ law. For the uniform law, this yields a Chen-Stein proof of the extreme-correlation point-process limit and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. We also identify two distinct scales: $η_n\asymp n$ alters the limiting spectrum, whereas $η_n\asymp n^2$ is needed to keep the Frobenius distance bounded. Finally, for a bounded, centered i.i.d. off-diagonal specification, projection to the nearest correlation matrix incurs a squared repair cost asymptotically at least one-half of the squared Frobenius norm of its off-diagonal part.

math.PR↗

arXiv2026

A variational approach to the generalized Fisher transformation of correlation matrices

The generalized Fisher transformation maps a non-singular correlation matrix to an unconstrained real vector through the off-diagonal elements of its matrix logarithm. We show that the inverse transformation is the unique minimizer of a smooth, strictly convex and coercive function of the diagonal elements of the matrix logarithm, which yields a new and short proof that the transformation is one-to-one and onto. The Hessian of this function is bounded, at every point, between the extreme eigenvalues of the associated matrix exponential and is dominated by the diagonal of that exponential, a bound that is attained. These bounds identify the standard fixed-point iteration for the inverse as a quasi-Newton method whose worst-case local factor can approach one for near-singular correlation matrices, and guarantee that the Newton system is never worse conditioned than the matrix exponential. We propose GFT-FP+N, which augments the fixed-point iteration with matrix-free Newton steps computed by conjugate gradients, with no explicit Jacobian. In benchmark experiments GFT-FP+N converged in all tested instances, including instances in which Broyden's method failed, and reduced computation time by up to a factor of thirty relative to the fixed-point iteration.

stat.CO↗

arXiv2026

The Quarter-Hour Effect: Periodic Algorithmic Trading and Return Predictability in Cryptocurrency Futures

Cryptocurrency markets exhibit periodic bursts in volatility and volume at one-minute, five-minute, and quarter-hour marks. Using trade data for six Binance perpetual contracts, we link these bursts to algorithmic participation: trade-size roundness declines sharply during them. The Autocorrelation Map, a clock-phase-resolved display, reveals serial dependence in order flow and returns at quarter-hour openings that conventional measures obscure. Opening returns are predictable out of sample, while opening order imbalance predicts returns over four to twelve hours, with much weaker effects at finer clock-time frequencies. Together, these findings characterize periodic algorithmic trading and its cross-frequency variation.

q-fin.TR↗

arXiv2026

Split-Session Cluster GARCH for Overnight and Intraday Returns: The Role of Tail Heterogeneity

We propose the Split-Session Cluster GARCH model for heavy-tailed multivariate dependence among asset returns decomposed into overnight and intraday components. The model uses convolution-$t$ distributions to allow tail behavior to differ across clusters defined by trading sessions and, within each session, by economic sectors. It also accommodates block-structured conditional correlation matrices, preserving parsimony and scalability in high-dimensional settings. The resulting likelihood remains tractable and yields a score-driven specification for dynamic correlations. We apply the model to U.S. equity returns in six-asset and 100-asset applications. The results reveal pronounced tail heterogeneity between overnight and intraday returns. Model comparisons show that session-specific tail parameters substantially improve fit relative to a common multivariate-$t$ specification, while sector-level tail partitioning delivers additional gains concentrated mainly in the overnight component. In the 100-asset application, asset-level tail heterogeneity delivers the strongest out-of-sample likelihood and global minimum-variance (GMV) portfolio performance.

econ.EM↗

arXiv2026

The Generalized Fisher Transformation: Finite-Sample Properties and Inference

We study the finite-sample behavior of the Generalized Fisher Transformation (GFT), the parametrization of a correlation matrix $C$ by $γ(C)=\operatorname{vecl}\log C$. The GFT coordinates extend Fisher's transformation to dimension $n>2$: for elliptical data their finite-sample distributions are close to Gaussian. More strikingly, the coordinates are nearly uncorrelated and their covariance is largely invariant to $C$. This approximate orthogonality and invariance make GFT-based inference far better behaved in finite samples than inference based on sample correlations or element-wise Fisher transformed correlations, yielding estimation errors that are approximately Gaussian, weakly dependent, and nearly pivotal.

econ.EM↗

arXiv2026

Moments by Integrating the Moment-Generating Function

We introduce a general integral framework for computing fractional, complex, absolute, and logarithmic moments from the moment-generating function (MGF) under explicit regularity conditions. By evaluating a complex extension of the MGF along a vertical contour, we obtain exact integral expressions that bypass the need for explicit probability densities and high-order derivatives. We establish conditions for negative fractional moments using the symmetric Cauchy principal value, including the requirement that the distribution have no point mass at the centering point. We demonstrate the theoretical scope and computational practicality of the framework through applications to the normal-inverse Gaussian distribution and a semicontinuous compound Poisson-Gamma distribution. In the latter case, the framework handles point masses at the boundary by evaluating conditional fractional moments.

econ.EM↗

arXiv2026

The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

We introduce the Parabolic Mellin Transform (PMT), defined by ${P}_σ[f](z)=\int_{-\infty}^{\infty}w^{2z}f(w^2)dt$, where $w=σ+it$ and $σ>0$. Under the substitution $u=w^2$, the vertical line $\operatorname{Re}(w)=σ$ is mapped to the parabolic contour $C_σ$ in the $u$-plane. For the Gaussian kernel, the PMT yields $\int_{-\infty}^{\infty}w^{2z}e^{w^2}dt=π/Γ(\tfrac{1}{2}-z)=\cos(πz)Γ(z+\tfrac{1}{2})$, a parabolic-contour form of the classical Hankel representation for the reciprocal Gamma function. The advantage of this parametrization is that the contour integral becomes a Gaussian-damped vertical-line integral. We develop scaling, differentiation, and Dirichlet-composition identities for the PMT and use them to derive integral representations of the Hurwitz zeta, Riemann zeta, and Dirichlet eta functions. The framework provides a unified transform dictionary for Gamma-type and zeta-type special functions and yields equivalent reformulations of the Riemann hypothesis and the Lindelöf hypothesis in terms of zeros and growth of parabolic-contour integrals.

math.NT↗

arXiv2026

Exact Likelihood Inference and Robust Filtering for Gauss-Cauchy Convolution Models

The convolution of a Gaussian and a Cauchy distribution, known as the Voigt distribution, is widely used in spectroscopy and provides a natural framework for modeling heavy-tailed measurement noise. We derive analytical expressions for its density, score, Hessian, Fisher information, and conditional moments using the scaled complementary error function, enabling stable maximum likelihood estimation without numerical convolution, finite-difference derivatives, or pseudo-Voigt approximations. The conditional expectation of the latent Gaussian component is governed by a redescending location score, so extreme observations are automatically discounted rather than propagated. This structure leads to the Gauss-Cauchy Convolution (GCC) filter for state-space models with Gaussian latent dynamics and Voigt measurement errors, where the Masreliez Gaussian prediction approximation preserves a Voigt prediction-error density. In an application to log realized volatility for the Technology Select Sector SPDR Fund, the GCC filter separates persistent latent variation from transient measurement noise and attains the highest implemented prediction-error criterion among the Gaussian, Student-$t$, Huber, and related filtering specifications considered.

econ.EM↗

arXiv2026

Tweedie's Formula and Score-Driven Updating

Score-driven models update time-varying parameters using conditional likelihood scores. This paper develops a Bayesian interpretation of such updates through Tweedie's formula, which connects posterior mean corrections with marginal scores. In Gaussian signal extraction, this gives an exact posterior-correction identity. For natural exponential families, related identities characterize posterior means in natural- and expectation-parameter spaces. Building on these identities, we show that conjugate Bayesian filtering in expectation space coincides exactly with an inverse-Fisher-scaled conditional score update under local precision discounting. For general conditional densities, the exact Bayesian correction involves a generally unavailable predictive-marginal score. A local Gaussian approximation shows that the conditional likelihood score provides the leading approximation to this posterior correction; under local precision discounting, the predictive covariance becomes proportional to inverse Fisher information, yielding the familiar inverse-Fisher-scaled score recursion. The results clarify when score-driven updates are exact Bayesian filters and when they should instead be viewed as tractable local approximations.

econ.EM↗

arXiv2026

Principled Identification of Structural Dynamic Models

We take a new perspective on identification in structural dynamic models: rather than imposing restrictions alone, we optimize an objective. While definitive structural identification ultimately requires exogenous economic insight, a weighted correlation-maximizing objective yields an Order- and Scale-Invariant Scheme (OASIS) that selects the orthogonal rotation most aligned with designated target variables. In traditional SVARs, these targets are the reduced-form innovations, making OASIS a natural reference rotation. We show that recursive Cholesky identification is a constrained version of the same objective and that OASIS is systematically closer to perfect correlation, closing roughly twice as much of the gap as recursive orderings, both theoretically and empirically. The same framework also provides a principled estimation strategy for Proxy VARs (IV-SVARs), where the weighted criterion is essential for resolving overdetermination in multi-proxy systems while symmetrically accommodating proxy leakage. Revisiting 22 published SVARs, we find that reduced-form innovations are typically only weakly correlated, helping explain the historical robustness of recursive schemes. Applying OASIS to seminal proxy applications, however, reveals economically important leakage across shocks and shows that accounting for such leakage can materially alter substantive conclusions.

econ.EM↗

arXiv2026

A Unifying Integral Representation of the Gamma Function and Its Reciprocal

We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/Γ(z)=G(z)/π$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma function and satisfies $G(1-z)=Γ(z)\sin(πz)$ for all $z\in\mathbb{C}$.

math.CV↗

arXiv2025

Option Pricing with Time-Varying Volatility Risk Aversion

We introduce a pricing kernel with time-varying volatility risk aversion to explain observed time variations in the shape of the pricing kernel. When combined with the Heston-Nandi GARCH model, this framework yields a tractable option pricing model in which the variance risk ratio (VRR) emerges as a key variable. We show that the VRR is closely linked to economic fundamentals, as well as sentiment and uncertainty measures. A novel approximation method provides analytical option pricing formulas, and we demonstrate substantial reductions in pricing errors through an empirical application to the S&P 500 index, the CBOE VIX, and option prices.

q-fin.PR↗

arXiv2025

Dynamic Factor Correlation Model

We introduce a new dynamic factor correlation model with a novel variation-free parametrization of factor loadings. The model is applicable to high dimensions and can accommodate time-varying correlations, heterogeneous heavy-tailed distributions, and dependent idiosyncratic shocks, such as those observed in returns on stocks in the same subindustry. We apply the model to a "small universe" with 12 asset returns and to a "large universe" with 323 asset returns. The former facilitates a comprehensive empirical analysis and comparisons and the latter demonstrates the flexibility and scalability of the model.

econ.EM↗

arXiv2025

A Multivariate Realized GARCH Model

We propose a novel class of multivariate GARCH models that incorporate realized measures of volatility and correlations. The key innovation is an unconstrained vector parametrization of the conditional correlation matrix, which enables the use of factor models for correlations. This approach elegantly addresses the main challenge faced by multivariate GARCH models in high-dimensional settings. As an illustration, we explore block correlation matrices that naturally simplify to linear factor models for the conditional correlations. The model is applied to the returns of nine assets, and its in-sample and out-of-sample performance compares favorably against several popular benchmarks.

econ.EM↗

arXiv2024

Cluster GARCH

We introduce a novel multivariate GARCH model with flexible convolution-t distributions that is applicable in high-dimensional systems. The model is called Cluster GARCH because it can accommodate cluster structures in the conditional correlation matrix and in the tail dependencies. The expressions for the log-likelihood function and its derivatives are tractable, and the latter facilitate a score-drive model for the dynamic correlation structure. We apply the Cluster GARCH model to daily returns for 100 assets and find it outperforms existing models, both in-sample and out-of-sample. Moreover, the convolution-t distribution provides a better empirical performance than the conventional multivariate t-distribution.

econ.EM↗

arXiv2024

Convolution-t Distributions

We introduce a new class of multivariate heavy-tailed distributions that are convolutions of heterogeneous multivariate t-distributions. Unlike commonly used heavy-tailed distributions, the multivariate convolution-t distributions embody cluster structures with flexible nonlinear dependencies and heterogeneous marginal distributions. Importantly, convolution-t distributions have simple density functions that facilitate estimation and likelihood-based inference. The characteristic features of convolution-t distributions are found to be important in an empirical analysis of realized volatility measures and help identify their underlying factor structure.

econ.EM↗

arXiv2023

Robust Estimation of Realized Correlation: New Insight about Intraday Fluctuations in Market Betas

Time-varying volatility is an inherent feature of most economic time-series, which causes standard correlation estimators to be inconsistent. The quadrant correlation estimator is consistent but very inefficient. We propose a novel subsampled quadrant estimator that improves efficiency while preserving consistency and robustness. This estimator is particularly well-suited for high-frequency financial data and we apply it to a large panel of US stocks. Our empirical analysis sheds new light on intra-day fluctuations in market betas by decomposing them into time-varying correlations and relative volatility changes. Our results show that intraday variation in betas is primarily driven by intraday variation in correlations.

econ.EM↗