Consistent ranking of multivariate volatility models

A large number of parameterizations have been proposed to model conditional variance dynamics in a multivariate framework. This paper examines the ranking of multivariate volatility models in terms of their ability to forecast out-of-sample conditional variance matrices. We investigate how sensitive the ranking is to alternative statistical loss functions which evaluate the distance between the true covariance matrix and its forecast. The evaluation of multivariate volatility models requires the use of a proxy for the unobservable volatility matrix which may shift the ranking of the models. Therefore, to preserve this ranking conditions with respect to the choice of the loss function have to be discussed. To do this, we extend the conditions defined in Hansen and Lunde (2006) to the multivariate framework. By invoking norm equivalence we are able to extend the class of loss functions that preserve the true ranking. In a simulation study, we sample data from a continuous time multivariate diffusion process to illustrate the sensitivity of the ranking to different choices of the loss functions and to the quality of the proxy. An application to three foreign exchange rates, where we compare the forecasting performance of 16 multivariate GARCH specifications, is provided.volatility, multivariate GARCH, matrix norm and loss function, norm equivalence

On the Forecasting Accuracy of Multivariate GARCH Models

This paper addresses the question of the selection of multivariate GARCH models in terms of variance matrix forecasting accuracy with a particular focus on relatively large scale problems. We consider 10 assets from NYSE and NASDAQ and compare 125 model based one-step-ahead conditional variance forecasts over a period of 10 years using the model confidence set (MCS) and the Superior Predictive Ability (SPA) tests. Model performances are evaluated using four statistical loss functions which account for different types and degrees of asymmetry with respect to over/under predictions. When considering the full sample, MCS results are strongly driven by short periods of high market instability during which multivariate GARCH models appear to be inaccurate. Over relatively unstable periods, i.e. dot-com bubble, the set of superior models is composed of more sophisticated specifications such as orthogonal and dynamic conditional correlation (DCC), both with leverage effect in the conditional variances. However, unlike the DCC models, our results show that the orthogonal specifications tend to underestimate the conditional variance. Over calm periods, a simple assumption like constant conditional correlation and symmetry in the conditional variances cannot be rejected. Finally, during the 2007-2008 financial crisis, accounting for non-stationarity in the conditional variance process generates superior forecasts. The SPA test suggests that, independently from the period, the best models do not provide significantly better forecasts than the DCC model of Engle (2002) with leverage in the conditional variances of the returns.Variance matrix, forecasting, multivariate GARCH, loss function, model confidence set, superior predictive ability

On Loss Functions and Ranking Forecasting Performances of Multivariate Volatility Models

A large number of parameterizations have been proposed to model conditional variance dynamics in a multivariate framework. However, little is known about the ranking of multivariate volatility models in terms of their forecasting ability. The ranking of multivariate volatility models is inherently problematic because it requires the use of a proxy for the unobservable volatility matrix and this substitution may severely affect the ranking. We address this issue by investigating the properties of the ranking with respect to alternative statistical loss functions used to evaluate model performances. We provide conditions on the functional form of the loss function that ensure the proxy-based ranking to be consistent for the true one – i.e., the ranking that would be obtained if the true variance matrix was observable. We identify a large set of loss functions that yield a consistent ranking. In a simulation study, we sample data from a continuous time multivariate diffusion process and compare the ordering delivered by both consistent and inconsistent loss functions. We further discuss the sensitivity of the ranking to the quality of the proxy and the degree of similarity between models. An application to three foreign exchange rates, where we compare the forecasting performance of 16 multivariate GARCH specifications, is provided.Volatility, multivariate GARCH, Matrix norm, Loss function, Model confidence set

Consistent ranking of multivariate volatility models

A large number of parameterizations have been proposed to model conditional variance dynamics in a multivariate framework. This paper examines the ranking of multivariate volatility models in terms of their ability to forecast out-of-sample conditional variance matrices. We investigate how sensitive the ranking is to alternative statistical loss functions which evaluate the distance between the true covariance matrix and its forecast. The evaluation of multivariate volatility models requires the use of a proxy for the unobservable volatility matrix which may shift the ranking of the models. Therefore, to preserve this ranking conditions with respect to the choice of the loss function have to be discussed. To do this, we extend the conditions defined in Hansen and Lunde (2006) to the multivariate framework. By invoking norm equivalence we are able to extend the class of loss functions that preserve the true ranking. In a simulation study, we sample data from a continuous time multivariate diffusion process to illustrate the sensitivity of the ranking to different choices of the loss functions and to the quality of the proxy. An application to three foreign exchange rates, where we compare the forecasting performance of 16 multivariate GARC

Dynamic conditional correlation models for realized covariance matrices

New dynamic models for realized covariance matrices are proposed. The expected value of the realized covariance matrix is specified in two steps: one for each realized variance, and one for the realized correlation matrix. The realized correlation model is a scalar dynamic conditional correlation model. Estimation can be done in two steps as well, and a QML interpretation is given to each step, by assuming a Wishart conditional distribution. The model is applicable to large matrices since estimation can be done by the composite likelihood method

Notes on the economy in the rural non-capitalist societies: the contribution of A.V. Čajanov to the historical studies and the current political debate

Il presente lavoro ha l’obiettivo di richiamare i nodi fondamentali del classico dibattito sulla questione agraria, ponendo in evidenza il contributo di Aleksandr Čajanov alla definizione del modo contadino di produzione. La sintesi delle principali tesi dell’economista russo e la ricostruzione bibliografica della loro fortuna nella letteratura economica e storiografica sulle società preindustriali forniscono l’opportunità di tracciare un percorso interdisciplinare utile alla riflessione contemporanea sulla nuova condizione contadina.This paper aims to recall the fundamental issues of the classic debate on the agrarian question, highlighting the contribution of Aleksandr Čajanov the definition of the peasant mode of production. The summary of the Russian economist’s main theses and the bibliographic reconstruction of their fortune in the economic literature and historiography about pre-industrial societies provide the opportunity to draw an interdisciplinary path, useful to contemporary reflection on the new peasant condition

Note sull’economia contadina nelle società non capitalistiche:il contributo di A.V. Čajanov agli studi storici e al dibattito politico attuale

Il presente lavoro ha l’obiettivo di richiamare i nodi fondamentali del classico dibattito sulla questione agraria, ponendo in evidenza il contributo di Aleksandr Čajanov alla definizione del modo contadino di produzione. La sintesi delle principali tesi dell’economista russo e la ricostruzione bibliografica della loro fortuna nella letteratura economica e storiografica sulle società preindustriali forniscono l’opportunità di tracciare un percorso interdisciplinare utile alla riflessione contemporanea sulla nuova condizione contadina

Campagne e società in Italia meridionale (secoli VI-XIII): note intorno all’opera di Jean-Marie Martin

Il contributo si occupa di fornire alcune chiavi di lettura storiografiche dell'opera di Jean-Marie Martin, eminente medievista francese scomparso ai primi del 2021, con particolare riguardo ai rapporti tra assetti sociali, quadri insediativi e contesti istituzionali in Italia meridionale tra alto e pieno medioevo