Multivariate GARCH estimation via a Bregman-proximal trust-region method

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the "curse of dimensionality". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem. Moreover, these parameters are subjected to convoluted nonlinear constraints necessary to ensure, for instance, the existence of stationary solutions and the positive semidefinite character of the conditional covariance matrices used in the model design. So far, this problem has been addressed in the literature only in low dimensional cases with strong parsimony constraints. In this paper we propose a general formulation of the estimation problem in any dimension and develop a Bregman-proximal trust-region method for its solution. The Bregman-proximal approach allows us to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Preliminary computational experiments are presented and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure

MGARCH models: tradeoff between feasibility and flexibility

The parameters of popular multivariate GARCH (MGARCH) models are restricted so that their estimation is feasible in large systems and covariance stationarity and positive definiteness of conditional covariance matrices are guaranteed. These restrictions limit the dynamics that the models can represent, assuming, for example, that volatilities evolve in an univariate fashion, not being related neither among them nor with the correlations. This paper updates previous surveyson parametric MGARCH models focusing on their limitations to represent the dynamics observed in real systems of financial returns. The conclusions are illustrated using simulated data and a five-dimensional system of exchange rate returns.The first author was supported by grants of 0969/13-3 CAPES, Coordination of Improvement of Higher Education Personnel. The second author acknowledges financial support from CAPES, grant 10600/13-2, São Paulo Research Foundation (FAPESP), grant 2013/00506-1, and Laboratory EPIFISMA. Financial support from ECO2012-32401 project by the Spanish Government is gratefully acknowledged by the third author

A closed-form estimator for the multivariate GARCH(1,1) model

We provide a closed-form estimator based on the VARMA representation for the unrestricted multivariate GARCH(1,1). We show that all parameters can be derived using basic linear algebra tools. We show that the estimator is consistent and asymptotically normal distributed. Our results allow also to derive a closed form for the parameters in the context of temporal aggregation of multivariate GARCH(1,1) by solving the equations as in Hafner [2008]

Time-domain Classification of the Brain Reward System: Analysis of Natural- and Drug-Reward Driven Local Field Potential Signals in Hippocampus and Nucleus Accumbens

Addiction is a major public health concern characterized by compulsive reward-seeking behavior. The excitatory glutamatergic signals from the hippocampus (HIP) to the Nucleus accumbens (NAc) mediate learned behavior in addiction. Limited comparative studies have investigated the neural pathways activated by natural and unnatural reward sources. This study has evaluated neural activities in HIP and NAc associated with food (natural) and morphine (drug) reward sources using local field potential (LFP). We developed novel approaches to classify LFP signals into the source of reward and recorded regions by considering the time-domain feature of these signals. Proposed methods included a validation step of the LFP signals using autocorrelation, Lyapunov exponent and Hurst exponent to assess the meaningful stability of these signals (lack of chaos). By utilizing the probability density function (PDF) of LFP signals and applying Kullback-Leibler divergence (KLD), data were classified to the source of the reward. Also, HIP and NAc regions were visually separated and classified using the symmetrized dot pattern technique, which can be applied in real-time to ensure the deep brain region of interest is being targeted accurately during LFP recording. We believe our method provides a computationally light and fast, real-time signal analysis approach with real-world implementation.Comment: 12 pages, 7 figures first two authors contributed equally to this wor

Essays on expected equity returns and volatility: modeling and prediction

Mención Internacional en el título de doctorThis thesis is based on modeling and predicting expected equity returns and volatility. In the first step, it focus on multivariate conditional volatility models, where multivariate GARCH (MGARCH) models are the most traditional approach considered in literature. However, the traditional MGARCH models need to be restricted so that their estimation is feasible in large systems and covariance stationarity and positive definiteness of conditional covariance matrices are guaranteed. To overcome this gap, this thesis analyzes the limitations of some very popular restricted parametric MGARCH models often implemented to represent the dynamics observed in real systems of financial returns. These limitations are illustrated using simulated data and a five-dimensional system of exchange rate returns. We show that the restrictions imposed by the BEKK model are very unrealistic generating potentially missleading forecasts of condicional correlations. On the contrary, models based on the DCC specification provide appropriate forecasts. Alternative estimators of the parameters are important to simplify the computations but do not have implications on the estimates of conditional correlations. In the second step, this thesis focus on predicting the mean of equity risk premium. In particular, we show that existing equity premium forecasts can be improved by combining parsimonious state-dependent regression models, where well-known macroeconomic predictors are interacted with an economic state variable based on technical indicators. The combining forecasts proposed deliver statistically and economically out-of-sample gains vis-a-vis the historical average, traditional univariate regressions and equal-weighted (EW) combination of macroeconomic forecasts. The EW combination is widely reported to be not worse than combining forecasts using estimated weights in equity-premium literature. However, given the relative large set of macroeconomic variables available as candidate predictors, we show that sparse combining method produces promising results for equity risk premium prediction.Esta tesis se basa en modelar y predecir el rendimiento esperado de las acciones y su volatilidad. En la primera etapa se analizan los modelos multivariantes de volatilidad condicional. Los modelos GARCH multivariantes (MGARCH) son los más utilizados en la literatura. Sin embargo, estos modelos necesitan ser restringidos para que su estimación sea factible en grandes sistemas y para que la estacionariedad de segundo orden así como la positividad de las matrices de covarianzas condicionales estén garantizadas. Para poder proponer una solución a este problema, la tesis analiza las limitaciones de algunos modelos MGARCH paramétricos restringidos, los cuales son muy utilizados para representar la dinámica observada en los sistemas reales de rentabilidad financiera. Estas limitaciones se ilustran usando datos simulados y un sistema de cinco series de rendimientos de tipos de cambio. Mostramos que las restricciones impuestas por el modelo BEKK son muy poco realistas, lo que puede generar una mala especificación de las previsiones de correlaciones condicionales. Por el contrario, los modelos DCC generan previsiones apropiadas. Los estimadores alternativos de parámetros son importantes para simplificar los cálculos, pero no tienen implicaciones en las estimaciones de las correlaciones condicionales. En la segunda etapa, la tesis estudia la predicción de la media de la prima de riesgo. En particular, se muestra que las previsiones de la prima de riesgo se pueden mejorar mediante la combinación de modelos de regresión state-dependent parsimoniosos, donde los predictores macroeconómicos interactúan con una variable de estado económico basada en indicadores técnicos. Las combinaciones de previsiones propuestas otorgan estadística y económicamente ganancias out-of-sample con relación a la media histórica, modelos de regresión univariantes y la media de la combinación de previsiones macroeconómicas utilizando iguales pesos (EW). La combinación EW es generalmente aceptada en la literatura de prima de riesgo por no ser peor que la combinación de pronósticos utilizando pesos estimados. Sin embargo, dado el gran conjunto de variables macroeconómicas disponibles como posibles predictores, demostramos que la combinación parsimoniosa del método produce resultados prometedores para la predicción de la prima de riesgo.Grants of 0969/13-3 CAPES, Coordination of Improvement of Higher Education Personnel (Brazil)Programa Oficial de Doctorado en Economía de la Empresa y Métodos CuantitativosPresidente: Fco. Javier Nogales Martín; Secretario: Cristina Amado; Vocal: José Olm