Giuseppe Storti

A GMM procedure for combining volatility forecasts

A novel approach to the combination of volatility forecasts is discussed. The proposed procedure makes use of the generalized method of moments (GMM) for estimating the combination weights. The asymptotic properties of the GMM estimator are derived while its finite sample properties are assessed by means of a simulation study. The results of an application to a time series of daily returns on the S&P500 are presented.

BL-GARCH models and asymmetries in volatility

In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the conditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index.

Minimum distance estimation of GARCH(1,1) models

A distribution free approach to the estimation of GARCH(1,1) models is presented. More specifically, the proposed method relies on a Minimum Distance Estimator (MDE) based on the autocovariance function of the squared observations. The asymptotic properties of the estimator are studied giving conditions for its consistency and asymptotic normality while its finite sample efficiency is assessed by means of a simulation study. Finally the proposed estimation method is applied to a time series of hourly returns on the FTSE100 index futures.

Combination of multivariate volatility forecasts

This paper proposes a novel approach to the combination of conditional covariance matrix forecasts based on the use of the Generalized Method of Moments (GMM). It is shown how the procedure can be generalized to deal with large dimensional systems by means of a two-step strategy. The finite sample properties of the GMM estimator of the combination weights are investigated by Monte Carlo simulations. Finally, in order to give an appraisal of the economic implications of the combined volatility predictor, the results of an application to tactical asset allocation are presented.

Likelihood inference in BL-GARCH models

SummaryThe paper presents a procedure based on the EM algorithm for the indirect estimation of the parameters of BiLinear GARCH (BL-GARCH) models. BL-GARCH generalize the class of GARCH models by considering interactions of past shocks and volatilities in the conditional variance equation. In this way the response of the conditional variance to past information becomes asymmetric allowing to account for the so called leverage effect, typically characterizing the behaviour of financial time series. The results of an application to a time series of stock market returns are presented.

Computationally efficient inference procedures for vast dimensional realized covariance models

This paper illustrates some computationally efficient estimation procedures for the estimation of vast dimensional realized covariance models. In particular, we derive a Composite Maximum Likelihood (CML) estimator for the parameters of a Conditionally Autoregressive Wishart (CAW) model incorporating scalar system matrices and covariance targeting. The finite sample statistical properties of this estimator are investigated by means of a Monte Carlo simulation study in which the data generating process is assumed to be given by a scalar CAW model. The performance of the CML estimator is satisfactory in all the settings considered although a relevant finding of our study is that the efficiency of the CML estimator is critically dependent on the implementation settings chosen by modeller and, more specifically, on the dimension of the marginal log-likelihoods used to build the composite likelihood functions.

A GARCH (1,1) estimator with (almost) no moment conditions on the error term

A least squares estimation approach for the estimation of a GARCH (1,1) model is developed. The asymptotic properties of the estimator are studied given mild regularity conditions, which require only that the error term has a conditional moment of some order. We establish the consistency, asymptotic normality and the law of iterated logarithm for our estimate. The finite sample properties are assessed by means of an extensive simulation study.

Least-squares estimation of GARCH(1,1) models with heavy-tailed errors

GARCH (1,1) models are widely used for modelling processes with time varying volatility. These include financial time series, which can be particularly heavy tailed. In this paper, we propose a log-transform-based least squares estimator (LSE) for the GARCH (1,1) model. The asymptotic properties of the LSE are studied under very mild moment conditions for the errors. We establish the consistency, asymptotic normality at the standard convergence rate of square root-of-n for our estimator. The finite sample properties are assessed by means of an extensive simulation study. Our results show that LSE is more accurate than the quasi-maximum likelihood estimator (QMLE) for heavy tailed errors. Finally, we provide some empirical evidence on two financial time series considering daily and high frequency returns. The results of the empirical analysis suggest that in some settings, depending on the specific measure of volatility adopted, the LSE can allow for more accurate predictions of volatility than the usual Gaussian QMLE.

Group Structured Volatility

In this work we investigate the presence of ‘group’ structures in financial markets. We show how this information can be used to simplify the volatility modelling of large portfolios. Our testing dataset is composed by all the stocks listed on the S&P500 index.

Combining Multivariate Volatility Models

Forecasting conditional covariance matrices of returns involves a variety of modeling options. First, the choice between models based on daily or intradaily returns. Examples of the former are the Multivariate GARCH (MGARCH) models while models fitted to Realized Covariance (RC) matrices are examples of the latter. A second option, strictly related to the RC matrices, is given by the identification of the frequency at which the intradaily returns are observed. A third option concerns the proper estimation method able to guarantee unbiased parameter estimates even for large (MGARCH) models. Thus, dealing with all these modeling options is not always straightforward. A possible solution is the combination of volatility forecasts. The aim of this work is to present a forecast combination strategy in which the combined models are selected by the Model Confidence Set (MCS) procedure, implemented under two economic loss functions (LFs).