The importance of modelling comovements of financial returns is well established in the literature. The knowledge of correlation structures is vital in many financial applications, including asset pricing, optimal portfolio allocation and risk management. Moreover, as the volatilities of different assets and markets move together, modelling volatility in a multivariate framework can lead to greater statistical efficiency. Starting from the Vector Autoregression model of Bollerslev, Engle and Wooldridge (1988), several multivariate conditional volatility models have been proposed in the literature and used extensively in applied work. Over the last few years, the literature on multivariate stochastic volatility models has also developed significantly, thanks to the availability of many new numerical estimation methods. Recently empirical studies found robust evidences of asymmetric response of volatilities to positive and negative returns in multivariate asset models. A number of conditional and stochastic volatility models have been proposed to capture this inherent characteristic of volatility in a multivariate context, such as the QARCH latent factor model of Sentana (1995), the MSV-Leverage model of Asai and McAleer (2005), the asymmetric dynamic covariance (ADC) model of Kroner and Ng (1998), the Matrix Exponential GARCH model of Kawakatsu (2006), and others. Another important property that characterizes the dynamic evolution of volatilities is that power transformations of absolute returns have significant autocorrelations that decay to zero at a slow rate. Many authors have argued that the slow decay of the autocorrelations of squared returns is consistent with the notion of long-memory, where the autocovariances are not absolutely summable. Univariate long-memory volatility models, such as the FIEGARCH model of Bollerslev and Mikklesen (1996), the nonlinear moving average model of Robinson and Zaffaroni (1996, 1997) and the long-memory stochastic volatility (LMSV) model of Ruiz and Veiga (2006), have received a great deal of attention. However, to my knowledge, no generalization to a multivariate long-memory volatility model has been attempted in the literature. The aim of this thesis is to fill in this gap. Chapter 4 introduces a new multivariate Exponential Volatility (MEV) model, which captures long-range dependence in certain nonlinear functions of the data, such as squares, while retaining the martingale difference assumption and short-memory dependence in the level. The multivariate Exponential Volatility model is an extension of the univariate exponential volatility model of Zaffaroni (2009). It nests "one-shock" conditional variance specifications and "two-shocks" stochastic volatility specifications. It captures cross-assets spillover effects, leverage and asymmetry. The choice of an exponential specification offers several advantages, the most relevant of which is that no further restriction is required to grant positive definiteness of the covariance matrix. Estimation of the MEV model by maximum likelihood methods is possible, however we advocate the use of the frequency domain Gaussian estimator in the sense of Whittle (1962). MLE estimation of nonlinear exponential models is computationally costly, and possibly unstable. Moreover its asymptotic properties depend on the invertibility of the model, which is not easy to establish in exponential models (see Straumann and Mikosh, 2006). These difficulties do not apply to the Whittle estimator, partly due to its frequency domain specification. We follow Harvey et al. (1994) and estimate a logarithmic transformation of the squared returns of the observations. The estimated model turns out to be a vector signal plus noise model, where the signal evolves according to an infinite order moving average process and the noise is an i.i.d shock. The dependence structure of the MEV model implies that the signal and the noise might be correlated. Statistical literature on Whittle estimation of signal plus noise models requires at least incoherent signal and noise. In fact all the available results deal with linearly regular signal plus noise processes with parameters that can be estimated directly on the factored representation of the process spectral density (see Dunsmuir, 1979, and Hosoya and Taniguchi, 1982). Such results do not readily apply to correlated signal plus noise processes, even when the processes are linearly regular. In chapter 4, we establish the strong consistency and asymptotic normality of the Whittle estimator when the signal coefficients have an exponential decay rate, following Robinson (1978). In chapter 5, we establish the properties of the estimator when the signal coefficients have an hyperbolic decay rate that imparts long-range dependence in the squares of the MEV model, relying the on the central limit theorem for the integrated weighted periodogram of Giraitis and Taqqu (1999). As expected, the asymptotic properties of the estimator do not depend on the degree of persistence of shocks to the conditional variances, thanks to a convenient feature of the Whittle function that allows to compensate for the possible lack of square integrability of the spectral density. The results are established under the conditions of strict stationarity, ergodicity and finite fourth moments and either absolute summability of the autocovariance function or standard smoothness assumptions for the spectral density and its higher order derivatives. Chapter 2 offers a detailed literature review of multivariate volatility models, with some discussion on the most relevant multivariate exponential model, i.e. the matrix exponential GARCH model of Kawakatsu. Chapter 3 introduces the multivariate Exponential Volatility model, discusses its estimation and establishes the asymptotic properties of the estimator under fairly general conditions suitable for both "one-shock" and "two-shocks" specification of the model. Chapter 4 extends these results to the long memory MEV model.

### Whittle Estimation of Multivariate GARCH models

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*MARCHESE, MALVINA*

##### In corso di stampa

#### Abstract

The importance of modelling comovements of financial returns is well established in the literature. The knowledge of correlation structures is vital in many financial applications, including asset pricing, optimal portfolio allocation and risk management. Moreover, as the volatilities of different assets and markets move together, modelling volatility in a multivariate framework can lead to greater statistical efficiency. Starting from the Vector Autoregression model of Bollerslev, Engle and Wooldridge (1988), several multivariate conditional volatility models have been proposed in the literature and used extensively in applied work. Over the last few years, the literature on multivariate stochastic volatility models has also developed significantly, thanks to the availability of many new numerical estimation methods. Recently empirical studies found robust evidences of asymmetric response of volatilities to positive and negative returns in multivariate asset models. A number of conditional and stochastic volatility models have been proposed to capture this inherent characteristic of volatility in a multivariate context, such as the QARCH latent factor model of Sentana (1995), the MSV-Leverage model of Asai and McAleer (2005), the asymmetric dynamic covariance (ADC) model of Kroner and Ng (1998), the Matrix Exponential GARCH model of Kawakatsu (2006), and others. Another important property that characterizes the dynamic evolution of volatilities is that power transformations of absolute returns have significant autocorrelations that decay to zero at a slow rate. Many authors have argued that the slow decay of the autocorrelations of squared returns is consistent with the notion of long-memory, where the autocovariances are not absolutely summable. Univariate long-memory volatility models, such as the FIEGARCH model of Bollerslev and Mikklesen (1996), the nonlinear moving average model of Robinson and Zaffaroni (1996, 1997) and the long-memory stochastic volatility (LMSV) model of Ruiz and Veiga (2006), have received a great deal of attention. However, to my knowledge, no generalization to a multivariate long-memory volatility model has been attempted in the literature. The aim of this thesis is to fill in this gap. Chapter 4 introduces a new multivariate Exponential Volatility (MEV) model, which captures long-range dependence in certain nonlinear functions of the data, such as squares, while retaining the martingale difference assumption and short-memory dependence in the level. The multivariate Exponential Volatility model is an extension of the univariate exponential volatility model of Zaffaroni (2009). It nests "one-shock" conditional variance specifications and "two-shocks" stochastic volatility specifications. It captures cross-assets spillover effects, leverage and asymmetry. The choice of an exponential specification offers several advantages, the most relevant of which is that no further restriction is required to grant positive definiteness of the covariance matrix. Estimation of the MEV model by maximum likelihood methods is possible, however we advocate the use of the frequency domain Gaussian estimator in the sense of Whittle (1962). MLE estimation of nonlinear exponential models is computationally costly, and possibly unstable. Moreover its asymptotic properties depend on the invertibility of the model, which is not easy to establish in exponential models (see Straumann and Mikosh, 2006). These difficulties do not apply to the Whittle estimator, partly due to its frequency domain specification. We follow Harvey et al. (1994) and estimate a logarithmic transformation of the squared returns of the observations. The estimated model turns out to be a vector signal plus noise model, where the signal evolves according to an infinite order moving average process and the noise is an i.i.d shock. The dependence structure of the MEV model implies that the signal and the noise might be correlated. Statistical literature on Whittle estimation of signal plus noise models requires at least incoherent signal and noise. In fact all the available results deal with linearly regular signal plus noise processes with parameters that can be estimated directly on the factored representation of the process spectral density (see Dunsmuir, 1979, and Hosoya and Taniguchi, 1982). Such results do not readily apply to correlated signal plus noise processes, even when the processes are linearly regular. In chapter 4, we establish the strong consistency and asymptotic normality of the Whittle estimator when the signal coefficients have an exponential decay rate, following Robinson (1978). In chapter 5, we establish the properties of the estimator when the signal coefficients have an hyperbolic decay rate that imparts long-range dependence in the squares of the MEV model, relying the on the central limit theorem for the integrated weighted periodogram of Giraitis and Taqqu (1999). As expected, the asymptotic properties of the estimator do not depend on the degree of persistence of shocks to the conditional variances, thanks to a convenient feature of the Whittle function that allows to compensate for the possible lack of square integrability of the spectral density. The results are established under the conditions of strict stationarity, ergodicity and finite fourth moments and either absolute summability of the autocovariance function or standard smoothness assumptions for the spectral density and its higher order derivatives. Chapter 2 offers a detailed literature review of multivariate volatility models, with some discussion on the most relevant multivariate exponential model, i.e. the matrix exponential GARCH model of Kawakatsu. Chapter 3 introduces the multivariate Exponential Volatility model, discusses its estimation and establishes the asymptotic properties of the estimator under fairly general conditions suitable for both "one-shock" and "two-shocks" specification of the model. Chapter 4 extends these results to the long memory MEV model.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.