THE SCORE OF CONDITIONALLY HETEROSKEDASTIC DYNAMIC REGRESSION MODELS WITH STUDENT T INNOVATIONS, AN LM TEST FOR MULTIVARIATE NORMALITY

We provide numerically reliable analytical expressions for the score of conditionally heteroskedastic dynamic regression models when the conditional distribution is multivariate $t$. We also derive one-sided and 2-sided LM tests for multivariate normality versus multivariate $t$ based on the first two moments of the (squared) norm of the standardised innovations evaluated at the Gaussian quasi-ML estimators of the conditional mean and variance parameters. We reinterpret them as specification tests for multivariate excess kurtosis, and show that they have power against leptokurtic alternatives. Finally, we analyse UK stock returns, and confirm that their conditional distribution has fat tails.Kurtosis, Inequality Constraints, ARCH, Financial Returns.

CONSTRAINED EMM AND INDIRECT INFERENCE ESTIMATION

We develop generalised indirect inference procedures that handle equality and inequality constraints on the auxiliary model parameters. We obtain expressions for the optimal weighting matrices, and discuss as examples an MA(1) estimated as AR(1), an AR(1) estimated as MA(1), and a log-normal stochastic volatility process estimated as a GARCH(1,1) with Gaussian or t distributed errors. In the first example, the constraints have no effect, while in the second, they allow us to achieve full efficiency. As for the third, neither procedure systematically outperforms the other, but equality restricted estimators are better when the additional parameter is poorly estimated.

Conditional heteroskedasticity in nonlinear simultaneous equations

We show in this paper that the treatment of conditional heteroskedasticity inside nonlinear systems of simultaneous equations is a sufficiently manageable matter for some types of multivariate ARCH error structures. Reparameterization makes it possible to estimate the model by means of the (nearly) standard algorithms developed in the past and widely used for estimating nonlinear simultaneous equations where the error structure is of the i.i.d. type with unrestricted contemporaneous covariance matrix. The method is discussed in this paper and empirical applications exemplify the efficiency gains.Nonlinear simultaneous equations; conditional heteroskedasticity; instrumental variables; nonlinear FIML; demand supply model, long term treasury bonds

Geomorphic signal of active faulting at the northern edge of Lut Block. Insights on the kinematic scenario of Central Iran

Recent works documented Neogene to Quaternary dextral strike-slip tectonics along the Kuh-e-Sarhangi and Kuh-e-Faghan intraplate strike-slip faults at the northern edge of the Lut Block of Central Iran, previously thought to be dominated by sinistral strike-slip deformation. This work focuses on the evidence of Quaternary activity of one of these fault systems, in order to provide new spatio-temporal constraints on their role in the active regional kinematic scenario. Through geomorphological and structural investigation, integrated with Optically Stimulated Luminescence (OSL) dating of three generations of alluvial fans and fluvial terraces (at ~53, ~25 and ~6 ka), this study documents (i) the topographic inheritance of the long-term (Myr) punctuated history of fault nucleation, propagation, and exhumation along the northern edge of Lut Block; (ii) the tectonic control on drainage network evolution, pediment formation, fluvial terraces, and alluvial-fan architecture; (iii) the minimum Holocene age of Quaternary dextral strike-slip faulting; and (iv) the evidence of Late Quaternary fault-related uplift localized along the different fault strands. The documented spatial and temporal constraints on the active dextral strike-slip tectonics at the northern edge of Lut Block provided new insights on the kinematic model for active faulting in Central Iran, which has been reinterpreted in an escape tectonic scenario

Estimating variances and covariances in a censored regression model

When the coefficients of a Tobit model are estimated by maximum likelihood their covariance matrix is typically, even if not necessarily, associated with the algorithm employed to maximize the likelihood. Covariance estimators used in practice are derived by: (1) the Hessian (observed information), (2) the matrix of outer products of the first derivatives of the log-likelihood (OPG version), (3) the expected Hessian (estimated information), (4) a mixture of 1 and 2 (White's QML covariance matrix). Significant differences among these estirnates are are usually interpreted as an indication of misspecification. From our Monte Carlo study this seems not to be true, unless the sample size is really very large. Even in absence of misspecification, large differences are encountered in small samples, and the sign of the differences is almost systematic. This suggests that the choice of the covariance estimator is not neutral and the results of hypotheses testing may be strongly affected by such a choice

Estimating variances and covariances in a censored regression model

When the coefficients of a Tobit model are estimated by maximum likelihood their covariance matrix is typically, even if not necessarily, associated with the algorithm employed to maximize the likelihood. Covariance estimators used in practice are derived by: (1) the Hessian (observed information), (2) the matrix of outer products of the first derivatives of the log-likelihood (OPG version), (3) the expected Hessian (estimated information), (4) a mixture of 1 and 2 (White's QML covariance matrix). Significant differences among these estirnates are are usually interpreted as an indication of misspecification. From our Monte Carlo study this seems not to be true, unless the sample size is really very large. Even in absence of misspecification, large differences are encountered in small samples, and the sign of the differences is almost systematic. This suggests that the choice of the covariance estimator is not neutral and the results of hypotheses testing may be strongly affected by such a choice

Conditional heteroskedasticity in nonlinear simultaneous equations

We show in this paper that the treatment of conditional heteroskedasticity inside nonlinear systems of simultaneous equations is a sufficiently manageable matter for some types of multivariate ARCH error structures. Reparameterization makes it possible to estimate the model by means of the (nearly) standard algorithms developed in the past and widely used for estimating nonlinear simultaneous equations where the error structure is of the i.i.d. type with unrestricted contemporaneous covariance matrix. The method is discussed in this paper and empirical applications exemplify the efficiency gains

Estimating variances and covariances in a censored regression model

When the coefficients of a Tobit model are estimated by maximum likelihood their covariance matrix is typically, even if not necessarily, associated with the algorithm employed to maximize the likelihood. Covariance estimators used in practice are derived by: (1) the Hessian (observed information), (2) the matrix of outer products of the first derivatives of the log-likelihood (OPG version), (3) the expected Hessian (estimated information), (4) a mixture of 1 and 2 (White's QML covariance matrix). Significant differences among these estirnates are are usually interpreted as an indication of misspecification. From our Monte Carlo study this seems not to be true, unless the sample size is really very large. Even in absence of misspecification, large differences are encountered in small samples, and the sign of the differences is almost systematic. This suggests that the choice of the covariance estimator is not neutral and the results of hypotheses testing may be strongly affected by such a choice.Tobit model, maximum likelihood, hessian matrix, outer products matrix, covariance estimators

Alternative estimators of the covariance matrix in GARCH models

With most of the available software packages, estimates of the parameter covariance matrix in a GARCH model are usually obtained from the outer products of the first derivatives of the log-likelihoods (BHHH estimator). However, other estimators could be defined and used, analogous to the covariance matrix estimators in maximum likelihood studies described in the literature for other types of models (linear regression model, linear and nonlinear simultaneous equations, Probit and Tobit models). These alternative estimators can be derived from: (1) the Hessian (observed information), (2) the estimated information (expected Hessian), (3) a mixture of Hessian and outer products matrix (White's QML covarjance matrix). Signifacant differences among these estimates can be interpreted as an indication of misspecification, or can be due to systematic inequalities between alternative estimators in small samples. Unlike other types of models, from our Monte Carlo study we do not encounter very large differences, presumably because GARCH estimation is usually applied when the sample size is rather large. However, analogously to otber types of models we find in this Monte Carlo study that, even in absence of misspecification, the sign of the differences between some estimators is almost systematic. This suggests that, as for other types of models, the choice of the covariance estimator is not neutral, but the results of hypotheses testing are not strongly affected by such a choice