Steffen Grønneberg

Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown

A general framework for the estimation and inference in univariate and multivariate Generalised log-ARCH-X (i.e. log-GARCH-X) models when the conditional density is unknown is proposed. The framework employs (V)ARMA-X representations and relies on a bias-adjustment in the log-volatility intercept. The bias is induced by (V)ARMA estimators, but the remaining parameters can be estimated in a consistent and asymptotically normal manner by usual (V)ARMA methods. An estimator of the bias and a closed-form expression for the asymptotic variance is derived. Adding covariates and/or increasing the dimension of the model does not change the structure of the problem, so the univariate bias-adjustment procedure is applicable not only in univariate log-GARCH-X models estimated by the ARMA-X representation, but also in multivariate log-GARCH-X models estimated by VARMA-X representations. Extensive simulations verify the properties of the log-moment estimator, and an empirical application illustrates the usefulness of the methods.

How general is the Vale-Maurelli simulation approach?

The Vale–Maurelli (VM) approach to generating non-normal multivariate data involves the use of Fleishman polynomials applied to an underlying Gaussian random vector. This method has been extensively used in Monte Carlo studies during the last three decades to investigate the finite-sample performance of estimators under non-Gaussian conditions. The validity of conclusions drawn from these studies clearly depends on the range of distributions obtainable with the VM method. We deduce the distribution and the copula for a vector generated by a generalized VM transformation, and show that it is fundamentally linked to the underlying Gaussian distribution and copula. In the process we derive the distribution of the Fleishman polynomial in full generality. While data generated with the VM approach appears to be highly non-normal, its truly multivariate properties are close to the Gaussian case. A Monte Carlo study illustrates that generating data with a different copula than that implied by the VM approach severely weakens the performance of normal-theory based ML estimates.

The Asymptotic Covariance Matrix and its Use in Simulation Studies

The asymptotic performance of structural equation modeling tests and standard errors are influenced by two factors: the model and the asymptotic covariance matrix Γ of the sample covariances. Although most simulation studies clearly specify model conditions, specification of Γ is usually limited to values of univariate skewness and kurtosis. We illustrate that marginal skewness and kurtosis are not sufficient to adequately specify a nonnormal simulation condition by showing that asymptotic standard errors and test statistics vary substantially among distributions with skewness and kurtosis that are identical. We argue therefore that Γ should be reported when presenting the design of simulation studies. We show how Γ can be exactly calculated under the widely used Vale–Maurelli transform. We suggest plotting the elements of Γ and reporting the eigenvalues associated with the test statistic. R code is provided.

Covariance Model Simulation Using Regular Vines

We propose a new and flexible simulation method for non-normal data with user-specified marginal distributions, covariance matrix and certain bivariate dependencies. The VITA (VIne To Anything) method is based on regular vines and generalizes the NORTA (NORmal To Anything) method. Fundamental theoretical properties of the VITA method are deduced. Two illustrations demonstrate the flexibility and usefulness of VITA in the context of structural equation models. R code for the implementation is provided.

Approximating Test Statistics Using Eigenvalue Block Averaging

We introduce and evaluate a new class of approximations to common test statistics in structural equation modeling. Such test statistics asymptotically follow the distribution of a weighted sum of i.i.d. chi-square variates, where the weights are eigenvalues of a certain matrix. The proposed eigenvalue block averaging (EBA) method involves creating blocks of these eigenvalues and replacing them within each block with the block average. The Satorra–Bentler scaling procedure is a special case of this framework, using one single block. The proposed procedure applies also to difference testing among nested models. We investigate the EBA procedure both theoretically in the asymptotic case, and with simulation studies for the finite-sample case, under both maximum likelihood and diagonally weighted least squares estimation. Comparison is made with 3 established approximations: Satorra–Bentler, the scaled and shifted, and the scaled F tests.

Why Has Econometric Inference Been Possible

In fields that are mainly nonexperimental, such as economics and finance, it is inescapable to compute test statistics and confidence regions that are not probabilistically independent from previously examined data. The Bayesian and Neyman-Pearson inference theories are known to be inadequate for such a practice. We show that these inadequacies also hold m.a.e. (modulo approximation error). We develop a general econometric theory, called the neoclassical inference theory, that is immune to this inadequacy m.a.e. The neoclassical inference theory appears to nest model calibration, and most econometric practices, whether they are labelled Bayesian or \`a la Neyman-Pearson. We derive a general, but simple adjustment to make standard errors account for the approximation error.In economics and finance, test statistics and confidence regions are typically probabilistically dependent on previously-used data. However, the Bayesian and classical justifications for usual tests and confidence regions are known to be invalid in this case. Given the success of empirical econometrics, this leads to the question: Why has econometric inference been possible? The paper provides a simple, yet general, answer to the question. We outline a novel econometric theory that i) is immune to conditioning on previously-used data, and that ii) nests model calibration, and most of econometric practices, whether they are labelled Bayesian or classical.

Testing Model Fit by Bootstrap Selection

Over the last few decades, many robust statistics have been proposed in order to assess the fit of structural equation models. To date, however, no clear recommendations have emerged as to which test statistic performs best. It is likely that no single statistic will universally outperform all contenders across all conditions of data, sample size, and model characteristics. In a real-world situation, a researcher must choose which statistic to report. We propose a bootstrap selection mechanism that identifies the test statistic that exhibits the best performance under the given data and model conditions among any set of candidates. This mechanism eliminates the ambiguity of the current practice and offers a wide array of test statistics available for reporting. In a Monte Carlo study, the bootstrap selector demonstrated promising performance in controlling Type I errors compared to current test statistics.

On the errors committed by sequences of estimator functionals

Consider a sequence of estimators

THE COPULA INFORMATION CRITERIA

\hat \theta _n

The Copula Information Criterion and Its Implications for the Maximum Pseudo-Likelihood Estimator

which converges almost surely to θ0 as the sample size n tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time