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The paper investigates the estimation of a wide class of multivariate volatility models. Instead of estimating an m-multivariate volatility model, a much simpler and numerically efficient method consists in estimating m univariate generalized auto-regressive conditional heteroscedasticity type models equation by equation in the first step, and a correlation matrix in the second step. Strong consistency and asymptotic normality of the equation-by-equation estimator are established in a very general framework, including dynamic conditional correlation models. The equation-by-equation estimator can be used to test the restrictions imposed by a particular multivariate generalized auto-regressive conditional heteroscedasticity specification. For general constant conditional correlation models, we obtain the consistency and asymptotic normality of the two-step estimator. Comparisons with the global method, in which the model parameters are estimated in one step, are provided. Monte Carlo experiments and applications to financial series illustrate the interest of the approach.

In high dimensional model selection problems, penalized least square approaches have been extensively used. The paper addresses the question of both robustness and efficiency of penalized model selection methods and proposes a data-driven weighted linear combination of convex loss functions, together with weighted L₁-penalty. It is completely data adaptive and does not require prior knowledge of the error distribution. The weighted L₁-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias that is caused by the L₁-penalty. In the setting with dimensionality much larger than the sample size, we establish a strong oracle property of the method proposed that has both the model selection consistency and estimation efficiency for the true non-zero coefficients. As specific examples, we introduce a robust method of composite L₁-L₂, and an optimal composite quantile method and evaluate their performance in both simulated and real data examples.

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1,1) models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1,1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.

Despite the growing interest in realized stochastic volatility models, their estimation techniques, such as simulated maximum likelihood (SML), are computationally intensive. Based on the realized volatility equation, this study demonstrates that, in a finite sample, the quasi-maximum likelihood estimator based on the Kalman filter is competitive with the two-step SML estimator, which is less efficient than the SML estimator. Regarding empirical results for the S&P 500 index, the quasi-likelihood ratio tests favored the two-factor realized asymmetric stochastic volatility model with the standardized t distribution among alternative specifications, and an analysis on out-of-sample forecasts prefers the realized stochastic volatility models, rejecting the model without the realized volatility measure. Furthermore, the forecasts of alternative RSV models are statistically equivalent for the data covering the global financial crisis.

In conditionally heteroscedastic models, the optimal prediction of powers, or logarithms, of the absolute value has a simple expression in terms of the volatility and an expectation involving the independent process. A natural procedure for estimating this prediction is to estimate the volatility in the first step, for instance by Gaussian quasi-maximum-likelihood or by least absolute deviations, and to use empirical means based on rescaled innovations to estimate the expectation in the second step. The paper proposes an alternative one-step procedure, based on an appropriate non-Gaussian quasi-maximum-likelihood estimator, and establishes the asymptotic properties of the two approaches. Asymptotic comparisons and numerical experiments show that the differences in accuracy can be important, depending on the prediction problem and the innovations distribution. An application to indices of major stock exchanges is given.

In this paper, we propose a novel spatial-temporal model with individual random effects characterized by a location-scale structure, which allows us to flexibly capture the pure influence of space-specific factors in a quantile regression framework. A hybrid two-stage estimation procedure is introduced for this model. The first stage proposes a Gaussian quasi-maximum likelihood estimator for the spatial-temporal effects, and the second constructs a weighted conditional quantile estimator, which we use to study the conditional quantiles of the random effects related to space-specific attributes. We verify the validity of the two-stage hybrid estimation, and establish the asymptotic properties of our estimators. The results of our simulation study indicate that the proposed estimation procedure performs well in different scenarios with finite-samples. Lastly, we apply the proposed method to data from a real case study on the air quality of China.

Systemic risk analysis reveals the interdependencies of risk factors especially in tail event situations. In applications the focus of interest is on capturing joint tail behavior rather than a variation around the mean. Quantile and expectile regression are used here as tools of data analysis. When it comes to characterizing tail event curves one faces a dimensionality problem, which is important for CoVaR (Conditional Value at Risk) determination. A projection-based single-index model specification may come to the rescue but for ultrahigh-dimensional regressors one faces yet another dimensionality problem and needs to balance precision versus dimension. Such a balance is achieved by combining semiparametric ideas with variable selection techniques. In particular, we propose a projection-based single-index model specification for very high-dimensional regressors. This model is used for practical CoVaR estimates with a systemically chosen indicator. In simulations we demonstrate the practical side of the semiparametric CoVaR method. The application to the U.S. financial sector shows good backtesting results and indicate market coagulation before the crisis period. Supplementary materials for this article are available online.

Since shipping connectivity reduces trade costs, which in turn improves trade, this paper aims to analyse the short- and long-run impacts of the liner shipping bilateral connectivity on South Africa’s trade flows. In addition to connectivity, measured by five separate components, we also consider the effects on trade of sailing distances, the direct (air) distance and the gross domestic product (GDP) of 142 trading partners. We apply the quasi-maximum likelihood method to estimate the parameters of a dynamic panel data model. The results show that GDP, the number of common direct connections and the level of competition have a positive and significant effect on trade flows, while the number of transshipments and the direct and sailing distances have a negative and significant impact, both in the short and long run. The estimated long-run effects are stronger than the short-run effects, suggesting that shippers take time to adjust their demand to changes in connectivity. The variable maximum ship size does not seem to have a positive bearing on trade, suggesting that countries may not need to try to accommodate ever larger ships to maintain their foreign trade competitiveness.

We first construct a new generalized Hausman test for detecting the structural change in a multiplicative form of covariance matrix time series model. This generalized Hausman test is asymptotically pivotal, and has nontrivial power in detecting a broad class of alternatives. Moreover, we propose a new semiparametric covariance matrix time series model. The proposed model has a time-varying long-run component that takes the structural change into account, and a BEKK-type short-run component that captures the temporal dependence. We propose a two-step estimation procedure to estimate this semiparametric model, and establish the asymptotics of the related estimators. Finally, the importance of the generalized Hausman test and the semiparametric model is illustrated by means of simulations and an application to realized covariance matrix data.

In this paper, we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We propose new portmanteau statistics for vector autoregressive moving average models with uncorrelated but nonindependent innovations by using a self-normalization approach. We establish the asymptotic distribution of the proposed statistics. This asymptotic distribution is quite different from the usual chi-squared approximation used under the independent and identically distributed assumption on the noise, or the weighted sum of independent chi-squared random variables obtained under nonindependent innovations. A set of Monte Carlo experiments and an application to the daily returns of the CAC40 is presented. Supplementary materials for this article are available online.