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The editors are pleased to offer the following papers to the reader in recognition and appreciation of the contributions to our literature made by Robert Engle and Sir Clive Granger, winners of the 2003 Nobel Prize in Economics. Please see the previous dedication page of this volume. This part of Volume 20 of Advances in Econometric focuses on volatility models. The contributions cover a variety of topics and are organized into three broad categories to aid the reader. The first five papers focus broadly on multivariate Generalised auto-regressive conditional heteroskedasticity (GARCH) models. The first four papers propose new models that enhance existing models, while the final paper proposes a test for multivariate GARCH in the models with non-stationary variables. The next three papers examine topics related to high frequency-data. The first of these papers compares asymptotically mean square error (MSE)-equivalent sampling frequencies and window lengths, while the other two papers in this group consider the problem of estimating volatility in the presence of microstructure noise. The last five papers are contributions relevant primarily to univariate volatility models. Of course, we are also pleased to include Rob's and Clive's remarks on their careers and their views on innovation in econometric theory and practice that were given at the third annual Advances in Econometrics Conference held at Louisiana State University, Baton Rouge, on November 5–7, 2004.

Existing multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models either impose strong restrictions on the parameters or do not guarantee a well-defined (positive-definite) covariance matrix. I discuss the main multivariate GARCH models and focus on the BEKK model for which it is shown that the covariance and correlation is not adequately specified under certain conditions. This implies that any analysis of the persistence and the asymmetry of the correlation is potentially inaccurate. I therefore propose a new Flexible Dynamic Correlation (FDC) model that parameterizes the conditional correlation directly and eliminates various shortcomings. Most importantly, the number of exogenous variables in the correlation equation can be flexibly augmented without risking an indefinite covariance matrix. Empirical results of daily and monthly returns of four international stock market indices reveal that correlations exhibit different degrees of persistence and different asymmetric reactions to shocks than variances. In addition, I find that correlations do not always increase with jointly negative shocks implying a justification for international portfolio diversification.

Empirical research on European stock markets has shown that they behave differently according to the performance of the leading financial market identified as the US market. A positive sign is viewed as good news in the international financial markets, a negative sign means, conversely, bad news. As a result, we assume that European stock market returns are affected by endogenous and exogenous shocks. The former raise in the market itself, the latter come from the US market, because of its most influential role in the world. Under standard assumptions, the distribution of the European market index returns conditionally on the sign of the one-day lagged US return is skew-normal. The resulting model is denoted Skew-GARCH. We study the properties of this new model and illustrate its application to time-series data from three European financial markets.

A new multivariate heavy-tailed distribution is proposed as an extension of the univariate distribution of Politis (2004). The properties of the new distribution are discussed, as well as its effectiveness in modeling ARCH/GARCH residuals. A practical procedure for multi-parameter numerical maximum likelihood is also given, and a real data example is worked out.

Capital markets thrive on information, and the information revolution has transformed these markets all over the world. Investors can now keep track of the movements of capital markets in real-time and they react to the flow of information from around the world. One of the concerns of stock market investors is whether the markets operate efficiently, independently, and with sound fundamentals. However, real market movements tend to exhibit a link as is evident from recent market movements across the world.

The assessment of interdependence between stock markets is an important aspect of international portfolio management. The aim of this chapter is to examine the shock and volatility spillover between the Standard and Poor’s 500 (S&P500) index from the United States (US) Stock Exchange and the Istanbul Stock Exchange 100 (BIST100) index from the Stock Exchange Istanbul.

S&P500 index, which is the most important index representing US markets, and BIST100 index, which is the index representing the Turkish market, were used as variables in this study. In the analysis, the causality in variance test was applied to determine the volatility spillover between these two markets. Later, multivariate GARCH (MGARCH) models were used to measure the volatility spillover in the markets. VAR(1)-GARCH (1,1)-Diagonal BEKK model was applied to the daily data to determine the shock and volatility spillover in the markets.

As a result of the variance causality test, it was found that there is a bi-directional volatility spillover between S&P500 index and BIST100 index. When the return spillover between the markets is examined, a one-way spillover from the S&P500 index to the BIST100 index emerged. Diagonal BEKK model results show that each market is affected by its own news (unexpected shocks) and volatility. Furthermore, the volatility is persistent for both markets. These findings demonstrate that the US market and the Turkish market interact with each other.

This chapter examines the dynamic linkages between the returns of Bitcoin, gold, and oil by using daily closing price data between July 17, 2010 and January 8, 2021. This study applies the diagonal BEKK–GARCH model for the purpose of analyzing a volatility spillover of variables in positive or negative ways. The empirical results show that the lagged returns inversely affect their current returns in oil. Based on the return spillovers between Bitcoin and gold, the empirical results indicate a unidirectional return spillover from Bitcoin to gold. Moreover, the authors found a unidirectional return transmission is observed from oil to Bitcoin, implying that oil returns are useful in forecasting Bitcoin returns. These findings are not only valuable for understanding of the interrelationships between the returns of Bitcoin, gold, and oil, but they are also of great interest to portfolio managers, investors, and investment funds that are actively dealing in Bitcoin, gold, and oil.

In this paper, we search to evaluate the systemic risk of the Moroccan banking sector. Indeed, we concentrate on the analysis and the evaluation on transverse dimension of the systemic. From this point of view, two approaches were used. First is based on the estimate on value at risk conditional allowing to measure the systemic importance of each banking institution. In addition, the second approach uses the heteroscedasticity models in order to consider the conditional correlations, making it possible, to measure the dependence between the Moroccan banks and with the whole of the financial system. The results obtained with through these two approaches confirm that ATW, BMCI and the BMCE are the most systemic banks in Moroccan banking system and who can initiate a systemic crisis. On another register and by using the conditional correlations of each bank we built an index of systemic risk. Moreover, a macrofinancial model was developed, connecting the index of the systemic risk and the principal macroeconomic variables. This model affirmed that the contagion dimension of systemic risk is procyclical.

It is well known that volatility asymmetry exists in financial markets. This paper reviews and investigates recently developed techniques for Bayesian estimation and model selection applied to a large group of modern asymmetric heteroskedastic models. These include the GJR-GARCH, threshold autoregression with GARCH errors, TGARCH, and double threshold heteroskedastic model with auxiliary threshold variables. Further, we briefly review recent methods for Bayesian model selection, such as, reversible-jump Markov chain Monte Carlo, Monte Carlo estimation via independent sampling from each model, and importance sampling methods. Seven heteroskedastic models are then compared, for three long series of daily Asian market returns, in a model selection study illustrating the preferred model selection method. Major evidence of nonlinearity in mean and volatility is found, with the preferred model having a weighted threshold variable of local and international market news.