Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.     

Temporal aggregation of multivariate GARCH processes

This paper derives results for the temporal aggregation of multivariate GARCH(1,1) processes in the general vector specification. It is shown that the class of weak multivariate GARCH(1,1) processes is closed under temporal aggregation. Fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables. In some aspects, the aggregation characteristics of multivariate GARCH processes are shown to be different from those of vector autoregressive moving average processes. A numerical example illustrates some of the results.     

Model selection criteria in multivariate models with multiple structural changes

This paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes. We develop a modified Akaike information criterion (AIC), a modified Mallows’ C_p criterion and a modified Bayesian information criterion (BIC). The penalty terms in these criteria are shown to be different from the usual terms. We prove that the modified BIC consistently selects the regressors and the number of breaks whereas the modified AIC and the modified C_p criterion tend to overfit with positive probability. The finite sample performance of these criteria is investigated through Monte Carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation.     

Forecasting multivariate realized stock market volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics.     

Density approximations for multivariate affine jump-diffusion processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.     

General-to-specific or specific-to-general modelling? An opinion on current econometric terminology

It is argued that what is the dominant approach to analyzing systems of cointegrated variables is not well described as general-to-specific (gets) modelling. The gets approach was developed during the last decades predominantly in a single equation framework. For multivariate modelling and especially cointegration analysis the leading approach is better classified as bottom-up or specific-to-general (spec) modelling.     

Modeling multivariate extreme events using self-exciting point processes

We propose a model that can capture the typical features of multivariate extreme events observed in financial time series, namely, clustering behaviors in magnitudes and arrival times of multivariate extreme events, and time-varying dependence. The model is developed within the framework of the peaks-over-threshold approach in extreme value theory and relies on a Poisson process with self-exciting intensity. We discuss the properties of the model, treat its estimation, and address testing its goodness-of-fit. The model is applied to the return data of two stock markets.     

Sequential estimation of shape parameters in multivariate dynamic models

Sequential maximum likelihood and GMM estimators of distributional parameters obtained from the standardised innovations of multivariate conditionally heteroskedastic dynamic regression models evaluated at Gaussian PML estimators preserve the consistency of mean and variance parameters while allowing for realistic distributions. We assess their efficiency, and obtain moment conditions leading to sequential estimators as efficient as their joint ML counterparts. We also obtain standard errors for VaR and CoVaR, and analyse the effects on these measures of distributional misspecification. Finally, we illustrate the small sample performance of these procedures through simulations and apply them to analyse the risk of large eurozone banks.     

The conditional autoregressive Wishart model for multivariate stock market volatility

We propose a Conditional Autoregressive Wishart (CAW) model for the analysis of realized covariance matrices of asset returns. Our model assumes an autoregressive moving average structure for the scale matrix of the Wishart distribution. It accounts for positive definiteness of covariance matrices without imposing parametric restrictions, and can be estimated by Maximum Likelihood. We also propose extensions of the CAW model obtained by including a Mixed Data Sampling (MIDAS) component and Heterogeneous Autoregressive (HAR) dynamics for long-run fluctuations. The CAW models are applied to realized variances and covariances for five New York Stock Exchange stocks.     

A spatial model for multivariate lattice data

In this article, we develop Markov random field models for multivariate lattice data. Specific attention is given to building models that incorporate general forms of the spatial correlations and cross-correlations between variables at different sites. The methodology is applied to a problem in environmental equity. Using a Bayesian hierarchical model that is multivariate in form, we examine the racial distribution of residents of southern Louisiana in relation to the location of sites listed with the U.S. Environmental Protection Agency's Toxic Release Inventory.     

Indirect estimation of large conditionally heteroskedastic factor models,with an application to the Dow 30 stocks

We derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a Kalman-filter approximation with inequality constraints on the auxiliary model parameters. We also propose alternative indirect estimators for large-scale models, and explain how to apply our procedures to many other dynamic latent variable models. We analyse the small sample behaviour of our indirect estimators and several likelihood-based procedures through an extensive Monte Carlo experiment with empirically realistic designs. Finally, we apply our procedures to weekly returns on the Dow 30 stocks.     

Methods for inference in large multiple-equation Markov-switching models

Inference for multiple-equation Markov-chain models raises a number of difficulties that are unlikely to appear in smaller models. Our framework allows for many regimes in the transition matrix, without letting the number of free parameters grow as the square as the number of regimes, but also without losing a convenient form for the posterior distribution. Calculation of marginal data densities is difficult in these high-dimensional models. This paper gives methods to overcome these difficulties, and explains why existing methods are unreliable. It makes suggestions for maximizing posterior density and initiating MCMC simulations that provide robustness against the complex likelihood shape.     

Model-based asymptotic inference on the effect of infrequent large shocks on cointegrated variables

Quasi-maximum-likelihood (QML) estimation of a model combining cointegration in the conditional mean and rare large shocks (outliers) with a factor structure in the innovations is studied. The goal is not only to robustify inference on the conditional-mean parameters, but also to find regularities and conduct inference on the instantaneous and long-run effect of the large shocks. Given the cointegration rank and the factor order, χ²${\chi }^2$ asymptotic inference is obtained for the cointegration vectors, the short-run parameters, and the direction of each column of both the factor loading matrix and the matrix of long-run impacts of the large shocks. Large shocks, whose location is assumed unknown a priori, can be detected and classified consistently into the factor components.     

Modelling security market events in continuous time: Intensity based,multivariate point process models

A continuous time econometric modelling framework for multivariate financial market event (or ‘transactions’) data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A bivariate point process model of the timing of trades and mid-quote changes is then presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes in continuous time is found to be important empirically.     

Cointegration,long-run structural modelling and weak exogeneity: Two models of the UK economy

Cointegration ideas as introduced by Granger in 1981 are commonly embodied in empirical macroeconomic modelling through the vector error correction model (VECM). It has become common practice in these models to treat some variables as weakly exogenous, resulting in conditional VECMs. This paper studies the consequences of different approaches to weak exogeneity for the dynamic properties of such models, in the context of two models of the UK economy, one a national-economy model, the other the UK submodel of a global model. Impulse response and common trend analyses are shown to be sensitive to these assumptions and other specification choices.     

On loss functions and ranking forecasting performances of multivariate volatility models

The ranking of multivariate volatility models is inherently problematic because when the unobservable volatility is substituted by a proxy, the ordering implied by a loss function may be biased with respect to the intended one. We point out that the size of the distortion is strictly tied to the level of the accuracy of the volatility proxy. We propose a generalized necessary and sufficient functional form for a class of non-metric distance measures of the Bregman type which ensure consistency of the ordering when the target is observed with noise. An application to three foreign exchange rates is provided.     

A two-step estimator for large approximate dynamic factor models based on Kalman filtering

This paper shows consistency of a two-step estimation of the factors in a dynamic approximate factor model when the panel of time series is large (n large). In the first step, the parameters of the model are estimated from an OLS on principal components. In the second step, the factors are estimated via the Kalman smoother. The analysis develops the theory for the estimator considered in Giannone et al. (2004) and Giannone et al. (2008) and for the many empirical papers using this framework for nowcasting.     

Estimating DSGE models using seasonally adjusted and unadjusted data

This paper evaluates the common practice of estimating dynamic stochastic general equilibrium (DSGE) models using seasonally adjusted data. The simulation experiment shows that the practice leads to sizable distortions in estimated parameters. This is because the effects of seasonality, which are magnified by the model’s capital accumulation and labor market frictions, are not restricted to the so-called seasonal frequencies but instead are propagated across the entire frequency domain.     

Methods for computing marginal data densities from the Gibbs output

We introduce two estimators for estimating the Marginal Data Density (MDD) from the Gibbs output. Our methods are based on exploiting the analytical tractability condition, which requires that some parameter blocks can be analytically integrated out from the conditional posterior densities. This condition is satisfied by several widely used time series models. An empirical application to six-variate VAR models shows that the bias of a fully computational estimator is sufficiently large to distort the implied model rankings. One of the estimators is fast enough to make multiple computations of MDDs in densely parameterized models feasible.     

A Reduced Rank Regression Approach to Coincident and Leading Indexes Building*

This paper proposes a reduced rank regression framework for constructing a coincident index (CI) and a leading index (LI). Based on a formal definition that requires that the first differences of the LI are the best linear predictor of the first differences of the CI, it is shown that the notion of polynomial serial correlation common features can be used to build these composite variables. Concepts and methods are illustrated by an empirical investigation of the US business cycle indicators.