Contemporaneous Aggregation of GARCH Processes

We study the impact of large cross-sections of contemporaneous aggregation of GARCH processes and of dynamic GARCH factor models. The results crucially depend on the shape of the cross-sectional distribution of the GARCH coefficients and on the cross-sectional dependence properties of the rescaled innovation. The aggregate maintains the core nonlinearity of a volatility model, uncorrelation in the levels but autocorrelation in the squares, when the rescaled innovation is common across units. The nonlinearity is, however, lost at the aggregate level, when the rescaled innovation is orthogonal across units. This is not a consequence of the usual result of a vanishing importance of purely idiosyncratic risk as, under appropriate conditions, this is simply not fully diversifiable in arbitrary large portfolios. Non-GARCH memory properties arise at the aggregate level. Strict stationarity, ergodicity and finite kurtosis might fail for the aggregate despite the micro GARCH do satisfy these properties. Under no conditions aggregation of GARCH induces long memory conditional heteroskedasticity.Contemporaneous aggregation, GARCH, conditionally heteroskedastic factor models, common and idiosyncratic risk, nonlinearity, memory

Stationarity and Memory of ARCH Models

Sufficient conditions for strict stationarity of ARCH(8) are established, without imposing covariance stationarity and for any specification of the conditional second moment coefficients. GARCH(p,q) as well as the case of hyperbolically decaying coefficients are included, such as the autoregressive coefficients of ARFIMA(p,d,q), once the non-negativity constraints are imposed. Second, we show the necessary and sufficient conditions for covariance stationarity of ARCH(8), both for the levels and the squares. These prove to be much stronger than the strict stationarity conditions. The covariance stationarity condition for the levels rules out long memory in the squares.ARCH(8), GARCH(p,q), nonlinear moving average representation, strict and weak stationarity, memory.

Optimal Asset Allocation with Factor Models for Large Portfolios

An electronic version of the paper may be downloaded • from the SSRN website: www.SSRN.com • from the RePEc website: www.RePEc.org • from the CESifo website: Twww.CESifo-group.org/wp

(Fractional) Beta Convergence

Unit roots in output, an exponential 2 per cent rate of convergence and no change in the underlying dynamics of output seem to be three stylized facts that cannot go together. This paper extends the Solow-Swan growth model allowing for cross-sectional heterogeneity. In this framework, aggregate shocks might vanish at a hyperbolic rather than at an exponential rate. This implies that the level of output can exhibit long memory and that standard tests fail to reject the null of a unit root despite mean reversion. Exploiting secular time series properties GDP, we conclude that traditional approaches to test for uniform (conditional and unconditional) convergence suit first step approximation. We show both theoretically and empirically how the uniform 2 per cent rate of convergence repeatedly found in the empirical literature is the outcome of an underlying parameter of fractional integration strictly between 1/2 and 1. This is consistent with both time series and cross-sectional evidence recently produced.growth model, convergence, long memory, aggregation

Model Averaging and Value-at-Risk Based Evaluation of Large Multi Asset Volatility Models for Risk Management

This paper considers the problem of model uncertainty in the case of multi-asset volatility models and discusses the use of model averaging techniques as a way of dealing with the risk of inadvertently using false models in portfolio management. In particular, it is shown that under certain conditions portfolio returns based on an average model will be more fat-tailed than if based on an individual underlying model with the same average volatility. Evaluation of volatility models is also considered and a simple Value-at-Risk (VaR) diagnostic test is proposed for individual as well as ‘average’ models and its exact and asymptotic properties are established. The model averaging idea and the VaR diagnostic tests are illustrated by an application to portfolios of daily returns based on twenty two of Standard & Poor’s 500 industry group indices over the period January 2, 1995 to October 13, 2003, inclusive.model averaging, value-at-risk, decision based evaluation

Optimal Asset Allocation with Factor Models for Large Portfolios

This paper characterizes the asymptotic behaviour, as the number of assets gets arbitrarily large, of the portfolio weights for the class of tangency portfolios belonging to the Markowitz paradigm. It is assumed that the joint distribution of asset returns is characterized by a general factor model, with possibly heteroskedastic components. Under these conditions, we establish that a set of appealing properties, so far unnoticed, characterize traditional Markowitz portfolio trading strategies. First, we show that the tangency portfolios fully diversify the risk associated with the factor component of asset return innovations. Second, with respect to determination of the portfolio weights, the conditional distribution of the factors is of second-order importance as compared to the distribution of the factor loadings and that of the idiosyncratic components. Third, although of crucial importance in forecasting asset returns, current and lagged factors do not enter the limit portfolio returns. Our theoretical results also shed light on a number of issues discussed in the literature regarding the limiting properties of portfolio weights such as the diversifiability property and the number of dominant factors.asset allocation, large portfolios, factor models, diversification

Pseudo-Maximum Likelihood Estimation of ARCH(8) Models

Strong consistency and asymptotic normality of the Gaussian pseudo-maximumlikelihood estimate of the parameters in a wide class of ARCH(8) processesare established. We require the ARCH weights to decay at least hyperbolically,with a faster rate needed for the central limit theorem than for the law of largenumbers. Various rates are illustrated in examples of particular parameteriza-tions in which our conditions are shown to be satisfied.ARCH(8,)models, pseudo-maximum likelihoodestimation, asymptotic inference

Optimality and Diversifiability of Mean Variance and Arbitrage Pricing Portfolios

This paper investigates the limit properties of mean-variance (mv) and arbitrage pricing (ap) trading strategies using a general dynamic factor model, as the number of assets diverge to infinity. It extends the results obtained in the literature for the exact pricing case to two other cases of asymptotic no-arbitrage and the unconstrained pricing scenarios. The paper characterizes the asymptotic behaviour of the portfolio weights and establishes that in the non-exact pricing cases the ap and mv portfolio weights are asymptotically equivalent and, moreover, functionally independent of the factors conditional moments. By implication, the paper sheds light on a number of issues of interest such as the prevalence of short-selling, the number of dominant factors and the granularity property of the portfolio weights.large portfolios, factor models, mean-variance portfolio, arbitrage pricing, market (beta) neutrality, well diversification