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1 Modelling multivariate returns Stefano Herzel Department ofeconomics, University of Perugia 1 Catalin Starica Department of Mathematical Statistics, Chalmers University of Technology Reha Tutuncu Department of Mathematics, Carnegie Mellon University Extended abstract Although the multivariate normality of stock returns is a crucial assumption in many assets pricing models, the modern econometric literature abounds with evidence against hypothesis. The prescriptions of modern risk management in particular depend critically on a correct description of the distribution of future returns. The current standard in the short-term risk management practice is that of RiskMetrics, a modelling approach that assumes the m-dimensional multivariate return vector to be conditionally normal: Currently, RiskMetrics methodology concentrates on the modelling the changing structure of conditional covariances without addressing the modelling of the conditional residuals t. These residuals have been shown by empirical evidence to be far from the multivariate normality assumption. The specicity of our approach is twofold. Firstly, unlike most of the current multivariate asset return models of the ARCH or stochastic volatility type, we do not assume the covariance structure to be (unconditionally) stationary. We acknowledge this way the fast and imprevisible shifts in the dependency structure of nancial instruments. Once we remove the changing covariance structure, the residuals can be modelled as a series of iid vectors that have independent coordinates with asymmetric heavy tails. Accuratelly 1 herzel@unipg.it starica@math.chalmers.se reha@math.cmu.edu

2 modelling the heavy tailed distribution of the residuals is the second contribution of the present work. We show that a simple univariate distribution with asymmetric heavy tails models exceptionally well the returns of a wide variety of nancial instruments: foreign exchanges, interest rates, market indexes. Hence, our approach decomposes the modelling of the multivariate process of returns into two steps: rst, that of estimating a changing, non-stationary covariance structure and second, that of modelling the relatively stable, quasi-stationary residuals. In the estimation phase of the changing covariance structure we document the fact that RiskMetrics methodology provides a sensible balance between technical simplicity and eciency. Eliminating the source of non-stationarity through the estimation in the rst phase, allows us to conduct a detailed analysis of the residual, analysis based on powerful statistical tools for modelling stationary time series. Describing the multivariate, heavy tail distribution of the residuals goes beyond the RiskMetrics methodology and in this sense, our modelling approach can be thought asan extension of it. We conduct an in-depth analysis on a three dimensional vector of daily log-returns spanning the period between the begining of 1 and the end of, a total of 1 observations. For the sake of diversity we chose three qualitatively dierent nancial instrumets: one foreign exchange rate, between the Australian dollar (AUS) and US dollar, an index, the S&P and an interest rate, the 1 year US T-bond. We restricted our analysis to a trivariate set up in order to be able to graphically display the results of the analysis. We start by investigating the multivariate distribution of our data. Figure takes a look at the marginal distributions and displays the QQ plots of the marginals against the normal. The graphs show that all six tails of the three marginal distributions are heavier than the normal tails Figure 1: QQ plots of the three series of data (AUD, SP, T-bond) agains the normal. Before saying more about the multivariate distribution of our returns, we need to present the following statistical test of ellipticity andmultivariate normality. Dene (j) i to be the angle ri makes with the j-th axis. Under the hypothesis of an elliptical distribution, (1) i are approximately uniformly distributed on [ ) while the remaining (m;) angles have probability distribution functions proportioned to j! sin m;1;j j, j =1 ::: m;. Hence plotting the n ordered values of the j-th angle transformed with the cdf of

3 this distribution against the corresponding quantiles of an uniform produces under the hypothesis of an elliptical distribution a straight line graph. Graphs that stray away from the staight line indicate that an elliptical distribution does not provide an accurate description of the data. Under the null hypothesis that the observations come from an multivariate normal with covariance S the radius d i satises Hence plotting sorted d i against the corresponding quantiles of m in the so-called squared radius plot produces, under the null hypothesis, i.e. normal tails for all coordinates a linear graph. This plot contains pooled information about the tails of the multivariate distribution. If the tails of the marginals are heavier than the normal ones the graph will have abent aspect. A graph that lies under the degree line signals that at least one of the tails is heavier than the normal. A graph which bends above the degree line is a sign of lighter tails Figure : Multivariate normality test for the three series of original data. For the sake of clarity and in order to help asses the amount of statistical error, our graphs will display the distance between the pairs of quantiles and the degree line together with condence intervals based on the asymptotic distribution of the supremum of this dierence, i.e. the supremum of a Brownian bridge. For the radius, besides the squared radius plot (which we plot mainly for traditional reasons) we produce also the type of graph described above, i.e. the distance between pairs of ordered values transformed with the cdf of a m and the corresponding uniform quantiles and the -degree line.

4 The serious violations of the condence intervals in the top two graphs of Figure show that, from a distributional point of view one cannot directly model the data as a stationary sequence drawn from an elliptical distribution. The bottom two graphs clearly indicate that the tails of the data are not normal, more specically that they are heavier than the normal tails. We continue with a brief analysis of the changes in the unconditional variance which we believe are the cause of the long memory eect present in the series of absolute values. For a more in-depth discussion we refer the reader to Mikosch and Starica ([1]). We show that accounting for the changing estimated covariance removes the heteroskedasticity which causes the long memory eect in the sample autocovariance function (SACF) of the absolute returns and produces residual vectors that are temporally uncorrelated. Our ndings support RiskMetrics (and ours) choice of modelling the residuals as a sequence of iid vectors. They also clearly show the inadequacy of the multivariate normality model for these residuals. To the positive and to the negative half of every marginal time series we t a simple heavy tail univariate distribution. Then the marginals are transformed rst to uniforms (by applying the estimated distribution function) and then into normals (by transforming these uniforms with the inverse normal cumulative distribution function). Figure and Figure show the coordinate and jointly normal t of these transformed residuals Figure : QQ plots of the three series of residuals transformed with our univariate distribution against the normal. Last two gures show an extremely good t of the model and we continue (omitted here) with an analysis of its out-of-sample performance. We then discuss its stability in time. Figure displays the results of our investigation on the stability of the tail index of the marginal distribution of the residuals. For all but one tail (the right tail of the rd coordinate) the hypothesis of constant tail index cannot be rejected. In all cases but the mentioned one, is it possible to nd values that are covered by all the condence intervals: around. for the rst and the second left tail, around -. for the third left tail and around. for the rst and second right tail. The right tail of the rd series seems to have changed drastically around 1. We see that the tails seem asymmetric with the left tail heavier than the right tail. At least in the case of the rst and third coordinate the patterns of change of the point estimates are diverse: the point estimates of the left tail indexes are going up, indicating possibly a process through

5 Figure : Multivariate normality test for the three series of residuals marginally transformed with our univariate distribution. which the tails are becoming lighter (although as we have already said the hypothesis of constant tail index is not rejected) while the point estimates of the right tails are going down possibly a signal that the tails are becoming heavier. Given the rather long period of observation (almost years) we nd these indications of tail stability quite reassuring. For all the series the tails are quite heavy. On the negative return side, using the information provided by the condence intervals, the rst series do not seem to have a nite th moment while for the third the th moment does not seem to exist. The positive return side seem to display lighter and more variable tails. The rst positive series does not seem to have th, the second 1th and the third th nite moment. Besides the left tail of the second coordinate all others seem to have nite variance. The left tail of the second coordinate has at least nite mean. We emphasise that such a precise tail analysis can be rarely done in the case of the nancial data due to the presence of both non-stationarities and unknown type of dependence. It is the very good t of a precise parametric model and the good approximation the hypothesis of independency provides that facilitates such a precise tail analysis of the residuals. We nish by demonstrating the succesful use of our approach for modelling lower frequency multivariate returns, more precisely weekly returns. Our analysis is important because it opens the perspective of improved high-dimensional modelling and out-of-sample forecasting of nancial returns, which in turn hold promise for the development of better decision making in practical situations of risk management, portofolio allocation and asset pricing.

6 Figure : The left tail index (Top) and the right tail index (Bottom) of the time series of residuals estimated monthly (every 1 days) on a sample of 1 (roughly years) past observations. References [1] Mikosch, T. and Starica, C. () Long memory and the ARCH models. Extremes and Integrated Risk Management, ed. P. Embrechts, Risk Books,.