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Properties of Estimated Characteristic Roots
Abstract
We introduce a new hybrid approach to joint estimation of Value at Risk (VaR) and Expected Shortfall (ES) for high quantiles of return distributions. We investigate the relative performance of VaR and ES models using daily returns for sixteen stock market indices (eight from developed and eight from emerging markets) prior to and during the 2008 financial crisis. In addition to widely used VaR and ES models, we also study the behavior of conditional and unconditional extreme value (EV) models to generate 99 percent confidence level estimates as well as developing a new loss function that relates tail losses to ES forecasts. Backtesting results show that only our proposed new hybrid and Extreme Value (EV)-based VaR models provide adequate protection in both developed and emerging markets, but that the hybrid approach does this at a significantly lower cost in capital reserves. In ES estimation the hybrid model yields the smallest error statistics surpassing even the EV models, especially in the developed markets.
... They speculate that such a strange pattern reflects the over-fitting rather than the persistence of the underlying series. Nielsen and Nielsen (2008) point out that the usual √ T rate of convergence slows down to T 1/2k for the roots of k-th order. They use this fact to provide a partial explanation of the 'halo phenomenon'. ...
We introduce a new hybrid approach to joint estimation of Value at Risk (VaR) and Expected Shortfall (ES) for high quantiles of return distributions. We investigate the relative performance of VaR and ES models using daily returns for sixteen stock market indices (eight from developed and eight from emerging markets) prior to and during the 2008 financial crisis. In addition to widely used VaR and ES models, we also study the behavior of conditional and unconditional extreme value (EV) models to generate 99 percent confidence level estimates as well as developing a new loss function that relates tail losses to ES forecasts. Backtesting results show that only our proposed new hybrid and Extreme Value (EV)-based VaR models provide adequate protection in both developed and emerging markets, but that the hybrid approach does this at a significantly lower cost in capital reserves. In ES estimation the hybrid model yields the smallest error statistics surpassing even the EV models, especially in the developed markets.
Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 and the initial conditions (Y−p+1,…, Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.
An exact discrete model is derived from a recursive model consisting of a set of rth order stochastic linear differential equations with constant coefficients such that observations generated at equidistant points of time by the differential system satisfy the discrete model irrespective of the length of sampling interval. The difficulty of estimating the exact model subject to a priori restrictions makes it necessary to approximate the differential system by a non-recursive discrete model that maintains the structural form. This discrete model has a moving average disturbance term of order r - 1, but the co-variance function of this process is approximated by a function that is independent of the parameters of the continuous model. The eigenvalues and eigenvectors of the approximations to the differential system as well as their asymptotic variance matrices are also derived but, like the approximate asymptotic variance matrices of the parameter estimates, these variances are about biased probability limit
One method of describing the properties of a fitted autoregressive model of order p is to show the p roots that are implied by the lag operator. Considering autoregressive models fitted to 215 US macro series, with lags chosen by either the Bayesian or Schwarz information criteria or Akaike information criteria, the roots are found to constitute a distinctive pattern. Later analysis suggests that much of this pattern occurs because of overfitting of the models. An extension of the results shows that they have some practical multivariate time-series modelling implications. Copyright 2006 Blackwell Publishing Ltd.
We show that the asymptotic distribution of the estimated stationary roots in a vector autoregressive model is Gaussian. A simple expression for the asymptotic variance in terms of the roots and the eigenvectors of the companion matrix is derived. The results are extended to the cointegrated vector autoregressive model and we discuss the implementation of the results for complex roots. Copyright 2003 Blackwell Publishing Ltd.