Figure 2: Estimated Quadratic Variation simulated with FFBS and G-FFBS and the true latent value vertical bar , over trading days, when the correlation between microstructure noise and financial latent return is On the other hand, we sample b1 t with a Hamiltonian step see Chapter 5 in Brooks et al.
The motivation for using this step is its ability to exploit the information in the full conditional gradient of b1 t , for a faster exploration of the parameter space, thus overcoming the random walk behavior of the Metropolis-Hastings step in a highly dimensional space. We refer the Reader to Appendix C for the details on the Hamiltonian step.
To summarize, the procedure for obtaining the IV estimator is the following: 1. Compute the estimator given in Equation Our methodology is compared with the estimators of Kalnina and Linton , Bandi and Russell and Jacod et al.
For completeness, we add to the comparison other popular estimators, as the quasi-maximum likelihood estimator of Xiu , the realized kernel of Barndorff-Nielsen et al. Alternative values of K are shown in Kalnina and Linton to perform worse and depend on unobservable quantities estimated with a slow-decaying bias.
For the estimator proposed in Bandi and Russell , the tuning parameters are chosen according to the rule of thumb proposed in Equation 26 of Bandi and Russell , in simulation computed using the true values and in the application below to Microsoft Corporation, using the corresponding values in Table 1 of Bandi and Russell The results are reported in Figure 3 and in Table 1: there is a clear advantage for our methodology in terms of dispersion and root mean square error RMSE.
The quasi-maximum likelihood estimator performs particularly well in terms of bias, even if it shows some relevant positive dispersion that contributes to increase the RMSE to a level higher than that of the method we propose. We also run the algorithm on 1-second frequency logarithmic prices of Microsoft Corporation, for the period April 1, - June 30, , and the estimated annualized quadratic variations are reported in Figure 4. Furthermore, such a dependence naturally arises in common microstructure models, as discussed in depth in Diebold and Strasser On the other hand, with the notable exceptions of Barndorff-Nielsen et al.
In the present paper we use the theoretical framework of the conditionally Gaussian random sequences of Liptser and Shiryayev , a,b , to propose a new integrated variance estimator that is robust to correlation between microstructure noise and latent returns.
To this aim, we adopt a Bayesian perspective and sample a posteriori the latent price process through a generalization of the Forward Filtering Backward Sampling algorithm of Fruwirth-Schnatter and Carter and Kohn An application to Microsoft 1-second logarithmic prices is provided, and a simulation study shows an improved performance of our estimator in terms of RMSE and dispersion, relative to the alternatives in the literature.
Our methodology can be implemented in other financial problems, for instance to generalize the framework of Barndorff-Nielsen to normal inverse Gaussian financial logarithmic returns with measurement error, or, following the approaches of Harvey et al. P1TIP1 Antonietta Mira also gratefully acknowledges the financial support by SNF. Appendix A: Proof of Proposition 3.
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The encoder and the jammer are assumed to know Rayleigh-fading channel considered in Section III, and show that for a only the distribution of H , i. We independent of H. We also make the natural assumption that n is independent of all the other random variables. We study a mutual information zero-sum game on this channel be- tween the communicator and the jammer. The payoff J is defined as Manuscript received November 12, ; revised April 16, The work of A.
We are The authors are with the Department of Electrical Engineering and the Co- interested in finding the saddle point of J. Medard, Associate Editor for Communications. Rician channel with side information at the receiver.
So we know that 3 We use h x to denote the differential entropy of x and I x ; y the input that maximizes I x; y is CSCG with a covariance Q that to denote the mutual information between x and y. So, we have reduced the problem to a game where the strategy of the definite. We list the reasoning for the sake of complete- ness.
Proof: See Appendix I. Now Lemma 2: [11, Lemma 5] For H 2 r2t with each entry H ij independent and identically distributed i. Now, for any function A H : r2t 7! Proof: See Appendix II. Moreover, any value of E log det 6 6ejH jH allowable by the 2An information-theoretic proof of this lemma appeared earlier in [5, Lemma constraints can also be achieved by a Gaussian distribution on v.
The purpose is to illustrate that for a constant channel, the jammer always makes use of the knowledge of the encoder output, as one would intuitively expect. It now remains to find the optimal 6z. To that end, note that This is essentially the same problem considered in [9]. Since for this v t r! But r1!
The knowledge of x is useless for the jammer. Therefore, x x 3 ; v 3 is the unique saddle point. For this case, we have reduced the problem to the single is convex and symmetric about the origin in xi. Use x1 ;. The saddle-point where X xk ; x0k is shorthand for [x1 ;.
It is the symmetric nature of fading that brings about this result. Multiplying both sides by the joint den- constant channel, the jammer always uses the knowledge of the channel sity of the temporarily fixed columns x1 ;. No credit is claimed for it, as it is a straightforward extension of [8, Theorem 1].
The proof Proving that f is convex over S is equivalent to proving that the function relies on the following two lemmas. Lemma 5: [8, Lemma 3] Let X 2 r2t. Then, using [6, Corollary 4. Here we analyze such systems at low SNR, which may find application in sensor networks and other n d2 g low-power devices.
We show that under various signaling constraints, e. Theory, vol. IT, pp. Olsder, Dynamic Noncooperative Game Theory. Borden, D. Mason, and R. Control Optimiz. Multiple-antenna wireless systems have been shown to provide high [5] S. Diggavi and T. However, Nov. Horn and C.
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