To guarantee the non-negativity of the conditional variance of the GARCH process, it is sufficient to assume the non-negativity of its parameters. This condition was empirically violated besides rendering the GARCH model more restrictive. It was subsequently relaxed for some GARCH orders by necessary and sufficient constraints. In this paper, we generalized an approach for the QML estimation of the GARCH(p,q) parameters for all orders $p\geq 1$ and $q\geq1$ using a constrained Kalman filter. Such an approach allows a relaxed QML estimation of the GARCH without the need to identify and/or apply the relaxed constraints to the parameters. The performance of our method is demonstrated through Monte Carlo simulations and empirical applications to real data.
The component GARCH (CGARCH) is suitable to better capture the short and long term of the volatility dynamic. Nevertheless, the parameter space constituted by the constraints of the non-negativity of the conditional variance, stationary and existence of moments, is only ex-post defined via the GARCH representation of the CGARCH. This is due to the lack of a general method to determine a priori the relaxed constraints of non-negativity of the CGARCH($N$) conditional variance for any $N\geq 1$. In this paper, a CGARCH parameter space constructed from the GARCH(1,1) component parameter spaces is provided a priori to identifying its GARCH form. Such a space fulfils the relaxed constraints of the CGARCH conditional variance non-negativity to be pre-estimated ensuring the existence of a QML estimation in the sense of the stochastic approximation algorithm. Simulation experiment as well as empirical application to the S\&P500 index are presented and both show the performance of the proposed method.
This paper provides algorithms for the numerical estimation of the log-GARCH model parameters with no assumptions on the existence of the log-moment orders greater than one. Our approach is based on the quasi-maximum likelihood estimation combined with the information filter. The proposed estimation is employed for two aims. The first is to treat the zero returns considered as missing values through an EM imputation algorithm. The second is to compute the kurtosis of the log-GARCH process by the so-called, right and left measures. A Monte Carlo simulation is performed to investigate the potential of the proposed algorithms to improve the accuracy of the quasi-maximum likelihood for the parameter estimation and the treatment of zero returns as well as to check the robustness of the used kurtosis measures.
Breast Cancer is a major public health problem and the most common diagnosed malignancy in woman. There have been significant developments in clinical approaches and theoretical experimental to understand the interactions of cancer cells dynamics with the immune system, also developments on analytical and computational models to help provide insights into clinical observations for a better understanding of cancer cells, but more are needed, especially at the genetic and molecular levels mathematically. Treatments such as immunotherapy, chemotherapy, hormone therapy, radiotherapy, and gene therapy are the main strategies in the fight against breast cancer. The present study aims at investigating the effects of estrogen derived from recent models, but this time combined with immunotherapy as a way to treat or inhibit the cancer growth by a mathematical model of breast cancer in situ, governed by a simplified model of nonlinear-coupled ordinary differential equations, that combines important interactions between natural cells, tumor cells, immune cells, ketogenic diet in the presence of an anticancer drug. Another contribution was to introduce the inhibition effect ǫ for new results and conclusions, A qualitative study was performed and biological interpretations were included to understand the conditions of stability in a realistic way.
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