英文标题：
《An Explicit Formula for Likelihood Function for Gaussian Vector
  Autoregressive Moving-Average Model Conditioned on Initial Observables with
  Application to Model Calibration》
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作者：
Du Nguyen
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最新提交年份：
2016
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英文摘要：
  We derive an explicit formula for likelihood function for Gaussian VARMA model conditioned on initial observables where the moving-average (MA) coefficients are scalar. For fixed MA coefficients the likelihood function is optimized in the autoregressive variables $\\Phi$\'s by a closed form formula generalizing regression calculation of the VAR model with the introduction of an inner product defined by MA coefficients. We show the assumption of scalar MA coefficients is not restrictive and this formulation of the VARMA model shares many nice features of VAR and MA model. The gradient and Hessian could be computed analytically. The likelihood function is preserved under the root invertion maps of the MA coefficients. We discuss constraints on the gradient of the likelihood function with moving average unit roots. With the help of FFT the likelihood function could be computed in $O((kp+1)^2T +ckT\\log(T))$ time. Numerical calibration is required for the scalar MA variables only. The approach can be generalized to include additional drifts as well as integrated components. We discuss a relationship with the Borodin-Okounkov formula and the case of infinite MA components.
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中文摘要：
我们推导了高斯VARMA模型在初始观测条件下的似然函数的显式公式，其中移动平均（MA）系数是标量。对于固定MA系数，自回归变量$\\Phi$中的似然函数通过一个封闭式公式进行优化，该公式推广了VAR模型的回归计算，并引入了由MA系数定义的内积。我们证明了标量MA系数的假设是不受限制的，并且VARMA模型的这种形式与VAR和MA模型有许多共同的优点。梯度和Hessian可以解析计算。似然函数保留在MA系数的根反转映射下。我们讨论了移动平均单位根似然函数梯度的约束条件。借助FFT，似然函数可以用$O（（kp+1）^2T+ckT\\log（T））$时间计算。仅标量MA变量需要进行数值校准。该方法可以推广到包括附加漂移和集成组件。我们讨论了Borodin-Okonkov公式与无限MA分量的关系。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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