The risk–return relation is an important topic in finance. To quantify such relation, the article introduces a new heteroscedastic volatility model, called the generalized-risk-in-mean model (GRM), and considers its entire statistical inference procedure. To adapt potential heavy-tailed phenomena in financial data or an over-parametrization problem in modeling, the self-weighted quasi-maximum likelihood estimation (S-QMLE) is studied and its asymptotics is established, which is non-standard when null volatility coefficients exist. Compared with GARCH-in-mean models (GARCH-M) in the literature, asymptotics of the S-QMLE of our model is much easier to be established under some tractable yet simple conditions. It is very difficult to obtain asymptotics of the QMLE in the GARCH-M, which requires many over-complicated assumptions that are hard to verify in practice. Further, the Wald, Lagrange multiplier, and quasi-likelihood ratio tests are proposed to test for coefficients, and their limiting distributions are derived. Simulation studies are conducted to assess the finite-sample performance of the entire statistical inference procedure and a real example is analyzed to illustrate the usefulness of the GRM.

中文翻译：

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经验风险回报关系的估计：广义平均风险模型

风险收益关系是金融领域的一个重要课题。为了量化这种关系，本文引入了一种新的异方差波动率模型，称为广义均值风险模型 (GRM)，并考虑了其整个统计推断过程。为了适应金融数据中潜在的重尾现象或建模中的过度参数化问题，研究了自加权拟极大似然估计（S-QMLE）并建立了它的渐近性，当波动系数为零时是非标准的存在。与文献中的 GARCH-in-mean 模型 (GARCH-M) 相比，我们模型的 S-QMLE 的渐近线在一些易于处理但简单的条件下更容易建立。在 GARCH-M 中很难获得 QMLE 的渐近线，这需要许多在实践中难以验证的过于复杂的假设。此外，还提出了 Wald、Lagrange 乘数和准似然比检验来检验系数，并推导出它们的极限分布。进行模拟研究以评估整个统计推断过程的有限样本性能，并分析一个真实示例以说明 GRM 的有用性。