Comments on: Reflections on the Probability Space Induced by Moment Conditions with Implications for Bayesian Inference
评论Gallant关于利用矩函数分布进行有效贝叶斯推断的理论，并将其应用于非高斯Ornstein-Uhlenbeck过程的估计函数，以分析能源现货价格。
Gallant’s interesting article provides some theoretical guidance on how to use the distribution of moment functions to conduct valid Bayesian inference. Let x denote the observed data, θ the parameters, and Z(x,θ) some transformation of moment functions. The distribution of Z induces a probability space (χ×Θ,C,P) . In this setting Gallant shows that when Z is semi-pivotal, imposing a proper prior on θ enlarges the σ -algebra C to C* that includes the rectangles (RK×B)∩(χ×Θ) . Thus, there exists a probability measure P* on C* such that (χ×Θ,C*,P*) can be used for Bayesian inference. We would like to discuss an application of Gallant’s idea to the class of estimation functions, which shares many features with generalized method of moments. Specifically, we illustrate the inference on the superposition of non-Gaussian Ornstein-Uhlenbeck (OU) processes. The non-Gaussian OU process is proposed by Barndorff-Nielsen and Shephard (2001) and it is commonly used for many time series, for example, the focus of this application, energy spot prices. While the non-Gaussian increments capture large price fluctuations that are often observed in energy prices, the OU specification reflects the mean-reverting property. Also, it allows for tractable derivative pricing. However, most of these models do not admit an analytical transitional density or likelihood function. Instead, we can use estimation functions.