Spatial and spatio-temporal GARCH models

该项目旨在开发空间统计的新模型，该模型处理空间随机过程的分析。这些过程在实证研究中非常重要，尤其是在计量经济学中。例如，空间统计涵盖了对地球表面或大气过程的分析，如空气污染物和颗粒物、建筑用地的区域价格或城市人口。一般来说，人们可以观察到，在空间中距离较近的观测结果，比在空间中距离较远的观测结果更相似；例如，如果一个城市的建筑用地价格很高，那么人们可能会预期邻近的城市也会有很高的价格。这种现象可以用空间自回归过程来模拟。除了观测值的这种空间依赖性之外，还可以观察到数据变化和条件异方差的模拟依赖性。在这个项目中，应该为显示这种行为的数据开发统计模型。特别是，空间模型的定义方式与Robert F. Engle（1982）发明的时间序列ARCH模型类似，Robert F. Engle因该理论于2003年获得诺贝尔经济学奖。此外，还应引入多元空间ARCH模型，使多个统计变量可以同时建模，例如几种环境污染物和颗粒物。在本例中，空间依赖性受到风向和风速的影响，而风向和风速实际上是随机变量。因此，该项目的另一个方面是对随机空间依赖性和加权方案的分析。

The project aims to develop new models in spatial statistics, which deals with the analysis of random processes in space. Such processes are highly important in empirical research and particularly in econometrics. For instance, spatial statistics covers the analysis of processes on the surface of the Earth or the atmosphere, like air pollutants and particulate matters, regional prices for building land, or the population in municipalities. Generally, one can observe that observations, which are close together in space, are more similar than observations that are more distant in space; e.g., if the prices for building land are high in one municipality, then one might expect high prices in the neighboring municipalities. This phenomenon can be modeled by spatial autoregressive processes.Beside this spatial dependence of the observed values, an analog dependence can be observed for the variation of the data and the conditional heteroscedasticity. In this project, statistical models should be developed for data showing this behavior. In particular, the spatial model is defined in an analogous manner to the time-series ARCH model invented by Robert F. Engle (1982), who won the Nobel Memorial Prize in Economics for this theory in 2003.In addition, a multivariate spatial ARCH model should be introduced, such that several statistical variables can be modeled simultaneously, like for instance, several environmental pollutants and particulate matters. For this example, the spatial dependence is influenced by the wind direction and speed, which are in fact stochastic variables. Thus, a further aspect of the project is the analysis of stochastic spatial dependence and weighting schemes.