Use of composite likelihood methods for the estimation of probit models

机动性需求模型通常基于离散选择模型，通常具有大量的选择选项。主要的模型类是由多项logit （MNL）和-probit （MNP）模型构成的。对于不同类型的（混合）MNL模型，存在基于（模拟）最大似然范式的数值和统计上有效的估计算法，然而，在不同替代方案的随机效用项之间的相关性表示以及混合分布的选择方面显示出缺点。对于大数据集，为了保证估计量的一致性和渐近效率，需要进行大量的模拟。另一方面，MNP模型，特别是在面板环境中，提供了良好的建模能力，然而，由于这些需要对高维高斯累积分布函数（CDF）进行评估，导致了数值上要求很高的估计问题。作为替代方案，钱德拉·巴特周围的小组提出了“最大复合边际似然”（MaCML）方法，将两个想法联系起来：可能性被所谓的“复合边际似然”（CML）取代，其次是高斯CDF的分析近似。到目前为止，Bhat的提议只是受到一些模拟演习的推动。目前还没有深入的理论研究。通过简单的例子可以很容易地验证MaCML方法不能保证一致的估计。此外，Bhat所使用的近似方法的特定选择以及所选择的CML也受到了批评。因此，本项目将处理基于MaCML思想的估计器属性的详细调查，检查CML函数和CDF近似的选择对(i)渐近偏差的影响，（ii）相对效率和（iii）基于MaCML估计的模型选择过程的属性。该项目的主要目标是为面板数据设置中的mnp模型开发数字上有效和统计上合理的估计程序（包括适当的初始化程序），显示大量替代方案。该项目中开发的方法将用于调查选择所谓的主题的决定因素（以有向图的形式表示个人一天的行程）。在不同城市的许多不同的数据集中，已经表明，在潜在的大量图案中，人们只从17个图案中选择一个。目前，人们对这种选择背后的决定因素以及对社会人口特征的依赖知之甚少。此外，相对选择频率的时间演变是未知的。这些知识对于基于活动的移动需求模型的开发非常重要。

Mobility demand models are usually based on discrete choice models with often a large number of choice alternatives. The predominant model classes are constituted by the multinomial logit (MNL) and -probit (MNP) models. For different flavors of (mixed) MNL model numerically and statistically efficient estimation algorithms exist based on the (simulated) maximum likelihood paradigm which, however, show disadvantages with respect to the representation of correlation between the random utility terms for different alternatives as well as the choice of the mixing distributions. For large data sets a large number of simulations are needed in order to ensure consistency and asymptotic efficiency of the estimators. MNP models on the other hand in particular in a panel context provide good modeling capabilities, however, lead to numerically demanding estimation problems as these necessitate the evaluation of high dimensional Gaussian cumulative distribution functions (CDF). As an alternative the group around Chandra Bhat proposed the "maximum composite marginal likelihood" (MaCML) approach linking two ideas: the likelihood is replaced by a so called "composite marginal likelihood" (CML) and second the Gaussian CDF is analytically approximated. Bhat's proposal has up to now only been motivated by a number of simulation exercises. A thorough theoretical investigation is currently not available. Investigating simple examples it can be verified easily that the MaCML approach does not guarantee consistent estimation. Also the particular choice of the approximation method used by Bhat as well as the chosen CML has been criticised. Thus this project will deal with a detailed investigation of the properties of estimator on the basis of the MaCML idea, examining the effects of the choice of the CML function and the CDF approximation with respect to (i) the asymptotic bias, (ii) the relative efficiency and (iii) the properties of model selection procedures based on the MaCML estimation. It is the main goal of this project to develop numerically efficient and statistically sound estimation procedures (including adequate initialisation routines) for MNP-models in panel data settings showing a large number of alternatives. The methods developed within the project will be used to investigate the determinants of the choice of so called motifs (representations of the trips of a day of an individual as directed graphs). In a number of different data sets in different cities it has been shown that out of a potentially large number of motifs people only choose one out of 17 motifs. Currently there is little knowledge as to the determinants underlying this choices as well as the dependence on sociodemographic characteristics. Additionally the temporal evolution of the relative frequency of choice are unknown. This knowledge is of importance for the development of activity based mobility demand models.