Use of composite likelihood methods for the estimation of probit models

移动性需求模型通常基于离散选择模型，通常有大量的选择方案。主要的模型类别由多项logit（MNL）和-probit（MNP）模型构成。对于不同口味的（混合）MNL模型的数值和统计上有效的估计算法存在的基础上（模拟）的最大似然范式，然而，表现出缺点的表示之间的相关性的随机效用条款为不同的替代品，以及选择的混合分布。对于大的数据集，需要大量的模拟，以确保一致性和渐近效率的估计。MNP模型在另一方面，特别是在面板上下文中提供了良好的建模能力，然而，导致数值上要求的估计问题，因为这些需要评估高维高斯累积分布函数（CDF）。作为替代方案，Chandra Bhat周围的小组提出了“最大复合边际似然”（MaCML）方法，该方法连接了两个想法：用所谓的“复合边际似然”（CML）代替似然，第二，高斯CDF被解析近似。到目前为止，巴特的提议只是受到一些模拟演习的推动。目前还没有彻底的理论研究。调查简单的例子，可以很容易地验证，MaCML方法并不能保证一致的估计。此外，特别选择的近似方法所使用的Bhat以及所选择的CML已受到批评。因此，本项目将处理的MaCML思想的基础上的估计量的属性的详细调查，检查CML函数和CDF近似的选择相对于（i）的渐近偏差的影响，（ii）的相对效率和（iii）基于MaCML估计的模型选择程序的属性。 这是这个项目的主要目标，制定数值效率和统计上合理的估计程序（包括适当的初始化例程）的MNP模型在面板数据设置显示了大量的替代品。在该项目中开发的方法将被用来调查所谓的图案的选择的决定因素（一个人的一天作为有向图的行程表示）。在不同城市的许多不同数据集中，人们在潜在的大量图案中只选择了17个图案中的一个。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性知之甚少。此外，选择的相对频率的时间演变是未知的。这些知识对于基于活动的移动性需求模型的开发具有重要意义。

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.