On the Asymptotics of Increasing Dimension Models : Methods for Complete or Incomplete Data

Sammanfattning: In some multivariate contexts there is a close relation between the number of parameters (p) and the number of observations (n). In a situation where p grows with n it frequently happens that the statistic does not converge to its true parameter. An additional issue is if the data set also contain missing observations and , for example, as p grows with n so does the number of missing observations. This situation arises in the context of empirical applications of the Arbitrage Pricing Theory model, where the data is incomplete due to the nature of stocks leaving or entering the stock exchange. In this thesis two situations of increasing dimension are considered: firstly, the case of complete data sets where the statistic of interest is the inverse covariance matrix where three types of shrinkage estimators of the inverse covariance matrix are investigated, particularly as an ingredient of a composite estimator, specifically Zellners seemingly unrelated regression  models and the Mahalanobis distance. Secondly, the case arising in empirical application of the APT model where the data set is incomplete and the interest is to model the underlying covariance structure among the variables by a few factors. Two possible solutions to the problem are considered and a case study using the Swedish OMX data is conducted for demonstration.