The Duality Between Unbiased Estimation/prediction In The Linear Model And The Associated Least Squares Problem, And The Theory Of Mean-variance Efficient Portfolios--two Applications Of Quadratic Minimization, G-inverses, And Deficiency Matrices (ridge Regression, Stochastic Parameters).

The first essay gives a unified theory of several applications of quadratic minimization subject to linear constraints in econometrics. In many linear models, the best linear unbiased estimator/predictor can be obtained by a procedure which seems unrelated with unbiasedness or minimization of the mean squared error matrix. That value of the unknown magnitude is the estimate which, together with the vector of observations, minimizes a certain positive semidefinite quadratic form, a weighted sum of squared errors. This GLS approach to the BLUE is possible not only in linear regression, but also for estimators in model with random effects and stochastic parameters, mixed estimation, linear prediction, Kalman filters, ridge regression, and others. The essay introduces a simple linear umbrella specification for all these, derives a general formalism for computing the best linear unbiased estimator/predictor (even with singular covariance matrices and deficient design matrices), and characterizes all those weighting matrices leading to exactly the same results as BLU estimation. The second essay is a critical review of the concept of mean-variance efficiency used in finance. The global minimum variance portfolio g and certain other portfolios on the rising branch of the minimum variance boundary should not be called efficient when there is no riskless asset, because they may be dominated by portfolios including the asset which has a zero payoff with certainty. Those derivations of efficient set mathematics in which g plays a central role are badly designed, unnecessarily complex, and valid only under restrictive assumptions. An alternative, simple and general treatment is given, which starts with three simple axioms. Also discussed is the puzzling result that mean variance investors do not always invest all their money--which is what makes g inefficient.PhDPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127751/2/8520891.pd