The Role of a Concentration Ellipsoid as a Geometric Tool

This dissertation, by making use of important geometric and econometric concepts such as a ‘linear manifold’, a ‘plane of support’, a ‘projection matrix’, ‘linearly estimable parametric functions’, ‘minimum variance estimators’ and ‘linear transformations’, seeks to explore the role of a Concentration Ellipsoid as a geometric tool in the interpretation of certain key econometric results connected with the efficiency of estimators of the linear regression model and thereby present these econometric results in a whole new light, namely through simple geometric interpretation.

The relationship between the Concentration Ellipsoid, its planes of support and the range space is pivotal in the geometric interpretation of econometric results connected with the efficiency of estimators of a linear regression model. The orientation of the planes of support (as determined by their normals), ultimately determines the angle at which observations of a random vector y (whilst undergoing a linear transformation) are projected on to the range space of X.

In a geometric context, it is essential that the size of the image of the Concentration Ellipsoid of a random vector y, projected on to the range space of X (as a result of the random vector y undergoing a linear transformation) is as small as it can possibly be, for the projection estimator Py to be deemed as an efficient estimator of X?. In this connection, the condition Va ? rg(X) is not only a necessary and sufficient condition for a linear estimator a? to be BLU for its expectation, but this condition essentially stresses on the importance of a plane of support being oriented in such a way that it is tangent to the Concentration Ellipsoid ? , such that this point of tangency is a point on the range space of X. The question of a minimum variance estimator can simply be narrowed down to the geometric idea that the Concentration Ellipsoid of the more efficient estimator is completely contained within the Concentration Ellipsoid of the less efficient estimator.

In the context of examining whether or not an efficiency loss occurs (in the context of the loss of information through discarding observations in model) by subjecting the linear regression model y = X? + u to a transformation, by means of a transformation matrix T, the condition VT? b ? rg(X) (similar to the necessary and sufficient condition Va ? rg(X)) is required to be satisfied in order for the linear estimator b? Ty to be BLUE for its expectation to conclude that no efficiency loss occurs as a result of such a transformation.

This dissertation has essentially explored the possibility of utilising the concept of a Concentration Ellipsoid, for the purpose of geometrically interpreting certain key econometric results, connected with the efficiency of estimators of a linear regression model, in the context of ‘projection estimators’, ‘minimum variance estimators’, transformations from the aspect of efficiency losses in terms of the potential loss of information by discarding observations of the model, etc. Geometric concepts such as a linear manifold (also known as the range space) and a plane support seem to have a crucial role to play in the geometric depiction of the efficiency of various econometric estimators. At this point, it would be appropriate to highlight the main conclusions which can be gathered from the geometric exposition of the econometric results examined in this dissertation.

We begin by recalling that the orientation of the planes of support is entirely determined by their normals, which happen to be the only vectors in the Euclidean space. It is the nature of the orientation of these planes of support which ultimately determine the angles at which observations of a random vector y (whilst undergoing a linear transformation) are projected on to the range space of X. In a geometric context, it is essential that the size of the image of the Concentration Ellipsoid of a random vector y, projected on to the range space of X (as a result of the random vector y undergoing a linear transformation) is as small as it can possibly be, for