The Role of a Concentration Ellipsoid as a Geometric Tool in Interpreting the Efficiency of Econometric Estimators
Title: The Role of a Concentration Ellipsoid as a Geometric Tool in Interpreting the Efficiency of Econometric Estimators
Author: Preetha Rajan
Supervisor: Dr. Alan J. Rogers
Date of submission: 21st November, 2011
A dissertation submitted in part fulfilment of the requirements for the degree of Bachelor of Commerce (Honours) in Economics, The University of Auckland, 2011.
Abstract
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 Xβ. In this connection, the condition Va ∈ rg(X) is not only a necessary and sufficient condition for a linear estimator a′y to be BLU for its expectation, but this condition essentially stresses on the
Author: Preetha Rajan
Supervisor: Dr. Alan J. Rogers
Date of submission: 21st November, 2011
A dissertation submitted in part fulfilment of the requirements for the degree of Bachelor of Commerce (Honours) in Economics, The University of Auckland, 2011.
Abstract
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 Xβ. In this connection, the condition Va ∈ rg(X) is not only a necessary and sufficient condition for a linear estimator a′y to be BLU for its expectation, but this condition essentially stresses on the
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