title: Evaluation of estimators for ill-posed statistical problems subject to multicollinearity

authors: Holland, Luke Murray

subjects: Multicollinearity

abstract: Multicollinearity is a significant problem in economic analysis and occurs in any situation where at least two of the explanatory variables in a model are related to one another. The presence of multicollinearity is problematic, as changes in the dependent variable cannot be accurately attributed to individual explanatory variables. It can cause estimated coefficients to be unstable and have high variances, and thus be potentially inaccurate and inappropriate to guide management or policy. Due to this problem, many alternative estimators have been developed for the analysis of multicollinear data. 
The primary objective of this thesis is to compare and contrast the performance of some of these common estimators, as well as a number of new estimators, and test their prediction accuracy and precision under various circumstances. Through the use of non-trivial Monte Carlo experiments, the estimators are tested under 10 different levels of multicollinearity, with regressors and errors drawn from different distributions (normal,  student t, chi-squared, and in the case of errors, mixed Gaussian). Insights are gained through response surface analysis, which is conducted to help summarise the output of these simulations.
A number of key findings are identified. The highest levels of mean square error (MSE) are generally given by a Generalised Maximum Entropy estimator with narrow support bounds defined for its coefficients (GMEN) and the One-Step Data Driven Entropy (DDE1) model. Yet, none of the estimators evaluated produced sufficiently high levels of MSE to suggest that they were inappropriate for prediction. The most accurate predictions, regardless of the distributions tested or multicollinearity, were given by Ordinary Least Squares (OLS). The Leuven-2 estimator appeared relatively robust in terms of MSE, being reasonably invariant to changes in condition number, and error distribution. However, it was unstable due to variability in error estimation arising from the arbitrary way that probabilities are converted to coefficient values in this framework. In comparison, MSE values for Leuven-1 were low and far more stable than those reported for Leuven-2.
The estimators that produced the least precision risk, as measured through mean square error loss (MSEL), were the GMEN and Leuven-1 estimators. However, the GMEN model requires exogenous information and, as such, is much more problematic to accurately apply in different contexts. In contrast, two models had very poor precision in the presence of multicollinear data, the Two-Step Data Driven Entropy (DDE2) model and OLS, rendering them inappropriate for estimation in such circumstances.
Overall, these results highlight that the Leuven-1 estimator is the most appropriate if a practitioner wishes to achieve high prediction accuracy and precision in the presence of multicollinearity. Nevertheless, it is critical that more attention is paid to the theoretical basis of the Leuven-1 estimator, as relating estimated probabilities to coefficients using concepts drawn from the theory of light appears highly subjective. This is illustrated through the differences in empirical results obtained for the Leuven-1 and Leuven-2 estimators.

text: Evaluation of estimators for ill-posed statistical problems subject to multicollinearity Multicollinearity is a significant problem in economic analysis and occurs in any situation where at least two of the explanatory variables in a model are related to one another. The presence of multicollinearity is problematic, as changes in the dependent variable cannot be accurately attributed to individual explanatory variables. It can cause estimated coefficients to be unstable and have high variances, and thus be potentially inaccurate and inappropriate to guide management or policy. Due to this problem, many alternative estimators have been developed for the analysis of multicollinear data. 
The primary objective of this thesis is to compare and contrast the performance of some of these common estimators, as well as a number of new estimators, and test their prediction accuracy and precision under various circumstances. Through the use of non-trivial Monte Carlo experiments, the estimators are tested under 10 different levels of multicollinearity, with regressors and errors drawn from different distributions (normal,  student t, chi-squared, and in the case of errors, mixed Gaussian). Insights are gained through response surface analysis, which is conducted to help summarise the output of these simulations.
A number of key findings are identified. The highest levels of mean square error (MSE) are generally given by a Generalised Maximum Entropy estimator with narrow support bounds defined for its coefficients (GMEN) and the One-Step Data Driven Entropy (DDE1) model. Yet, none of the estimators evaluated produced sufficiently high levels of MSE to suggest that they were inappropriate for prediction. The most accurate predictions, regardless of the distributions tested or multicollinearity, were given by Ordinary Least Squares (OLS). The Leuven-2 estimator appeared relatively robust in terms of MSE, being reasonably invariant to changes in condition number, and error distribution. However, it was unstable due to variability in error estimation arising from the arbitrary way that probabilities are converted to coefficient values in this framework. In comparison, MSE values for Leuven-1 were low and far more stable than those reported for Leuven-2.
The estimators that produced the least precision risk, as measured through mean square error loss (MSEL), were the GMEN and Leuven-1 estimators. However, the GMEN model requires exogenous information and, as such, is much more problematic to accurately apply in different contexts. In contrast, two models had very poor precision in the presence of multicollinear data, the Two-Step Data Driven Entropy (DDE2) model and OLS, rendering them inappropriate for estimation in such circumstances.
Overall, these results highlight that the Leuven-1 estimator is the most appropriate if a practitioner wishes to achieve high prediction accuracy and precision in the presence of multicollinearity. Nevertheless, it is critical that more attention is paid to the theoretical basis of the Leuven-1 estimator, as relating estimated probabilities to coefficients using concepts drawn from the theory of light appears highly subjective. This is illustrated through the differences in empirical results obtained for the Leuven-1 and Leuven-2 estimators.

year: 2014

title: Monitoring the landslide at Bramley Drive, Tauranga, NZ

authors: Moon, Vicki G.

subjects: landslide monitoring

abstract: Omokoroa Peninsula, Tauranga Harbour, is prone to landslides in sensitive pyroclastic soils, especially in coastal bluffs. The largest is the landslide at Bramley Drive that first occurred in 1979, and was reactivated in 2011 and 2012. Since 2012 the landslide has been monitored with laser scans, vibro-and static-CPT, pore water logging at 3 depths, and a borehole inclinometer. Laser scan results track degradation of the scarp and allow development of a preliminary magnitude-frequency curve for failure events on the scarp. To date the borehole inclinometer has shown no obvious shear surface development. However, deformations in phase with the solid earth tides are evident in the cumulative displacement plots. Layers of weak soils separated by sharp boundaries are believed to exaggerate the deformations of the solid earth tides to the extent that they are measurable with a simple inclinometer. Residual deformation after subtraction of the earth tide effects indicate some movement over winter of 2014 associated with sensitive soils at or near the failure surface. The depth of this movement corresponds with a zone of high induced pore water pressures under vibratory CPTu. Pore water pressures indicate two discrete aquifers: an upper aquifer in tephra layers high in the upper part of the sequence that responds to atmospheric pressures; and a second aquifer in the underlying ignimbrites. Pressures in the bottom aquifer are lower than in the overlying aquifer in summer and higher in winter. Large spikes in pore water pressure have been observed during winter of 2014; these coincide with the time of deformation noted in the inclinometer traces.

text: Monitoring the landslide at Bramley Drive, Tauranga, NZ Omokoroa Peninsula, Tauranga Harbour, is prone to landslides in sensitive pyroclastic soils, especially in coastal bluffs. The largest is the landslide at Bramley Drive that first occurred in 1979, and was reactivated in 2011 and 2012. Since 2012 the landslide has been monitored with laser scans, vibro-and static-CPT, pore water logging at 3 depths, and a borehole inclinometer. Laser scan results track degradation of the scarp and allow development of a preliminary magnitude-frequency curve for failure events on the scarp. To date the borehole inclinometer has shown no obvious shear surface development. However, deformations in phase with the solid earth tides are evident in the cumulative displacement plots. Layers of weak soils separated by sharp boundaries are believed to exaggerate the deformations of the solid earth tides to the extent that they are measurable with a simple inclinometer. Residual deformation after subtraction of the earth tide effects indicate some movement over winter of 2014 associated with sensitive soils at or near the failure surface. The depth of this movement corresponds with a zone of high induced pore water pressures under vibratory CPTu. Pore water pressures indicate two discrete aquifers: an upper aquifer in tephra layers high in the upper part of the sequence that responds to atmospheric pressures; and a second aquifer in the underlying ignimbrites. Pressures in the bottom aquifer are lower than in the overlying aquifer in summer and higher in winter. Large spikes in pore water pressure have been observed during winter of 2014; these coincide with the time of deformation noted in the inclinometer traces.

year: 2015

title: Reflections on the renewal of music: a polemic

authors: Lodge, Martin

abstract: Pluralism has been accepted as the best broad working definition of the current cultural and artistic situation in the West for some time now. In New Zealand, the debate over 'biculturalism' versus 'multiculturalism' drones on. Commercially this is reflected in both 'market segmentation' and its opposite-attempts at 'cross-over'. Artistically, it has made 'bicultural' works fashionable, as creative
attempts are made to fuse various traditions together, especially the Maori and Western, and nowhere more so than in music. The key to making successful 'fusion' pieces seems to lie in first grasping the essential differences between the musics, then trying to resolve the clashes or accommodate friendly juxtapositions.

text: Reflections on the renewal of music: a polemic Pluralism has been accepted as the best broad working definition of the current cultural and artistic situation in the West for some time now. In New Zealand, the debate over 'biculturalism' versus 'multiculturalism' drones on. Commercially this is reflected in both 'market segmentation' and its opposite-attempts at 'cross-over'. Artistically, it has made 'bicultural' works fashionable, as creative
attempts are made to fuse various traditions together, especially the Maori and Western, and nowhere more so than in music. The key to making successful 'fusion' pieces seems to lie in first grasping the essential differences between the musics, then trying to resolve the clashes or accommodate friendly juxtapositions.

year: 1998