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Spatially Varying Coefficient Mallows Model Averaging

Zhuang Yong, Lv Jing, Tingting Li

Abstract

Model averaging, as an appealing ensemble technique, strategically integrates all valuable information from candidate models to construct fast and accurate prediction. Despite of having been widely practiced in many fields such as cross-sectional data, censored data and longitudinal data, its application to spatial data characterized by inherent spatial heterogeneity remains surprisingly limited. To mitigate risk of model misspecification and enhance the flexibility of prediction, we propose a combined estimator constructed by computing the weighted average of estimators derived from a set of spatially varying coefficient candidate models. Herein, the model weights are determined via a Mallows-type criterion, which dynamically calibrates the relative importance of individual candidate models in the ensemble. Theoretically, we establish desirable asymptotic properties under two practical scenarios. First, in the case where all candidate models are misspecified, the proposed model averaging estimator attains asymptotic optimality in the sense that it minimizes the squared error loss function asymptotically. Second, when the candidate model set encompasses at least one quasi-correct model, the weights assigned by the Mallows-type criterion asymptotically concentrate on the quasi-correct models, and the resulting model averaging estimator converges in probability to the true conditional mean. Both simulation studies and a real-world empirical example demonstrate that the proposed method generally outperforms alternative comparative approaches in terms of predictive accuracy and robustness.

Spatially Varying Coefficient Mallows Model Averaging

Abstract

Model averaging, as an appealing ensemble technique, strategically integrates all valuable information from candidate models to construct fast and accurate prediction. Despite of having been widely practiced in many fields such as cross-sectional data, censored data and longitudinal data, its application to spatial data characterized by inherent spatial heterogeneity remains surprisingly limited. To mitigate risk of model misspecification and enhance the flexibility of prediction, we propose a combined estimator constructed by computing the weighted average of estimators derived from a set of spatially varying coefficient candidate models. Herein, the model weights are determined via a Mallows-type criterion, which dynamically calibrates the relative importance of individual candidate models in the ensemble. Theoretically, we establish desirable asymptotic properties under two practical scenarios. First, in the case where all candidate models are misspecified, the proposed model averaging estimator attains asymptotic optimality in the sense that it minimizes the squared error loss function asymptotically. Second, when the candidate model set encompasses at least one quasi-correct model, the weights assigned by the Mallows-type criterion asymptotically concentrate on the quasi-correct models, and the resulting model averaging estimator converges in probability to the true conditional mean. Both simulation studies and a real-world empirical example demonstrate that the proposed method generally outperforms alternative comparative approaches in terms of predictive accuracy and robustness.
Paper Structure (13 sections, 119 equations, 13 figures, 3 tables)

This paper contains 13 sections, 119 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Model Framework
  4. Model Averaging and Weight Choice Criterion
  5. Asymptotic Properties
  6. Asymptotic Optimality of the SVMMA Estimator
  7. Consistency of the Model Weights and SVMMA Estimator
  8. Simulation Study
  9. Data Generation Process
  10. Evaluation and Comparison
  11. Results
  12. Empirical Application
  13. Concluding Remarks

Figures (13)

  • Figure 1: Risk Comparisons under Design 1 for Case i. The relative risk is measured against the infeasible OLMA risk.
  • Figure 2: Risk Comparisons under Design 1 for Case ii. The relative risk is measured against the infeasible OLMA risk.
  • Figure 3: Risk Comparisons under Design 1 for Case iii. The relative risk is measured against the infeasible OLMA risk.
  • Figure 4: Risk Comparisons under Design 2 for Case i. The relative risk is measured against the infeasible OSVCMA risk.
  • Figure 5: Risk Comparisons under Design 2 for Case ii. The relative risk is measured against the infeasible OSVCMA risk.
  • ...and 8 more figures