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Focused Weighted-Average Least Squares Estimator

Shou-Yung Yin

Abstract

We propose a focused weighted-average least squares (FWALS) estimator that addresses the computational burden of focused model averaging. By semi-orthogonalizing auxiliary regressors, the weighting problem is reduced from $2^{k_2}$ sub-models to at most $k_2$ regressor-wise weights, yielding a tractable sub-optimal procedure. Under local-to-zero conditions, we derive the limiting distribution of FWALS for smooth focused functions and provide a plug-in AMSE criterion for data-driven weight selection. Simulations show that FWALS closely matches the focused information criterion (FIC) benchmark and delivers stable performance when focused function is designed for impulse response function. Prior-based WALS can be competitive in some settings, but its performance depends on the signal regime and the design of focused parameter. Overall, FWALS offers a practical and robust alternative with substantial computational savings.

Focused Weighted-Average Least Squares Estimator

Abstract

We propose a focused weighted-average least squares (FWALS) estimator that addresses the computational burden of focused model averaging. By semi-orthogonalizing auxiliary regressors, the weighting problem is reduced from sub-models to at most regressor-wise weights, yielding a tractable sub-optimal procedure. Under local-to-zero conditions, we derive the limiting distribution of FWALS for smooth focused functions and provide a plug-in AMSE criterion for data-driven weight selection. Simulations show that FWALS closely matches the focused information criterion (FIC) benchmark and delivers stable performance when focused function is designed for impulse response function. Prior-based WALS can be competitive in some settings, but its performance depends on the signal regime and the design of focused parameter. Overall, FWALS offers a practical and robust alternative with substantial computational savings.
Paper Structure (15 sections, 3 theorems, 45 equations, 13 figures, 1 table)

This paper contains 15 sections, 3 theorems, 45 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Specification
  3. Sub-optimal Averaging Estimator
  4. Focused Weighted-Average Least Squares Estimator
  5. The Asymptotic Risk of Focused WALS
  6. Asymptotic Mean Squared Error
  7. Plug-in Focused WALS
  8. Comparison with Bayesian Approaches
  9. Simulation Studies
  10. Basic Design
  11. Impulse Response Function Design
  12. Alternative Methods
  13. Small Sample Properties
  14. Computational Time Comparison
  15. Concluding Remarks

Key Result

Theorem 1

Under Assumptions ass:loca-ass:clt, and a non-stochastic $\tilde{\MBW}$: and where $\BXi=p\lim_{N\rightarrow\infty}\hat{\BXi}=p\lim_{N\rightarrow\infty}\hat{\MBQ}_{11}^{-1}\hat{\MBQ}_{12}=\MBQ_{11}^{-1}\MBQ_{12}$, $\MBC=p\lim_{N\rightarrow\infty}\hat{\BLambda}\hat{\MBP}^{-1/2}=\BLambda\MBP^{-1/2}$, $\MBB=\left[-\MBC^{\tp}\BXi^{\tp}\quad\MBC^{\tp}\right]$ and $\BPsi = \left[\

Figures (13)

  • Figure 1: Weights: Theoretical AMSE vs. Bayesian Priors
  • Figure 2: Risk with Different $k_2$s ($N=100$,$\tau=0.3$)
  • Figure 3: Risk with Different $k_2$s ($N=100$,$\tau=0.5$)
  • Figure 4: Risk with Different $k_2$s ($N=100$,$\tau=0.7$)
  • Figure 5: Risk with Different $k_2$s ($N=200$,$\tau=0.3$)
  • ...and 8 more figures

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • proof : Proof of Theorem 1
  • proof : Proof of Theorem 2 and Equation 14