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Estimation with Pairwise Observations

Felix Chan, Laszlo Matyas

Abstract

The paper introduces a new estimation method for the standard linear regression model. The procedure is not driven by the optimisation of any objective function rather, it is a simple weighted average of slopes from observation pairs. The paper shows that such estimator is consistent for carefully selected weights. Other properties, such as asymptotic distributions, have also been derived to facilitate valid statistical inference. Unlike traditional methods, such as Least Squares and Maximum Likelihood, among others, the estimated residual of this estimator is not by construction orthogonal to the explanatory variables of the model. This property allows a wide range of practical applications, such as the testing of endogeneity, i.e., the correlation between the explanatory variables and the disturbance terms.

Estimation with Pairwise Observations

Abstract

The paper introduces a new estimation method for the standard linear regression model. The procedure is not driven by the optimisation of any objective function rather, it is a simple weighted average of slopes from observation pairs. The paper shows that such estimator is consistent for carefully selected weights. Other properties, such as asymptotic distributions, have also been derived to facilitate valid statistical inference. Unlike traditional methods, such as Least Squares and Maximum Likelihood, among others, the estimated residual of this estimator is not by construction orthogonal to the explanatory variables of the model. This property allows a wide range of practical applications, such as the testing of endogeneity, i.e., the correlation between the explanatory variables and the disturbance terms.
Paper Structure (26 sections, 5 theorems, 87 equations, 6 figures, 14 tables)

This paper contains 26 sections, 5 theorems, 87 equations, 6 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Some Monte Carlo Simulation Results
  3. Some Analytical Results
  4. Notations and Definitions
  5. Partial Sum
  6. Adjacent Case
  7. Case 1: weight equals to Delta x
  8. Case 2: Weight equals to absolute Delta x
  9. Full-pairwise
  10. Case 1: Weight equals to Delta x
  11. Case 2: Weight equals to Absolute Delta x
  12. Confidence Intervals
  13. What about the Intercept?
  14. The adjacent case
  15. Loss Function
  16. ...and 11 more sections

Key Result

Proposition 1

Under Assumptions ass:moment_x to ass:mixingale with $\hat{\beta}$ as defined in Equation (eq:fullpw_estimator) and $\mathbf{w} = \{\Delta x_{ij} \}^n_{i,j=1}$:

Figures (6)

  • Figure 1: Sorted adjacent
  • Figure 2: Non-sorted full-pairwise
  • Figure 3: Simulated Sample of the Distribution
  • Figure 4: Distribution of the test-statistics, $x_i \sim$ N(0,5), $\Delta x$ weighted full-pairwise estimator, $n=50$
  • Figure 5: Distribution of the test-statistics, $x_i \sim$ N(0,5), $\Delta x$ weighted full-pairwise estimator, $n=500$
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof