# Bootstrap Tests for Overidentification in Linear Regression Models

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## Abstract

**:**

## 1. Introduction

## 2. Tests for Overidentification

## 3. Analysis Using a Simpler Model

**Theorem 1.**

**Proof:**

**Remark 1.**

## 4. Particular Cases

**Theorem 2.**

**Proof.**

**Remark 2.1.**

**Remark 2.2.**

**Remark 2.3.**

**Remark 2.4.**

**Remark 2.5.**

**Remark 2.6.**

**Remark 2.7.**

**Corollary 2.1.**

**Proof of Corollary 2.1.**

**Theorem 3.**

**Proof.**

**Remark 3.1.**

**Remark 3.2.**

**Remark 3.3.**

## 5. Bootstrap Tests

#### 5.1. Parametric Bootstraps

#### 5.2. Resampling

## 6. Performance of Asymptotic Tests: Simulation Results

**Figure 1.**Rejection frequencies for asymptotic tests as functions of n for $q=8$. (

**a**) $a=2$, $\rho =0.5$; (

**b**) $a=2$, $\rho =0.9$; (

**c**) $a=8$, $\rho =0.5$; (

**d**) $a=8$, $\rho =0.9$. Note that the scale of the vertical axis differs across panels.

**Figure 2.**Rejection frequencies for asymptotic tests as functions of q for $n=400$. (

**a**) $a=2$, $\rho =0.5$; (

**b**) $a=2$, $\rho =0.9$; (

**c**) $a=8$, $\rho =0.5$; (

**d**) $a=8$, $\rho =0.9$. Note that the scale of the vertical axis differs across panels.

**Figure 3.**Rejection frequencies for asymptotic tests as functions of ρ for $q=8$ and $n=400$. (

**a**) Very weak instruments: $a=2$; (

**b**) Weak instruments: $a=4$; (

**c**) Moderately strong instruments: $a=8$; (

**d**) Very strong instruments: $a=16$. Note that the scale of the vertical axis differs across panels.

#### 6.1. Near the Singularity

**Figure 5.**Contours of rejection frequencies for likelihood ratio (LR)${}^{\prime}$ tests with $q=8$ and $n=400$.

## 7. Performance of Bootstrap Tests

**Figure 6.**Rejection frequencies for Sargan tests as functions of ρ for $q=8$ and $n=400$. (

**a**) $a=2$; (

**b**) $a=4$; (

**c**) $a=8$; (

**d**) $a=16$. Note that the scale of the vertical axis differs across panels.

**Figure 7.**Rejection frequencies for LR tests as functions of ρ for $q=8$ and $n=400$. (

**a**) $a=2$; (

**b**) $a=4$; (

**c**) $a=8$; (

**d**) $a=16$. Note that the scale of the vertical axis differs across panels.

**Figure 8.**Rejection frequencies for Fuller LR tests as functions of ρ for $q=8$ and $n=400$. (

**a**) $a=2$; (

**b**) $a=4$; (

**c**) $a=8$; (

**d**) $a=16$. Note that the scale of the vertical axis differs across panels.

**Figure 9.**Contours of rejection frequencies for instrumental variables (IV)-R bootstrap Sargan tests.

**Figure 10.**Contours of rejection frequencies for limited information maximum likelihood (LIML)-ER bootstrap LR tests.

## 8. Power Considerations

**N**(0,1) variables ${\mathit{w}}_{j}{\phantom{\rule{-3.00003pt}{0ex}}}^{\top}\phantom{\rule{-0.166667em}{0ex}}{\mathit{v}}_{1}$, where the ${\mathit{w}}_{j}$ form an arbitrary orthonormal basis of the span of the columns of $\mathit{W}\phantom{\rule{-0.166667em}{0ex}}$.

#### 8.1. Finite-Sample Concerns

**N**(0,1). These represent the projections of ${\mathit{v}}_{1}$ and ${\mathit{v}}_{2}$ onto the orthogonal complement of the span of the instruments. For the ${P}_{ij}$, however, we decompose as follows:

#### 8.2. Simulation Evidence

**Figure 11.**Power of bootstrap tests as functions of δ with $n=400$, $q=8$, $\rho =0.5$, $\theta =0$, and $t=1$. (

**a**) $a=2$; (

**b**) $a=4$; (

**c**) $a=8$; (

**d**) $a=6$.

**Figure 12.**Power of bootstrap tests as functions of δ with $n=400$, $q=2$, $\theta =0$, and $t=1$. (

**a**) $a=2$, $\rho =0.5$; (

**b**) $a=4$, $\rho =0.5$; (

**c**) $a=4$, $\rho =0.1$; (

**d**) $a=4$, $\rho =0.9$.

**Figure 13.**Size-power curves for $q=2$, $n=400$, $\rho =0.5$, and $\theta =0$. (

**a**) LR${}^{\prime}$: $a=4$ and several values of δ; (

**b**) ${S}^{\prime}$: $a=4$ and several values of δ; (

**c**) LR${}^{\prime}$: $\delta =2$ and several values of a; (

**d**) ${S}^{\prime}$: $\delta =2$ and several values of a.

**Figure 14.**Power as a function of θ for $n=400$, $\rho =0.5$, and $\delta =4$. (

**a**) $q=2$; (

**b**) $q=8$; (

**c**) $q=2$; (

**d**) $q=8$

## 9. Relaxing the IID Assumption

## 10. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix: Proofs

**Lemma 1.**

**Proof of Lemma 1.**

**Proof of Theorem 1.**

**Proof of Theorem 2.**

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Davidson, R.; MacKinnon, J.G. Bootstrap Tests for Overidentification in Linear Regression Models. *Econometrics* **2015**, *3*, 825-863.
https://doi.org/10.3390/econometrics3040825

**AMA Style**

Davidson R, MacKinnon JG. Bootstrap Tests for Overidentification in Linear Regression Models. *Econometrics*. 2015; 3(4):825-863.
https://doi.org/10.3390/econometrics3040825

**Chicago/Turabian Style**

Davidson, Russell, and James G. MacKinnon. 2015. "Bootstrap Tests for Overidentification in Linear Regression Models" *Econometrics* 3, no. 4: 825-863.
https://doi.org/10.3390/econometrics3040825