# On the Stock–Yogo Tables

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

`ivregress`command or the

`ivreg2`package (Baum et al. 2010). The difficulty in the SY approach is that, in order to compute appropriate critical values, it is necessary to evaluate a complicated integral as an intermediate step. SY did this by Monte Carlo simulation and the tables of critical values they provided are widely used in practice.

## 2. An Analytic Development of Stock–Yogo

- The practitioner chooses a value for $\left|B\right|$, e.g., $\left|B\right|=0.1$, if an asymptotic relative bias of less than 10% is deemed acceptable.
- Given ${k}_{z}$ and $\left|B\right|$, ${\mu}_{0}^{2}={\lambda}_{0}^{\prime}{\lambda}_{0}$ is obtained on solving (8).
- Given ${\mu}_{0}^{2}$, critical values for F can be determined, which are proportional to those of the non-central chi-squared distribution as specified in (4).
- The null of weak instruments is then rejected for sufficiently large values of the first-stage F-statistic, and we conclude that $\left|B\right|$ is no larger than the value chosen in Step 1 above.

**Theorem**

**1.**

**Proof.**

## 3. Some Further Consequences of Theorem 1

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Theorem**

**2.**

**Proof.**

**Conjecture**

**1.**

**Remark**

**4.**

## 4. Some Monte Carlo Results

## 5. $\mathit{p}$-Values

## 6. Multiple Endogenous Regressors

## 7. The Wisdom of Hindsight: Some Historical Remarks

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Expectation of a Particular Function of Normal Random Variables

**Theorem**

**A1.**

**Proof.**

## Appendix B. Analysis of Table 1

#### Appendix B.1. Preliminaries

#### Appendix B.2. The Consequence of Varying B for Fixed ${k}_{z}\ge 2$

## Appendix C. Some Remarks on Computational Aspects

## Appendix D. Some Remarks on the Just-Identified Scalar Case

**Figure A1.**Plots of $B={}_{1}{F}_{1}^{}\phantom{\rule{-1.5pt}{0ex}}\left(1;\frac{1}{2};-\frac{{\mu}^{2}}{2}\right)$ and its absolute value against $\frac{{\mu}^{2}}{2}$.

B | ||||||||
---|---|---|---|---|---|---|---|---|

−0.01 | −0.05 | −0.10 | −0.20 | |||||

mean | std dev | mean | std dev | mean | std dev | mean | std dev | |

${\widehat{\beta}}_{OLS}$ | 1.4949 | 0.0086 | 1.4988 | 0.0086 | 1.4993 | 0.0086 | 1.4996 | 0.0087 |

${\widehat{\beta}}_{2SLS}$ | 0.9954 | 0.1003 | 0.9753 | 0.2327 | 0.9495 | 0.4389 | 0.8936 | 6.0492 |

F | 104.11 | 20.411 | 24.429 | 9.8352 | 14.821 | 7.5732 | 9.1757 | 5.8671 |

$relbias$ | −0.0092 | −0.0496 | −0.1011 | −0.2130 | ||||

${\mu}_{0}^{2}$ | 103.06 | 23.412 | 13.830 | 8.198 | ||||

cv F | 139.17 | 42.035 | 28.769 | 20.323 | ||||

rej freq F | 0.0496 | 0.0507 | 0.0485 | 0.0489 |

## Appendix E. Derivation of the O(${k}_{z}^{-2}$) Term in (16)

**Lemma**

**A1.**

**Proof.**

## References

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1. | Much of the later literature has focussed less on testing for the presence of weak instruments and more on the development of techniques that are robust to the presence of weak instruments. |

2. | A heuristically appealing aspect of using the first-stage F-statistic as a measure of instrument weakness, in the case of a single endogenous regressor, is its consistency with the well-known Staiger–Stock rule of thumb. Staiger and Stock (1997), p. 557, suggested that instruments be deemed weak if the first-stage F is less than 10. SY (pp. 101–2) observe that 10 corresponds closely to their tabulated critical values for a 5% test that the relative bias is 10% for all values of ${k}_{z}$, and concluded that ‘this provides a formal, and not unreasonable, testing interpretation of the the Staiger–Stock rule of thumb.’ |

3. | Through the use of more extensive simulation results than those used originally by SY, we are able to support the proposition that the numerical approximation errors inherent in the computation of our analytical results are less than those contained in the original SY tables. |

4. | Some references specify the non-centrality parameter for a non-central chi-squared distribution as $\delta $, whereas others specify it as $\delta /2$. We have adopted the former convention here. |

5. | The exact details of these arguments can be found in SY and will not be repeated here. |

6. | We thank an anonymous referee for bringing this subtlety to our attention. For a more complete discussion of this point, we refer the reader to the discussion of (Chao and Swanson 2007, pp. 518–19) and the references cited therein. |

7. | It should be noted that the proof provided is not the only one possible and we would like to thank helpful referees for drawing various alternatives to our attention. For example, in an elegant paper, Chao and Swanson (2007), Proposition 3.1 and Lemma 3.3, respectively, derive local to zero approximations for each of ${lim}_{n\to \infty}E\left[{\widehat{\beta}}_{2SLS,n}-\beta \right]$ and ${lim}_{n\to \infty}E\left[{\widehat{\beta}}_{OLS,n}-\beta \right]$, from whence derivation of the ratio is straightforward. Similarly, there are finite sample papers in the literature from which it would be possible to start a proof along the lines of the one presented but at a more advanced point (see, for example, Forchini and Hillier 2003, Equation B.13). However, we favour the proof presented for two reasons. First, it is a direct continuation of the developments of Stock and Yogo (2005), Equation 3.1, and the discussion immediately thereafter. Second, when viewed in the correct light, there are much earlier antecedents that take precedence over the two mentioned here. We discuss this further in Section 6. |

8. | |

9. | We have also computed simulated critical values from 20,000 random draws as in SY, but repeating the exercise 1000 times. The resulting mean critical values are virtually identical to those in Table 1, with the maximum difference being 0.02. |

10. | Theorem 2 is similar in spirit to Das Gupta and Perlman (1974), p. 180, Remark 4.1, although they only address the numerator of the ratio in Equation (5). Consequently, Das Gupta and Perlman are silent on the relative magnitudes of ${\chi}_{\alpha}$ and ${k}_{z}$ which, in essence, is the content of Theorem 2. |

11. | The complete set of assumptions are presented in SY (Section 2.4). |

12. | |

13. | Make the substitutions $\{{e}_{i},\gamma ,{k}_{z}\}$ for $\{\overline{\alpha},\beta ,\nu \}$, respectively, in Hillier et al. (1984), Equation (30). |

14. | Please note that the definition of $\mathsf{\Lambda}$ adopted here is slightly different from the definitions used in either Hillier et al. (1984) and SY. |

15. | |

16. | The zonal polynomials appearing in (14) adopt a normalisation due to Constantine (1963), which typically leads to more compact expressions than do the polynomials originally proposed by James (1961). |

17. | Some progress towards addressing the computational aspects of these polynomials has been made by Hillier et al. (2009, 2014). |

18. | Although the derivation of (15) is straightforward, this is less true for (16). A derivation of the terms in (16) that are in addition to those in (15) is provided in Appendix E. |

19. | Similarly, in the proof of Theorem A1, we established that $E\left[{(\xi -{\lambda}_{0})}^{\prime}\xi /{\xi}^{\prime}\xi \right]$ was unbounded when ${k}_{z}=1$. |

20. |

**Figure 1.**Plots of $B={}_{1}{F}_{1}\phantom{\rule{-1.5pt}{0ex}}\left(1;\frac{{k}_{z}}{2};-\frac{{\mu}^{2}}{2}\right)$ against $\frac{{\mu}^{2}}{2}$ for ${k}_{z}=2,3$ and 6.

**Table 1.**5% Critical values $\left(c{v}_{0.05}^{SW}\right)$ for single endogenous regressor, 2SLS bias.

${\mathit{k}}_{\mathit{z}}\setminus \mathit{B}$ | 0.01 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 |
---|---|---|---|---|---|---|---|

2 | 11.57 | 9.02 | 7.85 | 7.14 | 6.61 | 6.19 | 5.83 |

3 | 46.32 | 13.76 | 9.18 | 7.52 | 6.60 | 5.96 | 5.49 |

4 | 63.10 | 16.72 | 10.23 | 7.91 | 6.67 | 5.88 | 5.32 |

5 | 72.55 | 18.27 | 10.78 | 8.11 | 6.71 | 5.82 | 5.19 |

6 | 78.59 | 19.19 | 11.08 | 8.21 | 6.70 | 5.75 | 5.09 |

7 | 82.75 | 19.79 | 11.25 | 8.25 | 6.67 | 5.69 | 5.01 |

8 | 85.78 | 20.20 | 11.36 | 8.26 | 6.64 | 5.63 | 4.93 |

9 | 88.07 | 20.49 | 11.42 | 8.25 | 6.60 | 5.58 | 4.87 |

10 | 89.86 | 20.70 | 11.46 | 8.24 | 6.56 | 5.52 | 4.81 |

11 | 91.30 | 20.86 | 11.49 | 8.22 | 6.53 | 5.48 | 4.76 |

12 | 92.47 | 20.99 | 11.50 | 8.20 | 6.49 | 5.43 | 4.71 |

13 | 93.43 | 21.08 | 11.50 | 8.17 | 6.46 | 5.39 | 4.67 |

14 | 94.25 | 21.16 | 11.50 | 8.15 | 6.42 | 5.36 | 4.63 |

15 | 94.94 | 21.22 | 11.49 | 8.13 | 6.39 | 5.32 | 4.59 |

16 | 95.54 | 21.26 | 11.49 | 8.11 | 6.36 | 5.29 | 4.56 |

17 | 96.05 | 21.30 | 11.48 | 8.08 | 6.34 | 5.26 | 4.53 |

18 | 96.50 | 21.33 | 11.46 | 8.06 | 6.31 | 5.23 | 4.50 |

19 | 96.09 | 21.35 | 11.45 | 8.04 | 6.29 | 5.21 | 4.47 |

20 | 97.25 | 21.37 | 11.44 | 8.02 | 6.26 | 5.18 | 4.45 |

21 | 97.56 | 21.39 | 11.43 | 8.00 | 6.24 | 5.16 | 4.43 |

22 | 97.84 | 21.40 | 11.41 | 7.98 | 6.22 | 5.14 | 4.40 |

23 | 98.09 | 21.41 | 11.40 | 7.96 | 6.20 | 5.12 | 4.38 |

24 | 98.32 | 21.41 | 11.39 | 7.94 | 6.18 | 5.10 | 4.36 |

25 | 98.53 | 21.42 | 11.38 | 7.93 | 6.16 | 5.08 | 4.35 |

26 | 98.71 | 21.42 | 11.36 | 7.91 | 6.15 | 5.06 | 4.33 |

27 | 98.88 | 21.42 | 11.35 | 7.90 | 6.13 | 5.05 | 4.31 |

28 | 99.04 | 21.42 | 11.34 | 7.88 | 6.11 | 5.03 | 4.30 |

29 | 99.18 | 21.42 | 11.32 | 7.87 | 6.10 | 5.02 | 4.28 |

30 | 99.31 | 21.42 | 11.31 | 7.85 | 6.08 | 5.00 | 4.27 |

${\mathit{k}}_{\mathit{z}}\setminus \mathit{B}$ | 0.05 | 0.10 | 0.20 | 0.30 |
---|---|---|---|---|

3 | 0.15 | −0.10 | −0.14 | −0.10 |

4 | 0.13 | 0.04 | 0.04 | 0.02 |

5 | 0.10 | 0.05 | 0.06 | 0.06 |

6 | 0.09 | 0.04 | 0.06 | 0.06 |

7 | 0.07 | 0.04 | 0.06 | 0.07 |

8 | 0.05 | 0.03 | 0.05 | 0.06 |

9 | 0.04 | 0.04 | 0.05 | 0.05 |

10 | 0.04 | 0.03 | 0.05 | 0.05 |

11 | 0.04 | 0.01 | 0.03 | 0.04 |

12 | 0.02 | 0.02 | 0.04 | 0.04 |

13 | 0.02 | 0.02 | 0.03 | 0.04 |

14 | 0.02 | 0.02 | 0.03 | 0.04 |

15 | 0.01 | 0.02 | 0.03 | 0.04 |

16 | 0.02 | 0.01 | 0.03 | 0.03 |

17 | 0.01 | 0.01 | 0.02 | 0.03 |

18 | 0.01 | 0.02 | 0.02 | 0.03 |

19 | 0.01 | 0.01 | 0.02 | 0.04 |

20 | 0.01 | 0.01 | 0.02 | 0.03 |

21 | 0.00 | 0.01 | 0.02 | 0.03 |

22 | 0.00 | 0.01 | 0.02 | 0.03 |

23 | 0.00 | 0.01 | 0.02 | 0.03 |

24 | 0.00 | 0.01 | 0.02 | 0.03 |

25 | 0.00 | 0.00 | 0.02 | 0.02 |

26 | 0.00 | 0.01 | 0.01 | 0.02 |

27 | 0.00 | 0.01 | 0.01 | 0.03 |

28 | 0.00 | 0.02 | 0.02 | 0.02 |

29 | 0.00 | 0.01 | 0.01 | 0.03 |

30 | 0.00 | 0.01 | 0.01 | 0.02 |

B | ||||||||
---|---|---|---|---|---|---|---|---|

0.01 | 0.05 | 0.10 | 0.20 | |||||

${k}_{z}=3$ | mean | std dev | mean | std dev | mean | std dev | mean | std dev |

${\widehat{\beta}}_{OLS}$ | 1.4950 | 0.0086 | 1.4989 | 0.0087 | 1.4994 | 0.0087 | 1.4997 | 0.0087 |

${\widehat{\beta}}_{2SLS}$ | 1.0054 | 0.0998 | 1.0241 | 0.2222 | 1.0506 | 0.3161 | 1.1025 | 0.4276 |

F | 34.713 | 6.7626 | 8.0336 | 3.1828 | 4.7849 | 2.3952 | 3.0948 | 1.8630 |

rel bias | 0.0108 | 0.0482 | 0.1014 | 0.2052 | ||||

${\mu}_{0}^{2}/{k}_{z}$ | 33.674 | 7.0445 | 3.7754 | 2.0902 | ||||

cv F | 46.316 | 13.765 | 9.1815 | 6.5960 | ||||

rej freq F | 0.0515 | 0.0505 | 0.0508 | 0.0511 | ||||

${k}_{z}=2$ | mean | std dev | mean | std dev | mean | std dev | mean | std dev |

${\widehat{\beta}}_{OLS}$ | 1.4996 | 0.0087 | 1.4997 | 0.0087 | 1.4997 | 0.0087 | 1.4998 | 0.0087 |

${\widehat{\beta}}_{2SLS}$ | 1.0056 | 0.4398 | 1.0256 | 0.7195 | 1.0519 | 0.9651 | 1.0981 | 1.1404 |

F | 5.6124 | 3.1989 | 4.0004 | 2.6492 | 3.2963 | 2.3746 | 2.6011 | 2.0502 |

rel bias | 0.0111 | 0.0513 | 0.1039 | 0.1962 | ||||

${\mu}_{0}^{2}/{k}_{z}$ | 4.6052 | 2.9957 | 2.3026 | 1.6094 | ||||

cv F | 11.572 | 9.0232 | 7.8521 | 6.6087 | ||||

rej freq F | 0.0509 | 0.0507 | 0.0505 | 0.0498 |

${\mathit{k}}_{\mathit{z}}\setminus \mathit{B}$ | 0.01 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 |
---|---|---|---|---|---|---|---|

02 | 04.605 | 02.996 | 2.303 | 1.897 | 1.609 | 1.386 | 1.204 |

03 | 33.674 | 07.045 | 3.775 | 2.677 | 2.090 | 1.706 | 1.426 |

04 | 50.000 | 10.000 | 5.000 | 3.329 | 2.483 | 1.960 | 1.599 |

05 | 59.799 | 11.793 | 5.784 | 3.774 | 2.761 | 2.144 | 1.724 |

06 | 66.332 | 12.991 | 6.315 | 4.081 | 2.958 | 2.277 | 1.816 |

07 | 70.998 | 13.848 | 6.696 | 4.304 | 3.102 | 2.375 | 1.885 |

08 | 74.498 | 14.491 | 6.982 | 4.472 | 3.212 | 2.450 | 1.938 |

09 | 77.221 | 14.992 | 7.205 | 4.604 | 3.298 | 2.510 | 1.980 |

10 | 79.398 | 15.392 | 7.384 | 4.709 | 3.367 | 2.558 | 2.014 |

11 | 81.180 | 15.720 | 7.531 | 4.796 | 3.424 | 2.597 | 2.043 |

12 | 82.665 | 15.993 | 7.653 | 4.868 | 3.471 | 2.630 | 2.066 |

13 | 83.922 | 16.224 | 7.756 | 4.929 | 3.511 | 2.658 | 2.086 |

14 | 84.999 | 16.423 | 7.845 | 4.981 | 3.546 | 2.682 | 2.104 |

15 | 85.932 | 16.594 | 7.922 | 5.027 | 3.576 | 2.703 | 2.119 |

16 | 86.749 | 16.745 | 7.989 | 5.067 | 3.602 | 2.721 | 2.132 |

17 | 87.470 | 16.877 | 8.048 | 5.102 | 3.626 | 2.738 | 2.144 |

18 | 88.110 | 16.995 | 8.101 | 5.133 | 3.646 | 2.752 | 2.154 |

19 | 88.683 | 17.101 | 8.148 | 5.161 | 3.665 | 2.765 | 2.163 |

20 | 89.199 | 17.196 | 8.191 | 5.186 | 3.681 | 2.777 | 2.172 |

21 | 89.666 | 17.281 | 8.229 | 5.209 | 3.697 | 2.787 | 2.179 |

22 | 90.090 | 17.360 | 8.264 | 5.230 | 3.710 | 2.797 | 2.186 |

23 | 90.477 | 17.431 | 8.296 | 5.249 | 3.723 | 2.806 | 2.193 |

24 | 90.833 | 17.496 | 8.326 | 5.266 | 3.734 | 2.814 | 2.198 |

25 | 91.159 | 17.556 | 8.353 | 5.282 | 3.745 | 2.821 | 2.204 |

26 | 91.461 | 17.612 | 8.377 | 5.297 | 3.755 | 2.828 | 2.209 |

27 | 91.740 | 17.663 | 8.400 | 5.311 | 3.764 | 2.834 | 2.213 |

28 | 91.999 | 17.711 | 8.422 | 5.323 | 3.772 | 2.840 | 2.217 |

29 | 92.241 | 17.755 | 8.442 | 5.335 | 3.780 | 2.846 | 2.221 |

30 | 92.466 | 17.797 | 8.460 | 5.346 | 3.787 | 2.851 | 2.225 |

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Skeels, C.L.; Windmeijer, F. On the Stock–Yogo Tables. *Econometrics* **2018**, *6*, 44.
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Skeels, Christopher L., and Frank Windmeijer. 2018. "On the Stock–Yogo Tables" *Econometrics* 6, no. 4: 44.
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