Minimum eigenvalue statistic 2SLS interpretation

[Michael Frohn]

Hello,

I am currently conducting econometric research using an instrumental variable approach. In this context, I encountered the minimum eigenvalue statistic as a tool to assess instrument strength. I understand that if the statistic exceeds the relevant critical values, we can reject the null hypothesis that the instrument is weak.

However, I am still unclear about the exact interpretation of the test. Specifically, what does "relative bias" refer to in this context, and how should the critical value thresholds (e.g., 10%, 15%, 20%) be interpreted? Are these to be understood as confidence levels, or do they represent something else? I would appreciate clarification on how to interpret these results and what they imply about the strength or weakness of the instrument.

Thank you in addvance!

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Minimum eigenvalue statistic = 6.43682

Critical Values # of endogenous regressors: 1
Ho: Instruments are weak # of excluded instruments: 1
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| 5% 10% 20% 30%
2SLS relative bias | (not available)
-----------------------------------+---------------------------------
| 10% 15% 20% 25%
2SLS Size of nominal 5% Wald test | 16.38 8.96 6.66 5.53
LIML Size of nominal 5% Wald test | 16.38 8.96 6.66 5.53
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[George Ford]

Relative bias refers to the bias in your 2SLS estimator as a percentage of the bias in OLS when there's endogeneity (Bias(2SLS) / Bias(OLS)).
This ratio approaches zero if instruments are strong, but grows when instruments are weak. The 10%, 15%, 20% values represent maximum acceptable relative bias levels.

Your statistic rejects for 15%, but not 10%. So your instruments are moderately weak.
Roughtly, 10% is very strong, 15% acceptably strong, 20% borderline weak instruments, and 25% clearly weak.

So, your hypothesis tests will be over-sized (reject null too often), but not too badly.

[Michael Frohn]

Thank you, this has been very insightful. I’m wondering whether there is a formal way to estimate or approximate the bias of the OLS and 2SLS estimators. In my case, the statistic is 6.4, which exceeds the critical value for 25% (5.53).. Does this mean that the null gets wrongly rejected 25% and that the relative bias is 30%?

Additionally, I’ve noticed that the inclusion of control variables sometimes has little effect on the minimum eigenvalue, while in other cases it substantially reduces it. How should such differences be interpreted? Do degrees of freedom or the statistical relevance of these controls influence instrument strength?

[George Ford]

Sorry. I misread the result. The instruments are weak. Relative bias is between 20-25% and you're going to reject far too often. Need to think about your instrument(s).

Controls that are correlated with Z (or Y) will reduce the explanatory power of Z, so the EV changes. df probably not a big factor.