# Weak Instrument: Definition

Weak instruments can wreak havoc with your regression analysis. “Instruments” (instrumental variables) are a third variable, Z, used when you have endogenous variables—variables that are influenced by other variables in the model. Instruments are used to account for unexpected behavior between variables. You are using them to find the true correlation between the explanatory variable (x) and response variable, (y). Therefore, you want these instruments to be as strong as possible.

Weak instruments—instruments that are only marginally valid—can cause many problems, including:

## How to Find a Weak Instrument

There isn’t a single, agreed upon way to find weak instruments. Many different tests exist, depending on what type of regression you’re running. Even then, different authors have different definitions of what constitutes a “weak” instrument.

Stock & Watson (2003) suggest the following general rule of thumb for two-stage least squares (2SLS), useful only if the regression equation has a single variable on the right hand side. Use the F-statistic to test for the significance of excluded instruments. If the first-stage F-statistic is smaller than 10, this indicates the presence of a weak instrument.

For a scalar regressor (x) and scalar instrument (z), a small r squared (when x is regressed on z) indicates a weak instrument. However, if z is a vector of instruments in the same situation, a small r2 or small F-statistic also indicates a potential problem (Cameron et. al, 2005).

If you identify weak instruments in your 2SLS regression, Baltagi (2007) suggests the following choices:

• Find and use a stronger instrument, or
• Use an alternate estimator (such as LIML, which uses maximum likelihood instead of least squares, in place of 2SLS).

## References

Baltagi, B. (2007). Econometrics. Springer Science & Business Media.
Cameron et al. (2005). Microeconometrics: Methods and Applications. Cambridge University Press.
Stock, J. & Yogo, M. (2002). Testing for Weak Instruments in Linear IV Regression. National Bureau of Economic Research

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