I suspect that this is entirely possible since the endogenous variable coefficient can biased in many possible way, thus leading to a near 0 estimate despite having a real causal relationship.

More formally, let $Y$ be the dependent variable, $X$ be the endogenous independent variable, and $Z$ be the instrument for $X$.

We have the naive regression:

$$ Y = \beta_0 + \beta_1X $$

and the two-stage-least square regression:

$$ \hat X = \hat\delta_0 + \hat\delta_1Z \\ Y = \mu_0 + \mu_1\hat X $$

Is it possible for $\mu_1$ (2SLS estimate) to be significant whereas $\beta_1$ (naive regression estimate) is not?

  • $\begingroup$ You need to clarify your question. For instance, do you mean the IV is relevant, but the second stage coefficient on the endogenous variable is insignificant? $\endgroup$ – Dimitriy V. Masterov Nov 20 '14 at 6:30
  • $\begingroup$ @DimitriyV.Masterov I clarified my question in the edit. My original question was quite confusing indeed. $\endgroup$ – Heisenberg Nov 20 '14 at 15:30

Such cases can exist when the OLS coefficient is biased towards zero. This can happen for instance due to omitted variables (depending on the sign of the omitted variable in the long regression and its correlation with the endogenous variable), or when you have measurement error which always biases the OLS coefficient towards zero. For an excellent explanation of the latter problem see this answer by Dimitriy which also explains a way of dealing with such bias.

In this case, the OLS coefficient can be insignificant due to the attenuation bias given that you base inference on t-statistic which is the coefficient divided by its standard error. Then if you instrument and you get around the attenuation bias (see Dimitriy's answer linked above) the coefficient can be much larger and significant, i.e. your $\mu_1$ can be significant whereas $\beta_1$ is not.

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