Can the endogenous variable be insignificant while the instrument is significant?

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?

• 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? Commented Nov 20, 2014 at 6:30
• @DimitriyV.Masterov I clarified my question in the edit. My original question was quite confusing indeed. Commented Nov 20, 2014 at 15:30

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.