What is the relationship between the statistical significance of a reduced form coefficient and that of the IV estimate? Does one imply the other or could either be significant depending on data? I tried references but couldn't find anything, not to mention deriving it myself. Thanks.
Significance in the reduced form of an IV

Not a problem set, actual person trying to figure this out.
IV is reduced form over first stage
First stage is close to constant in large samples, hit the reduced form by sqrt (N)
The rest is left as an exercise for the readerThanks but could you be a bit more explicit? "First stage is close to constant?" what does that mean? "Hit the reduced form?

Op should take econometrics 101 before doing reg y x robust nonsense research
Classic EJMR...how about you get a life...
someone may have taken econ101 10 years ago and prefer to ask around to avoid spending a semester retaking it.Econ101 10 years ago is the extent of your econ background? No wonder you didn't get 8d3f's answer.

If you have good evidence and a solid theory you should avoid using instruments. In every other case, use as many or as little as you want because you will just be admitting that you lack foundations or good data.
It's a theoretical question for which one can be curious about the answer regardless of applications. MHE, etc.. don't have the answer so as far we are so far, it's maybe an (possibly) easy problem set for which it seems fair to ask for clues. This is as good a place to ask as any, but if this is the attitude, no wonder this place has a bad rep. 8d3f at least tried to answer.

What is the relationship between the statistical significance of a reduced form coefficient and that of the IV estimate? Does one imply the other or could either be significant depending on data? I tried references but couldn't find anything, not to mention deriving it myself. Thanks.
What the heII is a reduced form coefficient?

Seriously, you should be able to do it on your own if you are a grad student (let alone PhD).
IV asy. variance is Var(e)/Cov(x,z), while reduced form asy. variance is Var(e)/Var(z). If you rescale your instrument so that firststage coefficient of z is unity, then you can easily confirm that the asy. variances and the coefficients are same across IV and reduced form (thus the tstats). Just use the residuals instead of raw x and z in the derivation above if you have covariates. 
Seriously, you should be able to do it on your own if you are a grad student (let alone PhD).
IV asy. variance is Var(e)/Cov(x,z), while reduced form asy. variance is Var(e)/Var(z). If you rescale your instrument so that firststage coefficient of z is unity, then you can easily confirm that the asy. variances and the coefficients are same across IV and reduced form (thus the tstats). Just use the residuals instead of raw x and z in the derivation above if you have covariates.Actually, I was wrong (I am 0dbc but ID might have changed) reducedform residual is e + beta*u, where e is structural residual and u is firststage residual. So tstats are different in general.
79ca, "reducedform" refers to a regression of y on z, which is the most classic usage of the term 
Seriously, you should be able to do it on your own if you are a grad student (let alone PhD).
IV asy. variance is Var(e)/Cov(x,z), while reduced form asy. variance is Var(e)/Var(z). If you rescale your instrument so that firststage coefficient of z is unity, then you can easily confirm that the asy. variances and the coefficients are same across IV and reduced form (thus the tstats). Just use the residuals instead of raw x and z in the derivation above if you have covariates.Actually, I was wrong (I am 0dbc but ID might have changed) reducedform residual is e + beta*u, where e is structural residual and u is firststage residual. So tstats are different in general.
79ca, "reducedform" refers to a regression of y on z, which is the most classic usage of the termAlso, structural and reducedform residuals are equivalent if true beta is zero. It means testing IV significance is asymptomatically equivalent to testing reducedform significance, although they have algebraically different test statistics.