How to communicate "very negative" parameter estimate in paper? I'm currently writing a thesis and I have to interpret a coefficient estimate that's extremely negative compared to what I hypothesized.
How do I best communicate this? Describing it as "very negative" sounds a bit strange, but I can't think of anything better.
 A: If you expected the estimate to be negative, you could say something like "As expected, the coefficient estimate was negative, but its magnitude was much larger than expected".
A: I would just write almost exactly what you did: "the coefficient was much lower than expected". Or, "based on .... [previous research? pilot studies?], we expected the coefficient to be in range X-Y. However, the estimated coefficient turned out to be much lower".
Clearly, this is not statistics, and there is no statistical significance involved, but there you have it: statistical significance is not the same as significance, and even the best statistical tests need to be interpreted.
I would not go for testing the coefficient against the range X-Y; it might evoke the impression that you came up with it post hoc.
A: When the result is "non significant" you don't say "how much insignificant it is" besides quoting the p-value.
I think, that what you want to do, is to informally test another hypothesis, that the parameter is significantly smaller than some chosen  by you threshold. In that case you'd need to formally create another test for it (which in most cases would should be straightforward) and quote this new p-value (which we expect will yield significant result).
The bottom line is, that the classical statistics (contrary to common sense and every-day intuition) doesn't provide arguments for null hypothesis. It can only give arguments against it. 
Take a look at Bayesian Statistics. Although much more difficult to implement, in the end it does provide arguments both for or against any tested set of hypotheses.
