Sample size sensitivity in time series regression analysis

#1
I have a time series that I've built a regression on, say Y = X + Y. For a data set of size 300, the coefficients of X and Y are statistically significant. But, the coefficients are not statistically significant for a (sequential) subset of the dataset of, say, 150 points. Now, since the computation for this second significance does not include the first, is there some way to adjust the second significance computation to account for the first? That is, I want to compute the statistical significance of the second computation conditioned on the fact that the first computation shows significance.
 

noetsi

No cake for spunky
#2
I suspect, as is commonly the case, that the difference in statistical signficance is because your power went up having more cases. I don't think you can address this issue except simply to gather more cases.
 
#3
I think I already have enough cases - here is a rephrase of the question - suppose I am sampling 100 data points from a distribution and computing the confidence interval around the mean of a particular value, say, 10. Now, how much narrower can the confidence interval be if I add the precondition that at 200 data points, the mean of 11 is within the confidence interval?