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So I'm not sure if I 100% grasp confidence intervals. Say I have a huge data set of a bond prices from 1996 to present in MINUTES. Suppose I separate each data by day. If I were to use a Dickey Fuller Test (or any other t-test), on a particular day (thus a sample is drawn based on all points in one day) and the value I got rejected the null hypothesis, would it be conceptually right for me to conclude that because based on my sample I was able to reject the null, I am able to make a conclusion about the whole dataset (population)? In other words, would picking one day and not randomly picking datapoints throughout matter?

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Yes, it matters. And no, you cannot generalize, unless the parameter you are testing is some structural parameter, in which case you are really just using a lower-powered sample to test your time series model parameter.

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