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I understand statistical significance in the general sense: we take a sample from a population and compute some parameter from the sample to infer what is the propulsion parameter to some degree of confidence, usually 95%. So if I want to find the p-value for a slope between two financial datasets, say interest rates each quarter, then I’d compute the slope between the two sample datasets and find the p-value. Less than 5%, we conclude the slope arrived at is very unlikely to have occurred by random chance. Otherwise, we fail to reject the hypothesis that the population slope is 0.

Here’s my question… if we are dealing with 200 quarters of data, what exactly is our “population?” In the commonly used example of IQs, the population is easily defined as the IQs of ALL the people. With financial data, considering quarterly data, it isn’t clear to me what is the population. Is it all historical data dating back to as long as our interest rates, in our example, existed? So if there were actually 500 quarters where our interest rates existed, that’s the population of data? Is it all of the data measured more granularly? Our interest rate measured at every second, millisecond, etc.?

I’m asking because I’m a bit perplexed how a p-value can be interpreted if we find the p-value for a slope between two datasets of interest rates, both 200 quarters worth of data. The sampling distribution which is assumed to be normal makes sense when thinking of pulling a sample of IQs from the population of all IQs.. how does it work for pulling sample last from the population of our interest rate data?

Additionally and most importantly, the reason I’m interested in this is because I am calculating the rolling slopes 24 quarters at a time, moving by one quarter at a time. If I run a t test on each slope, most are highly insignificant. If I think of these slopes as not trying to ascertain the population slope, but as simply representing the sample slope, then I think the idea of significance goes out the window here. Right? Confusing!

Sorry for the long winded question… I want to be as clear as possible where I’m confused. As always thank you all.

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Welcome to the world of time series data.

Without knowing exactly what you did, your results are almost certainly spurious. There are only two cases when OLS is applicable:

  • cointegration
  • Classic linear assumptions for time series data: mainly no serial correlation needs to be satisfied

Instead of a random sample from a population, you have (many) observations of the same object over time. The observed data is also interpreted as one of infinitely many possible paths that never materialized. Many variables in finance are modelled as stochastics, and whether interest rates are stationary or not is disputed.

I can recommend a few brilliant books:

This answer has a few more details.

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  • $\begingroup$ Thank you for your answer! But the slopes are not “wrong” are they? They still represent the slope in that time window so certainly they aren’t “wrong?” $\endgroup$ Jun 29, 2021 at 1:10
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    $\begingroup$ Simple example. Catalytic converters and HIV started to become widespread at roughly the same time. If you regress that, it looks as if catalytic converters cause HIV. Obviously rubbish, but that is what happens with trending data. So your slopes are almost certainly wrong. $\endgroup$
    – AKdemy
    Jun 29, 2021 at 1:22
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    $\begingroup$ No, it simply is meaningless (provided it is spurious). $\endgroup$
    – AKdemy
    Jun 29, 2021 at 1:26
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    $\begingroup$ HIV and catalytic converters have a very high correlation. It simply is wrong. The slope mainly depends on correlation. I don't know what exactly you did but high significance in finance is almost always a sign something is wrong. $\endgroup$
    – AKdemy
    Jun 29, 2021 at 1:49
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    $\begingroup$ @user3138766, AKdemy is focusing on the causal interpretation, but there are things to be said about noncausal instances of significant relationships, too. Certain noncausal relationships can be highly valuable for prediction. You may want to check out this thread on Cross Validated. Some more relevant material is here. $\endgroup$ Jun 29, 2021 at 11:43

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