I have 2 sets of data:
- Set 1: Daily data of stock market returns (eg. [1%, 1.2%, -2%])
- Set 2: Those same stock market returns, multiplied by a number (eg. [2%, 0.6%, -1%] which equals [1% * 2, 1.2% * 0.5, -2% * 0.5])
I want to test the hypothesis that data set 1 has a mean that's equal to the mean of data set 2 when you adjust for the variance.
Can the dependent samples t test be used for this? If not, how can I do it?
I want to test the profitability's significance of a strategy that has a variable portfolio weight. That weight is dependent on the volatility of the past X days of data, and the adjustment is made so that the expected volatility is equal to a given target.
In the data set 1, I've got 20K+ days of data where the average daily return is ~0,025% with a daily standard deviation of ~0,63%.
On data set 2, when I adjust for the 0,63% standard deviation, the return is ~0,0024%. Intuitively, it would seem that with 20K points of data, the difference in the means would be pretty significant because the two data sets are always going to be very correlated (.8+ in this case). But the p-value is .50, random.
I would think that this t test isn't appropriate because the data set 1 has a direct influence on the results of the data set 2, in a way that doesn't happen in the other cases where this test is applied. I was thinking about making a Monte Carlo simulation where I run data set 1, multiply by random numbers with the same properties as the numbers that I used in data set 2 and then see the outcomes. I may be saying something silly though, tell me what you think!