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I have some financial data i'm trying to fit a random walk too, but the daily change increments have different distributions when studied over the last month, last year, last 5 years etc, along with heteroskedascity. What is the best way to proceed with modelling this? Techniques and interesting areas to look into further?

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if you have different distributions for each day, then you'll have different parameters to estimate for each day. if the data is at daily rate then it's impossible to do anything. you have to parameterize the change in distribution then. one way of doing this is GARCH, where you essentially are saying that the variance changes every day but otherwise it's random walk. you also express the functional form of this change. you'll end up estimating a couple more parameters of your random walk.

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  • $\begingroup$ Not sure I expressed this quite right - my 'data' is the logged daily change, then after this I have taken one month (the last 30 logged daily change), 1 year (last 365 etc.) then i've fit a different distribution to each time period. For example, the daily returns over a month follow a t-distribution, the daily returns over 1y follow a skew-normal etc. $\endgroup$ – user40124 Mar 5 '14 at 15:48
  • $\begingroup$ if your daily is t, then annual will be 12 dailies, which will tend to be closer to normal because of CLT. that's the problem with non-stable distributions: they don't have a very good theory for random walk when you change the time scale, e.g. what is a sum of t-distributions? the GARCH approach will give you different variances for different time scales, but in a systematic manner. $\endgroup$ – Aksakal Mar 5 '14 at 15:52

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