Setting aside questions about the stationarity and/or seasonality of the data, what are the benefits/pitfalls of using differencing versus the percentage change? When is it more ideal to use one transformation over the other?
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I would say it depends on the goal and your problems at hand.
For example, with some financial applications, you will consider returns when you aim at computing risk or investing, because returns is what is relevant to your portfolio and not really price changes (easier to think about returns, than price changes which need to be rescaled to take into account your AUM (asset under management)).
However, with some other financial applications such as predicting the next price for an asset, you may want to use price changes: in some cases, price changes can follow a multinomial and thus are easier to model/predict (and it may correspond to how market makers are really moving their prices, by increment or decrement), whereas returns will be more continuous.
From a statistical point of view, I think it is nice to have statistical methods which are invariant to monotone transforms of your variables. For example, you may want to obtain the same results using $P_{t+1}-P_t$ and the log-returns $\log P_{t+1} - \log P_t$, where $P_t$ is price at time $t$.
mic
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Why would you assume either is necessary let the data speak !. Both are forms of transformations and transformations like drugs can be both beneficial and hurtful. One can always take the forecast and express it in terms of a difference and/or a percent change but that is just window-dressing to deliver a palatable result.
See When (and why) should you take the log of a distribution (of numbers)? when to take power transforms . See the impact of taking unwarranted differences Variance of difference of $x_{i,t}$ and $x_{i,t+1}$ because you can actually inject structure by incorrect filtering ..see the Slutzky Effect in SE and here Analyzing up/down patterns in short time-series data .
