# Price Prediction Intervals from Forecasted Returns (ARIMA)

I have successfully fit an ARIMA model to a time series of the daily returns of power futures prices. The question I have is: How can I create a prediction interval for the prices? Or, alternatively, is there a nonparametric alternative to ARIMA that you would recommend? I understand econometric models are a bit iffy for long-term forecasts, and I would like this model to be extendable to at least six months (~120 periods).

Simply taking the cumulative product of the forecasted returns and the most recent historical price creates a band that diverges far too quickly to be realistic (sample image below for a 20-day futures price forecast using an ARIMA(2, 0, 2) model on the March 2021 contract as of December 2020). If it makes a difference, to make the data stationary and reduce the required "d" order, please note that the data were a) first standardized and b) then transformed via Yeo-Johnson before they were fed into the ARIMA model. Needless to say, I then reversed the transformations to scale the data. The ARIMA parameters were calculated using auto_arima in Python.

• Hi: if you extend an ARIMA prediction out 120 days where the data is estimated using daily data, unfortunately you'll probably be predicting the long term mean of the model + error because any of the AR effects will die out. Also, I wasn't clear if you are using ARIMA on the log prices Mar 16, 2021 at 1:51
• Thank you for your response, Mark. The underlying data are standardized daily geometric returns. They have been transformed through a Yeo-Johnson to “increase their normality” and make the residual terms in the ARIMA model approximately Gaussian. So, to summarize, given this mean reversion, I might as well stick to the traditional Brownian motion / Wiener process with log returns? Essentially, as long as the residuals aren’t auto correlated and the underlying data are stationary, I should be fine? (Going back to all models are wrong but some are useful.) Mar 16, 2021 at 4:15
• It sounds like you've definitely dealt with th various pitfalls ( I gotta check oput yeo-jonhson because I haven't heard of it ). But, if it's "pure arima", like I said, I don't think there's much you can do with it because you would need to know the future epsilon's. Mar 16, 2021 at 17:30