Stationarity and Differencing in ARIMA Models for Stock Data

Summary

The document explains the stationarity assumption behind ARIMA and the role of its integration component. ARIMA models are generally applied to a stationary series; when a series is integrated, differencing can transform it into a stationary one. The answer defines covariance stationarity in terms of a stable mean and covariances that depend on the time lag rather than the date.

For stock analysis, it recommends considering log returns, which are often treated as stationary, instead of modeling price levels directly. A nonstationary-looking ACF or PACF is a prompt to investigate transformations and model assumptions; apparently well-behaved residuals alone do not establish that the input series satisfies the model requirements. The response is a brief conceptual explanation rather than a diagnostic procedure, and its claim about returns is a generalization: stationarity should be assessed for the particular asset and sample.

Key ideas

• ARIMA typically requires a stationary series for modeling.
• The integration component allows differencing a series to seek stationarity.
• Covariance stationarity means a stable mean and lag-dependent covariance.
• Stock log returns are often treated as stationary, but this assumption should be checked for the data at hand.

Tags

# What do I need to do with my data before fitting the ARIMA model?


# What do I need to do with my data before fitting the ARIMA model?












I'm fitting a stock price time series data to ARIMA model and I have a question about the assumption.

Is it that ARIMA only applies to stationary data? The ACF and PACF of the data (and the logged return too)both show it's non-stationary. So can I still fit ARIMA to it?

The residuals of the fitted model actually shows it works quite and well. But I worry about the assumption.

Thanks!!

## Answer by Richi Wa (score 3)

https://quant.stackexchange.com/a/15803

What you should do:

- read a general introduction to time series analysis before you apply these methods otherwise you will misinterpret the results.

- time series are assumed to be covariance stationary. This is in short that their mean is the same for all points in time and that the covariance between two observations only depends on the lag.

- the "I" in ARIMA means that time series could be integrated. This means that you have to difference them in order to get something stationary.

You can start to read in this excellent online text book by Rob Hyndman and George Athana­sopou­los.

In the case of stock prices you should look at log-returns, which can usually be assumed to be a stationary time series.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.