How do you forecast future values of a differenced time series?

The essence of achieving stationarity in time series analysis cannot be overstated, with the order of differencing playing a crucial role. This practice, however, goes beyond mere statistical necessity; it embodies a strategic approach to data preprocessing. By precisely identifying the minimal differencing required, analysts can preserve the intrinsic properties of the data while removing non-stationary elements. This nuanced balance minimizes information loss and maintains the data's predictive power.

You can also do seasonal differencing to do away with seasonality as well as non-stationarity once. Or do a power transform to stabilize the variance and do differencing to stabilize the mean.

The first step in achieving stationarity in a time series is to determine how many times you need to difference the data. Stationarity implies that the mean, variance, and autocorrelation of the series remain constant over time. To assess stationarity, you can use tests like the augmented Dickey-Fuller test or the KPSS test. Alternatively, plotting the series along with its autocorrelation function (ACF) and partial autocorrelation function (PACF) allows for visual inspection of trends and seasonality. The order of differencing indicates the minimum number of times you need to difference the data to eliminate non-stationary components.

After differencing a time series to make it stationary, you can fit an ARIMA or SARIMA model to the differenced data. These models explicitly capture autocorrelation and seasonality to make forecasts. You would determine the AR and MA terms by inspecting the ACF and PACF plots. Then use cross-validation or holdout sets to evaluate prediction accuracy on unseen data. Choose the final model based on having good out-of-sample performance without overfitting.

After differencing the time series to make it stationary, I would fit an ARIMA or SARIMA model to the data. I'd determine the AR and MA terms by inspecting the ACF and PACF plots of the differenced series. I would then fit models with different parameter combinations and compare their AIC/BIC scores to choose the best one. I'd check that the residuals of the fitted model are white noise by plotting their ACF which should show no significant autocorrelation. If they aren't white noise, I may need to refine the model. I would simply accumulate the differenced forecasts over time by adding them recursively to the last actual observation. This allows forecasting nonstationary time series with models that require stationarity.

To forecast future values of a differenced time series, you start by predicting the differenced values using a suitable model, such as ARIMA or any other time series forecasting method. Once you have these differenced forecasts, reverse differencing is applied to convert them back to the original scale. This involves adding the differenced forecast to the last known value of the original series. If the time series was differenced more than once (e.g., second-order differencing), you reverse the differencing step-by-step, each time adding back the previously differenced values. This process ensures that the forecasts are in the same scale and context as the original time series, allowing for meaningful interpretation and analysis.

To invert differencing, I would take the predicted future values from my fitted model (i.e. SARIMA or ARIMA) on the differenced data and recursively add them back to the last actual observation. So if I have differenced the data once by taking first differences, I would take the model's one-step-ahead forecast and add it to the last actual value. For the next forecast, I'd add the two-step-ahead predicted difference to the previous forecast, and so on. This accumulates the differences over time to reconstruct the original scale. I'd use pandas shift() and cumsum() functions to align the indexes properly when adding the differences back. Care must be taken to initialize with first actual value and align indexes at each recursion step.

Some additional thoughts on forecasting a differenced time series: In my experience, out-of-sample model validation is critical before using forecasts for decision making. I would reserve some recent holdout data not used in model fitting to verify prediction accuracy. I prefer models that are interpretable, not just predictive black boxes. Understanding the dynamics driving the forecasts builds trust in the process. Purely statistical approaches can lead you astray if not grounded in subject matter knowledge. Understanding the genesis of the data guides appropriate preprocessing and modeling choices.