When would you use a second difference instead of a first difference?

Conditions

• The original series is non-stationary.
• First differencing ($d=1$) does not result in a stationary series.
• The modeler can inspect the stationarity of the differenced plots.

Reasoning, step by step

1. Apply first differencing to the original series.
2. Check if the resulting series is stationary (constant mean, constant variance).
3. If not stationary, apply second differencing to the first differenced series.
4. Check again for stationarity.
5. Continue increasing $d$ until stationarity is achieved, preferring the smallest $d$.
6. Example: If $Z_t = q_{t+1} - q_t$ is not stationary, try $W_t = Z_{t+1} - Z_t$.

Example

The speaker says, "you can instead want the second difference, creating a new series like $w_t = z_{t+1} - z_t$... though usually a first one suffices depending on the task." He also advises to "keep d as small as possible so the model remains simple."

Common misconceptions

• Always using the highest possible differencing order.
• Thinking second differencing is required for all trending series.
• Confusing second differencing with squaring the series.

Watch the explanation

Explore next

• What does the d in ARIMA(p,d,q) mean?
• Why does differencing help when the original time series has a linear trend?
• How do you recover the original time series $q_k$ from modeled differences $Z_t$?