Why does differencing help when the original time series has a linear trend?

Conditions

• The original series has an approximately linear trend with noise.
• The goal is to achieve stationarity for modeling.

Reasoning, step by step

1. Observe that the original series appears linear with noise.
2. Recall that a linear function changes by a constant amount at each step.
3. Apply the differencing operation $z_t = q_{t+1} - q_t$.
4. Conclude that the differences should hover around a constant level.
5. Interpret this constant mean as making the series stationary.

Example

The speaker says, "to get from one point of a linear function to the next, you just add some constant value," and therefore differences should hover around a constant. The lower plot shows the differenced series fluctuating around a horizontal dashed line.

Common misconceptions

• Thinking differencing amplifies the trend.
• Believing that differencing removes all noise, not just the trend.
• Assuming that any series becomes stationary after one difference.

Watch the explanation

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