Why are AR models described as long-memory models?

Conditions

• Applies to autoregressive time-series models
• Assumes recursive substitution backward in time

Reasoning, step by step

1. Start from the AR(1) equation $Y_t = \omega + \phi Y_{t-1} + e_t$.
2. Substitute the lagged dependence backward so that $Y_t$ is expressed using $Y_{t-2}$ and earlier errors.
3. Continue the same backward substitution repeatedly.
4. Observe that the current value inherits dependence farther and farther back in time.

Example

The slide shows $Y_t = \omega + \phi Y_{t-1} + e_t$ and then $Y_t = \omega^* + \phi^2 Y_{t-2} + \phi e_{t-1} + e_t$. The speaker explains that because $Y_{t-1}$ itself depends on $Y_{t-2}$, the current value inherits dependence farther and farther back in time.

Common misconceptions

• Thinking 'long memory' means the model only depends on one distant lag.
• Believing that AR models do not accumulate effects from past shocks.

Watch the explanation

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