What happens in an autoregressive model if β1 equals 0?

In an autoregressive model, the parameters represent the influence of the past values of a time series on its current value. If β1 equals 0, this signifies that the first lag of the time series has no effect on the current value. Consequently, the relationship captured by the lagged variable is nullified, leading to a situation where the correlation between the past value (lagged term) and the current value is effectively zero.

This directly impacts the model's behavior, as the lagged term that typically contributes to explaining the dynamics of the time series no longer has any significance. Instead, the current value is not influenced by its immediate past, which is an important aspect of autoregressive processes. As a result, the model reflects no lagged correlation, implying that previous values do not help predict the current value in any meaningful way.

This outcome clarifies why the assertion that the lagged correlation becomes zero is accurate in the context of an autoregressive model with β1 set to zero.