In an autoregressive model, if β1 is greater than or equal to 1, what can be said about the model?

In an autoregressive model, the coefficient β1 plays a crucial role in determining the stationarity of the process. When β1 is greater than or equal to 1, it indicates that the effect of shocks to the time series does not die out over time. Instead, it implies that the process exhibits a tendency to either explode or follow a unit root behavior, meaning that the time series can exhibit non-stationary characteristics.

Stationarity refers to the property of a stochastic process where its statistical properties, such as mean and variance, are constant over time. If β1 is less than 1, the process is stationary because shocks will eventually diminish. However, with β1 equal to or greater than 1, the model does not revert to a long-term mean, leading to non-stationarity. This can often manifest in trends or random walks, situations where past values have a lasting impact on future values.

Thus, when β1 is greater than or equal to 1, it is accurate to conclude that the autoregressive model is not stationary, and this understanding is pivotal in the context of time series analysis in risk modeling.