Understanding the AR(1) Model

In time-series analysis, autoregressive models (AR models) are commonly used to predict future values based on past values. An AR(1) model, for example, uses the previous period's value to predict the current period's value.

The general form of an autoregressive model of order p, denoted as AR(p), is given by the equation:

$$X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \ldots + \phi_p X_{t-p} + \epsilon_t$$

Where:

• $$X_t$$ is the current value of the time series.
• $$c$$ is a constant.
• $$\phi_1, \phi_2, \ldots, \phi_p$$ are the parameters of the model.
• $$\epsilon_t$$ is the error term.

Suppose an analyst is examining a time series of monthly sales data that they believe can be modeled using an autoregressive approach. They found that the past month's sales ($$X_{t-1}$$) provide significant predictive power for the current month's sales ($$X_t$$).

Which of the following statements correctly describes the AR(1) model?