Explanation:

Let's analyze each statement:

Statement I: "If the autoregressive (p) in an ARIMA model is 1, there is no autocorrelation in the series." - INCORRECT

• In ARIMA models, p represents the autoregressive component. If p = 1, it means the current value depends on the previous value (lag 1), indicating autocorrelation at lag 1.
• If p = 0, then there is no autoregressive component and no autocorrelation from the AR perspective.

Statement II: "If (d), the integrated component in an ARIMA model is 0, the series is not stationary" - INCORRECT

• The d parameter represents the number of differencing operations needed to make the series stationary.
• If d = 0, it means no differencing is required, suggesting the original series is already stationary (or at least not integrated).
• If d > 0, then the series is non-stationary and requires differencing to become stationary.

Statement III: "There is autocorrelation in a series with lag 1 if the moving average component (q) in an ARIMA model is 1" - CORRECT

• In ARIMA models, q represents the moving average component.
• If q = 1, it means the current value depends on the previous error term (lag 1), which indicates autocorrelation at lag 1.
• The moving average component captures the effect of past shocks/errors on the current value.

Therefore, only Statement III is correct, making option C the correct answer.

Key Concepts:

• ARIMA(p,d,q): Autoregressive Integrated Moving Average model
• p (AR): Number of autoregressive terms - captures relationship with past values
• d (I): Number of differencing operations - makes series stationary
• q (MA): Number of moving average terms - captures relationship with past error terms