CFA Level II ExamQuantitative MethodsHard
A quantitative analyst is building a time-series model for a country's quarterly inflation rate. After performing a unit root test, the analyst finds that the series is I(1), meaning it is integrated of order one. To achieve stationarity, the analyst decides to difference the series once. Subsequently, the analyst examines the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) of the differenced series and observes a significant spike at lag 1 in the PACF but not in the ACF, and then a rapid decline in the PACF. Based on these observations, which of the following ARIMA models is most appropriate for the differenced series?
- AARIMA(1,1,0)
- BARIMA(0,1,1)
- CARIMA(1,1,1)
- DARIMA(2,1,0)
Show answer & explanationAnswer & explanation
Correct answer: A. ARIMA(1,1,0)
The series is I(1), so d=1. A significant spike at lag 1 in the PACF, with no significant spikes in the ACF, and a rapid decline in PACF is characteristic of an AR(1) process for the differenced series. Thus, the model parameters would be p=1 (for AR), d=1 (for differencing), and q=0 (for MA). This corresponds to ARIMA(1,1,0).
Why the other options are wrong
- B. ARIMA(0,1,1) would imply a significant spike in the ACF at lag 1 and a rapid decline in PACF, characteristic of an MA(1) process for the differenced series, which contradicts the given observations.
- C. ARIMA(1,1,1) would imply significant spikes in both ACF and PACF, indicating a mixed ARMA process, which is not supported by the description.
- D. ARIMA(2,1,0) would imply significant spikes in the PACF at lags 1 and 2, which is not consistent with only 'a significant spike at lag 1'.
ARIMA Model Identification (ACF/PACF)
ARIMA model parameters (p, d, q) are identified by analyzing the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) of a stationary time series, where 'p' is the AR order, 'd' is the differencing order, and 'q' is the MA order.
- AR(p) process: PACF cuts off after lag p, ACF decays gradually.
- MA(q) process: ACF cuts off after lag q, PACF decays gradually.
- ARIMA(p,d,q): 'd' is determined by differencing to achieve stationarity.
Memory trick: ACF for MA, PACF for AR, and differencing for Integrated.