ACF and PACF Fundamentals: Understanding Time Series Dependencies Quiz

Explore the essentials of Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) in time series analysis. This quiz helps you identify dependencies, interpret plots, and distinguish AR, MA, and mixed processes using intuitive examples and key concepts.

1. Role of ACF

   What does the Autocorrelation Function (ACF) primarily measure in a stationary time series?

   1. The correlation of a series with its past values at different lags
   2. The difference between maximum and minimum values
   3. The trend component over time
   4. The average value of the time series

   Explanation: The ACF measures how current values of a series relate to its previous values at various time steps, quantifying dependence at different lags. The average value is not related to autocorrelation, while the difference between maximum and minimum is simply the range. The ACF does not specifically capture the trend but rather the time-based dependencies.

2. Interpreting ACF Cut-off

   If the ACF of a time series drops to zero after lag 1, which process is it most likely to represent?

   1. Integrated process of order 1 (I(1))
   2. Autoregressive process of order 1 (AR(1))
   3. Moving Average process of order 1 (MA(1))
   4. Seasonal process

   Explanation: An MA(1) process typically shows autocorrelation that drops to zero after lag 1 in its ACF plot. An AR(1) process displays a gradual decay in ACF, not a sharp cut-off. I(1) signifies differencing and is not identified by this pattern, while a seasonal process often shows repeating spikes, not a single cut-off.

3. PACF Identification

   What pattern in the Partial Autocorrelation Function (PACF) plot suggests an autoregressive process of order 2 (AR(2))?

   1. PACF values all near zero
   2. Significant PACF at every other lag
   3. Significant PACF only at lag 1
   4. Significant PACF at lags 1 and 2, near zero afterward

   Explanation: An AR(2) process will show significant PACF values at the first two lags, with the remaining PACF values dropping close to zero beyond lag 2. If only lag 1 is significant, it indicates AR(1), not AR(2). Every other lag suggests seasonality, not AR(2). All values near zero would show no autoregressive behavior.

4. MA vs AR Patterns

   Which statement best distinguishes the MA(1) from the AR(1) process using ACF and PACF plots?

   1. MA(1) shows ACF cut-off at lag 1; AR(1) shows PACF cut-off at lag 1
   2. MA(1) has PACF cut-off at lag 2; AR(1) has ACF cut-off at lag 2
   3. Both MA(1) and AR(1) show ACF cut-off at lag 1
   4. MA(1) shows both ACF and PACF with geometric decay

   Explanation: MA(1) processes have an ACF that cuts off sharply after lag 1, while AR(1) processes have a PACF cut-off at lag 1. The alternative patterns do not correctly describe the distinction; for example, cut-off at lag 2 or geometric decay in both plots are not characteristics of either MA(1) or AR(1) processes.

5. White Noise ACF

   What would you expect to see in the ACF plot of a white noise time series?

   1. A gradually decaying ACF
   2. All ACF values fall within the confidence bounds around zero
   3. A significant ACF spike at lag 1
   4. A regularly repeating seasonal pattern

   Explanation: White noise implies complete randomness, so the ACF plot will show all values within the bounds of random variation (generally no significant spikes). Significant spikes indicate dependency, as seen in the other options, whereas white noise lacks such dependencies or seasonality.

6. PACF Definition

   How does the Partial Autocorrelation Function (PACF) at lag k differ from the standard autocorrelation at lag k?

   1. PACF measures the correlation between current and lagged values, controlling for values in between
   2. PACF only measures the average of all prior lags
   3. PACF ignores intermediate lags completely
   4. PACF doubles the value of standard autocorrelation

   Explanation: PACF adjusts for the effects of all shorter lags, isolating the direct relationship at lag k. Ignoring intermediate lags describes neither ACF nor PACF. Averaging prior lags or doubling values is unrelated to the calculation or interpretation of PACF.

7. Interpreting Mixed ARMA

   Given a time series whose ACF decays gradually and PACF cuts off after lag 1, which model is it most likely to follow?

   1. White noise process
   2. Autoregressive process of order 1 (AR(1))
   3. Integrated process
   4. Moving Average process of order 1 (MA(1))

   Explanation: Gradual decay in the ACF coupled with PACF cut-off at lag 1 is typical of AR(1). An MA(1) would show cut-off in ACF, not PACF. White noise has no pattern or structure, and an integrated process refers to non-stationarity, which is not indicated here.

8. Lag Interpretation

   What does 'lag 2' mean in the context of ACF and PACF analysis?

   1. Comparing each value to the value two time periods earlier
   2. Doubling the values in the series
   3. Taking the second derivative of the series
   4. Summing values at two different time points

   Explanation: Lag 2 considers the relationship between a point and the point exactly two steps before it in time. It has nothing to do with doubling values, derivatives, or simply summing data points, which are different analytical operations.

9. MA(2) Recognition

   If a time series shows significant ACF values only at lags 1 and 2 with all further lags near zero, what model does this most likely indicate?

   1. Random walk
   2. Seasonal process with period 2
   3. Moving Average process of order 2 (MA(2))
   4. Autoregressive process of order 2 (AR(2))

   Explanation: An MA(2) process displays autocorrelation only up to lag 2, with the rest becoming negligible. AR(2) affects the PACF instead. Pure seasonality or random walks do not produce this exact ACF pattern.

10. Zero Lag in ACF

    What is the value of the ACF at lag 0 for any stationary time series?

    1. Exactly zero
    2. Exactly one
    3. Equal to the mean of the series
    4. Unpredictable

    Explanation: The ACF at lag 0 is always one because it represents the correlation of the series with itself at the same time point. Zero would mean no correlation, which is incorrect. The mean of the series or unpredictability are not descriptions of correlation at lag 0.