Time Series Analysis Autocorrelation and Partial Autocorrelation 2 — Questions and Answers
Question 1: What does the Partial Autocorrelation Function (PACF) measure at lag k?
- The total correlation between observations k periods apart
- The correlation between observations k periods apart after removing the effect of intervening lags (Correct answer)
- The average autocorrelation up to lag k
- The seasonal correlation at lag k
Correct answer: The correlation between observations k periods apart after removing the effect of intervening lags
The PACF at lag k measures the correlation between observations separated by k periods after controlling for all shorter lags.
Question 2: For a pure AR(p) process, which of the following correctly describes the PACF?
- It decays exponentially
- It cuts off to zero after lag p (Correct answer)
- It shows a slow linear decay
- It alternates in sign without cutting off
Correct answer: It cuts off to zero after lag p
The PACF of an AR(p) process cuts off sharply to zero after lag p, making the PACF the primary tool for identifying AR order.
Question 3: Which combination of ACF and PACF patterns suggests an AR(2) model is appropriate?
- ACF cuts off after lag 2; PACF decays gradually
- ACF decays gradually; PACF cuts off after lag 2 (Correct answer)
- Both ACF and PACF cut off after lag 2
- Both ACF and PACF decay gradually
Correct answer: ACF decays gradually; PACF cuts off after lag 2
An AR(p) model is indicated when the PACF cuts off after lag p while the ACF decays gradually (tails off).
Question 4: The Ljung-Box test statistic is used to:
- Test whether a single autocorrelation is significant
- Test whether a group of autocorrelations are jointly zero (Correct answer)
- Estimate the order of differencing needed
- Measure the strength of seasonal autocorrelation
Correct answer: Test whether a group of autocorrelations are jointly zero
The Ljung-Box Q statistic tests the null hypothesis that the first m autocorrelations are jointly equal to zero, useful for model diagnostics.
Question 5: In the Box-Jenkins methodology, which tools are primarily used in the model identification stage?
- Residual plots and histograms
- ACF and PACF plots (Correct answer)
- AIC and BIC information criteria
- Durbin-Watson and Breusch-Godfrey tests
Correct answer: ACF and PACF plots
The Box-Jenkins identification stage relies on examining the patterns in the ACF and PACF to determine the appropriate AR and MA orders.
Question 6: If the ACF of a time series shows a sinusoidal pattern with significant spikes at regular intervals, what does this most likely indicate?
- The series is integrated of order 1
- The series contains a seasonal component (Correct answer)
- The series is purely autoregressive
- The series exhibits heteroskedasticity
Correct answer: The series contains a seasonal component
Regular sinusoidal spikes in the ACF at fixed intervals (e.g., lags 12, 24, 36 for monthly data) indicate the presence of seasonality.
Question 7: Which of the following is a key difference between the ACF and the PACF?
- The ACF only applies to stationary series, while the PACF does not
- The PACF removes indirect effects of intermediate lags, while the ACF does not (Correct answer)
- The ACF is bounded between 0 and 1, while the PACF is not
- The PACF is always non-negative, while the ACF can be negative
Correct answer: The PACF removes indirect effects of intermediate lags, while the ACF does not
The key difference is that the PACF isolates the direct relationship at lag k by controlling for all shorter lags, whereas the ACF includes all indirect effects.
Time Series Analysis Autocorrelation and Partial Autocorrelation 2
What does the Partial Autocorrelation Function (PACF) measure at lag k?