Answer: Under both tests, a rejection of the null hypothesis implies that the model fails to capture some dynamics in a time series.

## Explanation Let's analyze each option: **Option A**: Incorrect. Both tests actually share the same null hypothesis that **all** autocorrelations are zero (no autocorrelation in residuals), not that at least one is non-zero. **Option B**: Incorrect. The Ljung-Box test is actually better for small samples because it has better small-sample properties compared to the Box-Pierce test. **Option C**: Incorrect. Both test statistics asymptotically follow chi-squared distributions, not different distributions. **Option D**: **CORRECT**. This is the accurate statement. Both tests examine whether there is autocorrelation in the residuals. If the null hypothesis is rejected, it means there is significant autocorrelation remaining in the residuals, indicating that the ARMA model has failed to capture some of the dynamics in the time series. ### Key Differences: - **Box-Pierce**: Q = n∑ρ²(k) ~ χ²(m) - **Ljung-Box**: Q* = n(n+2)∑ρ²(k)/(n-k) ~ χ²(m) - Ljung-Box has better small-sample properties - Both test for autocorrelation in residuals - Both use chi-squared distribution asymptotically - Rejection means model inadequacy

Author: LeetQuiz Editorial Team