Answer-first summary for fast verification
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 serial correlation), not that at least one autocorrelation is non-zero. **Option B**: Incorrect. The Ljung-Box test is actually better in small samples than the Box-Pierce test. The Ljung-Box test includes a finite-sample correction that makes it more accurate for smaller sample sizes. **Option C**: Incorrect. Both test statistics asymptotically follow a chi-squared distribution, not different distributions. The Ljung-Box test is a modified version of the Box-Pierce test with better small-sample properties. **Option D**: **Correct**. Both tests have the same interpretation: rejecting the null hypothesis (that all autocorrelations are zero) indicates that there is significant serial correlation in the residuals, meaning the ARMA model has failed to capture some 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 includes a finite-sample correction factor (n+2)/(n-k) that makes it more accurate for smaller samples - Both test the same null hypothesis: H₀: ρ₁ = ρ₂ = ... = ρₘ = 0 - Both have the same interpretation when rejecting the null: model inadequacy Therefore, option D correctly states the common interpretation of both tests.
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Testing autocorrelation in the residuals is a standard specification check applied after fitting an ARMA model. There are two closely related tests in this specification analysis, namely the Box-Pierce test and Ljung-Box test. Which of the following statements correctly distinguishes these two tests?
A
They share the same null hypothesis, which states that at least one autocorrelation is non-zero.
B
The Box-Pierce test works better in smaller samples compared to the Ljung-Box test.
C
Asymptotically, the Box-Pierce test-statistic follows a chi-squared distribution, while the Ljung-Box test-statistic follows a Levy distribution.
D
Under both tests, a rejection of the null hypothesis implies that the model fails to capture some dynamics in a time series.