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Answer: The Box-Pierce test statistic is equal to the sum of the squared autocorrelations scaled by the sample size.
**Explanation:** A is correct. The Box-Pierce test statistic is indeed equal to the sum of the squared autocorrelations scaled by the sample size. The formula for the Box-Pierce Q statistic is: \[ Q = n \sum_{k=1}^{m} \hat{\rho}_k^2 \] where: - n is the sample size - m is the number of lags being tested - \(\hat{\rho}_k\) is the sample autocorrelation at lag k B is incorrect because the Box-Pierce test statistic follows an asymptotic Chi-square distribution, not a "converging" Chi-square distribution. Both Box-Pierce and Ljung-Box statistics follow asymptotic Chi-square distributions under the null hypothesis of no autocorrelation. C is incorrect because the Ljung-Box test statistic also follows an asymptotic Chi-square distribution, not a standard normal distribution. The Ljung-Box test is actually a modified version of the Box-Pierce test that has better small-sample properties. D is incorrect because both tests are used to test the null hypothesis that the residuals follow a white noise process (i.e., all autocorrelations are zero). Both tests assume that the estimated model residuals should be white noise if the model is correctly specified.
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A newly hired analyst at an investment bank is using an autoregressive moving average (ARMA) process to model equity returns data. In addition to conducting a visual inspection to assess the model fit, the analyst decides to validate the model by testing for autocorrelation and considers using either the Ljung-Box test statistic or the Box-Pierce test statistic. Which of the following correctly describes these two test statistics?
A
The Box-Pierce test statistic is equal to the sum of the squared autocorrelations scaled by the sample size.
B
The Box-Pierce test statistic is the sum of autocorrelations that follow a converging Chi-square distribution.
C
Unlike the Box-Pierce test statistic, the Ljung-Box test statistic follows an asymptotic standard normal distribution.
D
Unlike the Box-Pierce test, the Ljung-Box test does not assume that the estimated model residuals follow a white noise process.
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