知识点

问题如下：

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?

选项：

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.

解释：

Explanation: A is correct. The Box-Pierce test statistic is the sum of the squared autocorrelations scaled by the sample size.

B is incorrect. It is the sum of asymptotic, not converging, Chi-square distributed autocorrelations.

C is incorrect. Just like the Box-Pierce, the Ljung-Box statistic also follows an asymptotic Chi-square distribution.

D is incorrect. Both tests verify if the key assumption that residuals are white noise distributed holds. That is, the two statistics are used to test the joint null hypothesis that all the autocorrelations are simultaneously zero.

Learning Objective: Describe the Box-Pierce Q statistic and the Ljung-Box Q statistic.

Reference: Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson, 2023, Chapter 10, Stationary Time Series [QA-10].