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# P1.T2.509. Box-Pierce and Ljung-Box Q-statistics

#### Nicole Seaman

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Learning outcomes: Calculate the sample mean and sample autocorrelation, and describe the Box-Pierce Q- statistic and the Ljung-Box Q-statistic. Describe sample partial autocorrelation

Questions:

509.1. If a time series is reasonably approximated as white noise, then each of the following is true EXCEPT which is not true of a white noise process?

a. Serial correlations (aka, autocorrelations) are zero
b. Observations in the time series are normally distributed
c. In a large sample, the distribution of the sample autocorrelations is approximately normal with mean of zero
d. In a large sample, the distribution of the sample autocorrelations is approximately normal with variance of 1/T

509.2. For a certain time series, you have produced a correlogram with an autocorrelation function that includes twenty four monthly observations; m = degrees of freedom = 24. Your calculated Box-Pierce Q-statistic is 19.50 and your calculated Ljung-Box Q-statistic is 27.90. You want to determined if the series is
white noise. Which is your best conclusion (please note this requires a lookup)?

a. With 95.0% confidence, you accept the series as white noise (more accurately, you fail to reject the null)
b. With 95.0% confidence, you accept the series as partial white noise (due to Box-Pierce) but reject the null (due to Ljung-Box)
c. With 95.0% confidence, you reject both null hypotheses and conclude the series is not white noise
d. With 95.0% confidence, you reject both null hypotheses but conclude the series is white noise because the sum of the statistics is greater than the critical value

509.3. In regard to the Box-Pierce and Ljung-Box Q-statistics, each of the following is TRUE except which is false?

a. The Box-Pierce Q-statistic is used to test whether the residuals in a time series are white noise
b. The Ljung-Box Q-statistic is used to test whether a time series exhibits a linear trend under the null hypothesis of a unit root
c. The Box-Pierce Q-statistic is approximately distributed as a chi-squared random variable under the null hypothesis that autocorrelations are jointly zero in a time series
d. Selection of the number of lags being tested (aka, maximum displacement, m) in the Ljung-Box test is a balance between conducting a joint test (i.e., can't be too small) and quality of the distribution approximations (i.e., can't be too large)

##### New Member
Hello, where could I find the answers to these questions? Thank you in advance! Kind regards

Learning outcomes: Calculate the sample mean and sample autocorrelation, and describe the Box-Pierce Q- statistic and the Ljung-Box Q-statistic. Describe sample partial autocorrelation

Questions:

509.1. If a time series is reasonably approximated as white noise, then each of the following is true EXCEPT which is not true of a white noise process?

a. Serial correlations (aka, autocorrelations) are zero
b. Observations in the time series are normally distributed
c. In a large sample, the distribution of the sample autocorrelations is approximately normal with mean of zero
d. In a large sample, the distribution of the sample autocorrelations is approximately normal with variance of 1/T

509.2. For a certain time series, you have produced a correlogram with an autocorrelation function that includes twenty four monthly observations; m = degrees of freedom = 24. Your calculated Box-Pierce Q-statistic is 19.50 and your calculated Ljung-Box Q-statistic is 27.90. You want to determined if the series is
white noise. Which is your best conclusion (please note this requires a lookup)?

a. With 95.0% confidence, you accept the series as white noise (more accurately, you fail to reject the null)
b. With 95.0% confidence, you accept the series as partial white noise (due to Box-Pierce) but reject the null (due to Ljung-Box)
c. With 95.0% confidence, you reject both null hypotheses and conclude the series is not white noise
d. With 95.0% confidence, you reject both null hypotheses but conclude the series is white noise because the sum of the statistics is greater than the critical value

509.3. In regard to the Box-Pierce and Ljung-Box Q-statistics, each of the following is TRUE except which is false?

a. The Box-Pierce Q-statistic is used to test whether the residuals in a time series are white noise
b. The Ljung-Box Q-statistic is used to test whether a time series exhibits a linear trend under the null hypothesis of a unit root
c. The Box-Pierce Q-statistic is approximately distributed as a chi-squared random variable under the null hypothesis that autocorrelations are jointly zero in a time series
d. Selection of the number of lags being tested (aka, maximum displacement, m) in the Ljung-Box test is a balance between conducting a joint test (i.e., can't be too small) and quality of the distribution approximations (i.e., can't be too large)