Question

Suppose that a researcher had estimated the first 5 autocorrelation coeffcients using a series of length 100 observations, and found them to be (from 1 to 5): 0.207, -0.013, 0.086, 0.005, -0.022.
Test each of the individual coefficient for significance, and use both the Box-Pierce and Ljung-Box tests to establish whether they are jointly significant.

Answer 1

Given:

Sample size = T = 100

The first 5 Autocorrelation coefficients are 0.207, -0.013, 0.086, 0.005, -0.022

Let the level of significance be 5% i.e., = 0.05

a] Test for significance of individual Autocorrelation coefficients:

Hypothesis: v/s  ; k = 1, 2 ,3 ,4 5

The 95% confidence interval is given as

Any Autocorrelation coefficient that belongs to this interval is significant.

The 95% confidence interval is

Decision: The first Autocorrelation coefficient (₁) = 0.207 does not belong to this interval. Hence it rejects H_0.

The rest belongs to this confidence interval. Hence they donot reject H₀.

Conclusion: (1st autocorrelation coefficient) is not significant. The rest are significant.

b]Test for the joint hypothesis of Autocorrelation coefficients:

Hypothesis:

v/s

1) L-Jung-Box test

Test statistic:    

m = 5

Q = (100*102*5.1506*10^-4) = 5.253612 [calculated]

Q ~

At 5% significance, Q = = 12.83 [tabulated]

Decision: Q_calculated < Q_tabulated . Hence we donot reject H₀.

Conclusion: The autocorrelation coefficients are jointly significant.

2] Box-Peirce test:

Test statistic:   

m = 5

Q = 100 * 0.050923 = 5.0923 [calculated]

Q ~

At 5% significance, Q =   = 12.83 [tabulated]

Decision: Q_calculated < Q_tabulated . Hence we donot reject H₀.

Conclusion: The autocorrelation coefficients are jointly significant.