Financial Risk Manager Part 1

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The sample autocorrelations for a time series are estimated to be $\hat{\rho}_1 = 0.20, \hat{\rho}_2 = -0.03$ and $\hat{\rho}_3 = 0.05$ from a sample size of 90. What is the value Ljung-Box Q statistic?_

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TTanishq

3.45

3.87

20.54

4.04

Explanation:

Explanation

The Ljung-Box Q statistic (also known as the modified Box-Pierce statistic) is calculated using the formula:

Q(m) = n(n + 2) \sum_{i=1}^{m} \left( \frac{\rho_i^2}{n - i} \right)

Where:

$n = 90$ (sample size)
$m = 3$ (time lag)
$\rho_1 = 0.20$ , $\rho_2 = -0.03$ , $\rho_3 = 0.05$

Let's calculate step by step:

Step 1: Calculate the individual terms

For $\rho_1 = 0.20$ :

\frac{0.20^2}{90 - 1} = \frac{0.04}{89} = 0.0004494

For $\rho_2 = -0.03$ :

\frac{(-0.03)^2}{90 - 2} = \frac{0.0009}{88} = 0.00001023

For $\rho_3 = 0.05$ :

\frac{0.05^2}{90 - 3} = \frac{0.0025}{87} = 0.00002874

Step 2: Sum the terms

\sum_{i=1}^{3} \left( \frac{\rho_i^2}{n - i} \right) = 0.0004494 + 0.00001023 + 0.00002874 = 0.00048837

Step 3: Multiply by $n(n + 2)$

n(n + 2) = 90 \times 92 = 8280

Q(3) = 8280 \times 0.00048837 = 4.04

Therefore, the Ljung-Box Q statistic is 4.04, which corresponds to option D.

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