Answer: 4.04

## 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.

Author: Tanishq Prabhu