Answer: 31

## Explanation The Ljung Box statistic, also known as the modified Box Pierce statistic, is a function of the accumulated autocorrelations, $\rho_i$, up to time lag $m$. It's calculated as: $$ Q(m) = n(n + 2)\sum_{i=1}^{m} \left( \frac{\rho_i^2}{n - i} \right) $$ **Given:** - Sample size (n) = 250 - Time lag (m) = 3 - Autocorrelations: ρ₁ = 0.3, ρ₂ = -0.15, ρ₃ = -0.10 **Calculation:** $$ Q(3) = 250 \times 252 \left( \frac{0.3^2}{249} \right) + 250 \times 252 \left( \frac{(-0.15)^2}{248} \right) + 250 \times 252 \left( \frac{(-0.1)^2}{247} \right) $$ Breaking it down: 1. First term: $250 \times 252 \times \frac{0.09}{249} = 63,000 \times 0.0003614 = 22.77$ 2. Second term: $250 \times 252 \times \frac{0.0225}{248} = 63,000 \times 0.0000907 = 5.71$ 3. Third term: $250 \times 252 \times \frac{0.01}{247} = 63,000 \times 0.0000405 = 2.55$ **Total:** $22.77 + 5.71 + 2.55 = 31.03$ The Ljung Box Q statistic is approximately **31**, which matches option B.

Author: Tanishq Prabhu