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.