Answer: 4.04

The Ljung-Box Q statistic is calculated using the formula: $$Q(m) = n(n + 2)\sum_{i=1}^{m} \left(\frac{\rho_i^2}{n - i}\right)$$ Given: - Sample size n = 90 - Autocorrelations: $\hat{\rho}_1 = 0.20$, $\hat{\rho}_2 = -0.03$, $\hat{\rho}_3 = 0.05$ - Time lag m = 3 Calculation: $$Q(3) = 90 \times 92 \left(\frac{0.2^2}{89}\right) + 90 \times 92 \left(\frac{(-0.03)^2}{88}\right) + 90 \times 92 \left(\frac{0.05^2}{87}\right)$$ Breaking it down: 1. First term: $90 \times 92 \times \frac{0.04}{89} = 8280 \times 0.0004494 = 3.72$ 2. Second term: $90 \times 92 \times \frac{0.0009}{88} = 8280 \times 0.00001023 = 0.0847$ 3. Third term: $90 \times 92 \times \frac{0.0025}{87} = 8280 \times 0.00002874 = 0.238$ Sum: $3.72 + 0.0847 + 0.238 = 4.0427 \approx 4.04$ Therefore, the Ljung-Box Q statistic is 4.04.

Author: Nikitesh Somanthe