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.