Answer: 11.7

## Box-Pierce Q-statistic Calculation The Box-Pierce Q-statistic is calculated using the formula: $$Q_{BP} = T \sum_{\tau=1}^{m} \hat{\rho}^2(\tau)$$ Where: - $T$ = Sample size = 200 - $\hat{\rho}(\tau)$ = Sample autocorrelation function for lag $\tau$ - $m$ = number of lags under observation = 4 **Step-by-step calculation:** 1. Square each autocorrelation coefficient: - Lag 1: $(0.15)^2 = 0.0225$ - Lag 2: $(-0.14)^2 = 0.0196$ - Lag 3: $(-0.1)^2 = 0.01$ - Lag 4: $(-0.08)^2 = 0.0064$ 2. Sum the squared autocorrelations: $$0.0225 + 0.0196 + 0.01 + 0.0064 = 0.0585$$ 3. Multiply by sample size: $$Q_{BP} = 200 \times 0.0585 = 11.7$$ Therefore, the Box-Pierce Q-statistic is **11.7**, which corresponds to option C. **Interpretation:** The Box-Pierce Q-statistic tests whether a time series is white noise. A large Q-statistic (compared to a chi-square distribution with m degrees of freedom) suggests the series is not white noise.

Author: Tanishq Prabhu