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: $$0.02`25 + 0.0196 + 0.01 + 0.0064 = 0.0585$$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.