Exercise: Christoffersen independence — a clustering counter-example

Consider a VaR model tested over $T = 250$ days at $\alpha = 4\%$. Suppose the observed exceedance pattern is clustered: exceedances happen on days $\{40, 41, 42, 100, 101, 200, 201, 202, 203, 204\}$ — ten exceedances total, all of them in three clusters.

Tasks

1. Verify that Kupiec's POF test fails to reject: the count matches $\alpha T = 10$ exactly, so $LR_{\text{POF}} = 0$.

2. Build the $2\times 2$ transition count matrix $n_{ij}$ for $(I_{t-1}, I_t)$.

3. Compute $\hat\pi_{01}$, $\hat\pi_{11}$, and the unconditional $\hat\pi$.

4. Compute $LR_{\text{ind}}$ and decide whether to reject independence at $5\%$ ($\chi^2_{1,0.95} = 3.841$).

5. Interpret what the clustered pattern implies about the VaR model's response to volatility regimes.

Hint

$n_{00}$ is the number of $(0, 0)$ transitions in the sequence; count exceedance runs carefully.