Im currently studying the Wald test statistic for testing $ H_0: \boldsymbol{R'\hat{\beta}} = \boldsymbol{r} $ given by $$ W = \frac{(\boldsymbol{R'\hat{\beta}} - \boldsymbol{r})'(\boldsymbol{R'(X'X)R)}^{-1}(\boldsymbol{R'\hat{\beta}} - \boldsymbol{r})}{\hat{\sigma}^2}, $$ where $\hat{\sigma}^2$ is a consistent estimator of $\sigma^2$. Now, under some assumptions, under the null: $$ W \overset{\alpha}{\sim} \chi^2_q, $$ where $q$ is the amount of restrictions imposed by $H_0$. Now my book says that the $(1-\alpha)$ confidence interval for $\boldsymbol{r}$ is given by $$ [\boldsymbol{r} | W \leq c^*_{1-\alpha}]. $$ Now my questions is: why is there only one critical value for this confidence interval? I would think the confidence interval would be given by $$ [\boldsymbol{r} | c^*_{\alpha/2} \leq W \leq c^*_{1-\alpha/2}]. $$ Does this have something to do with $W$ describing an ellipsoid?
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