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LetThen the null hypothesis is $R{\theta }_0=r$. Define the Wald statistic for linear restrictions

Under $H_0$, part (a) and the continuous mapping theorem givewhereThe matrix $V_0$ is positive definite because $I({\theta }_0)$ is positive definite and $R$ has full row rank. Consistency givesTherefore Slutsky theorem yieldsand applying the continuous squared-norm map proves rigorously that

Let $q_{k,1-\alpha }$ be the $(1-\alpha )$ quantile of ${\chi }_k^2$. Rejecting $H_0$ whenhas rejection probability tending to $\alpha$ under every fixed parameter satisfying $H_0$, and is therefore an asymptotically valid level-$\alpha$ test.