The test statistic in the likelihood ratio test for nested linear models Imagine that we have a family of probability disributions with p.d.f $f_{\theta}(z)$ where
$\theta \in \Theta$. We also know that there is a linear dependence between parameters. As a consequence we can restrict to a nested model with p.d.f $f_{\theta}(z)$, where $\theta \in \Theta_{0} \subseteq \Theta$.
Formally we have such a situation:
\begin{align}
\Theta \subseteq \mathbb{R}^p,~~~~ h:\mathbb{R}^p \to \mathbb{R}^{p-q},~~~~  \Theta_{0} = \{\theta \in \Theta : h(\theta)=0\}.
\end{align}
where h is a linear map onto $\mathbb{R}^{p-q}$ so we can say that:
\begin{align}
h(\theta) = A\theta  = 0,
\end{align}
where $A$ is a $(p-q) \times p$ matrix of a linear map $h$.
As a result we can say that $\Theta_{0} \subseteq \mathbb{R}^q$. HERE BEGINS MY PROBLEM.
I would be very grateful if someone could tell me why we can conclude now that 
\begin{align}
\sup \limits_{\theta \in \Theta_{1}}f_{\theta}(z) \overset{\huge{?}}{=} \sup \limits_{\theta \in \Theta}f_{\theta}(z). 
\end{align}
Consequently the test statistic of a likelihood ratio test is
\begin{align}
\lambda(z) = \frac{\sup_{\theta \in \Theta_{1}}f_{\theta}(z)}{\sup_{\theta \in \Theta_{0}}f_{\theta}(z)}\overset{\huge{?}}{=} \frac{\sup_{\theta \in \Theta}f_{\theta}(z)}{\sup_{\theta \in \Theta_{0}}f_{\theta}(z)}.
\end{align}
 A: After having thought about this a little and discussed with some peers, here's my take on it. 
The equality you are asking about is not necessarily true. The likelihood function is unimodal, so if the maximum likelihood estimate $\hat{\theta}$ is in $\Theta_0$, then obviously $\sup_{\theta\in \Theta_1}L(\theta|z)\neq \sup_{\theta\in\Theta}L(\theta|z)= \sup_{\theta\in\Theta_0}L(\theta|z)$, where $\Theta_0$ and $\Theta_1$ are disjoint subsets of $\Theta$. 
The issue here, I think, is that $H_1 : \theta \in \Theta_1$ does not mean that we should maximize the likelihood in $\Theta_1$. Instead, we conduct unrestricted maximization -- $\sup_{\theta\in\Theta_0\bigcup\Theta_1}L(\theta|z)=\sup_{\theta\in\Theta}L(\theta|z)$. See for example Definition 8.2.1 in Casella & Berger's Statistical Inference (p. 375). 
It might seem unintuitive why it is this way, and I haven't found any explanation why it actually is this way. However, two reasons may be that a) unrestricted maximization ensures that the ratio is always between 0 and 1 (having $H_0$ in the numerator) which is convenient, and b) it is easier calculating one unrestricted and one restricted maximization than two restricted maximizations. 
I hope this clears things up at least a little bit for you!
Edit: Here's the definition mentioned above.

