# Likelihood Ratio Test Involving Two Separate (simple) Regressions?

I'm having some trouble conceptualizing the following in a paper I've come across:

A remark is made on using a likelihood ratio test for the following hypothesis

\begin{align} H_0: \beta_{1,1} = \beta_{1,2} \\ H_{\alpha}: \beta_{1,1} \neq \beta_{1,2} \end{align}

where we have the model setup of

$$Y_{i,j} = \beta_{0,j} + \beta_{1,j}x_i + \epsilon_{i,j},$$

with $$\epsilon_{i,j} \sim N(0, \sigma^2)$$ for all $$i = 1, \dots, n$$ and $$j \in \{1,2\}$$.

I was under the impression that we could only do this for nested-models (i.e. one model has more parameters), or am I misinterpreting that/the model above? How can I specify the "full" model here to develop a test-statistic?

• Can you add a full reference to the paper? Jan 4, 2021 at 23:25