Linear Regression Question

Question

I'm struggling with the following question:

Consider the models

\begin{align} \ln(Y)=\beta_1+\beta_2 \ln(L) + \beta_3 \ln(K)+\epsilon \tag{1}\\ \ln(Y)-\ln(K)=\beta_1^*+\beta_2^*(\ln(L)-\ln(K))+\beta_3^*\ln(K)+\epsilon^* \tag{2}\\ \ln(Y)-\ln(K)=\beta_1^\# +\beta_2^\# (\ln(L) -\ln(K)) + \epsilon^\# \tag{3}\\ \end{align}

Find 3 relations between the parameters $(\beta_1,\beta_2,\beta_3)$ in model (1) and $(\beta_1^*,\beta_2^*,\beta_3^*)$ in model (2). Under which conditions can the model be rewritten as model (3), and what are then the relations between $(\beta_1,\beta_2,\beta_3)$ and $(\beta_1^\#,\beta_2^\#)$ in (3)?

By doing some rewriting, I can come up with a relation between the parameters in (1) and (2). But I don't understand what 3 relations they mean. I think that the condition for model 3 to hold is that $\beta_3=0$.

Could anyone please give me some more hints on this question, as I have no clue whether I'm interpreting things correctly.

Answer 1

You're close.

Models (2) and (3) have very similar forms, so it is easy to see how model (3) can be a special case of (2): to write a model of form (2) in form (3), you need $\beta_{3}^{\ast}=0$.

To see when a model of form (1) can be expressed in the form (3), you just need to rewrite this condition in terms of $\beta_{1}, \beta_{2}$, and $\beta_{3}$.