# How to Reshape Linear Regression Model using Correlation Coefficient instead of slope?

I try to reproduce Results from Harding and Pagan (2004), p.12, where they try to estimate the correlation coefficient $\rho_{S} = cor(S_{y,i},S_{x,i})$ using regression on

$$\frac{S_{y,t}}{\hat{\sigma_{S_x}}\hat{\sigma_{S_y}}} = \alpha_1 + \frac{\rho_S}{\hat{\sigma_{S_x}}\hat{\sigma_{S_y}}} S_{x,t} + \varepsilon_{1,t}$$.

But whatever I try to transform the equations for linear relation between $S_{x,t}$, $S_{x,t}$

$$S_{y,t} = \alpha_0+\beta S_{x,t}+\varepsilon_{0,t}$$

and the definition of the correlation coefficient $\hat{\beta} = \rho_S\frac{\hat{\sigma_{S_y}}}{\hat{\sigma_{S_x}}}$, I never manage to get the result in the first equation. Any ideas?

Thank's a lot!!

• This is correct if both the $\beta$ and $\rho$ are estimates. – Michael R. Chernick Apr 25 '17 at 20:04
• By "this" you mean the first equation? Can you point to any Source I can take further reading in? – EliteTUM Apr 25 '17 at 20:24
• I was talking about your claim regarding your final equation. I think you can find this in any regression book (e.g. Draper and Smith). – Michael R. Chernick Apr 25 '17 at 20:28