I have a - so I guess - a simple question: I am using Stata 13 and I am running a Tobit model to understand differences in firm performance. Among others, I am controling for firm types $T_i$- i.e. Single-Owner-Firms $SOF$vs. Multiple-Owner-Firms $MOF$.
So far I dummied $T_i$ so that $MOF=0$ and $SOF=1$. Here is why:
Given a simple relationship such as
$y_i= a + \beta_1x_i + \beta_2T_i +\beta_3x_iT_i$
We can show that
$y_i= (a + \beta_1x_i) + T_i(\beta_2+\beta_3x_i)$
The lower order coefficients $\beta_1$ shows the simple effect of $x_i$ on $y_i$ for $T_i=0$ since $y_i= (a + \beta_1x_i)$ for $T_i=0$. Similar, $\beta_2$ shows the simple effect of $T_i$ on $y_i$ since $y_i= a + T_i\beta_2$ for $x_i=0$.
Finally, and as far as I understand it, $\beta_3$ depicts how the slopes differ the firm types $T_i=0$ and $T_i=1$
However, I now got the advice to contrast code $T_i$ so that $MOF=-1$ and $SOF=1$.
I have three questions:
- Is the interpretation scheme I put forward above correct?
- Why and when should one use dummy and contrast (effect) coding?
- And if I use contrast coding, how would I interpret thet resulting coefficients?