Estimation of theta for IHS (inverse hyperbolic sine) transformation

I am trying to use the IHS transformation to correct for heteroskedasticity in a Tobit model. The main references have been Pence (2006) and Burbidge et al. (1988). I have noticed that the convention is to run a model without heteroskedasticity and another called "IHS Heteroscedastic Tobit" as exhibited in investigating the determinants of R&D cooperation by Carboni (2009).

I do understand the relation used is as follows

$\text{ihs}(y) = \log\big[y + \sqrt{y^2+1}\big]$

and may involve a theta value if the theta is not set to 1 as in Pence (2006) as

$θ^{-1}\sinh^{-1}(\theta y) = \theta^{-1}\log\bigg[\theta y + \sqrt{\theta^2 y^2 + 1}\bigg]$.

My questions are:

1. When do I set the value of theta to 1?

2. How do I obtain a theta value for IHS transformation if it should not be set to 1?

3. What is the dependent variable in the IHS Heteroscedastic Tobit?

I will be very grateful if the appropriate Stata codes and examples are included in your responses.

• References with just names and dates presuppose a minute audience in your particular specialism, which is not good practice. Please spell out full details as you would in a paper or thesis. (For some reason, I see this practice most frequently among economists.) Please see advice in the Help Center on software-related questions. Jan 21 '15 at 11:24
• This needs a different title for orientation to this forum. Flagging that you seek Stata code is the wrong emphasis. The statistical questions embedded here are pertinent. Jan 21 '15 at 11:27
• IHS here appears to mean "inverse hyperbolic sine". As asinh is already available as notation which is widely used in software, why do we need another term? (Digression: using "arc" in naming inverse hyperbolic functions rests on a misunderstanding and a false analogy with inverse trigonometric functions.) Jan 21 '15 at 11:38
• Thanks for the new title. I don't know the literature here, but $\theta$ is just a parameter to complicate the likelihood function, which you should maximise. If in practice it's close to 1, you might decide to do without the parameterisation. The response (you say "dependent") variable is surely $y$ in your notation; whether you transform it first or use asinh as a link function is secondary. Jan 21 '15 at 11:57