Dependent / conditional regression?

#1
Hi there!

Long time lurker, first time poster (first post in ANY board), so my sincerest apologies if I'm doing anything wrong! I have searched the forum, but as I do not know what model I’m looking for, I haven’t been able to find anything relevant.

I am trying to build a multiple linear regression model, but one of my independent variables can only take a value for a sub-sample of the observations. This sub-sample (roughly half of the observations) is identified by a binary variable.

My problem is, that if an observation belongs to said sub-sample, then the variable will have a value (it is a continuous variable), but if it does not, then the variable will not logically have any value. I cannot use zero as value, as it would completely bias the results.

So my question is: What regression model should I use, when an independent continuous variable only takes a value depending / conditional on the value of a different (binary) independent variable.

Further, if this requires some kind of exotic model (as opposed to e.g. transformation of vanilla linear regression), I would be very grateful if you could point me to a user-friendly modeling software, preferably with a free-to-try trial thingy (my school has just cancelled all SAS/SPSS licenses).

Thanking you all in advance!!

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For those interested in a little more detail, I am writing a thesis on the determinants of Underpricing in Danish Initial Public Offerings (IPOs). The IPOs can (generally speaking) be either Fixed Price or Tender Offer, in the former case the price is fixed in advance, in the latter the investment bank sets the final offer price considering investor demand, based on bids from potential investors.

For the Tender Offer IPOs, a minimum offer price is communicated in advance, and investors place bids at or above the minimum. After the subscription period has closed, a final offer price is set, and all bids at or above the offer price receive an allocation.

Using univariate analysis, I have found a statistically significant relationship between the level of Underpricing (dependent variable), and the distance from the minimum offer price to the final offer price (independent variable). I want to include this relationship in my overall multiple regression, but as this relationship does not logically exist in Fixed Price offerings, I don’t know how to include all the observations in one regression. Setting the “missing” observation to zero would, in my opinion, introduce significant bias, as the Fixed Price observations would equal a Tender Offer where the price was not revised upwards from the minimum offer price to the final offer price (which does happen quite often).

I could split the sample in two, and run separate regressions, but this would both decrease the overall power, as well as introduce a completely different set of independent variables. Further, I would highly appreciate having it all together in one model.

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