In Longstaff's original LSMC paper (Valuing American Options by Simulation: A Simple Least-Squares Approach, 2001 (link)), it is claimed that one should only use in-the-money paths for regression at each time step, in order to improve efficiency. It seems to be accompanied by no proof. The reason the author gives is that only in-the-money paths are relevant for comparing exercise vs continuation values.
Concerning the usage of ITM paths, there are two questions:
1). Would there be a serious problem if I have to use also OTM paths? For example, serious biasedness?
2). For some derivatives, it may be difficult to identify what exactly "in-the-money" even means. For example, consider a convertible bond with an embedded call option, then there is surely more than one way in which the convertible can be early terminated:
Voluntary conversion: when in the convertible period and conversion value exceeds continuation value.
Forced conversion: when in the convertible and callable period and call price is smaller than both continuation value and conversion value.
Call: when in the callable period and call value is smaller than continuation value but greater than conversion value.
In this case I don't think there is a clear definition of "in-the-money" or the "relevance" to exercising. Currently, I'm following the Tsiveriotis and Fernandes framework (Valuing Convertible Bonds with Credit Risk, Jrl Fixed Income, 1998) and splitting the convertible bond cashflows as equity-only (which arises from conversion only) and cash-only (which arises from call only), and I'm considering doing two regressions each step for the two parts of cashflows respectively. So now comes the problem, how to define "in-the-money" for the two respective cases? And for the equity-only part, should we distinguish between voluntary conversion and forced conversion when we regress?