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The mgcv package performs a linear extrapolation when the newdat argument of the predict function contains values outside the range of the training covariates. I am unable to find out what happens in this use case for the gamboostLSS package and the gamlss and mboost packages on which it is based.

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Just to close this issue an to let others know [I wasn't aware of this open question]:

We had an email exchange in 2015 and decided to use linear extrapolation for smooth effects (both, P-splines and constraint P-splines). This was implemented in mboost 2.5-0. See news(Version >= "2.5-0", package = "mboost") and package?mboost for more details.

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