As noticed by whuber the model I described seems to be just a variation of regression using linear splines. Such models are described, e.g. by James H. Steiger on those slides or on those notes by Frank Harrell. The model can be thought as a piecewise regression

$$ E(Y|X) = \beta_0 + \beta_1 X + \beta_2 (X-a)_+ + \beta_3 X (X-a)_+ + \dots $$

or regression using linear splines

$$ E(Y|X) = \beta_0 + \beta_1 X + \beta_2 (X-a)_+ + \dots $$

where $a$ is a knot and $(u)_+$ takes value of $1$ if $u$ is positive and $0$ otherwise.

So it seems that this problem reduces to asking Is there a problem with multicollinearity and for splines regression?

As noticed by whuber the model I described seems to be just a variation of regression using linear splines. Such models are described, e.g. by James H. Steiger on those slides or on those notes by Frank Harrell. The model can be thought as a piecewise regression

$$ E(Y|X) = \beta_0 + \beta_1 X + \beta_2 I_{\{X>a\}} + \beta_3 X \cdot I_{\{X>a\}} + \dots $$

or regression using linear splines

$$ E(Y|X) = \beta_0 + \beta_1 X + \beta_2 (X-a)_+ + \dots $$

where $a$ is a knot and $(u)_+$ takes value of $1$ if $u$ is positive and $0$ otherwise.

So it seems that this problem reduces to asking Is there a problem with multicollinearity and for splines regression?