Is there a collinearity issue when using: x, dummy indicating extreme negative value of x and their interaction? I was wondering whether I can build my baseline model using the following variables without incurring in any multicollinearity issue:


*

*$X_1$= Net capital flows over GDP (which may be positive and negative)

*$X_2$= Dummy variable tagging extreme negative values of $X_1$

*$X_3$= Interaction term $X_1*X_2$

*$X_4$= Net migration over population (which may be positive and
negative)

*$X_5$= Dummy variable tagging extreme positive values of $X_4$

*$X_6$= Interaction term $X_4*X_5$
I would be very grateful if you may help me understanding this.
 A: These won't be collinear (assuming you have the data).
A rank deficient data matrix $X$ would a screwy, degenerate case. For example, if $x_i$ is never extremely negative or if you have only 1 observation where $x_i$ is extremely negative, your $X$ matrix will be rank deficient.
A more likely problem is that you may have poor estimates of your crisis condition coefficients if the crisis condition (i.e. $x_i < c$) is rare.
Further Explanation
Your model is effectively estimating different intercept and slope depending on whether you're in the case $x_{i} < \gamma$ or case $x_{i} \geq \gamma$ where $\gamma$ is the cutoff for $x_{i}$ being "extremely negative."
Define dummy variable $d_i$ as:
$$d_{i} = \left\{ \begin{array}{rl} 1 & \text{ if }x_{i} < \gamma \\ 0 & \text{ otherwise} \end{array} \right.$$
Consider the model
$$y_i = b_0 + b_1 x_i + b_2 d_i + b_3 d_i x_i + \epsilon_i$$


*

*Q: What's the intercept if $x_i \geq c$? Answer: $b_0$

*Q: What's the intercept if $x_i < c$? Answer: $b_0 + b_2$

*Q: What's the slope on $x_i$ if $x_i \geq c$? Answer: $b_1$

*Q: What's the slope on $x_i$ if $x_i < c$? Answer: $b_1 + b_3$
