# 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:

1. $$X_1$$= Net capital flows over GDP (which may be positive and negative)
2. $$X_2$$= Dummy variable tagging extreme negative values of $$X_1$$
3. $$X_3$$= Interaction term $$X_1*X_2$$
4. $$X_4$$= Net migration over population (which may be positive and negative)
5. $$X_5$$= Dummy variable tagging extreme positive values of $$X_4$$
6. $$X_6$$= Interaction term $$X_4*X_5$$

I would be very grateful if you may help me understanding this.

• Can you reconstruct any of the variables with a linear combination of the others? I doesn't look like it to me. I'm not sure if that is really a conclusive check though. Are you clipping the extreme values of $X_1$? – Dan Nov 21 '18 at 10:32
• I am taking the lower 20 percent tail as extreme negative values of X1 (that is I am taking those values in order to construct X2). And I do the same for the other variable of interest (but taking the top 20 percentile values). – Macrina Nov 21 '18 at 10:45
• I think you'd need $X_3=X_2+X_1$ to get multicolinearity. That said, do you need both $X_2$ and $X_3$ in the model? – Dan Nov 21 '18 at 10:49
• I guess so. Because usually when you have a "crisis" dummy that you want to interact with another phenomenon you always keep: x, crisis dummy, x*crisis. In my case the crisis is referred to the x itself so it's a bit unusual. I don't really know. – Macrina Nov 21 '18 at 11:00

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: $$b_1 + b_3$$
• Many thanks for your explanation. I noticed that the coefficient attached to X1 and X3 have always opposite sign but more or less the same magnitude. This suggests that in absence of the condition represented by the dummy variable X2 (that is very negative values of X1 ), the variable X1 has a negative effect, but when the condition applies it has a positive effect of the same magnitude, resulting in an overall null effect. I am afraid that my results may be driven by how I created the condition, that is the dummy variable; in a sense it selects only extremely negative values of X1 – Macrina Nov 24 '18 at 11:24