add the sign of the independent variable in a linear regression

Question

I would like to include the sign of X in a linear regression to highlight the impact it has on Y (see the scatter plot below). I first thought of a dummy, taking the value of 1 if positive and 0 if negative but I had difficulties interpreting it, especially due to the dummy variable trap. So I finally just went with the following independent variables:

• absolute value of X
• sign of X

The results are as follow

                         OLS Regression Results                            
==============================================================================
Dep. Variable:                      Y   R-squared:                       0.255
Model:                            OLS   Adj. R-squared:                  0.254
Method:                 Least Squares   F-statistic:                     334.7
Date:                Sat, 25 Mar 2023   Prob (F-statistic):          6.51e-187
Time:                        09:08:30   Log-Likelihood:                 2567.9
No. Observations:                2938   AIC:                            -5128.
Df Residuals:                    2934   BIC:                            -5104.
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
====================================================================================
                       coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------
Intercept            0.2208      0.004     50.206      0.000       0.212       0.229
sign_X              -0.0700      0.004    -15.913      0.000      -0.079      -0.061
np.abs(X)            0.0088      0.003      2.818      0.005       0.003       0.015
sign_X:np.abs(X)    -0.0157      0.003     -4.987      0.000      -0.022      -0.010
==============================================================================
Omnibus:                     2593.479   Durbin-Watson:                   0.948
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           153874.664
Skew:                           3.917   Prob(JB):                         0.00
Kurtosis:                      37.577   Cond. No.                         20.2
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Am I allowed to do this? Shouldn't it be more 'clean' with a dummy? I feel quite unsure as the model will give important variation of Y when X is switching sign. I feel like it will not be the case with a dummy. Also I have seen that the errors are autocorrelated so I'll have to add variables to the model.

The equation of the regression line is as follow:

model.params[0] 
+ model.params[1]*np.sign(x_) 
+ model.params[2]*np.abs(x_) 
+ model.params[3]*np.abs(x_)*np.sign(x_)

Answer 1

This is as much a substantive issue as a statistical one.

The argument for such a parameterisation is that there could be a jump at zero. On the other hand, getting a jump out of a model fit is not especially convincing unless you explore other alternative parameterisations, even say linear or cubic splines or scatterplot smoothers.

How much sense that makes for your context would be something that people familiar with your field might be able to comment on if you explained what X is. Anonymising it doesn't make anything clearer.

My short answer to the question is that it is not a matter of whether it is allowed. The issue is what parameterisation fits the data and the application both accurately and parsimoniously, which is the trade-off in virtually any model-fitting.

Even without knowing your context, the model fit looks implausible: the fitted values fall into utterly distinct groups, but observed outcomes don't. And there appears to be some curvature that the lines aren't matching. If the outcome variable can't be negative, a model ever predicting negative values is qualitatively wrong.