If a residual plot shows mostly positive residuals on the left and right and mostly negative in the middle, what could improve the model?

When a residual plot reveals a pattern of mostly positive residuals on the left and right sides, and mostly negative residuals in the middle, this suggests that the model may be failing to capture a non-linear relationship between the independent and dependent variables. In this scenario, introducing a quadratic variable as a predictor can enhance the model's ability to fit the data more closely.

The rationale behind this is that a quadratic term allows for curvature in the regression function. When you add a quadratic term, you're enabling the model to adjust for changes in the rate of response at different levels of the predictor variable. This can better reflect the underlying relationship between the predictors and the response variable, as it can accommodate the pattern observed in the residuals, where extreme values (both high and low) lead to positive residual errors, while values in the middle lead to negative residual errors.

The choice of the quadratic term having a positive coefficient implies that the relationship is such that the response increases for both low and high values of the predictor, but decreases towards the middle of the range, which aligns with the pattern shown in the residual plot. By addressing the non-linear trend visible in the residuals, the model's fit can improve significantly, leading to more accurate predictions and a better understanding