I have recently remembered that $SSTot = SSRes + SSReg$ fails to hold in the case of nonlinear regression. 

$$ y_i-\bar{y} = (y_i - \hat{y_i} + \hat{y_i} - \bar{y}) = (y_i - \hat{y_i}) + (\hat{y_i} - \bar{y}) $$

$$( y_i-\bar{y})^2 = \Big[ (y_i - \hat{y_i}) + (\hat{y_i} - \bar{y}) \Big]^2  =  
(y_i - \hat{y_i})^2 + (\hat{y_i} - \bar{y})^2 + 2(y_i - \hat{y_i})(\hat{y_i} - \bar{y})
$$

$$ \sum_i ( y_i-\bar{y})^2 = \sum_i(y_i - \hat{y_i})^2 + \sum_i(\hat{y_i} - \bar{y})^2 + 2\sum_i\Big[ (y_i - \hat{y_i})(\hat{y_i} - \bar{y}) \Big]$$

Therefore, for a regression in general that predicts conditional expected value, $SSTot =$ $SSRes + SSReg + Other$. In the case of linear regression, the $Other$ term drops out due to orthogonality.

Consequently, I think that we can write the $R^2$ for nonlinear regression as:

$$R^2 =\dfrac{SSReg}{SSTot} = 1 - \dfrac{SSRes + Other}{SSTot}.$$

This becomes the regular $R^2$ when the regression is linear and $Other=0$. When the regression is nonlinear, we still have a calculation that R or Python can handle just fine.

However, I want something akin to $R^2$ for quantile regression. I have reasoned through a way to get something for the case of a linear quantile regression, which is what [Koenker and Machado][1] use in their 1999 JASA paper, according to [another post on CrossValidated][2]. (I can't access the paper, but this equation makes sense to me.) Let $q$ be the quantile being estimated by quantile regression, and let $\bar y$ be the marginal/pooled quantile (coming from an intercept-only quantile regression model, following Koenker and Machado (1999)).

$$R^2_q = 1 - \frac{\sum_{y_i \ge \hat{y_i}} q \vert y_i - \hat{y_i} \vert + \sum_{y_i < \hat{y_i}} (1-q) \vert y_i - \hat{y_i} \vert}{\sum_{y_i \ge \bar{y}} q \vert y_i - \bar{y} \vert + \sum_{y_i < \bar{y}} (1-q) \vert y_i - \bar{y} \vert}
$$

This uses the loss function for the model divided by the loss function for the naive model that takes the pooled $q^{th}$ quantile as the estimate, analogous to how $R^2$ divides the loss function of the model by the loss function of the naive model that takes the estimate as the pooled mean.

This assumes that the analogous $Other$ term is zero for the linear quantile regression, bringing me to my question.

**What is that $Other$ component for a nonlinear *quantile* regression?**

Proceeding analogous to my first set of equations for the regular regression case, I start with $\vert y_i - \bar{y} \vert = \vert y_i - \hat{y_i} + \hat{y_i} - \bar{y} \vert $. Breaking this into something related to $\vert y_i - \hat{y_i} \vert$ and $\vert \hat{y_i} - \bar{y} \vert$ is eluding me.

**REFERENCE**

Koenker, Roger, and Jose AF Machado. "Goodness of fit and related inference processes for quantile regression." Journal of the American Statistical Association 94.448 (1999): 1296-1310.

  [1]: https://amstat.tandfonline.com/doi/abs/10.1080/01621459.1999.10473882
  [2]: https://stats.stackexchange.com/questions/129200/r-squared-in-quantile-regression