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Are there regression models where variance is the outcome, not mean? For instance, for interquartile range I may use quantile regression. But is there something similar for variance?

For example, let us have observations $(X_1,Y_1),\dots,(X_N,Y_N)$, where $Y_j$ are sampled from the normal distributions $N(a+bX_j,\ c+dX_j)$, respectively. Are there regression methods that allow us to estimate the parameters $a,b,c,d$?

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For your example, for some parameter values $c,d$ the variance will be negative ... for that reason, in such models often is used a log link function for the variance. But such models (and many others) can be fitted with extensions of generalized linear models (glm's), also introducing link functions and linear predictors for the variance (and maybe even for other parameters.) One such family of models is known as gamlss see gamlss website for information.

For some examples Compare shape and scale parameters between Weibull distributions, Are there better approaches than the weighted mean?

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