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..for understanding the relationship between a dependent and independent variables, given that quantile regression makes no assumptions about the distribution of the residual.

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If you are really interested in determining how the conditional mean value of the dependent variable varies with the independent variables, then you would address this question by using:

  1. Ordinary least squares regression in the absence of heteroskedasticity;
  2. Generalized least squares regression or weighted least squares regression in the presence of heteroskedasticity.

On the other hand, if you are interested in determining how the quantiles of the conditional distribution of the dependent variable vary with the independent variables, then you would address that via quantile regression.

All of these regression techniques target some aspect(s) of the conditional distribution of the dependent variable given the independent variables. Usually, you would choose the aspect relevant to your study based on subject matter considerations.

In some cases, focusing on a single aspect of that distribution (e.g., conditional mean or conditional median) is sufficient given the study purposes.

In other cases, a more comprehensive look at the entire conditional distribution is necessary, which can be obtained by focusing on an appropriately selected set of quantiles of that distribution.

So what is appropriate depends primarily on the study question, though it also has to take into account features present in the data used to elucidate this question, such as presence/absence of heteroscedasticity when the study question involves the conditional mean of the dependent variable. Note that, if the study question concerns quantiles of the conditional distribution of the dependent variable, then quantile regression is appropriate whether or not heteroskedasticity is present.

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