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Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable.
It is also possible in some cases to fix the problem by applying a transformation to the response variable (e.g., fitting the logarithm of the response variable using a linear regression model, which implies that the response variable itself has a log-normal distribution rather than a normal distribution).
The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model.
When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated since independent variables might be (negatively or ...
Since the data in this context is defined to be (x, y) pairs for every observation, the mean response at a given value of x, say x d, is an estimate of the mean of the y values in the population at the x value of x d, that is ^ ^. The variance of the mean response is given by: [11]
If the response data used to estimate the model contain values that change sign, or if the lowest response value is far from zero (for example, when data are left-truncated), a location parameter, L, may be added to the response so that the expressions for the quantile function and for the median become, respectively: