Robust regression

In robust statistics, robust regression is a form of regression analysis designed to circumvent some limitations of traditional parametric and non-parametric methods. Regression analysis seeks to find the relationship between one or more independent variables and a dependent variable. Certain widely used methods of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results if those assumptions are not true; thus ordinary least squares is said to be not robust to violations of its assumptions. Robust regression methods are designed to be not overly affected by violations of assumptions by the underlying data-generating process.

In particular, least squares estimates for regression models are highly sensitive to (not robust against) outliers. While there is no precise definition of an outlier, outliers are observations which do not follow the pattern of the other observations. This is not normally a problem if the outlier is simply an extreme observation drawn from the tail of a normal distribution, but if the outlier results from non-normal measurement error or some other violation of standard ordinary least squares assumptions, then it compromises the validity of the regression results if a non-robust regression technique is used.

One instance in which robust estimation should be considered is when there is a strong suspicion of heteroscedasticity. In the homoscedastic model, it is assumed that the variance of the error term is constant for all values of x. Heteroscedasticity allows the variance to be dependent on x, which is more accurate for many real scenarios. For example, the variance of expenditure is often larger for individuals with higher income than for individuals with lower incomes. Software packages usually default to a homoscedastic model, even though such a model may be less accurate than a heteroscedastic model. One simple approach (Tofallis, 2008) is to apply least squares to percentage errors as this reduces the influence of the larger values of the dependent variable compared to ordinary least squares.

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