Median regression

The method of the smallest absolute deviations , also called median regression, is a robust estimation method for estimating unknown parameters of a linear regression . Such an estimator is called a least absolute deviations estimator ( LAD ). It minimizes the sum of the median of the absolute deviations .

history

Historically, the method of least absolute deviation is much older than the method of least squares . It was first proposed around 1760 by Rugjer Josip Bošković (1711–1787). In modern terminology, this approach is called median regression, since the result of the minimization leads to an estimate for the median of the dependent variable. The median regression is a special case of quantile regression .

The procedure

Instead of using the sum of the squared deviation for the minimization, a first obvious approach is to minimize the sum of the absolute deviations

${\displaystyle {\hat {\boldsymbol {\beta }}}={\underset {\boldsymbol {\beta }}{\rm {arg\,min}}}\,\sum _{t=1}^{n}\left|\varepsilon _{t}\right|={\underset {\boldsymbol {\beta }}{\rm {arg\,min}}}\,\sum _{t=1}^{n}\left|y_{t}-\mathbf {x} _{t}^{\top }{\boldsymbol {\beta }}\right|}$

with the vector of the regressors and the vector of the regression coefficients ${\displaystyle \mathbf {x} _{t}={\begin{pmatrix}\ x_{t1}\\\ x_{t2}\\\vdots \\\ x_{tk}\\\vdots \\\\x_{tK}\end{pmatrix}}_{(K\times 1)}\;\;\;}$ ${\displaystyle \;\;\;{\boldsymbol {\beta }}={\begin{pmatrix}\beta _{1}\\\beta _{2}\\\vdots \\\beta _{k}\\\vdots \\\beta _{K}\end{pmatrix}}_{(K\times 1)}}$

The result is a robust estimator compared to the least squares method .

Individual evidence

1. ^ Ludwig Fahrmeir , Thomas Kneib, Stefan Lang: Regression: Models, Methods and Applications. , P. 105