Some Aspects of Differences between L1 and L2 Criteria in the Linear Switching Regression

Polona Tominc and Lada Bele Tominc

The least squares procedure or L2 criterion is in theory and in practice generally used to estimate the regression coefficients. It is well known that given the assumptions of the classical linear regression model the least squares estimates posses some ideal properties. One of the assumptions underlying the L2 criterion is that the disturbance terms are normally distributed. But there are many cases where the disturbance terms are not normally distributed. Therefore, the use of some other criteria could be legitimate. As reported in the literature (for example Narula and Korhonen, 1994) the least absolute value or L1 criterion is less sensitive to outliers than the L2 criterion.

With the purpose to illustrate some aspects of differences between L2 and L1 criteria in the presence of switching regression function with a priori known switch, the Monte Carlo simulation was performed.

The least absolute value criterion has another advantage, especially in the cases, where the switch is not known in advance. Using the least absolute value criterion the estimation problem can be formulated and solved as a linear mixed integer optimisation model. If the switch is known in advance the optimisation model is linear.