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2015年3月31日 星期二

Internal Restriction on Residuals in Linear Regression Model

Mei-Yu Lee (2014) proposes a new concept about the central limit theorem of the residuals from the view of the restriction of the residuals.
Computer Simulates the Effect of Internal Restriction on Residuals in Linear Regression Model with First-order Autoregressive Procedures, Journal of Statistical and Econometric Methods, 3(3), 1-22.
The general concept of central limit theorem is about a group of variables, {X1, ..., XT}, whose average value approaches to the population mean when T is large enough.

Thus, when the number of the residuals becomes larger, each residual does not necessarily occur the property of central limit theorem, but the sample mean of the residuals converges to the mean of the errors.  

The fact is that the residuals are Normal distribution, but the errors are not necessary Normal distribution. 

Lee (2014) provides the evidences and gives an reason due to the conditions of the zero sum of the residuals and the zero sum of the residuals multiplied by the independent variables. The conditions are denoted as internal restriction of the residuals. Thus, the residuals have k+1 constraints. 

The evidences of the paper is from the computer simulation, not empirical method. 

The larger k is, that is linear combination of the residuals, the easier the residuals occur central limit theorem. The internal restriction is degenerated to form individual constraints, not viewed as a group of whole constraints, therefore, each residual converges to Normal distribution on the condition of larger k.

For instance, on the condition of T = 50 and k = 40, only 9 variables are flexed, that is degree of freedom. Thus, the residuals are Normal distribution and each residual is Normal distribution.

This paper also confirms that the model structure changes when we use Lagrange method showing the more constraints to do the linear algebra. Amazingly, internal restriction of the residuals leads that each residual becomes Normal distribution, not only the average.