Regression matrix

The matrix of regression coefficients (cf. Regression coefficient) , , , in a multi-dimensional linear regression model,

(*)

Here is a matrix with elements , , , where , , are observations of the -th component of the original -dimensional random variable, is a matrix of known regression variables , , , and is the matrix of errors , , , with . The elements of the regression matrix are unknown and have to be estimated. The model (*) is a generalization to the -dimensional case of the general linear model of regression analysis.

References

[1]	M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 3. Design and analysis, and time series , Griffin (1983)

Comments

In econometrics, for example, a frequently used model is that one has variables to be explained (endogenous variables) in terms of explanatory variables (exogenous variables) by means of a linear relationship . Given sets of measurements (with errors), , the matrix of relation coefficients is to be estimated. The model is therefore

With the assumption that the have zero mean and are independently and identically distributed with normal distribution, that is, the so-called standard linear multiple regression model or, briefly, linear model or standard linear model. The least squares method yields the optimal estimator:

where , . In the case of a single endogenous variable, , this can be conveniently written as

where is the column vector of observations and is the observation matrix consisting of the rows , . Numerous variants and generalizations are considered [a1], [a2]; cf. also Regression analysis.

References

[a1]	E. Malinvaud, "Statistical methods of econometrics" , North-Holland (1970) (Translated from French)
[a2]	H. Theil, "Principles of econometrics" , North-Holland (1971)