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The matrix  $  B $
of regression coefficients (cf. [[Regression coefficient|Regression coefficient]])  $  \beta _ {ji} $, 
$  j = 1 \dots m $, 
$  i = 1 \dots r $, 
in a multi-dimensional linear [[Regression|regression]] model,

$$ \tag{* }
X  =  B Z + \epsilon .
$$

Here  $  X $
is a matrix with elements  $  X _ {jk} $, 
$  j = 1 \dots m $, 
$  k = 1 \dots n $, 
where  $  X _ {jk} $, 
$  k = 1 \dots n $, 
are observations of the  $  j $-
th component of the original  $  m $-
dimensional random variable,  $  Z $
is a matrix of known regression variables  $  z _ {ik} $, 
$  i = 1 \dots r $, 
$  k = 1 \dots n $, 
and  $  \epsilon $
is the matrix of errors  $  \epsilon _ {jk} $, 
$  j = 1 \dots m $, 
$  k = 1 \dots n $, 
with  $  {\mathsf E} \epsilon _ {jk} = 0 $. 
The elements  $  \beta _ {ji} $
of the regression matrix  $  B $
are unknown and have to be estimated. The model (*) is a generalization to the  $  m $-
dimensional case of the general linear model of [[Regression analysis|regression analysis]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics" , '''3. Design and analysis, and time series''' , Griffin  (1983)</TD></TR></table>

====Comments====
In econometrics, for example, a frequently used model is that one has  $  m $
variables  $  y _ {1} \dots y _ {m} $
to be explained (endogenous variables) in terms of  $  n $
explanatory variables  $  x _ {1} \dots x _ {n} $(
exogenous variables) by means of a linear relationship  $  y= Ax $. 
Given  $  N $
sets of measurements (with errors),  $  ( y _ {t} , x _ {t} ) $, 
the matrix of relation coefficients  $  A $
is to be estimated. The model is therefore

$$ 
y _ {t}  =  A x _ {t} + \epsilon _ {t} .
$$

With the assumption that the  $  \epsilon _ {t} $
have zero mean and are independently and identically distributed with [[Normal distribution|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:

$$ 
\widehat{A}   =  M _ {yx} M _ {xx}  ^ {-} 1 ,
$$

where  $  M _ {xx} = N  ^ {-} 1 ( \sum _ {t=} 1  ^ {N} x _ {t} x _ {t}  ^ {T} ) $, 
$  M _ {yx} = N  ^ {-} 1 ( \sum _ {t=} 1  ^ {N} y _ {t} x _ {t}  ^ {T} ) $. 
In the case of a single endogenous variable,  $  y = a  ^ {T} x $, 
this can be conveniently written as

$$ 
\widehat{a}   =  ( X  ^ {T} X)  ^ {-} 1 X  ^ {T} Y ,
$$

where  $  Y $
is the column vector of observations  $  ( y _ {1} \dots y _ {N} )  ^ {T} $
and  $  X $
is the  $  ( N \times n ) $
observation matrix consisting of the rows  $  x _ {t}  ^ {T} $, 
$  t = 1 \dots N $. 
Numerous variants and generalizations are considered [[#References|[a1]]], [[#References|[a2]]]; cf. also [[Regression analysis|Regression analysis]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Malinvaud,   "Statistical methods of econometrics" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Theil,   "Principles of econometrics" , North-Holland  (1971)</TD></TR></table>