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Difference between revisions of "Regression matrix"

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

Latest revision as of 18:22, 18 December 2020


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

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 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, 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 [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)
How to Cite This Entry:
Regression matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regression_matrix&oldid=48475
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article