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1

ON THE Z. HELLWIG'S METHOD AND THE S. BARTOSIEWICZ'S METHOD OF EXPLANATORY VARIABLE SELECTION IN THE ECONOMETRIC MODEL

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 1
67-78
EN
In the paper, the Z. Hellwig's method and the S. Bartosiewicz's method are examined in the case where the potential explanatory variables are perfect multicollinear and all pairwise correlation coefficients of these variables are low in absolute values. It is proved that both methods may result in the set of explanatory variables with perfect multicollinearity.
2

A METHOD FOR CONSTRUCTING VECTORS OF OBSERVATIONS ON VARIABLES WITH THE GIVEN CORRELATION MATRIX

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 3
61-70
EN
In the paper, it was proposed a method for constructing the vectors of observations on the variables, such that the given positive semi-definite symmetric matrix, which has the following properties: - all elements on the main diagonal are units; - all elements outside the main diagonal are not greater than one in absolute value, is their correlation matrix.
3

ON USING THE GENERALIZED HELLWIG'S INEQUALITY FOR VERIFYING THAT A SYMMETRIC MATRIX IS A CORRELATION MATRIX

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 2
64-77
EN
Hellwig (1976) proposed an inequality concerning the relationship between all pairwise correlation coefficients in the case of three variables. The generalised Hellwig's inequality (hereafter GHI) was derived by Borowiecki, Kaliszyk and Kolupa (1984). They argued that any symmetric k x k matrix, whichhas the foliowing properties: (1) k is greater than 3; (2) all elements on the main diagonal are units; (3) all elements outside the main diagonal are not greater than one in absolute value; is a correlation matrix if GHI is fulfilled for every element above the main diagonal (hereafter GHI criterion). These results were used by Dudek (2003). Methods of verification that a symmetric matrix with properties (1)-(3) is a correlation matrix (hereafter CM verification) were also considered by Hauke and Pomianowska (1987). They derived conditions (hereafter HP conditions) of using GHI in CM verification for a symmetric matrix of certain type. They did not consider the GHI criterion. In the present paper new conditions of using GHI in CM verification were derived. It was proved that (a) the GHI criterion properly indicates the correlation matrices only for k = 3; (b) if k is greater than 3 then the fulfilment of the GHI criterion is not a sufficient condition for a symmetric matrix with properties (1)-(3) to be a correlation matrix; (c) HP conditions are not true.
4

PERFECT MULTICOLLINEARITY AND THE VALUES OF PAIRWISE CORRELATION COEFFICIENTS OF EXPLANATORY VARIABLES IN THE ECONOMETRIC MODEL

100%
Westa M.
Przegląd Statystyczny
|
2007
|
vol. 54
|
issue 1
78-93
EN
The authoress polemises with H. Dudek's paper (Ibid. 2003, no 2 pp. 41-51) and claims that the procedure applied is improper. From the theorems proved in the present paper it follows that the values of the pairwise correlation coefficients of explanatory variables with perfect multicollonearity may be analysed independently on both the variances and the parameters of linear functions concerning the relations between these variables. It was shown that perfect multicollinearity of explanatory variables in the econometric models may occur even though all pairwise correlation coefficients of these variables are very small in absolute values, e.g. when all pairwise correlation coefficients of 'k' explanatory variables are equal to ' -1/(k-1)' then these variables are perfectly multicollinear. This raises the question whether it is sensible to use those methods of explanatory variable selection in which low pairwise correlation of all explanatory variables is treated as one of the necessary conditions for the good quality explanatory variable set.
5

A NECESSARY AND SUFFICIENT CONDITION FOR A SYMMETRIC MATRIX TO BE A CORRELATION MATRIX

100%
Westa M.
Przegląd Statystyczny
|
2008
|
vol. 55
|
issue 3
33-46
EN
Borowiecki, Kolupa and Kaliszyk (1984) and Dudek (2003) proposed methods in which the generalised Hellwig's inequality is used for verifying that a symmetric matrix, which has the following properties: (1) - all elements on the main diagonal are units; (2) -all elements outside the main diagonal are not greater than one in absolute value, is a correlation matrix of certain variables. The authoress (see forthcoming paper) showed that this verification procedure may improperly indicate the correlation matrices. The theorems proved in the present paper define various forms of the necessary and sufficient condition for a symmetric matrix with properties (1)-(2) to be a correlation matrix. Among others things, it was shown that any symmetric 3x3 matrix with properties (1)-(2) is a correlation matrix if and only if its determinant is non-negative. Some results obtained generalize those given by Hauke and Pomianowska (1987) for correlation pair.
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