Sylvester Matrix and Common Factors in Polynomial Matrices

With the coefficient matrices of the polynomial matrices replacing the scalar coefficients in the standard Sylvester matrix, common factors exist if and only if this (generalized) Sylvester matrix is singular and the coefficient matrices commute. If the coefficient matrices do not commute, a necessary and sufficient condition for a common factor to exist is that a submatrix of the ratio (transfer) coefficient matrices is of less than full row rank. Whether coefficient matrices commute or not, a nonsingular (generalized) Sylvester matrix is always a sufficient condition for no common factors to exist. These conditions hold whether common factors are unimodular or not unimodular. These results follow from requiring that in the potential alternative pair of polynomial matrices with the same matrix ratio, i.e. with the same transfer function, all coefficient matrices beyond the given integers p and q are null matrices. Algebraically these requirements take the form of linear equations in the coefficient matrices of the inverse of the potential common factor. Lower block triangular Toeplitz matrices appear in these equations and the sequential inverse of these matrices generates sequentially the coefficient matrices of the inverse of the common factor. The conclusions follow from the properties of infinite dimensional diagonally dominant matrices.Sylvester matrix, common factor, polynomial matrix, transfer function, commutativity, Toeplitz matrix

Noncentral Student distributed LS and IV Estimators

The distribution of the least squares and the instrumental variable estimators of the coefficients in a linear relation is noncentral student when the data are normally distributed around possibly non-constant means. This is the claim of the paper and we show for what definition of a noncentral student density this claim to compact summary of the vast literature is justified. Unfortunately, the definition of the noncentral student density is as complicated as its parent, the noncentral Wishart density. Both are defined in terms of infinite series of zonal polynomials of all orders. We have developed however a recursive online method that generates these polynomials sequentially ad infinitum for bivariate and trivariate densities. The time is here that the practicality of the theory can be widened considerably.

Global Identifiability Under Uncorrelated Residuals

Suppose in each equation, not counting covariance restrictions, we need one more restriction to meet the order condition. If we now add to each equation a restriction that its structural residual is uncorrelated with the residual of some other equation, is the parameter of the new model identifiable globally? That is the question. In general the answer is no. The parameter could remain either not identifiable or is locally identifiable, possibly globally under additional inequality restrictions. In this paper we find families of models for which the answer to the question is yes without the help of inequalities. The families share common characteristics. First, the sufficient condition for local identifiability must hold. Secondly, the string of zero correlations between residuals contains a closed cycle of length at least four. Thirdly, with the variables, equations and residuals all numbered as they are in the cycle, the odd numbered variables must satisfy a kinship relationship and lastly, the structural residuals can not all be uncorrelated. There are also differences in families, but these come from the difference in the required kinship relationship. When there are four or more equations containing external variables, the variety of models with uniquely identifiable parameter under a string of uncorrelated residuals is considerable. In particular, when correlated inverse demand shocks are uncorrelated with correlated supply shocks, our results show that many flexible inverse demand and supply equations reproducing exactly the observed price and quantity moments are members of the above families.

Noncentral Student Distributed LS and IV Estimators

The distribution of the least squares and the instrumental variable estimators of the coefficients in a linear relation is noncentral student when the data are normally distributed around possibly non-constant means. This is the claim of the paper and we show for what definition of a noncentral student density this claim to compact summary of the vast literature is justified. Unfortunately, the definition of the noncentral student density is as complicated as its parent, the noncentral Wishart density. Both are defined in terms of infinite series of zonal polynomials of all orders. We have developed however a recursive online method that generates these polynomials sequentially ad infinitum for bivariate and trivariate densities. The time is here that the practicality of the theory can be widened considerably.

Sylvester Matrix and Common Factors in Polynomial Matrices

With the coefficient matrices of the polynomial matrices replacing the scalar coefficients in the standard Sylvester matrix, common factors exist if and only if this (generalized) Sylvester matrix is singular and the coefficient matrices commute. If the coefficient matrices do not commute, a necessary and sufficient condition for a common factor to exist is that a submatrix of the ratio (transfer) coefficient matrices is of less than full row rank. Whether coefficient matrices commute or not, a nonsingular (generalized) Sylvester matrix is always a sufficient condition for no common factors to exist. These conditions hold whether common factors are unimodular or not unimodular. These results follow from requiring that in the potential alternative pair of polynomial matrices with the same matrix ratio, i.e. with the same transfer function, all coefficient matrices beyond the given integers p and q are null matrices. Algebraically these requirements take the form of linear equations in the coefficient matrices of the inverse of the potential common factor. Lower block triangular Toeplitz matrices appear in these equations and the sequential inverse of these matrices generates sequentially the coefficient matrices of the inverse of the common factor. The conclusions follow from the properties of infinite dimensional diagonally dominant matrices.