Properties of the matrix A-XY

AbstractThe main topic of this paper is the matrix V=A−XY*, where A is a nonsingular complex k×k matrix and X and Y are k×p complex matrices of full column rank. Because properties of the matrix V can be derived from those of the matrix Q=I−XY*, we will consider in particular the case where A=I. For the case that Y*X=I, so that Q is singular, we will derive the Moore–Penrose inverse of Q. The Moore–Penrose inverse of V in case Y*A−1X=I then easily follows. Finally, we will focus on the eigenvalues and eigenvectors of the real matrix D−xy′ with D diagonal

Solving and Testing for Regressor-Error (in)Dependence When no Instrumental Variables are Available: With New Evidence for the Effect of Education on Income

This paper has two main contributions. Firstly, we introduce a new approach, the latent instrumental variables (LIV) method, to estimate regression coefficients consistently in a simple linear regression model where regressor-error correlations (endogeneity) are likely to be present. The LIV method utilizes a discrete latent variable model that accounts for dependencies between regressors and the error term. As a result, additional ‘valid’ observed instrumental variables are not required. Furthermore, we propose a specification test based on Hausman (1978) to test for these regressor-error correlations. A simulation study demonstrates that the LIV method yields consistent estimates and the proposed test-statistic has reasonable power over a wide range of regressor-error correlations and several distributions of the instruments.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47579/1/11129_2005_Article_1177.pd

The Correlation of Durations in Multivariate Hazard Rate Models

This empirical analysis of multiple durations using multivariate mixed proportional hazard rate models is widespread. In such models, the duration variable are dependent if their unobserved determinants are dependent on each other. In this paper it is shown that these models restrict the magnitude of the correlation of the duration variable. For example, if the baseline hazards are constant, then this correlation necessarily lies between -1/3 and 1/2. Similar results hold for more general models. The usefulness for empirical analysis is twofold. First, the results can be used to assess the ability of the model to describe certain phenomena, relative to the models that impose less restrictions on the values the correlation can attain. Secondly, they suggest that, in parametric analyses, it is important to take a family of heterogeneity distributions that is flexible in the sense that it does not restrict the values the correlation can attain either further. We show that some frequently used parametric families are much more restrictive than others.Multivariate hazard rate models, competing risks, proportional hazards, correlation of nonnegative random variables.