Common factors in conditional distributions for bivariate time series

A definition for a common factor for bivariate time series is suggested by considering the decomposition of the conditional density into the product of the marginals and the copula, with the conditioning variable being a common factor if it does not directly enter the copula. We show the links between this definition and the idea of a common factor as a dominant feature in standard linear representations. An application using a business cycle indicator as the common factor in the relationship between U.S. income and consumption found that both series held the factor in their marginals but not in the copula.