On Some Characterizations of Probability Distributions with Applications in Econometrics: A Centennial Tribute to CR Rao

CR Rao, the living legend in statistics, is celebrating his birth centenary year in 2020–21. One of the major areas in which Rao made significant contributions to statistics is characterization problems. The main objective of this paper is to connect the work of Rao and his followers on characterization to the underlying fundamental econometric problem. In doing so, the authors hope to draw the attention of the econometricians to the characterization method as a tool to further advance the less attempted problem of econometric modeling with errors in variables. The authors also hope that this paper would draw the attention of statisticians to the more interesting characterization problems arising in econometrics, there by providing fields of application to a class of characterization problems that otherwise remain mostly theoretical. Ragnar Frisch, the founder of econometrics, conceived the econometric model as a system of linear structural equations. Noting how one can use the reduced form equations to determine the joint distribution of the endogenous variables, given the distribution of the exogenous variables, Frisch asked the inverse question: under what distributional assumptions on the exogenous variables, does there exist a system of linear structural equations? Keeping this in the background, CR Rao answered that question in 1943 and 1947 using the characterization method and further advanced the characterization method in statistics. Rao and his followers, however, did not connect their work with the underlying econometric problem. The authors extend the econometric problem posed by Frisch to a two endogenous and two exogenous variables case and establish the existence of two linear structural equations and their identification, as an example of the potential benefits from this association between econometrics and statistics.