Chapter 1 Linear algebra and matrix methods in econometrics

Vectors and matrices played a minor role in the econometric literature published before Second World War, but they have become an indispensable tool in the past several decades. Part of this development results from the importance of matrix tools for the statistical component of econometrics; another reason is the increased use of matrix algebra in the economic theory underlying econometric relations. This chapter presents a selective survey of both areas. It reviews the concepts of linear dependence and orthogonality of vectors and the rank of a matrix. A major reason related to the usefulness of matrix methods is that many topics in econometrics have a multivariate character. The chapter illustrates the convenience of matrices for linear systems. The expression linear algebra should not be interpreted in the sense that matrices are useful for linear systems only. Vectors and matrices are important in the statistical component of econometrics. A general method of estimation is maximum likelihood (ML) that can be shown to have certain optimal properties for large samples under relatively weak conditions. The derivation of the ML estimates and their large sample covariance matrix involves the information matrix, which is (apart from sign) the expectation of the matrix of second-order derivatives of the log-likelihood function with respect to the parameters. The prominence of ML estimation in recent years has greatly contributed to the increased use of matrix methods in econometrics.