Reduced-Rank Regression Model

Abstract

The classical multivariate regression model presented in Chapter 1, as noted before, does not make direct use of the fact that the response variables are likely to be correlated. A more serious practical concern is that even for a moderate number of variables whose interrelationships are to be investigated, the number of parameters in the regression matrix can be large. For example, in a multivariate analysis of economic variables (see Example 2.2), Gudmundsson (1977) uses m = 7 response variables and n = 6 predictor variables, thus totaling 42 regression coefficient parameters (excluding intercepts) to be estimated, in the classical regression setup. But the number of vector data points available for estimation is only T = 36; these are quarterly observations from 1948 to 1956 for the United Kingdom. Thus, in many practical situations, there is a need to reduce the number of parameters in model (1.1) and we approach this problem through the assumption of lower rank of the matrix C in model (1.1). More formally, in the model Y _k = CX _k + ε_k we assume that

$$ {\mathop{\rm rank}\nolimits} (C) = r \le \min (m,n) $$

((2.1))

.