Vector regression estimation and linear transformation

In any regression model we usually deal with one response and one or more predictors. In real life situation we can find many separate models all with the same predictor. In this paper we are trying to combine many models with the same predictor in one model by using some basic concepts of linear algebra and in particular the linear transformation.

We discuss a linear transformation L: R³ → R² and L: R³ → R¹. Some application of the regression vector also discussed. Let X (t), Y (t) and Z (t) be random variables and as a function of t, respectively, t = 1,2,3,…, n. Let X (t) = T_1t (t)β₁ + ε_1t, Y (t) = T_2t (t)β₂ + ε_2t, and Z (t) = T_3t (t)β₃ + ε_3t. We assume that ε_it’s are independent and normally distributed with mean zero and variance σ², for i = 1,2,3. We define vector regression as By LS method, the estimator of vector regression is