Applications to Regression Analysis
5.1.1. The Linear Regression Model
Consider a random sample Zj = (Yj, Xj)T, j = 1, 2,…,n from a ^-variate, nonsingular normal distribution, where Yj є К, Xj є R-1. We have seen in Section 5.3 that one can write
Yj = a + Xj в + Uj, Uj – N (0, a2), j = 1,…,n, (5.31)
where Uj = Yj – E [Yj | Xj ] is independent of Xj. This is the classical linear regression model, where Yj is the dependent variable, Xj is the vector of independent variables, also called the regressors, and Uj is the error term. This model is widely used in empirical econometrics – even in the case in which Xj is not known to be normally distributed.
If we let
model (5.31) can be written in vector-matrix form as
Y = X00 + U, U|X – Nn [0, a2In],
where U|Xis a shorthand notation for “U conditional on X.”
In the remaining sections I will address the problems of how to estimate the parameter vector 00 and how to test various hypotheses about 0 0 and its components.