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.

Подпись: Y1 1 X1T Y= , X = Yn 1 XnT Подпись: 00 image357

If we let

model (5.31) can be written in vector-matrix form as

Подпись: (5.32)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.

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