By writing x, = (1, xf’)’, we can define Model 1 as follows. Assume
yt = x’fi+ut, t= 1,2,. . . ,T, (1.1.2)
where y, is a scalar observable random variable, fi is a AT-vector of unknown parameters, x( is a AT-vector of known constants such that x, xj is nonsin
gular, and u, is a scalar, unobservable, random variable (called the error term or the disturbance) such that Eut = 0, Vu, = a2 (another unknown parameter) for all t, and Eutus = 0 for t Ф s.
Note that we have assumed x* to be a vector of known constants. This is essentially equivalent to stating that we are concerned only with estimating the conditional distribution of у given x*. The most important assumption of Model 1 is the linearity of E(y,|xf); we therefore shall devote the next subsection to a discussion of the implications of that assumption. We have also made the assumption of homoscedasticity (Vu, = a2 for all t) and the assumption of no serial correlation (Eu, us = 0 for t Ф s), not because we believe that they are satisfied in most applications, but because they make a convenient starting point. These assumptions will be removed in later chapters.
We shall sometimes impose additional assumptions on Model 1 to obtain certain specific results. Notably, we shall occasionally make the assumption of serial independence of (ut) or the assumption that u, is normally distributed. In general, independence is a stronger assumption than no correlation, al
though under normality the two concepts are equivalent. The adr^b^P4 assumptions will be stated whenever they are introduced into Model ^