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Lag Operator Notation 631

g i∞=0(b i1)2 var(ut - i) = s2u g i∞=0(b1i )2 = su2 >(1 - b21), where the final equality follows from the
∞
fact that if ͉a͉ 6 1, g i= 0ai = 1 > (1 - a); thus var(Ys + 1) = var(Ys + 2) = su2 > (1 - b12),

which does not depend on s as long as ͉ b1 ͉ 6 1. Finally, because Ys + 2 = b1Ys + 1 + us + 2,

cov(Ys + 1,Ys + 2) = E(Ys + 1,Ys + 2) = E3Ys + 1(b1Ys + 1 + us + 2)4 = b1 var(Ys + 1) + cov(Ys+1, us + 2)

= b1 var(Ys + 1) = b1s2u>(1 - b21) .

The covariance does not depend on s, so Ys + 1 and Ys + 2 have a joint probability distri-

bution that does not depend on s; that is, their joint distribution is stationary. If ͉ b1 ͉ Ú 1,

this calculation breaks down because the infinite sum in Equation (14.36) does not converge,

and the variance of Yt is infinite. Thus Yt is stationary if ͉ b1 ͉ 6 1 but not if ͉ b1 ͉ Ú 1.
The preceding argument was made under the assumptions that b0 = 0 and ut is nor-

mally distributed. If b0 0, the argument is similar except that the means of Ys + 1 and
Ys + 2 are b0>(1 - b1), and Equation (14.36) must be modified for this nonzero mean. The
assumption that ut is i.i.d. normal can be replaced with the assumption that ut is stationary
with a finite variance because, by Equation (14.36), Yt can still be expressed as a function

of current and past ut’s, so the distribution of Yt is stationary, as long as the distribution of
ut is stationary and the infinite sum expression in Equation (14.36) is meaningful in the
sense that it converges, which requires that ͉ b1 ͉ 6 1.

A p p e n d i x

14.3 Lag Operator Notation

The notation in this and the next two chapters is streamlined considerably by adopting
what is known as lag operator notation. Let L denote the lag operator, which has the
property that it transforms a variable into its lag. That is, the lag operator L has the prop-
erty LYt = Yt - 1. By applying the lag operator twice, one obtains the second lag:
L2Yt = L(LYt) = LYt - 1 = Yt - 2. More generally, by applying the lag operator j times, one
obtains the jth lag. In summary, the lag operator has the property that

LYt = Yt - 1, L2Yt = Yt - 2, and LjYt = Yt - j . (14.37)

The lag operator notation permits us to define the lag polynomial, which is a polynomial
in the lag operator:

p

a(L) = a0 + a1L + a2L2 + g + apLp = a ajLj, (14.38)

j=0