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The ADL Model and Generalized Least Squares in Lag Operator Notation 681

Yt = b0 + b(L)Xt + ut . (15.40)

In addition, if the error term ut follows an AR(p), then it can be written as

f(L)ut = ut, (15.41)

where f(L) = g p 0 fjLj, where f0 = 1 and ut is serially uncorrelated [note that f1, c,
j=

fp as defined here are the negatives of f1, c, fp in the notation of Equation (15.31)].

To derive the ADL model, premultiply each side of Equation (15.40) by f(L) so that

f(L)Yt = f(L)3b0 + b(L)Xt + ut4 = a0 + d(L)Xt + ut, (15.42)

where p

a0 = f(1)b0 and d(L) = f(L)b(L), where f(1) = ja= 0fj . (15.43)

To derive the quasi-differenced model, note that f(L)b(L)Xt = b(L)f(L)Xt = b(L)Xt,
where Xt = f(L)Xt. Thus rearranging Equation (15.42) yields

Yt = a0 + b(L)Xt + ut, (15.44)

where Yt is the quasi-difference of Yt; that is, Yt = f(L)Yt.

The Inverse of a Lag Polynomial

Let a(x) = g p 0 ajxj denote a polynomial of order p. The inverse of a(x), say b(x), is a
j=

function that satisfies b(x)a(x) = 1. If the roots of the polynomial a(x) are greater than 1 in

absolute value, then b(x) is a polynomial in nonnegative powers of x: b(x) = g j∞= 0bj x j.
Because b(x) is the inverse of a(x), it is denoted as a(x)−1 or as 1>a(x).

The inverse of a lag polynomial a(L) is defined analogously: a(L) - 1 = 1>a(L) =

b(L) = g j∞= 0bjLj, where b(L)a(L) = 1. For example, if a(L) = (1 - fL), with 0 f 0 6 1,
∞
you can verify that a(L)-1 = 1 + fL + f2L2 + f3L3c = g j= 0f jLj. (See Exercise 15.11.)

The ADL and GLS Estimators

The OLS estimator of the ADL coefficients is obtained by OLS estimation of Equation
(15.42). The original distributed lag coefficients are b(L), which, in terms of the estimated
coefficients, is b(L) = f(L)-1d(L); that is, the coefficients in b(L) satisfy the restrictions