Lecture 9 – The Lag Operator
(Reference – 6.1, Hayashi)
Let {xt} denote a sequence of real numbers or
random variables.
The lag operator L is defined according to:
Lxt = xt-1
Thus, a single application of the lag operator to a
sequence {xt} defines a sequence {yt}, such that yt =
xt-1.
Similarly,
L2xt = L(Lxt) = xt-2
and, more generally,
Lsxt = xt-s , for s = 0, + 1, + 2, …
(so that, for example, L-2xt = xt+2).
We can define a (finite or infinite order) polynomial
in L or a filter according to:
a(L) = a0 + a1L + a2L2 + …
Then, for example, we can write the MA(q) model as:
yt = μ +   where θ0 = 1
q
s
s 0
t s
q
= μ+
 L 
s
s 0
s
t
q
= μ + θ(L)t , where θ(L) =   L
s
s
s 0
Similarly, we can write the MA(∞) model as

yt     ( L) t , where  ( L)   s Ls .
0
So, one benefit of introducing the lag operator is that
it provides us with a compact notation for writing
filters.
Another advantage is that the algebra of polynomials
can be applied to filters. This turns out to provide us
with a very useful way to study and manipulate the
behavior of covariance stationary processes, as we
will soon see.
The algebra of polynomials (in L) –
1.Addition
Let a(L) and b(L) be p-th and q-th order polynomials
in L, p < q < ∞:
a(L) = a0 + a1L +…+ apLp
b(L) = b0 + b1L+ …+ bqLq
Then c(L) = a(L)+b(L) is defined according to
c(L) = c0 + c1L + … + cqLq
where c0 = a0 + b0,…,cq = aq + bq (and ap+1 =…=aq=0
if p < q).
2. Multiplication
Let a(L) and b(L) be p-th and q-th order polynomials
in L, p < q < ∞:
a(L) = a0 + a1L +…+ apLp
b(L) = b0 + b1L+ …+ bqLq
Then c(L) = a(L)b(L) is defined according to
c(L) = c0 + c1L + … + cq+pLq+p
where c0, c1,… are defined as follows –
c(L) = (a0 + a1L +…+ apLp)( b0 + b1L+ …+ bqLq)
= a0b0 + (a1b0 + a0b1)L + (a0b2 + a1b1 + a2b0)L2 +…
We sometimes say that c(L) is the convolution of
a(L) and b(L).
Note that a(L)b(L)=b(L)a(L).
Suppose, for example, that

yt   h j xt  j  h( L) xt
j 0
Then,
zt ≡ b( L) yt  g ( L) xt , g(L)=b(L)h(L)
Fact – If {xt} is covariance stationary and {bj} and
{hj} are absolutely summable, then {yt} and {zt} are
covariance stationary. (We already know that if the
h’s are absolutely summable then yt will be
covariance stationary. Then, since yt is covariance
stationary and the b’s are absolutely summable, zt
will be covariance stationary, too.)
3. Inversion
Let a(L) be a finite or infinite order polynomial in L.
We define a(L)-1 to be the polynomial in L such that
a(L)-1a(L) = 1
That is,
a(L)-1a(L)yt = yt
For example –
Suppose a(L) = 1-L. Note that
(1+L+2L2+ …)(1-L) =
(1+L+2L2+3L3+ …)-(L+2L2+3L3+…) = 1
So 1+L+2L2+ … is the inverse of 1-L (and, vice
versa).
If, for example,



yt   s  t  s     t  s  (  s Ls ) t   ( L) t
s
0
0
0
then
(L)-1yt = (L)-1(L)t = t
i.e.,
(1- L)yt = yt - yt-1 = t
and, therefore,
yt = ρyt-1 + εt
Hayashi (pp.372-373) provides a very nice general
procedure to calculate the inverse of a polynomial in
L.
Absolutely Summable Inverses Consider the inverse of 1-L,
1+L+2L2+…
This inverse is defined, regardless of the value of .
However, the coefficients of this infinite-order
polynomial are absolutely (and square summable) if
and only if ││ < 1.

2
  = 1/(1- ) if ││ < 1
2s
0
= ∞ if ││ > 1
We will often be interested in inverses whose
coefficients are absolutely summable (so that
inverting covariance stationary processes maintain
stationarity).
A necessary and sufficient condition for an inverse to
meet the absolute summability condition –
Let a(L) = 1 - a1L - … - apLp
Define the polynomial in z:
a(z) = 1 – a1z - … - apzp
If the roots of a(z), i.e., the values of z that satisfy
a(z)=0 are all strictly greater than 1 in “absolute
value” (i.e., they all lie “outside of the unit circle”),
then the coefficient of a(L)-1 will be absolutely
summable. This condition is called the stability
condition.
Note that if a(L) = 1-L, this condition is the
condition that ││ < 1, since z = 1/ is the only
solution to the a(z) = 1-z = 0.
So, another form of the stability condtion –
Suppose a(L) = 1-a1L -…-apLp. Since the roots of 1a1z-…-apzp are the reciprocals of the roots of zp-a1zp1
-…-ap, the coefficients of the inverse of a(L) are
absolutely summable if and only if the roots of zpa1zp-1-…-ap lie inside the unit circle.