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
= μ +
= μ + θ(L)t , where θ(L) =
Similarly, we can write the MA(∞) model as
where .
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
Then,
zt≡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,
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.
= 1/(1-2) if ││ < 1
= ∞ 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 1-a1z-…-apzp are the reciprocals of the roots of zp-a1zp-1-…-ap, the coefficients of the inverse of a(L) are absolutely summable if and only if the roots of zp-a1zp-1-…-ap lie inside the unit circle.