The Lag operator and functions of lag operators

Summary

  • Given: a stochastic process \(\{ Y_t \}\)
  • The lag operator \(L\) is defined by

\[LY_t=Y_{t-1}\]

  • Think of \(L\) as a function. You “feed” observation at time \(t\) and \(L\) returns the observation in the previous period. We could write \(L\left( Y_t \right)=Y_{t-1}\) but it is more common to drop the parenthesis.
  • \(L^2\) is defined as a composite function

\[L^2Y_t=L\left( L\left( Y_t \right) \right)=L\left( Y_{t-1} \right)=Y_{t-2}\]

  • If \(p\) is a natural number then, similarly,

\[L^pY_t=Y_{t-p}\]

  • \(1-L\) is another operator that “operates” on \(Y_t\) :

\[\left( 1-L \right)Y_t=Y_t-LY_t=Y_t-Y_{t-1}\]

  • \(1-L\) creates first differences.
  • We can extend this idea to polynomials of order 2. \(a_0+a_1L+a_2L^2\) is another operator where \(a_0,a_1,a_2\) are constants.

\[\left( a_0+a_1L+a_2L^2 \right)Y_t=a_0Y_t+a_1LY_t+a_2L^2Y_t=a_0Y_t+a_1Y_{t-1}+a_2Y_{t-2}\]

  • Extended to general polynomials,

\[f\left( L \right)=a_0+a_1L+ \ldots +a_kL^k\]

  • is called an operator polynomial (or a polynomial in the lag operator). \(f\left( L \right)\) operates on \(Y_t\) :

\[f\left( L \right)Y_t=\left( a_0+a_1L+ \ldots +a_kL^k \right)Y_t=a_0Y_t+a_1Y_{t-1}+a_2Y_{t-2}+ \ldots +a_kY_{t-k}\]

  • We can add and multiply operator polynomials. For example

\[\left( 1-L \right)+\left( 2L+L^2 \right)=1+L+L^2\]

  • and

\[\left( 1-L \right)\left( 1+L \right)=1-L^2\]

  • An operator polynomial \(f\left( L \right)\) is said to be the inverse of another operator polynomial \(g\left( L \right)\) if

\[f\left( L \right)g\left( L \right)=1\]

  • The inverse of the lag operator \(L\) is denoted by \(L^{-1}\) , \(LL^{-1}=1\) . Since

\[LY_t=Y_{t-1}\]

  • we have

\[L^{-1}LY_t=L^{-1}Y_{t-1}\]

  • or

\(L^{-1}Y_{t-1}=Y_t\)

  • \(L^{-1}\) is therefore a “forward operator”.
  • \(L^{-2}={\left( L^{-1} \right)}^2\) is defined as the composite function

\[L^{-2}Y_t=L^{-1}\left( L^{-1}\left( Y_t \right) \right)=L^{-1}\left( Y_{t+1} \right)=Y_{t+2}\]

  • \(L^{-p}\) is defined similarly, \(L^{-2}Y_t=Y_{t+p}\) for natural numbers \(p\) .
  • By convention, \(L^0\) is defined as 1.