8 Lag polynomials
A \(p\)-th degree lag polynomial is given by:
\[ \alpha(L) = \alpha_0 + \alpha_1L + \cdots + \alpha_pL^p, \] where \(L\) is the lag operator, defined by the relation \(L^jx_t=x_{t-j}.\)
We define a filter given by \(\alpha(L)\) to an input process \(\{x_t\}\), we get a weighted average of the current and \(p\) most recent values of the process:
\[ \begin{aligned} \alpha(L)x_t &= \alpha_0x_t + \alpha_1Lx_t + \alpha_2L^2x_t + \cdots + \alpha_pL^px_t \\ &= \alpha_0x_t + \alpha_1x_{t-1}+ \alpha_2x_{t-2} + \cdots + \alpha_px_{t-p} \\ &= \sum_{j=0}^p \alpha_jx_{t-j} \end{aligned} \]
8.1 Product of Filters
Let \(\{\alpha_j\}\) and \(\{\beta_j\}\) be two arbitrary sequences of real numbers and define the sequence \(\{\delta_j\}\) by the relation
\[ \begin{gathered} \delta_0 = \alpha_0\beta_0, \\ \delta_1 = \alpha_0\beta_1 + \alpha_1\beta_0, \\ \delta_2 = \alpha_0\beta_2 + \alpha_1\beta_1 + \alpha_2\beta_0, \\ \vdots \\ \delta_j = \alpha_0\beta_j + \alpha_1\beta_{j-1} + \alpha_2\beta_{j-2} + \cdots + \alpha_{j-1}\beta_{1} + \alpha_{j}\beta_{0}, \\ \vdots \\ \end{gathered} \tag{8.1}\]
The sequence \(\{\delta_j\}\) created from this convoluted formula is called the convolution of \(\{\alpha_j\}\) and \(\{\beta_j\}.\)
For example, for \(\alpha(L)=1+\alpha_1L\) and \(\beta(L)=1+\beta_1L\), we have
\[ \delta(L)=(1+\alpha_1L)(1+\beta_1L) = 1+ (\alpha_1+\beta_1)L + \alpha_1\beta_1L^2. \]
Filters are commutative:
\[ \alpha(L)\beta(L) = \beta(L)\alpha(L) \]
8.2 Inverses
The inverse of \(\alpha(L)\) is denoted as \(\alpha(L)^{-1}\) or \(1/\alpha(L)\):
\[ \alpha(L)\alpha(L)^{-1}=1 \] Define a \(p\)-th degree lag polynomial \(\phi(L)\)
\[ \phi(L) = 1-\phi_1L-\phi_2L^2-\cdots-\phi_pL^p. \tag{8.2}\]
Equation 8.2 is often used to construct AR processes.
Now let’s calculate its inverse, \(\psi(L) = \phi(L)^{-1}.\)
\[ \psi(L) = \psi_0 + \psi_1L + \psi_2L^2 + \cdots \]
By the convolution formula (8.1), we have
\[ \begin{aligned} \text{constant}:&\quad \psi_0 =1 \\ L: &\quad \psi_1-\psi_0\phi_1 = 0 \Longrightarrow \psi_1 = \phi_1 \\ L^2: &\quad \psi_2-\psi_1\phi_1-\psi_0\phi_2 = 0 \Longrightarrow \psi_2 = \phi_1^2 + \phi_2 \\ \vdots \\ L^p: &\quad \psi_p - \psi_{p-1}\phi_1 - \psi_{p-2}\phi_2 - \cdots - \psi_{1}\phi_{p-1} - \psi_{0}\phi_p = 0 \\ L^{p+1}: &\quad \psi_{p+1} - \psi_{p}\phi_1 - \psi_{p-1}\phi_2 - \cdots - \psi_{2}\phi_{p-1} - \psi_{1}\phi_p = 0 \\ \vdots \end{aligned} \]
Example 8.1 Consider a 1st degree lag polynomial \(\phi(L)=1-\phi L\), its inverse \(\psi(L)\) can be calculated as
\[ \begin{aligned} \text{constant}: &\quad \psi_0 =1 \\ L: &\quad \psi_1 - \psi_0\phi = 0 \Longrightarrow \psi_1 = \phi \\ L^2: &\quad \psi_2 - \psi_1\phi = 0 \Longrightarrow \psi_2 = \phi^2 \\ L^3: &\quad \psi_3 - \psi_2\phi = 0 \Longrightarrow \psi_3 = \phi^3 \\ \vdots \end{aligned} \] Hence
\[ \begin{aligned} \psi(L) &= (1-\phi L)^{-1} \\ &=1 + \phi L + \phi^2 L^2 + \phi^3 L^3 + \cdots \\ &= \sum_{j=0}^\infty \phi^jL^j. \end{aligned} \]
8.3 Stability Condition
The solution sequence \(\{\psi_j\}\) eventually starts declining at a geometric rate if the stability condition holds. The condition states:
All the roots of the \(p\)-th degree polynomial equation in \(z\)
\[ \phi(z) = 0 \text{ where } \phi(z) \equiv 1-\phi_1z-\phi_2z^2-\cdots-\phi_pz^p \] are greater than 1 in absolute value (lie outside the unit circle).
Equivalently, we can consider the roots of the reciprocal polynomial defined as (basically this means inverting the order of the coefficients)
\[ \phi^*(z) = z^p\phi(z^{-1}) = z^p - \phi_1z^{p-1} - \dots - \phi_p. \] The stability condition can be stated as:
All the roots of
\[ \phi^*(z) \equiv z^p - \phi_1z^{p-1} - \dots - \phi_p =0 \] are less than 1 in the absolute value (i.e., lie inside the unit circle).