I'm studying time series from the Hamilton's book.
I have the following question:
Consider a 2nd order difference equation:
$y_t=\phi_1 y_{t-1}+\phi_2 y_{t-2}+w_t$
By using the lag-operator, I write: $(1-\phi_1L-\phi_2L^2)y_t=w_t$
I can factorize my polynomial in the lag operator as
$(1-\phi_1 L -\phi_2 L^2)=(1-\lambda_1 L)(1- \lambda_2 L)=[1- (\lambda_1+\lambda_2)L+(\lambda_1 \lambda_2)L^2]$ $~~~~~~~~~~~~~~~~ [1]$
with: $\lambda_1 +\lambda_2=\phi_1$, and $\lambda_1 \lambda_2=-\phi_2$
I want to find the values for $\lambda_1$ and $\lambda_2$ such that the left hand side of $[1]$ is equal to the right hand side.
Then, the book says to replace the values of $L$, with a scalar $z$ as follows
$(1-\phi_1 z -\phi_2 z^2)=(1-\lambda_1 z)(1- \lambda_2 z) $ $~~~~~~~~~~~~~~~~ [2]$
Then the values making the right hand side of $[2]$ equal to zero are $z=\lambda_1^{-1}$ or $z=\lambda_2^{-1}$.
The question is why the right hand side of $[2]$ should be zero? Is that because $(1-\phi_1 z -\phi_2 z^2)=0$ is the characteristic equation of the my polynomial in the lag operator and by construction it is zero? Indeed, $z$ should be the roots of the characteristic equation. Am I right?
