I'm following http://faculty.chicagobooth.edu/john.cochrane/research/papers/time_series_book.pdf to ramp up on time series (my background is Elec Eng). I'm having confusion about the meaning of some notation: When a variable that represents a lag polynomial is supplied with an integer argument. Can anyone clarify what is the prevailing interpretation?
In the above link, this occurs on page 112, where the cumulative sum of an impulse response is equated to $a(1)$. As shown in the last paragraph, $a(L)$ is a lag polynomial, so it looks like he means to take $L=1$ as the argument. I'm not sure what that means, since $L$ is not a real number.
I have a similar confusion on pages 48-49, where lag polynomials $A(L)$ and $B(L)$ are referred to with $L=0$. From equation at the top of page 51, it seems that the argument $L=0$ extracts out the coefficient for the $L^0$ term, so it presumes a specific form of the polynomial, i.e. unfactored form. If this is correct, it probably doesn't apply to the $A(L=1)$ on page 112, because the coefficient is already referred to as $a_1$, as shown at the top of page 112.
I guess a related question to the meaning of $a(1)$ on page 112 is: Assuming that it is the coeffecient for the lag-1 term (whatever the reason), why is $a(1)=0$ for a trend-stationary process? The polynomial $a(L)$ applies to the noise source (bottom of page 110), not the output signal $y$.
It might be related to the fact that the lag-1 coefficient is the long-run response, according to the last paragraph on page 52 (again, in the context of an adjacent differencing equation). But there isn't an explanation of why this is the case.