Shape of Impulse Responses of $ARMA(p,q)$ Processes

Question

Suppose that $x_t$ is an $ARMA(p,q)$ stochastic process,

$$ \phi(L)x_t = \theta(L)\varepsilon_t ,$$

where $\varepsilon_t \sim N(0,\sigma^2)$, and $\phi(L)$ and $\theta(L)$ are lag-polynomials given by

\begin{align} \phi(L) &\equiv 1 - \phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p \\[2ex] \theta(L) &\equiv 1 + \theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q \\ \end{align}

Are there conditions on the roots of these lag-polynomials that imply that the impulse response (to a unit shock to $\varepsilon_t$) is monotonic or hump shaped?

Answer 1

Your notation $\phi(L)x_t = \theta(L)\varepsilon_t$ looks slightly strange to me because it mixes the "time" domain (variable $t$) with the transform domain (variable $L$). But I'm guessing that's just a notation issue, and we are speaking here of a LTI causal stable filter, in discrete time, with impulse response $h_t$, so that in the transformed domain

$$h_t \leftrightarrow H(L)=\frac{\theta(L)}{\phi(L)}$$ Here $L$ plays the same role as $z^{-1}$ in the Z transform.

Now, let's look at particular cases:

If $\phi(L)=1$, (constant denominator), $H$ "all-zeros" (no poles), the filter is FIR and the process is MA. In this case, $h_t$ is (loosely) "hump shaped" : more precisely, it has finite support (values given by the coefficients of $\theta(L)$ )

If $\theta(L)=1$, (constant numerator), $H$ "all-poles" (no zeros), the filter is IIR and the process is AR. In this case, $h_t$ is a (loosely) a "infinitely tail"; more precisely, it's a linear combination of decaying exponentials - but including here complex exponentials, that is, damped sinusoids. That the exponential is monotonic roughly corresponds to real roots of the polynomial; complex roots give rise to oscillations, less dampened as the roots approach the unit circle.

This is very broad picture, the detailes picture is more complicated. More so, if we consider the general (ARMA) case. To this, it might help to regard the ARMA filter as a concatenation of two AR/MA filters, and remember that the total impulse response is the convolution of the two inner filters.