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

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?

• If someone has suggestions for additional tags, I would appreciate it.
– mzp
Commented Mar 19, 2016 at 18:51
• I'm not sure if we are speaking of continous or discrete (in time) processes... Commented Mar 28, 2016 at 14:19
• I'm mainly interested in results in discrete time. But you know of something for continuous time that could be helpful also.
– mzp
Commented Mar 28, 2016 at 14:23

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.
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.