I have the following time series
$$x_t = \epsilon_t + a \epsilon_{t-1}^3,$$
where $0<a<1$, and $\epsilon_t$ is an i.i.d sequence of standard normal random variables. I have to obtain the IRF. By definition, the IRF of at a future time $j$ is of the form $\Psi_j = \frac{\partial x_{t+j}}{\partial \epsilon_t}$. Then,
$$\Psi_0 = \frac{\partial x_{t}}{\partial \epsilon_t}=\frac{\partial(\epsilon_{t} + a \epsilon_{t-1}^3)}{\partial \epsilon_t} = \frac{\partial \epsilon_t}{\epsilon_t}=1.$$
The problem starts when $j=1$,
$$\Psi_1 = \frac{\partial x_{t+1}}{\partial \epsilon_t}=\frac{\partial(\epsilon_{t+1} + a\epsilon_{t}^3)}{\partial \epsilon_t} = \frac{\partial \epsilon_t}{\epsilon_t}=3a \epsilon_t^2,$$ as I end up with a result that depends on $\epsilon_t$, which seems not right.