Let $X_t$ be an ARIMA(1,1,1) process and $Y_t = Y_{t-1} + X_t$. What kind of process is $Y_t$?

Question

Q: Let $X_t$ be an ARIMA(1,1,1) process and $Y_t = Y_{t-1} + X_t$. What kind of process is $Y_t$?

$X_t$ is an ARIMA(1,1,1), i.e $\nabla X_t = X_t - X_{t-1} = Z_t $ where $Z_t$ is a casual ARMA(1,1) process and satisfies $(1-\phi_1 B)Z_t = (1+\theta_1B)\epsilon_t. $ Since $Z_t$ is casual, we may write $Z_t = \frac{1 + \theta_1 B}{1-\phi_1 B} \epsilon_t $.

Then $X_t = \frac{1}{1-B} \frac{1 + \theta_1 B}{1-\phi_1 B} \epsilon_t$, and

\begin{equation} \begin{split} (1-B)Y_t &= X_t \\ (1-B)Y_t &= \frac{1}{1-B} \frac{1 + \theta_1 B}{1-\phi_1 B} \epsilon_t \\ (1-B)^2(1-\phi_1 B)Y_t &= (1+\theta_1B)\epsilon_t \end{split} \end{equation}

So $Y_t$ is an ARMA(3,1) process?

Answer 1

The process $Y_t$ is the integration of $X_t$

$$Y_t = \sum_{-\infty}^t X_t$$

So you have one more integration step and that means that it is an ARIMA(1,2,1) process.

Indeed, the recursive formula can also be rewritten as the recursive function of an ARMA(3,1) process, as you deduced, but it won't be a stable ARMA process. (see for more about that here: ARMA vs ARIMA Models)