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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?

Oskar
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1 Answers1

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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)

Sextus Empiricus
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  • Why can the OP not difference the series twice and then estimate an ARMA(1,1) on that series ? Where does the instability and lack of solution arise ? Thanks. – mlofton Nov 10 '23 at 17:19
  • @mlofton sure, taking twice the difference $Z_t := (Y_{t} - Y_{t-1}) - (Y_{t-1} - Y_{t-2})$ is an ARMA(1,1) process, but $Y_t$ is not that ARMA(1,1) process in the same way as a random walk is not equivalent to Gaussian white noise (instead it is the sum of that noise). – Sextus Empiricus Nov 10 '23 at 17:30
  • Right. It's not. But as far as estimation, can one not take $Z_t$ and estimate the parameters of the ARMA(1,1) process ? Maybe I didn't understand your comment about stable process and no solution. Thanks. – mlofton Nov 11 '23 at 02:19
  • @mlofton If I have a random walk with the given recursive formula $X_t = X_{t-1} + \epsilon$ then, conditional on the parameters, you do not have a unique distribution for the vector $\mathbf{X}$ (you would need some anchor point as well, e.g. given that $X_0 = 0$, you can describe the distribution of $X_t$ for $t \geq 0$). I am not sure, but I believe that you can always (without problems) solve the parameters for the recursive formula by using a least squares formulation. But it is not possible to solve the equations with maximum likelihood / method of moments. – Sextus Empiricus Nov 11 '23 at 07:43
  • And besides using least squares, the parameters can also be 'solved'/found by differencing twice and finding the parameters for the ARMA(1,1) process (as you mentioned in the first comment). I have removed the 'has no solution' part which has been confusing. You do have a solution for finding the parameters in the problem... but, it is in finding the likelihood function that there is no solution. – Sextus Empiricus Nov 11 '23 at 07:52
  • I agree that the random walk needs an initial value and so will the $Z_t$. But there are "methods" ( I don't know how well they work ) that generate a pseudo initial value. All I'm saying is that I don't think estimating the parameters of the model has any unusual problems that a standard ARIMA estimation problem doesn't have. – mlofton Nov 11 '23 at 12:03
  • It's an interesting coincidence because, recently, I was reading a 2006 interview of Manfred Deistler ( top researcher dealing in systems theory which has a lot of overlap with econometerics and stats ) and he said that a colleague showed him that, empirically, the standard optimization algorithms for ARIMA estimation ( BFGS etc ) return the wrong ARIMA parameter estimates 70 percent of the time. He has developed improved methods ( coordinate methods and subspace approaches ) that improve on the convergence issue. The interview is in econometric theory 2006 if you're interested. – mlofton Nov 11 '23 at 12:09
  • I don't know if it's freely available ( I found it one of my boxes when I was looking for something else cause I moved ) but here's the interview. Definitely worth a read if you can get your hands on it. https://www.cambridge.org/core/journals/econometric-theory/article/abs/et-interview-professor-manfred-deistler-interviewed-by-benedikt-m-potscher/DE6E4100776D6136CD9739721E8430C1 – mlofton Nov 11 '23 at 12:11