I have this ARMA(1,1) process where $\epsilon_t$ is the classical White Noise process

$$X_t=\epsilon_t +\alpha_{t-1}\epsilon_{t-1}+\theta_{t-1}X_{t-1}$$

and I have to write its Wold representation. Using the lag operator I get

$$\epsilon_t=\frac{1-\theta_{t-1}L}{1+\alpha_{t-1}L}X_{t}$$ Assuming the process is stationary and invertible, how can I recover the Wold representation?

  • 1
    $\begingroup$ In the Wold representation $X_t$ is expressed as an infinite weighted sum of the current and past innovations $\epsilon_t$. You should rearrange the polynomials in your equation since there you have the innovations as a sum of past values of $X_t$. In the definition of the ARIMA(1,1) process, move the terms depending on $X_t$ to one side of the equation and those depending on $\epsilon_t$ to the other, then use the lag operator $L$ as you did and solve for $X_t$ (leave $X_t$ alone on one side of the equation). $\endgroup$
    – javlacalle
    Oct 2, 2014 at 15:16
  • $\begingroup$ It seems there a type it should be X(t-1) on the right not X(t) of the first equation. $\endgroup$ Oct 2, 2014 at 16:35
  • $\begingroup$ You may be interested in this post. $\endgroup$
    – javlacalle
    Oct 2, 2014 at 20:37
  • $\begingroup$ Thanks, I fixed the error @Cagdas Ozgenc. So if I revert the formula what should i get? I'm sorry I don't know exactly what the result should be $\endgroup$
    – Marco
    Oct 3, 2014 at 13:21
  • $\begingroup$ The common arrangements of terms is $X_t = \frac{\theta(L)}{\phi(L)} \epsilon_t$. See my answer here for details on the notation. $\endgroup$
    – javlacalle
    Oct 3, 2014 at 16:43

1 Answer 1


The Wold's theorem states that every covariance-stationary time series $X_T$ can be written as the sum of two time series, one deterministic and one stochastic. What we are searching for is something of the following form:

$$X_t = \sum_{t=0}^{\infty} \gamma^j \varepsilon_{t - j} +\eta_t . $$

How can we achieve this if we are given the process $y_t = \phi y_{t-1} -\theta \varepsilon_{t-1} +\varepsilon_t?$ By substitution! Write

\begin{align*} y_{t} =& \phi( \phi y_{t-2} - \theta\varepsilon_{t-2}+\varepsilon_{t-1}) - \theta \varepsilon_{t-1}+\varepsilon_t,\\ =& \phi^2 y_{t-2} - \phi\theta \varepsilon_{t-2} + \phi\varepsilon_{t-1} - \theta \varepsilon_{t-1} +\varepsilon_t,\\ =&\phi^2( \phi y_{t-3} - \theta\varepsilon_{t-3}+\varepsilon_{t-2}) - \phi\theta \varepsilon_{t-2} + \phi\varepsilon_{t-1} - \theta \varepsilon_{t-1} +\varepsilon_t,\\ \vdots& \end{align*}

We see that a pattern emerges and that this process can be written as an infinite sum of error terms:

$$ y_t = \sum_{j=1}^{\infty}\phi^{j-1} ( \phi - \theta)\varepsilon_{t-j} + \varepsilon_t. $$

That is, we have rewritten the ARMA(1,1) as an MA($\infty$) process.


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