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 '14 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. $\endgroup$ – Cowboy Trader Oct 2 '14 at 16:35
  • $\begingroup$ You may be interested in this post. $\endgroup$ – javlacalle Oct 2 '14 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 '14 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 '14 at 16:43

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