I have the following first-order difference equation

$$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$

where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution to this difference equation heuristically, but I am wondering if there is a general procedure. A solution (the solution?) is

$$y_{t} = \sum_{i = 0}^{\infty} \rho^{i}X_{t-i} + \epsilon_{t}$$

I looked in Goldberger's text and couldn't find it there. Any reference is appreciated also. Thanks.


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