2 versions of the koyck distributed lag In econometrics, ( this material is in a lot of the literature but, IMHO, can most clearly be found in Harvey's text, "econometric analysis of time series"), there are generally two versions of a specific lagged dependent variable which is usually  termed the koyck distributed lag. ( It's a very special case of a distributed lag ). 

I write down version 1 and version 2 below.

Version 1 is shown below.

1):   $y_t = \rho \times y_{t-1} + \beta \times x_t + \epsilon_t$

where $\epsilon_t = (v_t - \rho v_{t-1}) \sim N(0, \sigma^2) $.

There is also version 2 which is

2):     $y_t = \rho \times y_{t-1} + \beta \times x_t + \epsilon_t$

where $\epsilon_t \sim N(0, \sigma^2) $.

So, in version 1, the error term is such that $v_t$ is AR(1) and in version 2), the error term is pure noise.

My question is the following: Is there a way to write to a version where the error term is MA(1) ? I initially figured there must be some kind of symmetry because  we have the pure noise case and the AR(1) case so I figured there must be a version with an MA(1) error term. But now I'm not so sure and actually don't think do.  Thanks for any insights.

EDIT BASED ON GOOD QUESTION IN COMMENT:

Hi: $\frac{\epsilon_t}{(1 - \rho L)} = v_t$ where $L$ is the lag operator.

Therefore, $v_{t}$ can be thought of as an exponentially smoothed average of the past white noise error terms $\epsilon_{i}$.

Now, version 1) of the model can be re-written in the following way:

$y_t = \rho \times y_{t-1} + \beta \times x_t + (1-\rho L) v_{t} $

which can be re-written as

$y_{t}(1 - \rho L) = \beta \times x_t + (1 - \rho L) v_{t} $

Then, dividing the whole the equation by $(1 - \rho L)$ results in what I call the long form of version 1 of the model:

$y_{t} =  \beta \times \sum_{i=0}^{\infty} \rho^{i} x_{t-i} +  v_{t}$

So, the model implies that response is $\beta$ times an exponentially smoothed average of the past $x_{i}$  plus v_{t}. But, we can re-write the last term which results in

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

So, in version 1), the response can be thought of as an exponentially smoothed version of the past $x_{t}$ plus the exponentially smoothed version of the past error terms, namely the $\epsilon_{i}$.

I won't write it out, but, in  the long form of version 2, the error term is not an exponentially smoothed average of the past $\epsilon_{i}$. The past error terms  are not involved and the error term is just $\epsilon_{t}$.

Thanks for good question.

SECOND EDIT BASED ON GOOD QUESTION IN COMMENT:

When your question made me write it out in the long form, it hit me that
an MA(1) structure could just be:

$y_{t} =  \beta \times \sum_{i=0}^{\infty} \rho^{i} \times x_{t-i} + \epsilon_t + \rho \epsilon_{t-1} $

This would be an MA(1) error term for the koyck distributed lag. I don't think it simplifies in any way by writing it in the short form but it's still a nice way of thinking of a third possibility. In one case, the error term lasts on period. in another case, the error term is  is an exp smoothed average of past errors and in the last case, the error is linear combination of the last two errors. Clearly your question totally led to this and I think what you're saying in your comment is the same as here, so thanks much for insight.
As I explained in the edited question, the comment-question made by @caverac made me write the models out in their respective long form versions. Doing this made it obvious that an MA(1) structure can be imposed and the model is shown below.  As far as I can tell, the long form of this model cannot be transformed to a lagged dependent variable model (what I refer to as the short form ) nicely like can be in the  AR(1) and pure noise cases.

Long Form of Koyck Distributed Lag Model With MA(1) error term:

$y_{t} = \beta \times \sum_{i=0}^{\infty} \rho^{i} x_{t-i} + \epsilon_t + \rho \times \epsilon_{t-1}$