Hi Neil,

Sorry to flood the thread - just hoping to minimize your efforts and time spent at my expense. I was completely wrong in my previous response to which I say your code was outputting a 'wrong' solution I was looking for. It's indeed exactly correct and just specified in an unfamiliar way. It required some simple clerical simplifications and the solutions maintain the same solution.

$ \left\{\left\{a(n)\to (-3)^n 2^{1-2 n}\right\}\right\} $ is actually identical to $ a(n) \rightarrow (2)(-0.75)^t $ after simplification. My apologies for that.

And again thanks so much for the help.

Best, Ali

I'm not sure what's going on here but you might have some insight on this:

Following you code above using RSolve. I used it to find the solution to virtually the same problems with different coefficients.

RSolve[{4 a[n] + 3 a[n - 1] == 0, a[0] == 2}, a[n], n]

The (latex) output Mathematica gives is:

$ \left\{\left\{a(n)\to (-3)^n 2^{1-2 n}\right\}\right\}$

I'm not sure what exactly is going on here because that isn't exactly a solution I can make any sense of. The correct solution to this equation given initial conditions would be:

$ y^h_t = 2(-0.75)^t $ representing the homogenous portion of the solution. At least this is my understanding and am fairly confident is the correct solution. Oddly I get this solution when inputting the following text into Mathematica and ignoring the initial condition parameter:

RSolve[4 a[n] + 3 a[n - 1] == 0, a[n], n]

Gives me: $ \left\{\left\{a(n)\to c_1 \left(-\frac{3}{4}\right)^{n-1}\right\}\right\} $

Following setting up the initials I would get to my answer. Not sure you're able to see where my misunderstanding may lie, but I appreciate the time.

Best, Ali

Sorry! After closing Mathematica and re-opening it seems it works again.

Ali,

Have you looked at the function ARMAProcess? The documentation is here. I think this is what you want. They show how to compute stationarity, etc.

Regards,

Neil

Ali,

My previous post seems to have the "updating name" problem. I hope this helps -- Was this what you were asking?

Regards,

Neil

Thanks for the suggestion Neil,

I apologize for any possible confusions. I’ll post an example. The nature of the question is to solve difference equations, in time, with a stochastic disturbance. In other words just solving for the particular solutions and all homogenous solutions to an SDE (stochastic difference equation). I believe on Mathematica this is referenced instead as a recurrence equation. As an example: Suppose an AR(1):

$$ \alpha_o + \beta_1 X_{t-1} + \epsilon_t = y_t $$

The solution to this would follow the null solution and the particular solutions. Regardless of which however I have to have a way to set the random parameter $ \epsilon_t $.

This is of course for pure conceptual and analytical understanding of the solutions behind ARMA process'. As far as I know, I'm not sure to it's direct application. But functionally an ARMA or AR process has a solution. I'm looking for how to derive to that solution but with an error added.

As an added example of what I managed to get correct. I got a solution to a difference equation specified as follows:

$$ RSolveValue[ y[t] == \alpha + \beta_1 x[t-1] + \beta_2 x[t-2], y[t], t] $$

This actually got me very close, if not exactly what I was looking for. But I had to truncate the equation so as to not contain the random disturbance term $ \epsilon_t $

Again any help is greatly appreciated. Thanks, Ali