In case you need any further examples I bring up an explicit problem. I included one above in a reply btw hopefully that adds some clarification. For clarify though here's the explicit form of the exact question I'm seeking an answer to:

I) Solve the following initial-value difference equations. Plot the time path of Yt and discuss the convergence or divergence of the solutions as $ t \rightarrow \infty $

$$ y_t = 1.6 + 0.75 y_{t-1} + \epsilon_t $$ $$y_0 = 2 $$

At first it was an issue with the error term. But now that you showed me the syntax for that -- all that would remain is how to specify the initial conditions correctly. What's odd is I'm not getting an error and your code is directly from the references so I can't see it being incorrect. But the solution it gives me when I use your code snippet on these problems with the initial conditions specified I get an unfamiliar answer as you see above.

Hope this clears some things up.

Thanks again.

So after trying the code again for some odd reason it's now outputting an error:

Inputting RSolve[{a[n + 1] - 2 a[n] + eps[n] == 1, a[0] == 1}, a[n], n]

Out: RSolve::deqn: Equation or list of equations expected instead of False in the first argument {-2 a[n]+a[1+n]+eps[n]==1,False}.

I've looked at the ARMA reference in the past and its the closest I've gotten to getting my actual specification correctly. Primarily since it includes the stochastic error term correctly. But as far as then providing an functional solution to the associated recurrence equation, I haven't had any luck. Clearly I think there is a function required to transform an ARMA process to it's associated functional form. If I could do that then I imagine I could bracket everything in RSolve[] and find the functional solution.

That aside -- the original code you gave me seemed to work I'm really struggling to see what went wrong.

Best, Ali

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

Hi Neil,

I'm so sorry I don't know why I wasn't getting notified to a response. Hopefully you're able to see my messages; but I appreciate your help it's still very much an inquiry, as I'm not able to do the calculation in Mathematica. Fortunately I've been using R with no problems. That being said I know once I get it in Mathematica there'll be some really nice functions.

From the example it seems eps is correct thanks for that. But I was really trying to avoid creating seeded random distributions; I was hoping for the explicit analytic solutions. So is there no direct way to transform an ARMA specification to an objective function, to then have solved?

For instance instead of specifying the recurrence equation:

$$ y_t = \alpha_0 + \beta_1 y_{t-1} + \beta_2 y_{t-2} + \epsilon_t + \epsilon_{t-1} $$

Then having it solve it as done above, is there a way to instead specify that equation as an: ARMA{2, 1}? Aside from that obviously inputting the coefficients as well. The use to this would be to test for stationarity, invertibility of the time series.

Thanks again

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

Ali,

I do not understand from your post what you want to do. I am guessing from your post that you have either a dynamic system with a random input or a random process. You really need to post some example code for anyone to help because it is not clear what "solve" means in your context. For example, are you trying to simulate the process or are you trying to obtain statistics (ie correlations or covariances) or are you trying to use ARMAProcess[] or a related function to create a model? If you post an example, people here can be more helpful.

Regards,

Neil