I want to calculate the (moduli of the) inverse roots of an autoregressive process of order greater than 2 using coefficient values that I specify and that are not estimated. This is to determine whether the process is stationary or not. I can calculate these for AR(1) and AR(2) processes using standard analytical expressions. Whilst econometric software like EViews will automatically calculate the (moduli of the) inverse roots of an autoregressive process of order greater than 2 where the coefficients are estimated it will not allow me to do this for coefficient values that I specify (as far as I am aware). For example, I can estimate the AR(3) process and obtain the inverse roots (using EViews) based on the estimated coefficients for a, b and c:
Y(t) = a * Y(t-1) + b * Y(t-2) + c * Y(t-3) + u(t)
However, I cannot calculate the inverse roots of this AR(3) process (in EViews) with user-specified coefficients such as a = 0.6, b= 0.4 and c = -0.2. It is this that I want to do.
Hence, my question is, does anyone know how I can calculate ithe inverse roots of a higher -order autoregressive process with user-specified coefficients using software like EViews or Stata or other readily available software?
Hi, here is an idea: Why dont you create a long time series process in excel or in eviews with the population parameters that you want and with a very small variance in the error term. If you generate a series like this with 10.000 observations, and with a variance of the error term equal to 0.0001, I bet eviews will estimate de AR(p) process almost to perfection. You can then ask eviews to give you the roots of the polynomial you are interested at.
If you have doubts, then you can repeat the same procedure generating a new independent data series with the same population parameters. I bet e-views will give you the same estimates of the parameters.
Please try this out and let me know how it goes!!
Good Luck!!
Dear Pablo,
Thanks for your suggestion. I had previously tried this (except with 1000 observations rather than 10000) and found that the estimates were typically a little inaccurate. For example, if I choose coefficients for an AR(3) process to be 0.5, -0.3 and 0.1 the resulting estimates would be in the region of 0.53, -0.28 and 0.12. Even when I tried different sets of random numbers to simulate the processes with very small variances. When a nonstationary AR(3) process is estimated the estimates can be very inaccurate. A true value of 0.7 can be estimated as 0.4! I will try a larger sample and see if this improves the accuracy of estimation. I would loike accuracy to at least two decimal places.
Best wishes,
Chris.
You need to know the roots of the "characteristic equation". See explanation of how to interpret these at https://davegiles.blogspot.com/2013/06/when-is-autoregressive-model.html You can also read there how to translate your specified AR model into a characteristic equation
Once you have the characteristic equation then use an online polyomial roots calculator such as https://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php
Vince, Thanks for the links, especially for Dave Giles blog. It was the generalisation for the characteristic equation to AR(3) that I needed to identify and it is in this blog. I have now calculated the inverse roots for some AR(3) processes with user specified coefficients (and checked them with EViews).
have a look at Maxima ( http://maxima.sourceforge.net/ ). This is a symbolic mathematical program which can be downloaded from the link above. It works in Windows, Linux, Apple and even on Android. For example the allroots function below returns the roots of the equation in the argument of the function.
allroots(4*x^4+5*x^5-6*x^2-2*x+4);
[x=0.248542040386245*%i+0.7253631665409909,x=0.7253631665409909-0.248542040386245*%i,x=-0.9420481191786501,x=1.008099982067917*%i-0.6543391069516659,x=-1.008099982067917*%i-0.6543391069516659]
If you use the graphical interface, wxmaxima, you can access this function from the menu | equations | Roots of polynomial | and fill in the polynomial in the dropdown box. You might also find the differentiation and integration functions under the calculus menu item useful.
John, thanks for directing me to the program, it looks very useful and, as you say, can do more than just find inverse roots.
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