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This is a mathematical challenge, thought it is related to statistical parameter estimation in the context of time series / auto-regressive processes, such as ARMA. No prior advanced calculus knowledge necessary - smart high school kids can find the solution, thought it's not trivial!
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Let's say that we have the model X(t) = a X(t-1) + b X(t-2) + e, where e is a white, independent noise (random variable) with zero mean, and t is the time. In short, a basic auto-regressive process or time series. More complex models are considered below.
The questions are as follows:
Example: Let's assume that X(0) = 1, X(1) = 1, and for the sake of simplicity, let's assume that e = 0. Clearly if a=0.5 and b=0.5, then X(t) is constant, always equal to 1 no matter the value of t. If a=1 and b=1, then X(t) quickly becomes infinite as t grows.
We have the following potential cases for X(t), depending on a and b:
Question: what are the parameter sets driving stability?
The model X(t) = a X(t-1) + b X(t-2) + e has the following characteristic equation:
x^2 - a*x - b = 0.
The solutions to this equation (as well as initial conditions X(0) and X(1)) entirely determines whether X(t) is stable or not. Let's denote as r and s the two solutions of this characteristic equation:
Perform monte carlo simulations with various values of a, b, X(0) and X(1) to simulate these auto-regressive time series (can be done in Excel, R, Perl, Matlab or Python), to confirm your findings.
Former weekly challenge