The equation or its variable is said to be stable if from any set of initial conditions the variable's limit as time goes to infinity exists; this limit is called the steady state. Difference equations are used in a variety of contexts, such as in economics to model the evolution through time of variables such as gross domestic product , the inflation rate , the exchange rate , etc. They are used in modeling such time series because values of these variables are only measured at discrete intervals. In econometric applications, linear difference equations are modeled with stochastic terms in the form of autoregressive AR models and in models such as vector autoregression VAR and autoregressive moving average ARMA models that combine AR with other features.
If the solution is to be used numerically, all the roots of this characteristic equation can be found by numerical methods. However, for use in a theoretical context it may be that the only information required about the roots is whether any of them are greater than or equal to 1 in absolute value.
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It may be that all the roots are real or instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs.
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If no characteristic roots share the same value, the solution of the homogeneous linear difference equation. Specifically, for each time period for which an iterate value is known, this value and its corresponding value of t can be substituted into the solution equation to obtain a linear equation in the n as-yet-unknown parameters; n such equations, one for each initial condition, can be solved simultaneously for the n parameter values.
If all characteristic roots are real, then all the coefficient values c i will also be real; but with non-real complex roots, in general some of these coefficients will also be non-real.
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If there are complex roots, they come in pairs and so do the complex terms in the solution equation. Using this in the last equation gives this expression for the two complex terms in the solution equation:. Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicity involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots.
To solve this equation it is convenient to convert it to homogeneous form, with no constant term.
On the periodicity of a max-type rational difference equation
Given the steady state, the difference equation can be rewritten in terms of deviations of the iterates from the steady state, as. This is the homogeneous form. If the absolute value of the root is greater than 1 the term will become larger and larger over time. A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the absolute value of the modulus M of the roots is less than 1; if the modulus equals 1 then constant amplitude fluctuations in the combined terms will persist; and if the modulus is greater than 1, the combined terms will show fluctuations of ever-increasing magnitude.
Thus the evolving variable x will converge to 0 if all of the characteristic roots have magnitude less than 1. If the largest root has absolute value 1, neither convergence to 0 nor divergence to infinity will occur. If all roots with magnitude 1 are real and positive, x will converge to the sum of their constant terms c i ; unlike in the stable case, this converged value depends on the initial conditions: different starting points lead to different points in the long run.
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