Linear difference equation
In mathematics and in particular dynamical systems, a linear difference equation or linear recurrence relation equates 0 to a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1. Usually the context is the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t–1, one period later as t+1, etc.
An n-th order linear difference equation is one that can be written in terms of parameters ai and b as
or equivalently as
Since the longest time lag between iterates appearing in the equation is n, this is an n-th order equation, where n could be any positive integer. When the longest lag is specified numerically so n does not appear notationally as the longest time lag, n is occasionally used instead of t to index iterates.
In the most general case the coefficients ai and b could themselves be functions of time; however, this article treats the most common case, that of constant coefficients.
The solution of such an equation is a function of time, and not of any iterate values, giving the value of the iterate at any time. To find the solution it is necessary to know the specific values (known as initial conditions) of n of the iterates, and normally these are the n iterates that are oldest. 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 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.
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