Difference Equations

• Phoebus J. Dhrymes

Abstract

In this chapter we deal with econometric applications of (vector) difference equations with constant coefficients. This occurs in connection with dynamic models, particularly in the context of the general linear structural econometric model (GLSEM). To begin, recall that the second-order difference equation
$${a_0}{y_t} + {a_1}{y_{t + 1}} + {a_2}{y_{t + 2}} = g\left( {t + 2} \right)$$
(6.1)
where y t is the scalar dependent variable, the a i , i = 0, 1, 2, are the (constant) coefficients, and g(t) is the real-valued “forcing function”, is solved in two steps. First, we consider the homogeneous part
$${a_0}{y_t} + {a_1}{y_{t + 1}} + {a_2}{y_{t + 2}} = 0,$$
and find the most general form of its solution, called the general solution to the homogeneous part. Then, we find just one solution to Eq. (6.1), called the particular solution. The sum of the general solution to the homogeneous part and the particular solution is said to be the general solution to the equation. What is meant by the “general solution”, denoted, say, by $$y_t^*$$, is that $$y_t^*$$ satisfies Eq. (6.1) and that it can be made to satisfy any prespecified set of “initial conditions”. To appreciate this aspect, rewrite Eq. (6.1) as
$${y_{t + 2}} = \bar g\left( {t + 2} \right) + {\bar a_1}{y_{t + 1}} + {\bar a_0}{y_t},$$
(6.2)
where, assuming a2 ≠ 0,
$$\bar g(t + 2) = \frac{1}{{{a_2}}}g(t + 2),{\bar a_1} = - \frac{{{a_1}}}{{{a_2}}},{\bar a_0} = - \frac{{{a_0}}}{{{a_2}}}.$$

Keywords

General Solution Difference Equation Unit Circle Trivial Solution Characteristic Root
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