## Abstract

In this chapter we deal with econometric applications of (vector) difference equations where and find the most general form of its solution, called the where, assuming a

**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)

*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$${a_0}{y_t} + {a_1}{y_{t + 1}} + {a_2}{y_{t + 2}} = 0,$$

**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)

_{2}≠ 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 2000