## 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## Preview

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## Copyright information

© Springer Science+Business Media New York 2000