## Appendix A The Future Component

The future component we used was a fourth-order polynomial in the state variables. Below, in the interest of space and clarity, we will develop that polynomial only up to its third-order terms. The extension to the higher-order terms is obvious. From equation (22.12), the future component is the flexible functional form

F(Xilt + x( j = 1), Xn + x( j = 2), Sit + x( j = 3), t + 1, x( j = 3))

Define ik = x( j = k). Then, to third-order terms, we used the following polynomial to represent this function.

F(X1 + lV X2 + ^ S + ^ t + 1 l3) = P1 + P2(X1 + l1) + P3(X2 + l2) + P4(S + l3)

+ P5(t + 1) + P6(X1 + I1)2 + P7(X2 + I2)2 + P8(S + І3)2 + P9(t + 1)2 + P10X1 + І1)3 + Pu(X2 + I2)3 + P12(S + I3)3 + P13(t + 1)3 + P14(X1 + l1)2(X2 + I2) + P15(X1 + l1)2(S + 13)

+ P16(X1 + l1)2(t + 1) + P17(X2 +...

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