A COMPANION TO Theoretical Econometrics

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 + l2)2(X1 + l1) + P18(X2 + l2)2(S + l3) + P19(X2 + l2)2 (t + 1)

+ P20(S + l3)2(X1 + l1) + Pn(S + l3)2 (X2 + l2) + P22(S + l3)2 (t + 1) + P*(f + 1)2(X1 + l1)

+ P24(t + 1)2(X2 + l2) + P25(t + 1)2(S + l3) + P26l3 + P27l3(X1 + l1) + P28l3(X2 + l2)

+ P29l3(S + l3) + P30l3(t + 1) + P31l3(X1 + l1)2 + P32l3 (X2 + l2)2 + P33l3(S + l3)2 + P34l3(t + 1)2

The differenced future components used above are defined by f(I*t, j) = F(I*t, j) - F(I*, 4). Several of the parameters of the level future component drop out due to differencing. For instance, the intercept P1 and all coefficients of terms involving only (t + 1) vanish. Simple algebra reveals the differenced future components have the following forms.

f(I *, 1) = n + n2g(Xi) + n3h(X1) + niX2g(X1) + ^gX) + n6(t + 1)g(X1)

+ n 7X2 + n Б + n 9(t + 1)2.

f(I*, 2) = nX 12 + П7X1 g(X2) + П10 + nng(X2) + ni2h(X2) + n^SgX + nM(f + 1)g(X2) + П15Б2 + П1б(і + 1)2.

f(I*, 3) = n5X2 + n 8X1 g(S) + n^X2 + n^X^S) + П17 + n18g(S) + %19h(S) + n^X?

+ n21X 2 + n22(t + 1)g(S) + П 23(t + 1)2 + n24X1 + n25X2 + n26(t + 1).

where g(x) = 2x + 1, and h(x) = 3x2 + 3x + 1. Several of the parameters appear in multiple equations. Such cross equation restrictions reflect the specification's logical consistency. The future components' asymmetry arises since choosing school both augments school experience and removes the cost of returning to school that one would otherwise face. In contrast, choosing alternative one or two only augments experience within that alternative.