Dear R-helpers,
After having search everywhere on the R documentation and on R-help forum I
finally come to ask you some help.
I am willing to estimate a model with constraints on parameters. This model is
built from econometrics theory of jumping process and takes the following
reduced form of a linear regression:
dYt= par1_1 * a1a0_1 + par1_2 * a1a0_2 + par1_3 * a1a0_3 + par1_4 * a1a0_4
+ par2_1 * b0_1 + par2_2 * b0_2 + par2_3 * b0_3 + par2_4 * b0_4
+ par3_1 * c0_1 + par3_2 * c0_2 + par3_3 * c0_3 + par3_4 * c0_4
+ par4_1 * d0_1 + par4_2 * d0_2 + par4_3 * d0_3 + par4_4 * d0_4
+ par5_1 * b1_1 + par5_2 * b1_2 + par5_3 * b1_3 + par5_4 * b1_4
+ par6_1 * c1_1 + par6_2 * c1_2 + par6_3 * c1_3 + par6_4 * c1_4
+ par7_1 * d1_1 + par7_2 * d1_2 + par7_3 * d1_3 + par7_4 * d1_4 + residual
(and no constant)
where :
a1a0 =dZt
b0=1 - Zt - Xt*pidZt
c0=2*Xt - 2*Xt*Zt - Xtsq*dZt
d0=3*Xtsq - 3*Xtsq*Zt - Xtcub*dZt
b1=Zt + Xt*dZt
c1=2*Xt*Zt + Xtsq*dZt
d1=3*Xtsq*Zt + Xtcub*dZt
The parameters are thus estimated for diffrent pieces of the intervall of Xt.
The constraints are related to a cubic spline estimation:
constraint 1 a1a0_1 + (b1_1-b0_1)*1.0 + (c1_1-c0_1)*(1.0*1.0) +
(d1_1-d0_1)*(1.0*1.0*1.0) = a1a0_2 + (b1_2-b0_2)*1.0 + (c1_2-c0_2)*(1.0*1.0)
+ (d1_2-d0_2)*(1.0*1.0*1.0)
constraint 2 a1a0_2 + (b1_2-b0_2)*3.0 + (c1_2-c0_2)*(3.0*3.0) +
(d1_2-d0_2)*(3.0*3.0*3.0) = a1a0_3 + (b1_3-b0_3)*3.0 + (c1_3-c0_3)*(3.0*3.0)
+ (d1_3-d0_3)*(3.0*3.0*3.0)
constraint 3 a1a0_3 + (b1_3-b0_3)*9.0 + (c1_3-c0_3)*(9.0*9.0) +
(d1_3-d0_3)*(9.0*9.0*9.0) = a1a0_4 + (b1_4-b0_4)*9.0 + (c1_4-c0_4)*(9.0*9.0)
+ (d1_4-d0_4)*(9.0*9.0*9.0)
constraint 4 b0_1 + 2*c0_1*1.0 + 3*d0_1*(1.0*1.0) = b0_2 + 2*c0_2*1.0 +
3*d0_2*(1.0*1.0)
constraint 5 b0_2 + 2*c0_2*3.0 + 3*d0_2*(3.0*3.0) = b0_3 + 2*c0_3*3.0 +
3*d0_3*(3.0*3.0)
constraint 6 b0_3 + 2*c0_3*9.0 + 3*d0_3*(9.0*9.0) = b0_4 + 2*c0_4*9.0 +
3*d0_4*(9.0*9.0)
constraint 7 b1_1 + 2*c1_1*1.0 + 3*d1_1*(1.0*1.0) = b1_2 + 2*c1_2*1.0 +
3*d1_2*(1.0*1.0)
constraint 8 b1_2 + 2*c1_2*3.0 + 3*d1_2*(3.0*3.0) = b1_3 + 2*c1_3*3.0 +
3*d1_3*(3.0*3.0)
constraint 9 b1_3 + 2*c1_3*9.0 + 3*d1_3*(9.0*9.0) = b1_4 + 2*c1_4*9.0 +
3*d1_4*(9.0*9.0)
constraint 10 c0_1 + 3*d0_1*1.0 = c0_2 + 3*d0_2*1.0
constraint 11 c0_2 + 3*d0_2*3.0 = c0_3 + 3*d0_3*3.0
constraint 12 c0_3 + 3*d0_3*9.0 = c0_4 + 3*d0_4*9.0
constraint 13 c1_1 + 3*d1_1*1.0 = c1_2 + 3*d1_2*1.0
constraint 14 c1_2 + 3*d1_2*3.0 = c1_3 + 3*d1_3*3.0
constraint 15 c1_3 + 3*d1_3*9.0 = c1_4 + 3*d1_4*9.0
How can I estimate it with R, knowing that it is not a simple spline regression
but a complex built model based partly on it ? The model has to be this way and
no other reduced form is possible.
Thank you very much for your help.
Sophie
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