QPROG.RPF

QPROG.RPF demonstrates the use of quadratic programming (LQPROG instruction) for constrained least squares. It estimates a regression subject to non-negativity constraints and an adding up constraint.

In matrix form, the sum of squared residuals can be written

\begin{equation} \left( {{\bf{y}} - {\bf{X}}\beta } \right)^\prime \left( {{\bf{y}} - {\bf{X}}\beta } \right) = \beta '{\kern 1pt} {\bf{X'}}{\kern 1pt} {\bf{X}}\beta - {\bf{2y'}}{\kern 1pt} {\bf{X}}\beta + {\bf{y'y}} \end{equation}

The last term doesn’t involve \(\beta\) and so can be ignored in finding the optimal coefficients. If we multiply this by 1/2 we can read off \({\bf{Q}} = {\bf{X'}}{\kern 1pt} {\bf{X}}\) and \({\bf{c}} = - {\bf{X'y}}\). For illustration, we will add to this the constraint that \({\bf{1}} \bullet \beta \le 1\). \({\bf{X'}}{\kern 1pt} {\bf{X}}\) and \({\bf{X'y}}\) can be obtained as submatrices of the cross-moment matrix of the regressors and \(y\). The CMOMENT instruction with the LASTREG option produces just such a matrix (regressors first, then dependent variable):

linreg longrate

# shortrate{0 to 12}

cmom(lastreg)

compute [symmetric] q = %xsubmat(%cmom,1,%ncmom-1,1,%ncmom-1)

compute [vector] c = -1.0*%xsubmat(%cmom,%ncmom,%ncmom,1,%ncmom-1)

This sets up the \(\bf{A}\) matrix:

compute a=%ones(1,%ncmom-1)

This does the optimization (which, with these options, will have all coefficients non-negative and summing to one):

lqprog(c=c,q=q,a=a,b=1.0) x

and this uses REPORT to display the OLS estimates (which are in %BETA) and the constrained estimates (in X) in parallel columns:

report(action=define,hlabels=||"Label","OLS","Constrained"||)
report(fillby=cols,atrow=1,atcol=1) %eqnreglabels(0)
report(fillby=cols,atrow=1,atcol=2) %beta
report(fillby=cols,atrow=1,atcol=3) x
report(action=show)

Full Program

open data haversample.rat

calendar(m) 1947

data(format=rats) 1947:1 2007:4 fltg ftb3

set shortrate = ftb3

set longrate = fltg

linreg longrate

# shortrate{0 to 12}

cmom(lastreg)

compute [symmetric] q = %xsubmat(%cmom,1,%ncmom-1,1,%ncmom-1)

compute [vector] c = -1.0*%xsubmat(%cmom,%ncmom,%ncmom,1,%ncmom-1)

compute a=%ones(1,%ncmom-1)

lqprog(c=c,q=q,a=a,b=1.0) x

report(action=define,hlabels=||"Label","OLS","Constrained"||)

report(fillby=cols,atrow=1,atcol=1) %eqnreglabels(0)

report(fillby=cols,atrow=1,atcol=2) %beta

report(fillby=cols,atrow=1,atcol=3) x

report(action=show)

Output

Linear Regression - Estimation by Least Squares

Dependent Variable LONGRATE

Monthly Data From 1948:01 To 2007:04

Usable Observations 712

Degrees of Freedom 699

Centered R^2 0.7128498

R-Bar^2 0.7079201

Uncentered R^2 0.9545383

Mean of Dependent Variable 6.0815870787

Std Error of Dependent Variable 2.6394706845

Standard Error of Estimate 1.4264864346

Sum of Squared Residuals 1422.3696202

Log Likelihood -1256.6368

Durbin-Watson Statistic 0.0526

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. SHORTRATE 0.705702074 0.134743997 5.23735 0.00000022

2. SHORTRATE{1} -0.215702353 0.225120830 -0.95816 0.33831207

3. SHORTRATE{2} 0.120288070 0.234432537 0.51310 0.60804130

4. SHORTRATE{3} -0.119402443 0.234465771 -0.50925 0.61073548

5. SHORTRATE{4} 0.013673999 0.234970463 0.05819 0.95361030

6. SHORTRATE{5} 0.031937149 0.236038455 0.13530 0.89240980

7. SHORTRATE{6} 0.180818041 0.240205942 0.75276 0.45184604

8. SHORTRATE{7} -0.038395627 0.235983733 -0.16270 0.87079809

9. SHORTRATE{8} 0.046579619 0.234967765 0.19824 0.84291623

10. SHORTRATE{9} -0.105761855 0.234483504 -0.45104 0.65209933

11. SHORTRATE{10} 0.141730720 0.234480817 0.60444 0.54574407

12. SHORTRATE{11} -0.180278995 0.225195680 -0.80054 0.42366782

13. SHORTRATE{12} 0.587076076 0.134843579 4.35376 0.00001538

QPROG converges at 9 iterations

minimum = -14621.17962725

Solution x =

0.4847369835 0.0000000000 0.0000000000 0.0000000000 0.0000000000

0.0267942351 0.0794425993 0.0196099240 0.0000000000 0.0000000000

0.0000000000 0.0000000000 0.3894162584

Lagrange Multipliers

3673.1565043

Label OLS Constrained

SHORTRATE 0.705702 0.484737

SHORTRATE{1} -0.215702 0.000000

SHORTRATE{2} 0.120288 0.000000

SHORTRATE{3} -0.119402 0.000000

SHORTRATE{4} 0.013674 0.000000

SHORTRATE{5} 0.031937 0.026794

SHORTRATE{6} 0.180818 0.079443

SHORTRATE{7} -0.038396 0.019610

SHORTRATE{8} 0.046580 0.000000

SHORTRATE{9} -0.105762 0.000000

SHORTRATE{10} 0.141731 0.000000

SHORTRATE{11} -0.180279 0.000000

SHORTRATE{12} 0.587076 0.389416