CONSUMER.RPF

CONSUMER.RPF is an example of estimation of a system of equations by multivariate least squares. It uses the NLSYSTEM instruction to estimate equations from an expenditure system given in Theil (1971), Section 7.6. It works with four groups of commodities: Food, Vice, Durable Goods and Remainder. Data for each consists of its value share $\times$ (percent) change in quantity (WQ...) and the (percent) change in price (DP...). For commodity group $i$, we have the equation

\begin{equation} WQ_{it} = u_i DQ_t + \sum\limits_j {\pi _{ij} DP_{jt} + \varepsilon _{it} } \end{equation}

This system of equations is redundant, as the sum of the $\varepsilon$ over i must be zero. We thus estimate only three of the four equations. We are interested in two constraints:

\begin{equation} \sum\limits_j {\pi _{ij} } = 0 \text{(homogeneity)} \label{eq:consumer_homogeneity} \end{equation}

\begin{equation} \pi _{ij} = \pi _{ji} \text{(symmetry)} \end{equation}

To simplify the former, we parameterize the equations using an “excess” parameter (called EP...) for $\pi_{i4}$. This parameter has a value of 0 if constraint \eqref{eq:consumer_homogeneity} holds.

We use different PARMSETs to handle the different problems. For instance, the PARMSET RELAX, when added in with the base parameter set, relaxes the homogeneity assumptions by allowing the $\pi_{i4}$ to be estimated. The PARMSET SYMMETRY, when added to the others, imposes the symmetry constraints:

nonlin(parms=base) mu1 mu2 mu3 p11 p12 p13 p21 p22 p23 p31 p32 p33

nonlin(parms=relax) ep14 ep24 ep34

nonlin(parms=symmetry) p12=p21 p13=p31 p23=p32

The three FRML’s for the three equations we will estimate are:

frml ffood wqfood = mu1*dq+p11*(dpfood-dprem)+$

p12*(dpvice-dprem)+p13*(dpdura-dprem)+ep14*dprem

frml fvice wqvice = mu2*dq+p21*(dpfood-dprem)+$

p22*(dpvice-dprem)+p23*(dpdura-dprem)+ep24*dprem

frml fdura wqdura = mu3*dq+p31*(dpfood-dprem)+$

p32*(dpvice-dprem)+p33*(dpdura-dprem)+ep34*dprem

Because the equations are linear, the guess values aren’t particularly important, and we start everything at 0. The ones that are really important to be zero are the EP14, EP24 and EP34 since zeroing those imposes homogeneity. That’s why those get reset before the third NLSYSTEM:

compute mu1=mu2=mu3=0.0

compute p11=p12=p13=p21=p22=p23=p31=p32=p33=0.0

compute ep14=ep24=ep34=0.0

nlsystem(parmset=base) / ffood fvice fdura

nlsystem(parmset=base+relax) / ffood fvice fdura

compute ep14=ep24=ep34=0.0

nlsystem(parmset=base+symmetry) / ffood fvice fdura

Full Program

cal(a) 1922
all 1962:1
open data consumer.wks
data(format=wks,org=cols) / wqfood wqvice wqdura wqrem dpfood dpvice dpdura dprem
set dq = wqfood + wqvice + wqdura + wqrem
nonlin(parmset=base) mu1 mu2 mu3 $
   p11 p12 p13 p21 p22 p23 p31 p32 p33
nonlin(parmset=relax) ep14 ep24 ep34
nonlin(parmset=symmetry) p12=p21 p13=p31 p23=p32
*
frml ffood wqfood = mu1*dq+p11*(dpfood-dprem)+$
   p12*(dpvice-dprem)+p13*(dpdura-dprem)+ep14*dprem
frml fvice wqvice = mu2*dq+p21*(dpfood-dprem)+$
   p22*(dpvice-dprem)+p23*(dpdura-dprem)+ep24*dprem
frml fdura wqdura = mu3*dq+p31*(dpfood-dprem)+$
   p32*(dpvice-dprem)+p33*(dpdura-dprem)+ep34*dprem
*
compute mu1=mu2=mu3=0.0
compute p11=p12=p13=p21=p22=p23=p31=p32=p33=0.0
compute ep14=ep24=ep34=0.0
*
nlsystem(parmset=base,sigma) / ffood fvice fdura
nlsystem(parmset=base+relax) / ffood fvice fdura
compute ep14=ep24=ep34=0.0
nlsystem(parmset=base+symmetry,iters=500) / ffood fvice fdura
*
* Additional code for non-linear maximum likelihood from section 4.11.
*
dec frml[vect] ufrml
frml ufrml = ||wqfood-ffood,wqvice-fvice,wqdura-fdura||
dec symm sigma(3,3)
nonlin(parmset=sigmaparms) sigma
frml mvlikely = %logdensity(sigma,ufrml)
compute sigma=%identity(3)
maximize(parmset=base+sigmaparms) mvlikely

Output

Non-Linear System Estimation

Convergence in 5 Iterations. Final criterion was 0.0000000 <= 0.0000100

Annual Data From 1922:01 To 1952:01

Usable Observations 31

Log Likelihood -24.6331

Dependent Variable WQFOOD

Mean of Dependent Variable 0.3564516129

Std Error of Dependent Variable 0.7798570313

Standard Error of Estimate 0.4708680856

Sum of Squared Residuals 6.8732193747

Durbin-Watson Statistic 1.6916

Dependent Variable WQVICE

Mean of Dependent Variable 0.1632258065

Std Error of Dependent Variable 0.3317568095

Standard Error of Estimate 0.1829687560

Sum of Squared Residuals 1.0378045355

Durbin-Watson Statistic 2.1291

Dependent Variable WQDURA

Mean of Dependent Variable 0.6574193548

Std Error of Dependent Variable 1.5693097585

Standard Error of Estimate 0.4976413368

Sum of Squared Residuals 7.6770539027

Durbin-Watson Statistic 1.7114

Variable Coeff Std Error T-Stat Signif

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

1. MU1 0.153224487 0.032500016 4.71460 0.00000242

2. MU2 0.086039027 0.012628776 6.81293 0.00000000

3. MU3 0.489684409 0.034347946 14.25659 0.00000000

4. P11 -0.104267263 0.024568831 -4.24388 0.00002197

5. P12 -0.039179986 0.049319948 -0.79440 0.42696001

6. P13 0.043645731 0.022959760 1.90097 0.05730642

7. P21 0.022834432 0.009546896 2.39182 0.01676518

8. P22 -0.030968853 0.019164623 -1.61594 0.10610761

9. P23 0.002120576 0.008921647 0.23769 0.81212247

10. P31 0.052775172 0.025965799 2.03249 0.04210429

11. P32 0.015375630 0.052124248 0.29498 0.76800888

12. P33 -0.046514680 0.024265237 -1.91693 0.05524725

Covariance\Correlation Matrix of Residuals

WQFOOD WQVICE WQDURA

WQFOOD 0.470868086 0.19671098 -0.66920363

WQVICE 0.056319556 0.174085247 -0.55618678

WQDURA -0.284033825 -0.143536976 0.382581786

Non-Linear System Estimation

Convergence in 2 Iterations. Final criterion was 0.0000000 <= 0.0000100

Annual Data From 1922:01 To 1952:01

Usable Observations 31

Log Likelihood -22.8544

Dependent Variable WQFOOD

Mean of Dependent Variable 0.3564516129

Std Error of Dependent Variable 0.7798570313

Standard Error of Estimate 0.4517981656

Sum of Squared Residuals 6.3277690566

Durbin-Watson Statistic 1.9743

Dependent Variable WQVICE

Mean of Dependent Variable 0.1632258065

Std Error of Dependent Variable 0.3317568095

Standard Error of Estimate 0.1792821023

Sum of Squared Residuals 0.9964042383

Durbin-Watson Statistic 2.2702

Dependent Variable WQDURA

Mean of Dependent Variable 0.6574193548

Std Error of Dependent Variable 1.5693097585

Standard Error of Estimate 0.4764026232

Sum of Squared Residuals 7.0357432418

Durbin-Watson Statistic 1.9323

Variable Coeff Std Error T-Stat Signif

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

1. MU1 0.147835996 0.031357522 4.71453 0.00000242

2. MU2 0.084554489 0.012443261 6.79520 0.00000000

3. MU3 0.495527247 0.033065220 14.98636 0.00000000

4. P11 -0.106171762 0.023602578 -4.49831 0.00000685

5. P12 -0.012806948 0.049997083 -0.25615 0.79783200

6. P13 0.039531361 0.022173215 1.78284 0.07461186

7. P21 0.022309739 0.009365952 2.38200 0.01721869

8. P22 -0.023703039 0.019839793 -1.19472 0.23219564

9. P23 0.000987060 0.008798753 0.11218 0.91067925

10. P31 0.054840255 0.024887950 2.20349 0.02756050

11. P32 -0.013221128 0.052719872 -0.25078 0.80198365

12. P33 -0.042053396 0.023380745 -1.79863 0.07207665

13. EP14 0.042114704 0.025763248 1.63468 0.10211584

14. EP24 0.011602668 0.010223347 1.13492 0.25640934

15. EP34 -0.045665727 0.027166288 -1.68097 0.09276865

Non-Linear System Estimation

Convergence in 4 Iterations. Final criterion was 0.0000023 <= 0.0000100

Annual Data From 1922:01 To 1952:01

Usable Observations 31

Log Likelihood -25.7915

Dependent Variable WQFOOD

Mean of Dependent Variable 0.3564516129

Std Error of Dependent Variable 0.7798570313

Standard Error of Estimate 0.4841485646

Sum of Squared Residuals 7.2663948093

Durbin-Watson Statistic 1.7725

Dependent Variable WQVICE

Mean of Dependent Variable 0.1632258065

Std Error of Dependent Variable 0.3317568095

Standard Error of Estimate 0.1831803921

Sum of Squared Residuals 1.0402067375

Durbin-Watson Statistic 2.1368

Dependent Variable WQDURA

Mean of Dependent Variable 0.6574193548

Std Error of Dependent Variable 1.5693097585

Standard Error of Estimate 0.4998142698

Sum of Squared Residuals 7.7442434345

Durbin-Watson Statistic 1.6721

Variable Coeff Std Error T-Stat Signif

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

1. MU1 0.179071334 0.027533501 6.50376 0.00000000

2. MU2 0.087927604 0.012223275 7.19346 0.00000000

3. MU3 0.482091683 0.028434138 16.95468 0.00000000

4. P11 -0.107306957 0.024968933 -4.29762 0.00001726

5. P12 0.021986050 0.000000000 0.00000 0.00000000

6. P13 0.043381733 0.000000000 0.00000 0.00000000

7. P21 0.021986050 0.009121665 2.41031 0.01593892

8. P22 -0.026206200 0.017468176 -1.50023 0.13355605

9. P23 0.002226149 0.000000000 0.00000 0.00000000

10. P31 0.043381733 0.019552510 2.21873 0.02650513

11. P32 0.002226149 0.008913365 0.24975 0.80277754

12. P33 -0.044386294 0.023563419 -1.88370 0.05960624

MAXIMIZE - Estimation by BFGS

Convergence in 43 Iterations. Final criterion was 0.0000055 <= 0.0000100

Annual Data From 1922:01 To 1962:01

Usable Observations 31

Skipped/Missing (from 41) 10

Function Value -24.6331

Variable Coeff Std Error T-Stat Signif

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

1. MU1 0.153224533 0.032435002 4.72405 0.00000231

2. MU2 0.086039029 0.013164255 6.53581 0.00000000

3. MU3 0.489684251 0.035342158 13.85553 0.00000000

4. P11 -0.104267487 0.026956216 -3.86803 0.00010972

5. P12 -0.039179519 0.051896479 -0.75496 0.45027584

6. P13 0.043645811 0.025627600 1.70308 0.08855341

7. P21 0.022834437 0.009649813 2.36631 0.01796645

8. P22 -0.030968525 0.021129682 -1.46564 0.14274615

9. P23 0.002120522 0.008807082 0.24077 0.80972982

10. P31 0.052775153 0.027347386 1.92981 0.05363086

11. P32 0.015374354 0.058659560 0.26209 0.79324851

12. P33 -0.046514296 0.025907692 -1.79539 0.07259229

13. SIGMA(1,1) 0.221716843 0.056878825 3.89806 0.00009697

14. SIGMA(2,1) 0.026518937 0.016725003 1.58559 0.11283316

15. SIGMA(2,2) 0.033477335 0.008881499 3.76933 0.00016368

16. SIGMA(3,1) -0.133742641 0.050021016 -2.67373 0.00750130

17. SIGMA(3,2) -0.040984184 0.019469504 -2.10505 0.03528738

18. SIGMA(3,3) 0.247647120 0.065670535 3.77105 0.00016256