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The SYSLIN Procedure

Example 19.1: Klein's Model I Estimated with LIML and 3SLS

This example uses PROC SYSLIN to estimate the classic Klein Model I. For a discussion of this model, see Theil (1971). The following statements read the data.


   *--------------------Klein's Model I--------------------------*
   | By L.R. Klein, Economic Fluctuations in the United States,  |
   | 1921-1941 (1950), NY: John Wiley.   A macro-economic model  |
   | of the U.S. with three behavioral equations, and several    |
   | identities. See Theil, p.456.                               |
   *-------------------------------------------------------------*;
   data klein;
      input year c p w i x wp g t k wsum;
      date=mdy(1,1,year);
      format date monyy.;
      y   =c+i+g-t;
      yr  =year-1931;
      klag=lag(k);
      plag=lag(p);
      xlag=lag(x);
      label year='Year'
            date='Date'
            c   ='Consumption'
            p   ='Profits'
            w   ='Private Wage Bill'
            i   ='Investment'
            k   ='Capital Stock'
            y   ='National Income'
            x   ='Private Production'
            wsum='Total Wage Bill'
            wp  ='Govt Wage Bill'
            g   ='Govt Demand'
            i   ='Taxes'
            klag='Capital Stock Lagged'
            plag='Profits Lagged'
            xlag='Private Product Lagged'
            yr  ='YEAR-1931';
      cards;
   1920     .  12.7     .    .  44.9    .     .     .  182.8     .
   1921  41.9  12.4  25.5 -0.2  45.6  2.7   3.9   7.7  182.6  28.2
   1922  45.0  16.9  29.3  1.9  50.1  2.9   3.2   3.9  184.5  32.2
   1923  49.2  18.4  34.1  5.2  57.2  2.9   2.8   4.7  189.7  37.0
   1924  50.6  19.4  33.9  3.0  57.1  3.1   3.5   3.8  192.7  37.0
   1925  52.6  20.1  35.4  5.1  61.0  3.2   3.3   5.5  197.8  38.6
   1926  55.1  19.6  37.4  5.6  64.0  3.3   3.3   7.0  203.4  40.7
   1927  56.2  19.8  37.9  4.2  64.4  3.6   4.0   6.7  207.6  41.5
   1928  57.3  21.1  39.2  3.0  64.5  3.7   4.2   4.2  210.6  42.9
   1929  57.8  21.7  41.3  5.1  67.0  4.0   4.1   4.0  215.7  45.3
   1930  55.0  15.6  37.9  1.0  61.2  4.2   5.2   7.7  216.7  42.1
   1931  50.9  11.4  34.5 -3.4  53.4  4.8   5.9   7.5  213.3  39.3
   1932  45.6   7.0  29.0 -6.2  44.3  5.3   4.9   8.3  207.1  34.3
   1933  46.5  11.2  28.5 -5.1  45.1  5.6   3.7   5.4  202.0  34.1
   1934  48.7  12.3  30.6 -3.0  49.7  6.0   4.0   6.8  199.0  36.6
   1935  51.3  14.0  33.2 -1.3  54.4  6.1   4.4   7.2  197.7  39.3
   1936  57.7  17.6  36.8  2.1  62.7  7.4   2.9   8.3  199.8  44.2
   1937  58.7  17.3  41.0  2.0  65.0  6.7   4.3   6.7  201.8  47.7
   1938  57.5  15.3  38.2 -1.9  60.9  7.7   5.3   7.4  199.9  45.9
   1939  61.6  19.0  41.6  1.3  69.5  7.8   6.6   8.9  201.2  49.4
   1940  65.0  21.1  45.0  3.3  75.7  8.0   7.4   9.6  204.5  53.0
   1941  69.7  23.5  53.3  4.9  88.4  8.5  13.8  11.6  209.4  61.8
   ;

The following statements estimate the Klein model using the limited information maximum likelihood method. In addition, the parameter estimates are written to a SAS data set with the OUTEST= option.


   proc syslin data=klein outest=b liml;
      endogenous c p w i x wsum k y;
      instruments klag plag xlag wp g t yr;
      consume: model c = p plag  wsum;
      invest:  model i = p plag  klag;
      labor:   model w = x xlag  yr;
   run;
   
   proc print data=b; run;

The PROC SYSLIN estimates are shown in Output 19.1.1.

Output 19.1.1: LIML Estimates
 
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation

Model CONSUME
Dependent Variable c
Label Consumption
 
Analysis of Variance
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 854.3541 284.7847 118.42 <.0001
Error 17 40.88419 2.404952    
Corrected Total 20 941.4295      
 
Root MSE 1.55079 R-Square 0.95433
Dependent Mean 53.99524 Adj R-Sq 0.94627
Coeff Var 2.87209    
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 17.14765 2.045374 8.38 <.0001 Intercept
p 1 -0.22251 0.224230 -0.99 0.3349 Profits
plag 1 0.396027 0.192943 2.05 0.0558 Profits Lagged
wsum 1 0.822559 0.061549 13.36 <.0001 Total Wage Bill

 
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation

Model INVEST
Dependent Variable i
Label Taxes
 
Analysis of Variance
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 210.3790 70.12634 34.06 <.0001
Error 17 34.99649 2.058617    
Corrected Total 20 252.3267      
 
Root MSE 1.43479 R-Square 0.85738
Dependent Mean 1.26667 Adj R-Sq 0.83221
Coeff Var 113.27274    
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 22.59083 9.498146 2.38 0.0294 Intercept
p 1 0.075185 0.224712 0.33 0.7420 Profits
plag 1 0.680386 0.209145 3.25 0.0047 Profits Lagged
klag 1 -0.16826 0.045345 -3.71 0.0017 Capital Stock Lagged

 
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation

Model LABOR
Dependent Variable w
Label Private Wage Bill
 
Analysis of Variance
Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 696.1485 232.0495 393.62 <.0001
Error 17 10.02192 0.589525    
Corrected Total 20 794.9095      
 
Root MSE 0.76781 R-Square 0.98581
Dependent Mean 36.36190 Adj R-Sq 0.98330
Coeff Var 2.11156    
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 1.526187 1.320838 1.16 0.2639 Intercept
x 1 0.433941 0.075507 5.75 <.0001 Private Production
xlag 1 0.151321 0.074527 2.03 0.0583 Private Product Lagged
yr 1 0.131593 0.035995 3.66 0.0020 YEAR-1931

The OUTEST= data set is shown in part in Output 19.1.2. Note that the data set contains the parameter estimates and root mean square errors, _SIGMA_, for the first stage instrumental regressions as well as the parameter estimates and {\sigma} for the LIML estimates for the three structural equations.

Output 19.1.2: The OUTEST= Data Set
 
Obs _TYPE_ _STATUS_ _MODEL_ _DEPVAR_ _SIGMA_ Intercept klag plag xlag wp g t yr c p w i x wsum k y
1 LIML 0 Converged CONSUME c 1.55079 17.1477 . 0.39603 . . . . . -1 -0.22251 . . . 0.82256 . .
2 LIML 0 Converged INVEST i 1.43479 22.5908 -0.16826 0.68039 . . . . . . 0.07518 . -1 . . . .
3 LIML 0 Converged LABOR w 0.76781 1.5262 . . 0.15132 . . . 0.13159 . . -1 . 0.43394 . . .

The following statements estimate the model using the 3SLS method. The reduced form estimates are produced by the REDUCED option; IDENTITY statements are used to make the model complete.


   proc syslin data=klein 3sls reduced;
      endogenous c p w i x wsum k y;
      instruments klag plag xlag wp g t yr;
      consume: model    c = p plag wsum;
      invest:  model    i = p plag klag;
      labor:   model    w = x xlag yr;
      product: identity x = c + i + g;
      income:  identity y = c + i + g - t;
      profit:  identity p = y - w;
      stock:   identity k = klag + i;
      wage:    identity wsum = w + wp;
   run;

The preliminary 2SLS results and estimated cross-model covariance matrix are not shown. The 3SLS estimates are shown in Output 19.1.3. The reduced form estimates are shown in Output 19.1.4.

Output 19.1.3: 3SLS Estimates
 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

System Weighted MSE 5.9342
Degrees of freedom 51
System Weighted R-Square 0.9550
 
Model CONSUME
Dependent Variable c
Label Consumption
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 16.44079 1.449925 11.34 <.0001 Intercept
p 1 0.124890 0.120179 1.04 0.3133 Profits
plag 1 0.163144 0.111631 1.46 0.1621 Profits Lagged
wsum 1 0.790081 0.042166 18.74 <.0001 Total Wage Bill

 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Model INVEST
Dependent Variable i
Label Taxes
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 28.17785 7.550853 3.73 0.0017 Intercept
p 1 -0.01308 0.179938 -0.07 0.9429 Profits
plag 1 0.755724 0.169976 4.45 0.0004 Profits Lagged
klag 1 -0.19485 0.036156 -5.39 <.0001 Capital Stock Lagged

 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Model LABOR
Dependent Variable w
Label Private Wage Bill
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard Error t Value Pr > |t| Variable
Label
Intercept 1 1.797218 1.240203 1.45 0.1655 Intercept
x 1 0.400492 0.035359 11.33 <.0001 Private Production
xlag 1 0.181291 0.037965 4.78 0.0002 Private Product Lagged
yr 1 0.149674 0.031048 4.82 0.0002 YEAR-1931

Output 19.1.4: Reduced Form Estimates
 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Endogenous Variables
  c p w i x wsum k y
CONSUME 1 -0.12489 0 0 0 -0.79008 0 0
INVEST 0 0.013079 0 1 0 0 0 0
LABOR 0 0 1 0 -0.40049 0 0 0
PRODUCT -1 0 0 -1 1 0 0 0
INCOME -1 0 0 -1 0 0 0 1
PROFIT 0 1 1 0 0 0 0 -1
STOCK 0 0 0 -1 0 0 1 0
WAGE 0 0 -1 0 0 1 0 0

 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Exogenous Variables
  Intercept plag klag xlag yr g t wp
CONSUME 16.44079 0.163144 0 0 0 0 0 0
INVEST 28.17785 0.755724 -0.19485 0 0 0 0 0
LABOR 1.797218 0 0 0.181291 0.149674 0 0 0
PRODUCT 0 0 0 0 0 1 0 0
INCOME 0 0 0 0 0 1 -1 0
PROFIT 0 0 0 0 0 0 0 0
STOCK 0 0 1 0 0 0 0 0
WAGE 0 0 0 0 0 0 0 1

 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Inverse Endogenous Variables
  CONSUME INVEST LABOR PRODUCT INCOME PROFIT STOCK WAGE
c 1.634654 0.634654 1.095657 0.438802 0.195852 0.195852 0 1.291509
p 0.972364 0.972364 -0.34048 -0.13636 1.108721 1.108721 0 0.768246
w 0.649572 0.649572 1.440585 0.576943 0.072629 0.072629 0 0.513215
i -0.01272 0.987282 0.004453 0.001783 -0.0145 -0.0145 0 -0.01005
x 1.621936 1.621936 1.10011 1.440585 0.181351 0.181351 0 1.281461
wsum 0.649572 0.649572 1.440585 0.576943 0.072629 0.072629 0 1.513215
k -0.01272 0.987282 0.004453 0.001783 -0.0145 -0.0145 1 -0.01005
y 1.621936 1.621936 1.10011 0.440585 1.181351 0.181351 0 1.281461

 
The SYSLIN Procedure
Three-Stage Least Squares Estimation

Reduced Form
  Intercept plag klag xlag yr g t wp
c 46.7273 0.746307 -0.12366 0.198633 0.163991 0.634654 -0.19585 1.291509
p 42.77363 0.893474 -0.18946 -0.06173 -0.05096 0.972364 -1.10872 0.768246
w 31.57207 0.596871 -0.12657 0.261165 0.215618 0.649572 -0.07263 0.513215
i 27.6184 0.744038 -0.19237 0.000807 0.000667 -0.01272 0.014501 -0.01005
x 74.3457 1.490345 -0.31603 0.19944 0.164658 1.621936 -0.18135 1.281461
wsum 31.57207 0.596871 -0.12657 0.261165 0.215618 0.649572 -0.07263 1.513215
k 27.6184 0.744038 0.80763 0.000807 0.000667 -0.01272 0.014501 -0.01005
y 74.3457 1.490345 -0.31603 0.19944 0.164658 1.621936 -1.18135 1.281461

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