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

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