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'; datalines; 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 ... more lines ...
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 29.1.1 through Output 29.1.3.
Output 29.1.1: LIML Estimates for Consumption
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 |
Output 29.1.2: LIML Estimates for Investments
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 |
Output 29.1.3: LIML Estimates for Labor
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 29.1.4. Note that the data set contains the parameter estimates and root mean squared errors, _SIGMA_, for the first-stage instrumental regressions as well as the parameter estimates and for the LIML estimates for the three structural equations.
Output 29.1.4: 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 29.1.5 through Output 29.1.7. The reduced form estimates are shown in Output 29.1.8 through Output 29.1.11.
Output 29.1.5: 3SLS Estimates for Consumption
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 |
Output 29.1.6: 3SLS Estimates for Investments
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 |
Output 29.1.7: 3SLS Estimates for Labor
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 29.1.8: Reduced Form Estimates
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 |
Output 29.1.9: Reduced Form Estimates
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 |
Output 29.1.10: Reduced Form Estimates
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 |
Output 29.1.11: Reduced Form Estimates
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 |