The SIMLIN Procedure

## Example 23.1 Simulating Klein’s Model I

In this example, the SIMLIN procedure simulates a model of the U.S. economy called Klein’s Model I. The SAS data set KLEIN is used as input to the SYSLIN and SIMLIN procedures.

```   data klein;
input year c p w i x wp g t k wsum;
date=mdy(1,1,year);
format date year.;
y   = c + i + g - t;
yr  = year - 1931;
klag = lag( k );
plag = lag( p );
xlag = lag( x );
if year >= 1921;
label 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'
t   ='taxes'
klag='capital stock lagged'
plag='profits lagged'
xlag='private product lagged'
yr  ='year-1931';
datalines;
1920     .  12.7     .    .  44.9    .     .     .  182.8     .

... more lines ...

```

First, the model is specified and estimated using the SYSLIN procedure, and the parameter estimates are written to an OUTEST= data set. The printed output produced by the SYSLIN procedure is not shown here; see Example 26.1 in Chapter 26 for the printed output of the PROC SYSLIN step.

```   title1 'Simulation of Klein''s Model I using SIMLIN';
proc syslin 3sls data=klein outest=a;

instruments klag plag xlag wp g t yr;
endogenous c p w i x wsum k y;

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 = x - w - t;
stock:   identity k = klag + i;
wage:    identity wsum = w + wp;
run;
```

The OUTEST= data set A created by the SYSLIN procedure contains parameter estimates to be used by the SIMLIN procedure. The OUTEST= data set is shown in Output 23.1.1.

Output 23.1.1 The OUTEST= Data Set Created by PROC SYSLIN
 Simulation of Klein's Model I using SIMLIN

Obs _TYPE_ _STATUS_ _MODEL_ _DEPVAR_ _SIGMA_ Intercept klag plag xlag wp g t yr c p w i x wsum k y
1 INST 0 Converged FIRST c 2.11403 58.3018 -0.14654 0.74803 0.23007 0.19327 0.20501 -0.36573 0.70109 -1 . . . . . . .
2 INST 0 Converged FIRST p 2.18298 50.3844 -0.21610 0.80250 0.02200 -0.07961 0.43902 -0.92310 0.31941 . -1.00000 . . . . . .
3 INST 0 Converged FIRST w 1.75427 43.4356 -0.12295 0.87192 0.09533 -0.44373 0.86622 -0.60415 0.71358 . . -1 . . . . .
4 INST 0 Converged FIRST i 1.72376 35.5182 -0.19251 0.92639 -0.11274 -0.71661 0.10023 -0.16152 0.33190 . . . -1 . . . .
5 INST 0 Converged FIRST x 3.77347 93.8200 -0.33906 1.67442 0.11733 -0.52334 1.30524 -0.52725 1.03299 . . . . -1.00000 . . .
6 INST 0 Converged FIRST wsum 1.75427 43.4356 -0.12295 0.87192 0.09533 0.55627 0.86622 -0.60415 0.71358 . . . . . -1.00000 . .
7 INST 0 Converged FIRST k 1.72376 35.5182 0.80749 0.92639 -0.11274 -0.71661 0.10023 -0.16152 0.33190 . . . . . . -1 .
8 INST 0 Converged FIRST y 3.77347 93.8200 -0.33906 1.67442 0.11733 -0.52334 1.30524 -1.52725 1.03299 . . . . . . . -1
9 3SLS 0 Converged CONSUME c 1.04956 16.4408 . 0.16314 . . . . . -1 0.12489 . . . 0.79008 . .
10 3SLS 0 Converged INVEST i 1.60796 28.1778 -0.19485 0.75572 . . . . . . -0.01308 . -1 . . . .
11 3SLS 0 Converged LABOR w 0.80149 1.7972 . . 0.18129 . . . 0.14967 . . -1 . 0.40049 . . .
12 IDENTITY 0 Converged PRODUCT x . 0.0000 . . . . 1.00000 . . 1 . . 1 -1.00000 . . .
13 IDENTITY 0 Converged INCOME y . 0.0000 . . . . 1.00000 -1.00000 . 1 . . 1 . . . -1
14 IDENTITY 0 Converged PROFIT p . 0.0000 . . . . . -1.00000 . . -1.00000 -1 . 1.00000 . . .
15 IDENTITY 0 Converged STOCK k . 0.0000 1.00000 . . . . . . . . . 1 . . -1 .
16 IDENTITY 0 Converged WAGE wsum . 0.0000 . . . 1.00000 . . . . . 1 . . -1.00000 . .

Using the OUTEST= data set A produced by the SYSLIN procedure, the SIMLIN procedure can now compute the reduced form and simulate the model. The following statements perform the simulation.

```   title1 'Simulation of Klein''s Model I using SIMLIN';
proc simlin data=klein
est=a type=3sls
estprint
total interim=2
outest=b;
endogenous c p w i x wsum k y;
exogenous  wp g t yr;
lagged  klag k 1   plag p 1   xlag x 1;
id year;
output out=c p=chat phat what ihat xhat wsumhat khat yhat
r=cres pres wres ires xres wsumres kres yres;
run;
```

The reduced form coefficients and multipliers are added to the information read from EST= data set A and written to the OUTEST= data set B. The predicted and residual values from the simulation are written to the OUT= data set C specified in the OUTPUT statement.

The SIMLIN procedure first prints the structural coefficient matrices read from the EST= data set, as shown in Output 23.1.2 through Output 23.1.4.

Output 23.1.2 SIMLIN Procedure Output – Endogenous Structural Coefficients
 Simulation of Klein's Model I using SIMLIN

The SIMLIN Procedure

Structural Coefficients for Endogenous Variables
Variable c p w i x wsum k y
c 1.0000 -0.1249 . . . -0.7901 . .
i . 0.0131 . 1.0000 . . . .
w . . 1.0000 . -0.4005 . . .
x -1.0000 . . -1.0000 1.0000 . . .
y -1.0000 . . -1.0000 . . . 1.0000
p . 1.0000 1.0000 . -1.0000 . . .
k . . . -1.0000 . . 1.0000 .
wsum . . -1.0000 . . 1.0000 . .

Output 23.1.3 SIMLIN Procedure Output – Lagged Endogenous Structural Coefficients
Structural Coefficients for Lagged Endogenous
Variables
Variable klag plag xlag
c . 0.1631 .
i -0.1948 0.7557 .
w . . 0.1813
x . . .
y . . .
p . . .
k 1.0000 . .
wsum . . .

Output 23.1.4 SIMLIN Procedure Output – Exogenous Structural Coefficients
Structural Coefficients for Exogenous Variables
Variable wp g t yr Intercept
c . . . . 16.4408
i . . . . 28.1778
w . . . 0.1497 1.7972
x . 1.0000 . . 0
y . 1.0000 -1.0000 . 0
p . . -1.0000 . 0
k . . . . 0
wsum 1.0000 . . . 0

The SIMLIN procedure then prints the inverse of the endogenous variables coefficient matrix, as shown in Output 23.1.5.

Output 23.1.5 SIMLIN Procedure Output – Inverse Coefficient Matrix
Inverse Coefficient Matrix for Endogenous Variables
Variable c i w x y p k wsum
c 1.6347 0.6347 1.0957 0.6347 0 0.1959 0 1.2915
p 0.9724 0.9724 -0.3405 0.9724 0 1.1087 0 0.7682
w 0.6496 0.6496 1.4406 0.6496 0 0.0726 0 0.5132
i -0.0127 0.9873 0.004453 -0.0127 0 -0.0145 0 -0.0100
x 1.6219 1.6219 1.1001 1.6219 0 0.1814 0 1.2815
wsum 0.6496 0.6496 1.4406 0.6496 0 0.0726 0 1.5132
k -0.0127 0.9873 0.004453 -0.0127 0 -0.0145 1.0000 -0.0100
y 1.6219 1.6219 1.1001 0.6219 1.0000 0.1814 0 1.2815

The SIMLIN procedure next prints the reduced form coefficient matrices, as shown in Output 23.1.6.

Output 23.1.6 SIMLIN Procedure Output – Reduced Form Coefficients
Reduced Form for Lagged Endogenous Variables
Variable klag plag xlag
c -0.1237 0.7463 0.1986
p -0.1895 0.8935 -0.0617
w -0.1266 0.5969 0.2612
i -0.1924 0.7440 0.000807
x -0.3160 1.4903 0.1994
wsum -0.1266 0.5969 0.2612
k 0.8076 0.7440 0.000807
y -0.3160 1.4903 0.1994

Reduced Form for Exogenous Variables
Variable wp g t yr Intercept
c 1.2915 0.6347 -0.1959 0.1640 46.7273
p 0.7682 0.9724 -1.1087 -0.0510 42.7736
w 0.5132 0.6496 -0.0726 0.2156 31.5721
i -0.0100 -0.0127 0.0145 0.000667 27.6184
x 1.2815 1.6219 -0.1814 0.1647 74.3457
wsum 1.5132 0.6496 -0.0726 0.2156 31.5721
k -0.0100 -0.0127 0.0145 0.000667 27.6184
y 1.2815 1.6219 -1.1814 0.1647 74.3457

The multiplier matrices (requested by the INTERIM=2 and TOTAL options) are printed next, as shown in Output 23.1.7 and Output 23.1.8.

Output 23.1.7 SIMLIN Procedure Output – Interim Multipliers
Interim Multipliers for Interim 1
Variable wp g t yr Intercept
c 0.829130 1.049424 -0.865262 -.0054080 43.27442
p 0.609213 0.771077 -0.982167 -.0558215 28.39545
w 0.794488 1.005578 -0.710961 0.0125018 41.45124
i 0.574572 0.727231 -0.827867 -.0379117 26.57227
x 1.403702 1.776655 -1.693129 -.0433197 69.84670
wsum 0.794488 1.005578 -0.710961 0.0125018 41.45124
k 0.564524 0.714514 -0.813366 -.0372452 54.19068
y 1.403702 1.776655 -1.693129 -.0433197 69.84670

Interim Multipliers for Interim 2
Variable wp g t yr Intercept
c 0.663671 0.840004 -0.968727 -.0456589 28.36428
p 0.350716 0.443899 -0.618929 -.0401446 10.79216
w 0.658769 0.833799 -0.925467 -.0399178 28.33114
i 0.345813 0.437694 -0.575669 -.0344035 10.75901
x 1.009485 1.277698 -1.544396 -.0800624 39.12330
wsum 0.658769 0.833799 -0.925467 -.0399178 28.33114
k 0.910337 1.152208 -1.389035 -.0716486 64.94969
y 1.009485 1.277698 -1.544396 -.0800624 39.12330

Output 23.1.8 SIMLIN Procedure Output – Total Multipliers
Total Multipliers
Variable wp g t yr Intercept
c 1.881667 1.381613 -0.685987 0.1789624 41.3045
p 0.786945 0.996031 -1.286891 -.0748290 15.4770
w 1.094722 1.385582 -0.399095 0.2537914 25.8275
i 0.000000 0.000000 -0.000000 0.0000000 0.0000
x 1.881667 2.381613 -0.685987 0.1789624 41.3045
wsum 2.094722 1.385582 -0.399095 0.2537914 25.8275
k 2.999365 3.796275 -4.904859 -.2852032 203.6035
y 1.881667 2.381613 -1.685987 0.1789624 41.3045

The last part of the SIMLIN procedure output is a table of statistics of fit for the simulation, as shown in Output 23.1.9.

Output 23.1.9 SIMLIN Procedure Output – Simulation Statistics
Fit Statistics
Variable N Mean Error Mean Pct
Error
Mean Abs Error Mean Abs
Pct Error
RMS
Error
RMS Pct
Error
Label
c 21 0.1367 -0.3827 3.5011 6.69769 4.3155 8.1701 consumption
p 21 0.1422 -4.0671 2.9355 19.61400 3.4257 26.0265 profits
w 21 0.1282 -0.8939 3.1247 8.92110 4.0930 11.4709 private wage bill
i 21 0.1337 105.8529 2.4983 127.13736 2.9980 252.3497 investment
x 21 0.2704 -0.9553 5.9622 10.40057 7.1881 12.5653 private production
wsum 21 0.1282 -0.6669 3.1247 7.88988 4.0930 10.1724 total wage bill
k 21 -0.1424 -0.1506 3.8879 1.90614 5.0036 2.4209 capital stock
y 21 0.2704 -1.3476 5.9622 11.74177 7.1881 14.2214 national income

The OUTEST= output data set contains all the observations read from the EST= data set, and in addition contains observations for the reduced form and multiplier matrices. The following statements produce a partial listing of the OUTEST= data set, as shown in Output 23.1.10.

```   proc print data=b;
where _type_ = 'REDUCED' | _type_ = 'IMULT1';
run;
```

Output 23.1.10 Partial Listing of OUTEST= Data Set
 Simulation of Klein's Model I using SIMLIN

Obs _TYPE_ _DEPVAR_ _MODEL_ _SIGMA_ c p w i x wsum k y klag plag xlag wp g t yr Intercept
9 REDUCED c   . 1.63465 0.63465 1.09566 0.63465 0 0.19585 0 1.29151 -0.12366 0.74631 0.19863 1.29151 0.63465 -0.19585 0.16399 46.7273
10 REDUCED p   . 0.97236 0.97236 -0.34048 0.97236 0 1.10872 0 0.76825 -0.18946 0.89347 -0.06173 0.76825 0.97236 -1.10872 -0.05096 42.7736
11 REDUCED w   . 0.64957 0.64957 1.44059 0.64957 0 0.07263 0 0.51321 -0.12657 0.59687 0.26117 0.51321 0.64957 -0.07263 0.21562 31.5721
12 REDUCED i   . -0.01272 0.98728 0.00445 -0.01272 0 -0.01450 0 -0.01005 -0.19237 0.74404 0.00081 -0.01005 -0.01272 0.01450 0.00067 27.6184
13 REDUCED x   . 1.62194 1.62194 1.10011 1.62194 0 0.18135 0 1.28146 -0.31603 1.49034 0.19944 1.28146 1.62194 -0.18135 0.16466 74.3457
14 REDUCED wsum   . 0.64957 0.64957 1.44059 0.64957 0 0.07263 0 1.51321 -0.12657 0.59687 0.26117 1.51321 0.64957 -0.07263 0.21562 31.5721
15 REDUCED k   . -0.01272 0.98728 0.00445 -0.01272 0 -0.01450 1 -0.01005 0.80763 0.74404 0.00081 -0.01005 -0.01272 0.01450 0.00067 27.6184
16 REDUCED y   . 1.62194 1.62194 1.10011 0.62194 1 0.18135 0 1.28146 -0.31603 1.49034 0.19944 1.28146 1.62194 -1.18135 0.16466 74.3457
17 IMULT1 c   . . . . . . . . . . . . 0.82913 1.04942 -0.86526 -0.00541 43.2744
18 IMULT1 p   . . . . . . . . . . . . 0.60921 0.77108 -0.98217 -0.05582 28.3955
19 IMULT1 w   . . . . . . . . . . . . 0.79449 1.00558 -0.71096 0.01250 41.4512
20 IMULT1 i   . . . . . . . . . . . . 0.57457 0.72723 -0.82787 -0.03791 26.5723
21 IMULT1 x   . . . . . . . . . . . . 1.40370 1.77666 -1.69313 -0.04332 69.8467
22 IMULT1 wsum   . . . . . . . . . . . . 0.79449 1.00558 -0.71096 0.01250 41.4512
23 IMULT1 k   . . . . . . . . . . . . 0.56452 0.71451 -0.81337 -0.03725 54.1907
24 IMULT1 y   . . . . . . . . . . . . 1.40370 1.77666 -1.69313 -0.04332 69.8467

The actual and predicted values for the variable C are plotted in Output 23.1.11.

```   title2 'Plots of Simulation Results';
proc sgplot data=c;
scatter x=year y=c;
series x=year y=chat / markers markerattrs=(symbol=plus);
refline 1941.5 / axis=x;
run;
```

Output 23.1.11 Plot of Actual and Predicted Consumption

 Previous Page | Next Page | Top of Page