PROC SIMLIN: Multipliers for a Third-Order System

The SIMLIN Procedure

Example 23.2 Multipliers for a Third-Order System

This example shows how to fit and simulate a single equation dynamic model with third-order lags. It then shows how to convert the third-order equation into a three equation system with only first-order lags, so that the SIMLIN procedure can compute multipliers. (See the section "Multipliers for Higher Order Lags" earlier in this chapter for more information.)

The input data set TEST is created from simulated data. A partial listing of the data set TEST produced by PROC PRINT is shown in Output 23.2.1.

Output 23.2.1 Partial Listing of Input Data Set

Simulate Equation with Third-Order Lags

Listing of Simulated Input Data

Obs	y	ylag1	ylag2	ylag3	x	n
1	8.2369	8.5191	6.9491	7.8800	-1.2593	1
2	8.6285	8.2369	8.5191	6.9491	-1.6805	2
3	10.2223	8.6285	8.2369	8.5191	-1.9844	3
4	10.1372	10.2223	8.6285	8.2369	-1.7855	4
5	10.0360	10.1372	10.2223	8.6285	-1.8092	5
6	10.3560	10.0360	10.1372	10.2223	-1.3921	6
7	11.4835	10.3560	10.0360	10.1372	-2.0987	7
8	10.8508	11.4835	10.3560	10.0360	-1.8788	8
9	11.2684	10.8508	11.4835	10.3560	-1.7154	9
10	12.6310	11.2684	10.8508	11.4835	-1.8418	10

The REG procedure processes the input data and writes the parameter estimates to the OUTEST= data set A.

   title2 'Estimated Parameters';
   proc reg data=test outest=a;
      model y=ylag3 x;
   run;
   
   title2 'Listing of OUTEST= Data Set';
   proc print data=a;
   run;

Output 23.2.2 shows the printed output produced by the REG procedure, and Output 23.2.3 displays the OUTEST= data set A produced.

Output 23.2.2 Estimates and Fit Information from PROC REG

Simulate Equation with Third-Order Lags

Estimated Parameters

The REG Procedure

Model: MODEL1

Dependent Variable: y

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	2	173.98377	86.99189	1691.98	<.0001
Error	27	1.38818	0.05141
Corrected Total	29	175.37196

Root MSE	0.22675	R-Square	0.9921
Dependent Mean	13.05234	Adj R-Sq	0.9915
Coeff Var	1.73721

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	1	0.14239	0.23657	0.60	0.5523
ylag3	1	0.77121	0.01723	44.77	<.0001
x	1	-1.77668	0.10843	-16.39	<.0001

Output 23.2.3 The OUTEST= Data Set Created by PROC REG

Simulate Equation with Third-Order Lags

Listing of OUTEST= Data Set

Obs	_MODEL_	_TYPE_	_DEPVAR_	_RMSE_	Intercept	ylag3	x	y
1	MODEL1	PARMS	y	0.22675	0.14239	0.77121	-1.77668	-1

The SIMLIN procedure processes the TEST data set using the estimates from PROC REG. The following statements perform the simulation and write the results to the OUT= data set OUT2.

   title2 'Simulation of Equation';
   proc simlin est=a data=test nored;
      endogenous y;
      exogenous  x;
      lagged ylag3 y 3;
      id n;
      output out=out1 predicted=yhat residual=yresid;
   run;

The printed output from the SIMLIN procedure is shown in Output 23.2.4.

Output 23.2.4 Output Produced by PROC SIMLIN

Simulate Equation with Third-Order Lags

Simulation of Equation

The SIMLIN Procedure

Fit Statistics
Variable	N	Mean Error	Mean Pct Error	Mean Abs Error	Mean Abs Pct Error	RMS Error	RMS Pct Error
y	30	-0.0233	-0.2268	0.2662	2.05684	0.3408	2.6159

The following statements plot the actual and predicted values, as shown in Output 23.2.5.

   title2 'Plots of Simulation Results';
   proc sgplot data=out1;
      scatter x=n y=y;
      series x=n y=yhat / markers markerattrs=(symbol=plus);
   run;

Output 23.2.5 Plot of Predicted and Actual Values

Next, the input data set TEST is modified by creating two new variables, YLAG1X and YLAG2X, that are equal to YLAG1 and YLAG2. These variables are used in the SYSLIN procedure. (The estimates produced by PROC SYSLIN are the same as before and are not shown.) A listing of the OUTEST= data set B created by PROC SYSLIN is shown in Output 23.2.6.

   data test2;
      set test;
      ylag1x=ylag1;
      ylag2x=ylag2;
   run;

   title2 'Estimation of parameters and definition of identities';
   proc syslin data=test2 outest=b;
      endogenous y ylag1x ylag2x;
      model y=ylag3 x;
      identity ylag1x=ylag1;
      identity ylag2x=ylag2;
   run;
   
   title2 'Listing of OUTEST= data set from PROC SYSLIN';
   proc print data=b;
   run;

Output 23.2.6 Listing of OUTEST= Data Set Created from PROC SYSLIN

Simulate Equation with Third-Order Lags

Listing of OUTEST= data set from PROC SYSLIN

Obs	_TYPE_	_STATUS_	_MODEL_	_DEPVAR_	_SIGMA_	Intercept	ylag3	x	ylag1	ylag2	y	ylag1x	ylag2x
1	OLS	0 Converged	y	y	0.22675	0.14239	0.77121	-1.77668	.	.	-1	.	.
2	IDENTITY	0 Converged		ylag1x	.	0.00000	.	.	1	.	.	-1	.
3	IDENTITY	0 Converged		ylag2x	.	0.00000	.	.	.	1	.	.	-1

The SIMLIN procedure is used to compute the reduced form and multipliers. The OUTEST= data set B from PROC SYSLIN is used as the EST= data set for the SIMLIN procedure. The following statements perform the multiplier analysis.

   title2 'Simulation of transformed first-order equation system';
   
   proc simlin est=b data=test2 total interim=2;
      endogenous y ylag1x ylag2x;
      exogenous  x;
      lagged  ylag1 y 1  ylag2 ylag1x 1  ylag3 ylag2x 1;
      id n;
      output out=out2 predicted=yhat residual=yresid;
   run;

Output 23.2.7 shows the interim 2 and total multipliers printed by the SIMLIN procedure.

Output 23.2.7 Interim 2 and Total Multipliers

Simulate Equation with Third-Order Lags

Simulation of transformed first-order equation system

The SIMLIN Procedure

Interim Multipliers for Interim 2
Variable	x	Intercept
y	0.000000	0.0000000
ylag1x	0.000000	0.0000000
ylag2x	-1.776682	0.1423865

Total Multipliers
Variable	x	Intercept
y	-7.765556	0.6223455
ylag1x	-7.765556	0.6223455
ylag2x	-7.765556	0.6223455

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