Example 14.6: General-Form Equations

The MODEL Procedure

Example 14.6: General-Form Equations

Data for this example are generated. General-form equations are estimated and forecast using PROC MODEL. The system is a basic supply-demand model. Portions of the output from the following code is shown in Output 14.6.1 through Output 14.6.4.

   title1 "General Form Equations for Supply-Demand Model";
   
   proc model;
      var price quantity income unitcost;
      parms d0-d2 s0-s2;
      eq.demand=d0+d1*price+d2*income-quantity;
      eq.supply=s0+s1*price+s2*unitcost-quantity;
   
   /* estimate the model parameters */
      fit supply demand / data=history outest=est n2sls;
      instruments income unitcost year;
   run;
   
   /* produce forecasts for income and unitcost assumptions */
      solve price quantity / data=assume out=pq;
   run;
   
   /* produce goal-seeking solutions for income and quantity assumptions*/
      solve price unitcost / data=goal out=pc;
   run;
   
   title2 "Parameter Estimates for the System";
   proc print data=est;
   run;
   
   title2 "Price Quantity Solution";
   proc print data=pq;
   run;
   
   title2 "Price Unitcost Solution";
   proc print data=pc;
   run;

Three data sets were used in this example. The first data set, HISTORY, was used to estimate the parameters of the model. The ASSUME data set was used to produce a forecast of PRICE and QUANTITY. Notice that the ASSUME data set does not have to contain the variables PRICE and QUANTITY.

   data history;
      input year income unitcost price quantity;
      datalines;
   1976    2221.87    3.31220     0.17903    266.714
   1977    2254.77    3.61647     0.06757    276.049
   1978    2285.16    2.21601     0.82916    285.858
   1979    2319.37    3.28257     0.33202    295.034
   1980    2369.38    2.84494     0.63564    310.773
   1981    2395.26    2.94154     0.62011    319.185
   1982    2419.52    2.65301     0.80753    325.970
   1983    2475.09    2.41686     1.01017    342.470
   1984    2495.09    3.44096     0.52025    348.321
   1985    2536.72    2.30601     1.15053    360.750
   ;
   
   data assume;
      input year income unitcost;
      datalines;
   1986    2571.87    2.31220
   1987    2609.12    2.45633
   1988    2639.77    2.51647
   1989    2667.77    1.65617
   1990    2705.16    1.01601
   ;

The output produced by the first SOLVE statement is shown in Output 14.6.3.

The third data set, GOAL, is used in a forecast of PRICE and UNITCOST as a function of INCOME and QUANTITY.

   data goal;
      input year income quantity;
      datalines;
   1986    2571.87     371.4
   1987    2721.08     416.5
   1988    3327.05     597.3
   1989    3885.85     764.1
   1990    3650.98     694.3
   ;

The output from the final SOLVE statement is shown in Output 14.6.4.

Output 14.6.1: Printed Output from the FIT Statement

General Form Equations for Supply-Demand Model

The MODEL Procedure

The 2 Equations to Estimate
supply =	F(s0(1), s1(price), s2(unitcost))
demand =	F(d0(1), d1(price), d2(income))
Instruments	1 income unitcost year

General Form Equations for Supply-Demand Model

The MODEL Procedure

Nonlinear 2SLS Summary of Residual Errors
Equation	DF Model	DF Error	SSE	MSE	Root MSE	R-Square	Adj R-Sq
supply	3	7	3.3240	0.4749	0.6891
demand	3	7	1.0829	0.1547	0.3933

Nonlinear 2SLS Parameter Estimates
Parameter	Estimate	Approx Std Err	t Value	Approx Pr > \|t\|
d0	-395.887	4.1841	-94.62	<.0001
d1	0.717328	0.5673	1.26	0.2466
d2	0.298061	0.00187	159.65	<.0001
s0	-107.62	4.1780	-25.76	<.0001
s1	201.5711	1.5977	126.16	<.0001
s2	102.2116	1.1217	91.12	<.0001

Output 14.6.2: Listing of OUTEST= Data Set Created in the FIT Statement

General Form Equations for Supply-Demand Model

Parameter Estimates for the System

Obs	_NAME_	_TYPE_	_STATUS_	_NUSED_	d0	d1	d2	s0	s1	s2
1		2SLS	0 Converged	10	-395.887	0.71733	0.29806	-107.620	201.571	102.212

Output 14.6.3: Listing of OUT= Data Set Created in the First SOLVE Statement

General Form Equations for Supply-Demand Model

Price Quantity Solution

Obs	_TYPE_	_MODE_	_ERRORS_	price	quantity	income	unitcost	year
1	PREDICT	SIMULATE	0	1.20473	371.552	2571.87	2.31220	1986
2	PREDICT	SIMULATE	0	1.18666	382.642	2609.12	2.45633	1987
3	PREDICT	SIMULATE	0	1.20154	391.788	2639.77	2.51647	1988
4	PREDICT	SIMULATE	0	1.68089	400.478	2667.77	1.65617	1989
5	PREDICT	SIMULATE	0	2.06214	411.896	2705.16	1.01601	1990

Output 14.6.4: Listing of OUT= Data Set Created in the Second SOLVE Statement

General Form Equations for Supply-Demand Model

Price Unitcost Solution

Obs	_TYPE_	_MODE_	_ERRORS_	price	quantity	income	unitcost	year
1	PREDICT	SIMULATE	0	0.99284	371.4	2571.87	2.72857	1986
2	PREDICT	SIMULATE	0	1.86594	416.5	2721.08	1.44798	1987
3	PREDICT	SIMULATE	0	2.12230	597.3	3327.05	2.71130	1988
4	PREDICT	SIMULATE	0	2.46166	764.1	3885.85	3.67395	1989
5	PREDICT	SIMULATE	0	2.74831	694.3	3650.98	2.42576	1990

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