SAS/ETS 9.22 User''''s Guide 127 ppt

title2 "Price Unitcost Solution"; /* produce goal-seeking solutions for income and quantity assumptions*/ proc model model=model; solve price unitcost / data=goal out=pc; run; proc print

Trang 1

Output 18.6.3 Listing of OUTEST= Data Set Created in the FIT Statement

General Form Equations for Supply-Demand Model

Price Quantity Solution _

1 2SLS 0 Converged 10 -395.887 0.71733 0.29806 -107.620 201.571 102.212

Output 18.6.4 Listing of OUT= Data Set Created in the First SOLVE Statement

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

The following statements produce the goal-seeking solutions for PRICE and UNITCOST by using the GOAL dataset

title2 "Price Unitcost Solution";

/* produce goal-seeking solutions for

income and quantity assumptions*/

proc model model=model;

solve price unitcost / data=goal out=pc;

run;

proc print data=pc;

run;

The output data set produced by the final SOLVE statement is shown inOutput 18.6.5

Trang 2

Output 18.6.5 Listing of OUT= Data Set Created in the Second SOLVE Statement

Price Unitcost Solution Obs _TYPE_ _MODE_ _ERRORS_ price quantity income unitcost year

Example 18.7: Spring and Damper Continuous System

This model simulates the mechanical behavior of a spring and damper system shown inFigure 18.92

Figure 18.92 Spring and Damper System Model

A mass is hung from a spring with spring constant K The motion is slowed by a damper with damper constant C The damping force is proportional to the velocity, while the spring force is proportional

to the displacement

This is actually a continuous system; however, the behavior can be approximated by a discrete time model We approximate the differential equation

@ d i sp

@ t i me D velocity

Trang 3

with the difference equation

d i sp

t i me D velocity

This is rewritten as

d i sp LAG.d i sp/

where dt is the time step used In PROC MODEL, this is expressed with the program statement

disp = lag(disp) + vel * dt;

or

dert.disp = vel;

The first statement is simply a computing formula for Euler’s approximation for the integral

d i spD

Z veloci ty dt

If the time step is small enough with respect to the changes in the system, the approximation is good Although PROC MODEL does not have the variable step-size and error-monitoring features of simulators designed for continuous systems, the procedure is a good tool to use for less challenging continuous models

The second form instructs the MODEL procedure to do the integration for you

This model is unusual because there are no exogenous variables, and endogenous data are not needed Although you still need a SAS data set to count the simulation periods, no actual data are brought in Since the variables DISP and VEL are lagged, initial values specified in the VAR statement determine the starting state of the system The mass, time step, spring constant, and damper constant are declared and initialized by a CONTROL statement as shown in the following statements:

title1 'Simulation of Spring-Mass-Damper System';

/*- Data to drive the simulation time periods -*/

data one;

do n=1 to 100;

output;

end;

run;

proc model data=one outmodel=spring;

var force -200 disp 10 vel 0 accel -20 time 0;

force = -k * disp -c * vel;

disp = lag(disp) + vel * dt;

Trang 4

vel = lag(vel) + accel * dt;

accel = force / mass;

time = lag(time) + dt;

run;

The displacement scale is zeroed at the point where the force of gravity is offset, so the acceleration

of the gravity constant is omitted from the force equation The control variable C and K represent the damper and the spring constants respectively

The model is simulated three times, and the simulation results are written to output data sets The first run uses the original initial conditions specified in the VAR statement In the second run, the initial displacement is doubled; the results show that the period of the motion is unaffected by the amplitude In the third run, the DERT syntax is used to do the integration Notice that the path of the displacement is close to the old path, indicating that the original time step is short enough to yield an accurate solution These simulations are performed by the following statements:

proc model data=one model=spring;

title2 "Simulation of the model for the base case";

control run '1';

solve / out=a;

run;

title2 "Simulation of the model with twice the initial displacement"; control run '2';

var disp 20;

solve / out=b;

run;

data two;

do time = 0 to 10 by 2; output;end;

run;

title2 "Simulation of the model using the dert syntax";

proc model data=two;

var force -200 disp 10 vel 0 accel -20 time 0;

control run '3' ;

force = -k * disp -c * vel;

dert.disp = vel ;

dert.vel = accel;

accel = force / mass;

solve / out=c;

id time ;

run;

The output SAS data sets that contain the solution results are merged and the displacement time paths for the three simulations are plotted The three runs are identified on the plot as 1, 2, and 3 The following statements produceOutput 18.7.1throughOutput 18.7.5

data p;

set a b c;

run;

Trang 5

title2 'Overlay Plot of All Three Simulations';

proc sgplot data=p;

series x=time y=disp / group=run lineattrs=(pattern=1);

xaxis values=(0 to 10 by 1);

yaxis values=(-20 to 20 by 10);

run;

Output 18.7.1 Model Summary

Simulation of Spring-Mass-Damper System Simulation of the model for the base case

The MODEL Procedure

Model Summary

Model Variables 5 Control Variables 5

Number of Statements 6 Program Lag Length 1

Model Variables force(-200) disp(10) vel(0) accel(-20) time(0) Control Variables mass(9.2) c(1.5) dt(0.1) k(20) run(1)

Equations force disp vel accel time

Output 18.7.2 Printed Output Produced by PROC MODEL SOLVE Statements

Simulation of Spring-Mass-Damper System Simulation of the model for the base case

The MODEL Procedure Dynamic Simultaneous Simulation

Data Set Options

DATA= ONE

Solution Summary

Simulation Lag Length 1 Solution Method NEWTON

Trang 6

Output 18.7.2 continued

Observations Processed

Solved 99

Variables Solved For force disp vel accel time

Simulation of Spring-Mass-Damper System Simulation of the model with twice the initial displacement

The MODEL Procedure Dynamic Simultaneous Simulation

DATA= ONE

Simulation Lag Length 1 Solution Method NEWTON

Solved 99

Variables Solved For force disp vel accel time

Trang 7

Simulation of Spring-Mass-Damper System Simulation of the model using the dert syntax

The MODEL Procedure Simultaneous Simulation

DATA= TWO

Solution Method NEWTON Maximum Iterations 0

Solved 51

Variables Solved For force disp vel accel

Auxiliary Equations force accel

Trang 8

Output 18.7.5 Overlay Plot of Three Simulations

Example 18.8: Nonlinear FIML Estimation

The data and model for this example were obtained from Bard (1974, p.133–138) The example is a two-equation econometric model used by Bodkin and Klein to fit U.S production data for the years 1909–1949 The model is the following:

g1D c110c2 z 4.c5z c4

1 C 1 c5/z c4

2 / c3 =c 4 z3D 0

g2D Œc5=.1 c5/.z1=z2/ 1 c4 / z5D 0 where z1is capital input, z2is labor input, z3is real output, z4is time in years with 1929 as year zero, and z5is the ratio of price of capital services to wage scale The ci’s are the unknown parameters z1 and z2are considered endogenous variables A FIML estimation is performed by using the following statements:

Trang 9

data bodkin;

input z1 z2 z3 z4 z5;

datalines;

1.33135 0.64629 0.4026 -20 0.24447

1.39235 0.66302 0.4084 -19 0.23454

more lines

title1 "Nonlinear FIML Estimation";

proc model data=bodkin;

parms c1-c5;

endogenous z1 z2;

exogenous z3 z4 z5;

eq.g1 = c1 * 10 **(c2 * z4) * (c5*z1**(-c4)+

(1-c5)*z2**(-c4))**(-c3/c4) - z3;

eq.g2 = (c5/(1-c5))*(z1/z2)**(-1-c4) -z5;

fit g1 g2 / fiml ;

run;

When FIML estimation is selected, the log likelihood of the system is output as the objective value The results of the estimation are shown inOutput 18.8.1

Output 18.8.1 FIML Estimation Results for U.S Production Data

Nonlinear FIML Estimation

The MODEL Procedure

Nonlinear FIML Summary of Residual Errors

Equation Model Error SSE MSE Root MSE R-Square R-Sq

Nonlinear FIML Parameter Estimates

Parameter Estimate Std Err t Value Pr > |t|

Trang 10

Output 18.8.1 continued

Number of Observations Statistics for System

Example 18.9: Circuit Estimation

Consider the nonlinear circuit shown inFigure 18.93

Figure 18.93 Nonlinear Resistor Capacitor Circuit

The theory of electric circuits is governed by Kirchhoff’s laws: the sum of the currents flowing to a node is zero, and the net voltage drop around a closed loop is zero In addition to Kirchhoff’s laws, there are relationships between the current I through each element and the voltage drop V across the elements For the circuit inFigure 18.93, the relationships are

Cd V

dt D I

for the capacitor and

V D R1C R2.1 exp V///I

for the nonlinear resistor The following differential equation describes the current at node 2 as a function of time and voltage for this circuit:

Cd V2

dt

V1 V2

R1C R2.1 exp V// D 0 This equation can be written in the form

d V2

.R1C R2.1 exp V///C

Định dạng
Số trang	10
Dung lượng	289,21 KB