**Tilburg University**

**MØDSCØ, a computer programm for the revised method of scoring**

### Vandaele, W.H.

*Publication date:*

### 1970

*Document Version*

### Publisher's PDF, also known as Version of record

### Link to publication in Tilburg University Research Portal

### Citation for published version (APA):

### Vandaele, W. H. (1970). MØDSCØ, a computer programm for the revised method of scoring. (EIT Research

### Memorandum). Stichting Economisch Instituut Tilburg.

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**CBM**

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### 1970

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### EIT

### 13

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### --

### - -.

### Walter H. Vandaele

### M~DSC~, a computer programm

### for the revised method of scoring

**i I!~NIINNII~IUIII~NII~!pI~IR~I~IÍp**

**Research Memorandum**

### ~ ~,~

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### TILBURG INSTITUTE OF ECONOMICS

**K.U.B.**

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M~DSC~, A computer program for the Revised Method of Scorinq

1. Purpose

**To obtain maximum likelihood estimates of parameters**
in a non-linear model by maximizing the logarithm of the
**li-kelihood function of the parametervector e-(91,** **e2,..., en).**

**The method of the iterative technigue used is described in**
VANDAELE, Y;alter H, and S. R.CHOP7DHURY, "A Revised Method cf
Scorinc?" I 3 ] .

In this research memorandum a comparison is made with the normal Fisher Method of Scoring t. So this computer program will also be discussed.

The programs are written in F~6RTRAN II Language for anIBM 1620-II computer with F and K standard 8 and 9, and with a 1311 2 disk unit.

2. In order to allow the greatest possible flexibility in handling the input of the data and in specifying the

ma-The author likes to thank the University of Wisconsin Computing Center for providing him a non-linear least

. sguares subroutine (GAUSHAUS). The GAUSHAUS program is based on the iterative technique of MARQUARDT, D.L. [ 1 J. There can be drawn an analogue between the structure of

these computer programs.

-4-thematical model to be fitted on the data, M~DSC~ is writ-ten in the form of a subroutine. Normally the User must pro-vide subroutines to generate the inverse of the Information matrix and to calculate value of the log likelihood function. Because in our research memorandum we have applied the Revi-sed Method of Scoring on an autocorrelated model, with the usual assumption of a first order autoregressive scheme, the mentioned subroutines are made for that particular si-tuation.

The following chart summarizes the function of the mair. program, the subroutine D4~DSC~ and the User's

-5-MAIN - programs
(See Section 3)
CALL
**REGRES** MAIN1
~
MAIN2
CALL
MfbDSC~
(SCOR)

### ~

Read observed function values Y, ir.dependent variables X, and starting parameter values Th

(if to calculate: call REGRES). Certain constants NP, NT, MIT, EPS1, EPS2 ... are initialized. All these values are put in CSdM-M~DDI, and MQJDSC~D (or SCSdR) is called. When control is returned

to MAIP~2, _{another problem may be}
begun or the run terminated.
~~

**-SUBROUTINE M(11DSC~ó** **( SCSdR)**

**(See Section** _{4.5)}

Performs maximum-likelihood estimates and provides printed
output. _{At various times during the calculation, it calls}
the M~DDT, _{MRES or INF~ subroutine. At the completion of the}
estimation procedure, control is returned to MAIN2.

SUBROUTINES

M DT subroutine determines the calculated residuals (with
the help of MRES) and _{the value of the log likelihood }
func-tion for _{a choice of the parameters transmitted to it by}
M~JDSC~1 (SC~6R). Subroutine _{INFqJ calculates the inverse of the}

Information Matrix. Control is _{then returned to M~DSC~D (SC~R).}
The independent variable values, together with the needed

information are obtained from C~MM~N.

----~--

----C~M.~I,IdN

-~-3 MAIN - programs

3.1 MAIN1-program

The functions of "MAIN1" program are

1. to supply the data and store them in C(óMM~N; 2. to supply or calculate(via subroutine REGRES)

the initial parameter value for the iterati-ve~procedure;

3. to inïtialize certain constants;

4. to call "MÁIN2" where certain other constants will be initialized.

Read statements

a) READ 200, NP, IY, IRED, MIT, ISC 200 F~RMAT (10 I5)

NP is an integer constant which is the number of parameters of the likelihood function to estimate without the o(the standard-deviation of the residual term) and p(the autocorrelation parameter). Further in the program (see calling sequence of M~óDSC(d), NK - NP t 2: the number of all the

parame-ters, a and p included.

(See 4.1. The M~6DSC~6 calling sequence). NP must satisfy 0 ~ NP ~ S, NP ~ NT

-~-IY

IRED

MIT

is an integer constant which is the number of the sector on the disk where the Y-variable data can be found. t If the data are supplied by cards, any number can be given to IY.

is an integer constant:

- 0 . the Y and X variables must be read from disk

- 1 . the Y and X variables are pun-ched on cards.

**If IRED - 0** **or** **1,** **then** **in the MAIN1**
**program,** **with the help of a** **subroutine**
**REGRES,** **the initial value of the **

**para-meters are calculated**

- 2 . the Y and X variables are on
disk, **and the initial **
parame-ter values are put on cards
- 3 . all information is put on cards:

Y, X-values and initial value of the parameters

is an integer constant (0 c MIT ~ 1000) which is the maximum number of itera-tions to be performed. If the calcula-tions have not been terminated for some other reason, they will be stopped when the number of iterations equals MIT.

t _{It is only necessary to indicate the sectornumber where}
the first 10 observations are on disk. In the program
automatically the subroutir.e F2TCH is called 3 times,

-8-ISC - 0 indicates `hat the Revised Method of Scoring will be used

- 1 the Fisher Method of Scoring will be applied.

b) READ **200, M, N**

200 F~6RMAT (1 0 I5 )

M, N integer constants which indicate res-pectively the first and the last ob-servation of the timeseries to be used.

c) READ 200, (IX(I), I- 1, NP) 20 0 FC~2MAT (1 0 I 5)

IX is an integer one-dimensional array

containing the sector numbers on the disk of the explanatory variables. The same remark as that mentioned

with IY, when IRED - 1 or 3, is Valued. d) READ 204, SH2, RQ)

READ 204,(BETA (I), I- 1, NP) 204 FS6RMAT (6 F 13.8)

BETA _{is a real one-dimensional array }
con-taining the initial parameter values,
except of a2 (SH2) and p(R~6). The a,
p and BETA's are then stored in that
order in the real one-dimensional
ar--ray TH before M~DSC~d (SC~áR) is called.
During the execution of M~óDSC~d (SC~dR) ,
the most current parameter estimates
are kept in TB, the previous values

-9-e) If IRED - 1 or 3 READ DO 9 9 READ 201 , (Y (J) , J - M, N) I - 1, NP 201 , (X (J, I) , J- M, N) 201 F~DRMAT (6 F 13.5) Y 3.2 Subroutine REGRES

is a real one-dimensional array con-taining the vector of observed func-tion values.

is a real matrix (NT x NP) containing the values of the explanatory varia-bles.

REGRES is _{called from the main program when IRED is}
either 0 or 1. _{The F~RTRAN statement used is:}

**CALÍ, REGRES** **( NP,** **NT,** **Y,** **X,** **SH2,** **RfD,** **BETA,** **A,** **E,** **V,** **TE)**

NP, NT,Y,X, SH2, R~, BETA _{have the same meaning as that}
explained in MAIi11 .

A

V

TE

is a real one-dimensional array containing the vector of calculated Y-values.

is a real _{one-dimensional array of calculated}
disturbances: _{E - Y - A.}

is a real one-dimensional array of normalized calculated disturbances.

-10-3.3 MAIN2-program

The MAIN1 program is connected with MAIN2 by a CALL LINK-statement. This program will initialize still some additional constants:

EPS1 is a real constant which is the value of the log
likelihood function (without constant part t) L(6)_{T}
convergence criterion and is used to terminate the
calculations based on the relative change of that
value from one iteration to the next. More precisely,
if at the completion of the i-th iteration, it is
true that

L_{T}(A(i)) - L_{T}(6(i-1))
L (8(i-1))

T

~ EPS1 ,

then the calculations are terminated. In the pro-gram EPS1 - 10.0E - 05. If EPS1 is set equal to zero, this feature is disabled.

EPS2 is a real constant which is the parameter conver-gence criterion and is used to terminate the cal-culations based on the relative change in the pa-rameter values from one iteration to the next one. If, at the completion of the i-th iteration, the following holds: e (i) - e (i-1 ) 7 J e(i-1) 7 ~ EPS 2

for all j- 1(1) NT, then the calculations are terminated. EPS2 is set equal to EPS1. This feature is disabled if EPS2 is set equal to zero.

-11-4. Subroutine M~DSC~

4.1 M~DSC~ calling sequence

M~DSC~ is called from the MAIN2 program with a F~RTRAN statement of the form

CALL **M~DSC~** **(NK,** **EPS1,** **EPS2)**

NK is an integer constant which is the total number of parameters of the likelihood function to es-timate: a(standarddevíation of the disturbance terms) and p(autocorrelation parameter) included

EPS1, EPS2 see 3.3 MAIN2-program.

In addition to the information transmitted through these arguments, the values of X, Y, TH, NP, NT, MIT, TE are assumed to be available from C~MM~N where they were put by the MAIN-programs.

4.2 Since the value of the likelihood function is needed at various times during the calculations, we provi-de subroutines to compute ít. Whenever that value is neeprovi-ded for a particular value of the parameters TH, M~DSC~ simply transmits TH to that subroutine called M~DT.

5UBR~UTINE M~DT (NK, PAR, VAL)

NK as above

PAR is a one-dimensional array containing the para-meter values for which the log likelihood is to be evaluated

12

-A subroutine used in the M~DT subroutine is

SUBR~UTINE MRES (NK, PAR, A)

This program calculates the estimated y-values (A vector, a real one-dimensional array), as well as the estimated disturbances (E vector, a real one-dimensional array) for the values of PAR.

This subroutine is also separately used at the end of the M~DSC~ program, to calculate the final additio-nal information (see 4.4).

4.3 Information matrix

During each iteration, the inverse of the Infor-mation matrix has to be evaluated. For that purpose a

sepa-rate subroutine is used which is called in the MSdDSC~6 pro-gram with the calling statement.

CALL INFQ! **( NK,** **AA,** **B)**

**NK** **as above** **.**

AA **is the 2 x 2 matrix of a and p(See [** **3, p. 11** **])**

**B** **is a NP x NP covariance matrix of regression **
**para-meters** **( BETA-parameters).**

13

-4.4 _{Final additional information}

Before control _{is returned to MAIN2,} _{the M~DSC~7}
subroutine supplies still some _{interestinq additional }
in-formation.

- _{Y-values: observed, and calculated ones, together with}
the residuals. The values are calculated in the MRES
subroutine (See above);

~ Flow ~nart of h1~DSCQ subroutine

14

19

-6. SC~6R prograr.i

This subroutine calculates the parameter value with the Fisher's Method of Scoring.

**The calling sequence is the following:**
CALL SC~R (NK, EPS2).

23

-8. _{Other subroutines used in the program}

These programs are all available at the Computer

**Center of the Tilburg Scilool of Economics, Social Sciences**
**and Law:**

F2TCH _{A subroutine which brings 10 floating-point numbers}
from a certain sector of the disk to the central
memory.

MAPRI A _{vector and màtrix printing subroutine.}

INVE05 An _{inverse subroutine for a 5 x 5 matrix.}

F'LEM _{A subroutine used for multiplícation of }
floating-point variables in double r~recision.

9. REFERENCES

**I** **1** **1** **MARQUARDT,** **Donald W.** _{"An algorithm for least-squares}**estimation of Nonlinear parameters", Journal**
**Soc.** **Indust.** **Appl.** **M~~th** , Vol. **11,** **June,** **1963,**
**nr.** **2,** **pp.** _{431-441.}

### I 2 1

VANDAELE, Walter H._{Heteruscedastic Errors in }Re-gression Analysis, 'I'cr discussie nr. 6902, Katholieke Hogeschool, Economische Faculteit, May, 1969, 38 pp.

l 3] VAt7DAELE, [Jalter }:. and :. f.('HOWDHURY. **F~** **I2evised**

Method of Scor~ nq, ** _{Rc s. arch Memorandu7i EIT 1 1,}**
Tilhurq

_{Instit.utr}

_{c~f f,-,~., c.ur,ics,}

### APPENDIX:

### Listing of the programs

### C

### MAIN1 PROGRAM - MODSCO-AUTOCORRELATED MODEL

### C

### FIRST PART

### C

### C

### IREO - 0

### X,Y FROM DISK

### C

### IRED - 1

### X,Y FROM CARDS

### C

### IRED - 2

### X,Y FROM DI SK AND PARAMETERS FROM CARD

### C

### IRED - 3

### X,Y

### ANO PARAMETERS FROM CARDS

### C

### C

### I SC - 1 GO TO SCOR 1

### C

### ISC - 0 GO TO MAIN2

### C

**DIMENSION X(70,5),Y(70),Et70),TH(7),TE(5)**

**-DIMENSION IXt10),JXt10)**

**DIMENSION Z(70),A(70),BETA(5),V(70)**

### C

### COMMON X,Y,E,G,H,FGH,TH,NP,NT,MIT,TE,ISC

### 100 FORMAT (1N1,30H

### ORIGINAL Y- AND X-VALUEStMa,I2,5H

### ,N-,I2,1H))

### 101 FORMAT (I)

### 102 FORMAT t1N1)

**K-0**

**1 READ**

**200,NP,IY,IRED,MIT,ISC**

**IF**

**(NP)**

**2,3,2**

**2 READ**

**200,M,N**

**4 READ**

**200,(IXfI)~I-1,NP)**

**IF**

**(IX(1))**

**5,1,5**

**5 IF (IRED-1) 16,16,17**

### 17 K-1

### READ 204,SH2,R0

**READ 204~(BETA(I),I-1~NP)**

**IRED-IRED-2**

**16 IF ( IRED)**

**6,7~6**

### 7 CALL F2TCH(IY,YfI))

### JY-IYf1

### CALL F2TCH(JY,Yfll))

**DO 8 I-1,NP**

**CALL F2TCH(IX(I),X(1,I))**

**JX(I)-IX(I)fl**

### 12 Z(JJ)-X(J,I)

### DO 13 J-1,NT

### Y(J)-A(J1

### 13 XtJ,I)-Z(J)

### 11 CONTINUE

**CALL MAPRI**

**(4,NP,NT,Y,TEMP,X,70)**

### C

### C

### IF tK) 18,18,19

### 18 CALL REGRES (NP,NT~Y,X~SH2,RO,BETA,A,E,V,TE)

### PRINT 103

### DO 14 I-1,NT

### 14 PRINT 2t)3,Y(I ),A(I),E(I )rV(I )

### C

**C**

**INITI AL PARAMETER VECTOR**

### C

### C

### 19 SDEV-SORTtSH2)

### TH(1)-SDEV

**TH12)-R0**

**DO 15 I-1,NP**

### JsIt2

### 15 TH(J)xBETA(I)

### C

**CALL LINK tMAIN2)**

### C

### 3 CALL EXIT

**C**

_{MAIN2 PROGRAM - MODSCO-AUTOCORRELATED MODEL}

_{MAIN2 PROGRAM - MODSCO-AUTOCORRELATED MODEL}

**C**

**SECOND PART**

### C

### C

**DIMENSION X(70,5),Y(70),E(70),TH(7),TE(5)**

**COMMON X,Y,E,G,H,FGH,TH,NP,NT,MIT,TE, ISC**

### CONSTANTS

### C

### C

### C

### EPS1-1.I10.~~5

### EPS2-EPS1

**NK-NPf2**

### C

**IF (ISC) 21,20,21**

### 20

_{CALL MODSC 0 ( NK,EPSI,EPS2)}

_{CALL MODSC 0 ( NK,EPSI,EPS2)}

### GO TO 22

### 21

### CALL

**SCOR (NK,EPS2)**

### 22

**CALL LINK (MAIN1)**

### C

### C

### SUBROUTINE MODSCO (NK,EPSI,EPS2)

**DIMENSION X(70,5),Y(70),E(70),TH(7),TE(5)**

**DIMENSION A(70),TB1(7),TB(7),SD(7),SC(7),AA(2,2),B(5,5),BATA(5)**

### C

### COMMON X,Y,E,G,H,FGH,TH,NP,NT,MIT,TE

### C

### 1000 FORMAT (1H1,44H REV. METH. OF SCORING APPLIEO ON AN AUTOCOR,I3HREL

### lATED MODEL,~I1H

### ,I5,14H OBSERVATIONS,I5,34H PARAMETERS

### (SDEV AND R

### 20 INCLUDED))

### 1001 FORMAT (I20H INIT. PARAM. VALUES)

### I002 FORMAT (~21H INIT. VALUE CONC. LF,F19.10)

### 1003 FORMAT (~~I~I45X,13HITERATION NO., I4)

### 1004 FORMAT (23H TEST POINT PAR. VALUES)

### 1005 FORMAT ( II30H TEST POI NT VALUE OF C ONC . LF.,F 15.8 )

*1006 FORMAT (II27H PAR. VALUES VIA REGRESSION)*

### 1007 FORMAT ( II23H NUMBER

### OF STEP CHANGES, 20X, I15)

### 1008 FORMAT (~~11H FINAL STEP,32X,F15.8)

### 1009 FORMAT (II36H VALUE OF CONC. LF. AFTER REGRESSION,F15.8)

### 1010 FORMAT (II39H RELATIVE CHANGE IN EACH PAR. LESS THAN

### ,E12.4)

*1011 FORMAT (II48H RELATIVE CHANGE*

### IN VALUE OF CONC. LF. LESS THAN,E12.

### 14)

### 1020 FORMAT (1H1,25H FINAL ADDITIONAL

### INFORM.)

### 1021 FORMAT ( 11H

### Y-VALUES ,7X,8HOBSERVED,9X, IOHCALCULATED, lOX,9HRESIDU

### lALS)

### ,

### 1022 FORMAT (I,21H FINAL INFORM. MATRIX)

*1023 FORMAT (1II)*

### 1024 FORMAT (~,33H COVARIANCE MATRIX OF SDEV AND RO)

### 1025 FORMAT ( ~, 33H COVARI ANC E MATRI X OF BETA-C OEFF. 1

*1026 FORMAT (II,4H END)*

### 1027 FORMAT (II,11H PAR. ERROR)

### 1028 FORMAT (II,19H IDEC OR IINC ERROR)

*1029 FORMAT ( II,48H THE VALUE OF THE r ONC. LF. CANNOT BE REDUCED T0,37H*

### 2THAT AT THE END OF THE LAST ITERATION)

### 1030 FORMAT (II,30H A(RELATIVE) MAX.

### IS OBTAINED)

*1031 FORMAT (II,9H T-VALUES)*

### C

**2000 FORMAT ( 14X,2(F12.6,7X),F12.6)**

### C

### ~

### EPS3-1.I10.~~5

### T-NT

### PR1NT 1000,NT,NK

### PRINT 1001

### 30 IF(EPS1) 80,80,70

### 80 JORDAN-2

### 70 CALL MODT (NK,TH,VAL1)

### PRINT 1002,VAL1

### NIT-1

### C

### BEGIN ITERATION

### 100 PRINT 1003,NIT

### C

**C**

**INFORMATION MATRIX**

**CALL INFO (NK,AA,B)**

### C

**C**

**SCORE VECTOR**

### C

### SH2-TH(1)~~2

### R02-TH12)~~2

### SC(1)--TITH(1)tFGHITH(1)~~3

### SC(2)--TH(2)~(1.-R02)-(TH(1)~G-H)ISH2

**DO 111 I-1rNP**

**Q -0.**

**Q3-0.**

**Q 2~0.**

**Q 1-0 .**

**QT-O.**

**FF-O.**

**HH 1-0.**

**~**

**HH-O.**

**B1L0.**

**~**

**BT~O.**

**DO 109 L-1,NT**

**DO 110 L- 2,NT**

### JJ-L-1

**CALI FLEM ( HH1,Q3,XtL,I),EtJJ)1**

**110 CALL FLEM ( WM,Q2,EtL),XíJJ,I ))**

**CALL FLEM (B1,Q1,Et11,XI1,I))**

**CALL FLEM ( BT,QT~E(NT),X(NT,I))**

**J-I t2**

**SC(J)-tFFfR02~tFF-B1-BT)-THt21~(NHtHH11)ISH2**

**111 C ONTINUE**

### C

### C

### C

**STEEPEST ASCENT**

**DO 112 I-1,2**

**DO 112 J-1,2**

**QzO.**

**SDí I )~.**

**112 CALL FLEM i5D(I),Q,AAtI,J),SC(J))**

**DO 113 I-1,NP**

**Q-O.**

### JJ-I }2

**SDtJJ)-0.**

**DO 113 L~1,NP**

**JzLf2**

**CALL FLEM (SO(JJ),Q,BtI,L),SCtJ))**

**113 C ONTINUE**

### C

### C

**C**

**COMPUTE OPTIMAL STEP**

### C

### STEPI-STEP

### 169 IDEC-O

**IINC-O**

### GO TO 171

### 170 SAI-VAL

**DO 850**

**I-1,NK**

**850 TB 1( I 1-TB ( I)**

### 171 DO 220

### I-1,NK

### 220 TB(I )-TH(I )}STEP-xSD(I )

### PRINT 1004

**CALL MAPRI**

### (1,NK,ITEMP,TB,TEMP,TEMP,ITEMP)

### C

### C

### C

### 150 STEP-O.

### GO TO 280

### 6b4 DO

### 668 I-1,NK

### 668 TB ( I )-TB1 ( I )

### IC-IC1

### VAL-SAL

### STEP-STEP1

### ~

### 662 PRINT 1006

**CALL MAPRI**

**( 1,NK,ITEMP,TH,TEMP,TEMP,ITEMP)**

**DO 180 J-1,NP**

### JJ-Jt2

**180 BATA (J)-TH(JJ)~(TH(1)~SQRT(TE(J)))**

**PRINT 1031**

**CALL MAPRI ( 1,NP,ITEMP,BATA,TEMP,TEMP,ITEMP)**

**PRINT 1009,VAL1**

**PRINT 1020**

**PRINT 1021**

**CALL MRES (NK,TH,A)**

### DO 143

### I-1,NT

**143 PRINT 2000,Y(I),A(I),E(I)**

### C

**C**

**COVARIANCE-MATRIX VIA INFORMATION MATRIX**

### C

### CALL INFO (NK,AA,B)

### PRINT 1022

### PRINT 1023

### PRINT 1024

**C ALL M APRI**

### (3,2,ITEMP,TEMP,TEMP,AA,2)

### PRINT 1025

### SUBROUTINE MODT (NK,PAR,VAL)

**DIMENSION X(70,5),Y(70),E(70),TH(7)**

**DIMENSICIN A(70),PAR(7)**

### COMMON

### X,Y,E,G~H,FGH,TH,NP,NT,MIT

### CALL MRES (NK,PAR,Q)

### CALL HULP (PAR)

### T-NT

### VAL-(LOG((1.-PAR(2)~~2)IPAR(11~~(2.~T)))I2.--FGHIl2.~(PAR(1)~~2!)

### RETURN

### f,

**SUBROUTINE MRFS ( NK,PAR,A)**

**DIMENSION X(70,5),Y(70),E(70),TH(7)**

**DIMENSION A(70),PAR(7)**

### COMMQN X,Y,E,G,H,FGH,TH,NP,NT,MIT

### DO 1 J-1~NT

### A(J)-0.

**Q-O.**

**DO 2 Ix1,NP**

**K-I t2**

### C

### SUBROUTINE HULP (PAR)

**DIMENSION X(70,5),Y(70),E(70),TH(7)**

**DIMENSION PAR f7)**

### COMMON X,Y,E,G,H,FGH,TH,NP,NT,MIT

### H-0 .

### F -0.

### QF -0 .

### QH -0.

### DO 1

### J:1,NT

### 1 CALI FLEM (F,QF,E(J),E(J))

### DO 2

### J-2,NT

### K-J-1

### 2 CALL FLEM (H,QH,EfJ),E(K))

### G 1-0.

### GT-O.

### Q 1-0.

### QT-O.

### CALL FLEM (G1,Q1,E(1),E(I)1

### CALL FLEM (GT,QT,E(NT),E(NT))

### G-F-GI-GT

### FGH-Ff(PAR(2)~~2)~G-2.~PAR(2)~H

### RETURN

### C

### St"BROI~TINE

**INFO ( NK,A,B)**

### DIMENSION X(70,5),Y(70),E(70),TH(7)

**DIMENSION A(2,2),E~(5,5)**

### C

### C

### COMMON X,Y,F,G,H,FGH,TH,NP,NT,MIT

**100 FORMAT (I23H NO INVERSE OF B-MATRIX)**

**l0I FORMAT (~17H NP IN INFO WRONGI**

### Q1-0.

### Qrzo.

### c~,.

### DD~O.

D s0 . ,### B1s0.

### BTzO.

### DO 2 Ls1,NT

### 2 CALL FLEM (C,QC,X(L,I),X(L,J1)

### DO 3 La2,NT

### K~L-1

### CALL FLEM (DD,Q3,XlK,I),X(L~J)1

### 3 CALL FLEM (O,QD,X(LrI),XtK~J)1

### CALI FLEM (81,Q1,X(1,I1,X(1,J)1

### CALL FIEM tBT,QT,X(NT,I),X(NT,JI)

### B(I,J)slC~R02~tC-B1-BT)-THt2)~tOfDD))ISH2

### 1 B(J,I1sBtI,JI

### C

### C

**INVERSE B-MATRIX**

### C

### IF (NP-1 ) 4,7,8

### 4 PRINT 101

### GO TO 9

### 7 B(1,1)-1.I6(1,1)

### GO TO 6

### 8 DET-0.0000001

**SUBROUTINE SC OR ( NK,EPS1)**

### DIMENSION X(70,5),Y(701,E(70),TH(7),TE(5)

### DIMENSION A(70),TB1171,TBí7),SD(7),SC(7),AA(2,2),B(5,5),BATAl5)

### C

**COMMON X,Y,E,G,H,FGH,TH,NP,NT,MIT,TE**

### 1000 FORMAT ( 1H 1, 39H ME TH. OF SC OR I NG A PPL I ED ON AN AUT OC OR, 13HRE LATED

### 1MODEL,IIIH ,I5,14H OBSERVATIONS,,I5,34H PARAMETERS (SDEV AND RO INC

### 2LUDED))

### 1001 FORMAT (I20H INIT. PARAM. VALUES)

### 1003 FORMAT (IIIII45X,13HITERATION NO.,I4)

### 1004 FORMAT (23H TEST POINT PAR. VALUES)

### 1005 FORMAT (II30H TEST POINT VALUE OF C ONC. LF.,F15.81

*1006 FORMAT (II27H PAR. VALUES VIA REGRESSION)*

### 1009 FORMAT ( I136H VALUE OF C ONC. LF. AFTE.R REGRESSION,F 15.8 )

### 1010 FORMAT (II39H RELATIVE CHANGE IN EACH PAR. LESS THAN ,E12.4)

### 1020 FOP,MAT (1H1,25H FINAL ADDITIONAL

### INFORM.1

### 1021 FORMAT (11H

### Y-VALUES ,7X,SHOBSERVED,9X~lOHCALCULATED,lOX,9HRESIDU

### lAlSl

### 1022 FORMAT (I,21H FINAL INFORM. MATRIX)

*1023 FORMAT (III)*

*1024 FORMAT (I,27H COV. MATRIX OF SDEV AND RO)*

### 1025 FORMAT ( I, 27H COV. MATRI X OF BETA-C OEFF. )

*1026 FORMAT (II,4H END)~*

### .

*1027 FORMAT ( II, 11H PAR. ERROR)*

### 1030 FORMAT (II,30H A(RELATIVE) MAX. IS OBTAINED)

*1031 FORMAT (II,9H T-VAIUES)*

### 1032 FORMAT ( II, 19H VALUE OF C ONC . LF.,F15.8 )

### 2000 ~-QRMAT

_{(14?r2!~i2.a:;'7X),F12.6)}

### T-NT

**PRINT 1000,NT,NK**

**PR1NT**

**1001**

### CALL

**MAPRI**

_{i1,NK,ITEMP,TH,TEMP,TEMP,ITEMP)}

_{i1,NK,ITEMP,TH,TEMP,TEMP,ITEMP)}

**IF (NT-NK) 99,99,15**

**15 IF ( NK-7) 16,16,99**

**16 IF ( NK-1)**

**99,17,17**

**17 IF ( MIT-1) 99,18~,18**

**18 IF ( MIT-1000) 19,19,99**

**I9 DO 20 I-1,NK**

**IF (TH(I)) 24,99,20**

**20 CONTINUE**

**CALL MODT tNK,TH,VAL)**

**PRINT 1032,VAL**

### C

### NIT-1

### 100 PRINT 1003,NIT

### C

**C**

**INFORMATION MATRIX**

**CALL INFO ( NK,AA,B)**

### C

**C**

**SCORE VEC TOR**

### C

### SC(2)--TH(2)I(1.-R02)-(THtl)~G-H)ISH2

### DO 111 I -1,NP

### Q -0.

### Q3-0.

### Q2-0.

### Q1-0.

### QT-O.

### FF-O.

### HH 1-0 .

### HH - 0 .

### 81-0.

### 6T-0.

### DO 109 L-1~NT

### 109 CALL FLEM (FF,Q~E (L ),X tL, I) 1

### DO 110 L-2,NT

### JJ-L-1

### CALL FLEM (HH1~Q3,X(L,I),E(JJ))

### 110 CALL FLEM (HH,Q2,E(L),X(JJ,I))

**I12 i.ALL FLEM (SD(I),9,AA(I,J),SC(J))**

**DO 113 I -1,NP**

**Q-O.**

### JJ-I t2

**SD(JJ)-0.**

**DO 113 L-1,NP**

### J-Lf2

**CALI FLEM (SD(JJ1,Q,B(I,L),SC(J))**

**113 CONTINUE**

### C

### c

### ~

**r**

**COMPUTE OPTIMAL STEP**

### STEP-1.

**DO 220 I-1,NK**

### 220 TB(I)-TH(I1fSTEP~SD(I)

**PRINT 1004**

**CALL MAPRI**

**( 1,NK,ITEMP,TB,TEMP,TEMP,ITEMP)**

**CALL MODT ( NK,TB,VAL)**

PRINT 100E~

CALL **MAPRj** **(1.NK.ITFMP.TH.TEMP.TFMP,jTEMD)**

t)n 1AO J-t.NP

JJ-Ji2

**lA0 BATA (J1-THfJJ1I(THI1)~SURTITF(;1111**

PRINT 1031

**CALL MAPRI** **(1.NP,ITENP.RATA,TEMV,TEMP,iTEMP)**
CALL M~~fiT (NK.TH,VAL 1

PRI~lT 1~09.VA~. PR1NT 1020 PR1NT 10?1

**CALL MRES ( NK,TH,AI**
r1r7 143 I -1,NT

**143 PRINT 2000,Y111,A(11~E(II**

C

C **COVARIANCE-MATRiX VIA iNFORMATI(1rJ MATRiX.**
C

CALL INFfI **( NK,AA,B)**
PRTNT 1022

PRINT 1023 PR1NT 1024

CALL MAPRI **( 3,2.1TEMP,TEMP,TEMP,AA,2)**
PRINT 10?5

**CALL NAPR1** **(3,NP,ITtëMP~TEMP~TEMP~H,51**

410 PR INT 102h

RFTl1RN ~

9y PRINT 1027