MØDSCØ, a computer programm for the revised method of scoring

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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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1970

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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

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K.U.B.

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

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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.

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-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

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-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

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-~-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

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-~-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,

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-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

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-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.

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-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.

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-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

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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).

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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);

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~ Flow ~nart of h1~DSCQ subroutine

14

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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).

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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,}

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APPENDIX:

Listing of the programs

C

MAIN1 PROGRAM - MODSCO-AUTOCORRELATED MODEL

C

FIRST PART

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

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)

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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

(28)

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

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

INITI AL PARAMETER VECTOR

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

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C

_{MAIN2 PROGRAM - MODSCO-AUTOCORRELATED MODEL}

C

SECOND PART

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

EPS1-1.I10.~~5

EPS2-EPS1

NK-NPf2

C

IF (ISC) 21,20,21

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

GO TO 22

21 CALL

SCOR (NK,EPS2)

22 CALL LINK (MAIN1)

C

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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)

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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

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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

INFORMATION MATRIX

CALL INFO (NK,AA,B)

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

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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

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

COMPUTE OPTIMAL STEP

C

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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

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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

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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

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

(37)

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

(38)

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

(39)

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

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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

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

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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

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

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MØDSCØ, a computer programm for the revised method of scoring

Tilburg University