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  • eM '- t~~2 Discussionp `~ papermicResearch~seo

    ~~~~I IN~~I~II~H~ ~~~ól~~l,~~l l~ ~~~ lltll ~~l l~~ll

  • No. 9021

    EVALUATION OF MOMENTS OF QUADRATICFORMS IN NORMAL VARIABLES .

    i~. -,~

    by Jan R. Magnus '~ U. yzand Bahram Pesaran ~

    April 1990

    ISSN o924-7815

  • Evaluatlon of moments of quadrattc

    forms In normal veriables

    by

    Jan R. MagnusLondon School of Economics

    andCentER, Tilburg University

    and

    Bahram PesaranEconomics Division, Bank of Engtand

    February 1990

    Keywords: Quadratic forms; Moments; Cumulants

  • ABSTRACT

    We present and describe a subroutine, called CUM, which calculates the

    cumulan[s and momen[s of a quadratic form in normal variables. That ís,

    if we let Q- x'Ax, then CUM can calculate E(Qs), where x is an n x 1

    vector of normally distributed variables with some mean and a positive

    definite (hence non-singular) covaríance matríx , and A is an n x n

    symmetric matríx. A diskette containing the Fortran code of CUM and

    some test programmes is available on request.

  • 2

    Language

    Fortran 77

    Descriptlon and Purpose

    The subroutine CUM calculates the exact cumulants and moments of the quadraticform x'Ax, where x is an nx1 vector of normally distributed variables with some mean uand a positive definite (hence non-singular) variance-covariance matrix n and A is annxn symmetric matrix. The routine is based on theory developed by Magnus(1978,1979) for the central case where u-0, and Magnus(1986, Lemmas 2 and 3) for thegeneral case.

    Parameter statements

    The following parameters have been set in subroutines CUM and PARINT :

    ISPAR-77 The dimensions of the array tSPRTN ( ISPARxISDIM) where all

    8 possible partitions tor a particular s are stored. This two

    ISDIM-12 dimensional array is set up by subroutine PARINT.

    MAXMOM-24 The maximum of s allowed.

    If 12~ss24 is to be calculated, then both ISPAR and ISDIM should be increased.

    Working out cumulants and moments for s~24 is possible. In that case not only ISPAR

    and ISDIM should be changed in the pa~ameter statements, but also MAXMOM.

    Furthermore, the DATA statement in subroutine PARINT should be extended to include

    MAXMOM numbers. In the data statement the vector NUM has been set up to contain

    the number of all possible partitions of numbers up to 24. If a MAXMOM bigger than 24

  • 3

    is specified, the data statement for NUM should be extended accordingly. For a table

    containing the partitions for integers up to 100 see Hall(1986, p.38).

    Structure

    SUBROUTINE CUM(IMOM, N, LS, A, IEMU, EMU, IOMEGA, OMEGA, RKUM, RMOM,WORK1, WORK2, WORK3, VEC, IFAIL)

    Formal Parameters

    IMOM integer input: -0 if only cumulants are requiredL1 if both cumulants and momentsare required.

    N integer input: No of observations n

    LS integer input: order of the highest cumulant ormoment required.

    output: unchanged unless LS~M whereM-MIN(MAXMOM,ISDIM) in whichcase LS is set equal to M.

    A real array of dimension input: symmetric matrix in the quadraticat least Nx(Nf 1)l2 form x'Ax. Only the lower part of

    A is stored as:a11,a21,a22,a31,a32,ag3 etc.

    IEMU integer input: -0 if N-0.~0 it u~0

    EMU real array of dimension input: vector u. Values required onlyat least N if IEMUrO though storage should be

    allocated to it.

    IOMEGA integer input: ~-1 if L"1 is supplied wherect - LL', L lower triangular

    - 1 if L is supplied wheren- LL', L lower triangular

    - 2 if p is supplied

    OMEGA real array of dimension input: either L or L-1 or tl where onlyat least Nx(Nt1)l2 lower part of these are stored

  • 4

    RKUM real array of dimensionat least LS

    output: the required cumulants with i-thcumulant stored in RKUM(i).

    RMOM real array of dimensionat least LS

    WORK1 real array of dimensionat least Nx(Nt1)!2

    WORK2 real array of dimensionat least Nx(Nt1)!2

    WORK3 real array of dimensionat least Nx(Nt1)!2

    VEC real array of dimensionat least N

    IFAIL iMeger

    output: the required moments with i-thmoment stored as RMOM(i).

    work space

    work space

    work space

    work space

    output: a fault indicator where:

    0: no error1:Ns12: IOMEGA out of range3:LSc14: if IOMEGA-2 sZ not pos.

    definite

    if IOMEGA-1 diagonal elementsof L not all positive

    if IOMEGA- 1 diagonal elementsof L-~ not all positive

    5: if IOMEGA-2 or 1, L can't beinverted

    if IOMEGA--1, L-1 can't beinverted

    6: ISPAR in the parameterstatement is too small

    Auxlliary Algorithms

    CUM calls various functions and subroutines listed below:

  • 5

    (F1) FUNCTION INX(I,J): picks out the appropriate element of a

    symmetric matrix stored in lower triangular

    form

    (F2) REAL'8 FUNCTION FACT(N): calculates N!

    A real'8 function is used because large

    values of N are passed as the argument.

    (S1) SUBROUTINE PARINT: constructs the matrix containing allpartitions of an integer

    (S2) SUBROUTINE SEP: decomposes A(pos. def.) into LL' (L lower

    triangular) and replaces A by L

    (S3) SUBROUTINE LOWINV: inversion ( in place) of a lower tríangular

    matrix

    (S4) SUBROUTINE MULT1: computes C-8'AB (A symmetric, B lower

    triangular)

    Constants

    The DATA statement in CUM sets EPS - 1.0-11 as a small number. Any number with

    an absolute value below EPS will be treated as zero.

    Precision

    The version ot the routines listed below is in double precision (Real'8). In order to

    change the program to single precision the following changes should be made:

    (1) Change all IMPLICIT REAL'8 to IMPLICIT REAL'4

  • 6

    (2) Change the constants in the DATA statements to single precision versions.

    (3) Change DSQRT, DABS to SORT, ABS, in the statement functions appearing inthe beginning of routines SEP and LOWINV.

    Tlme

    Using the VAX 6330 at the London School of Economics, typical CPU times for thedouble precision version are reported in Table 1 for different combinations of LS(4, 8and 12) and n(10, 20, 30, 40 and 50). Other parameters were kept fixed atIMOM~1 and IEMU-1 as the variations in these did not affect the CPU timesignificantly. It should be pointed out that the effect of increase in LS is cumulative, i.e.when LS-4 all of the first 4 cumulants and moments are calculated while for LS-B allfirst 8 cumulants and moments are evaluated by CUM.

    Table 1. Typical CPU times

    n LS-4 LS-B LS-12

    10 0.30 0.32 0.47

    20 1.07 1.47 1.98

    30 3.74 5.17 6.39

    40 10.43 13.24 16.35

    50 23.81 29.52 35.26

    Accuracy

    In order to check the accuracy of calculations we used CUM to evaluate the first 4

    cumulants and moments ot the x2 distribution with 4 degrees of freedom. This was

    achieved by using an arbitrary (20x4) matrix R and by setting A-R(R'R)"1 R' and 52-1.

    Setting IEMU-O the first 4 cumulants and moments of x'Ax were calculated.

  • The exact rth cumulant of a x2 distribution with v degrees of freedom can be calculatedusing the formula

    (1) Kr - v 2r-1 (r -1)!

    while the moments of a x2 distribution with v degrees ot freedom can be worked outusing its moment generating function which is

    (2) MGM(t) - (1 - 2t )-~~z

    Table 2 allows the comparison between the exact results using (1) and (2) and thecalculated values using CUM.

    Table 2. Accuracy of cumulants and moments of a x2 distrtbution calculatedby CUM .

    Cumulants Moments

    r Exact Calculated Exact Calculated

    1 4 4.00000000 4 4.0000000

    2 8 8.00000000 24 24.00000000

    3 32 32.00000000 192 192.00000000

    4 192 192.00000000 1920 1920.00000000

  • 8

    Examining Table 2 shows that in this example an accuracy of at least eight decimal

    points is achieved.

    Related Algorithms

    The algorithm for the evaluation of moments of ratios of quadratic forms in normal

    variables and related statistics by Magnus and Pesaran (1990).

  • 9

    REFERENCES

    Hall, M.(1986), Combinatorial Theorv(2nd edition), John Wiley and Sons, New York.

    Magnus, J. R.(1978), 'The moments of products of quadratic forms in normal variables',Statistica Neerlandica, 32, 201-210.

    Magnus, J. R.(1979), 'The expectation of products of quadratic forms in normal

    variables: the practice', Statistica Neerlandica, 33, 131-136.

    Magnus, J. R.(1986), 'The exact moments of a ratio of quadratic forms in normal

    variables', Annales d'Economie et de Statistigue, 4, 95-109.

    Magnus, J. R. and Pesaran, B.(1990), 'Evaluation of moments of ratios of quadraticforms in normal variables and related statistics', submitted to Applied Statistics.

  • 10

    SUBROUTINE CUM(IMOM,N,LS,A,IEMU,EMU,IOMEGA,tOMEGA,RKUM,RMOM,WORKI,WORK2,WORK3,VEC,IFAIL)

    C

    C ALGORITHM AS??? APPL. STATIST. (1990).C CALCULATES CUMULANTS AND MOMENTS OF A

    C QUADRATIC FORM IN NORMAL VARIABLES

    CIMPLICIT REAL~B(A-H,O-Z)

    PARAMETER(ISDIM-I2,ISPAR~77,MAXMOMa24)DIMENSION A(~),OMEGA(t),WORK1('),WORK2('),woRx3(Y)DIMENSION EMU(~),RKUM(`),RMOM(~),VEC(~)DIMENSION ISPRTN(ISPAR,ISDIM),TR(ISDIM)

    DATA ZERO,ONE,TWO,EPS~O.ODO,I.OD0,2.ODO,1.D-11~IFAIL-1

    IF(N.LE.1) RETURNIFAILa2

    IF(IOMEGA.NE.-1.AND.IOMEGA.NE.I.AND.IOMEGA.NE.2) RETURN

    IFAIL-3

    IF(LS.LT.1) RETURN

    NN~N~ (Ntl) ~2LS~MIN(LS,MAXMOM,ISDIM)

    C

    C SET WORK2 ~ L WHERE OMEGA ~ LL'

    C

    DO 10 IS1,NN

    10 WORK2(I)~OMEGA(I)

    IF(IOMEGA.EQ.I.OR.IOMEGA.EQ.-1) THEN

    IFAIL-4

    DO 20 I-1,N

    RsWORK2(INX(I,I))

    IF(R.LT.EPS) RETURN

    20 CONTZNUE

    END IF

  • 11

    IFAIIrOZF(IOMEGA.EQ.2) CALL SEP(WORK2,N,ONE,EPS,IFAIL)IF(IOMEGA.EQ.-I) CALL LOWINV(WORK2,N,ZERO,ONE,EPS,IFAIL)IF(IFAIL.NE.O) THEN

    IF(ZOMEGA.EQ.2) IFAIL-4IF(IOMEGA.EQ.-1) IFAIL~SRETURN

    END IFC

    C

    C

    CC

    C

    SET UP L'AL IN WOAK1

    CALL MULT1(A,WORK2,WORKI,N,2ER0)

    CALCULATE LINVREMU IN VEC IF EMU IS NOT ZERO

    IF(IEMU.NE.O) THENIF(ZOMEGA.GT.O) THEN

    CALL LOWINV(WORK2,N,ZERO,ONE,EPS,IFAIL)

    IFAILaIFAILt4

    IF(ZFAIL.NE.9) RETURNEND IF

    DO 40 I~1,NSUMaZERO

    DO 30 J~1,IIF(IOMEGA.GT.O) THENR-WORK2(INX(I,J))

    ELSE

    R~OMEGA(INX(I,J))

    END IF

    30 SUM~SUMtR~EMU(J)

    40 VEC(I)aSUM

    END IF

  • 12

    C

    C CALCULATE CUMULANTS OF QUADRATIC FORM IN RKUMC

    DO 130 ISz1,LSIF(IS.EQ.1) THENDO 50 I~1,NN

    50 WOAK2(I)~WORK1(I)

    ELSE

    DO 70 Iz1,N

    DO 70 J-1,IIJ~INX(I,J)

    SUMIIZERO

    DO 60 K~1,N

    IKzINX(I,K)KJ~INX(K,J)

    60 SUMaSUMtWORKl(IK)~WORK2(KJ)70 WORK3(IJ)-SUM

    DO 80 I~1,NN80 WORK2(I)~WORK3(I)

    END IF

    SUM~ZERO

    DO 90 I~1,N

    RI~WORK2(INX(I,Z))

    90 SUMaSUMtRI

    TR(IS)zSUM

    IF(IEMU.NE.O) THENSUMzZERO

    DO 100 I31,N

    TT-TWO

    DO 100 Je1,I

    IJzZNX(I,J)IF(I.EQ.J) TT-ONE

    100 SUM-SUMtTT'VEC(Z)~WORK2(INX(I,J))~VEC(J)RSzZS

  • 13

    TR(IS)~TR(IS)tRS"SUMEND IF

    TT~2""(IS-1)

    RKUM(IS)~TT`FACT(IS-1)'TR(IS)

    C

    C CALCULATE MOMENTS OF QUADRATIC FORM ( IF REQUIRED) IN RMOMC

    IF(IMOM.EQ.1) THEN

    CALL PARINT(IS,ISPRTN,ISPAR,ISDIM,ISROW,IFAIL)

    IF(IFAIL.NE.O) THEN

    IFAILa6

    RETURN

    END IF

    SUM~ZERO

    DO 120 Ig1,ISROW

    RB~ONE

    RA-ONE

    DO 110 L~1,IS

    R2L~2`L

    KSISPRTN(I,L)

    IF(K.NE.O) THEN

    RBaRB'FACT(K)"(R2L"'K)

    RA~RA'(TR{L)i"K)

    END IF

    110 CONTINUE

    RB~(FACT(IS)"(TWOx'IS))~RB

    120 SUMzSUMtRA"RB

    RMOM(IS)~SUM

    END IF

    130 CONTINUE

    IFAIL~O

    RETURN

    END

  • 14

    SUBROUTINE PARINT(M,MPRTN,MRDZM,MDIM,MR,IFAIL)

    C

    C CONSTRUCTS THE MR X M MATRIX "MPRTN" CONTAININGC ALL MR PARTITIONS OF THE INTEGER MC

    PARAMETEA(MAXMOM-24)

    DIMENSION MPRTN(MRDIM,MDIM)

    DIMENSION NUM(MAXMOM),IWORK(MAXMOM)

    DATA NUM~1,2,3,5,7,11,15,22,30,42,56,77,301,135,176,231,t297,385,990,627,792,1002,1255,1575~

    IFAIL-1IF(M.LT.I.OR.M.GT.MAXMOM.OR.M.GT.MDZM) RETURNIFAIL-2

    IFINUM(M).GT.MRDIM) RETURN

    IFAIL-0

    N1-0

    N2~1

    N3-0

    MR-1

    M1-1

    L-0

    MPRTN(1,1)-1

    IF(M.EQ.1) RETURN

    DO 10 J-2,M

    10 MPRTN(1,J)-0

    DO 99 K-2,M

    IF(N2.NE.0) THEN

    DO 30 I-1,N2

    MPRTN(MRfI,l)a0

    DO 20 Jv2,M

    20 MPRTN(MRtI,J)-MPRTN(ItN1,J)

    30 MPRTN(MRtI,2)-MPRTN(MRfI,2)tl

    END IF

    IF(N3.NE.0) THEN

  • is

    L-0DO 80 I~N1tN2f1,MRDO 80 J~2,K-1IF(MPRTN(I,J).EQ.O) GO TO 80DO 90 JJ~1,M

    90 IWORK(JJ)~MPRTN(I,JJ)IWORK(J)~IWORK(J)-1IWORK(Jtl)~IWORK(Jtl)tl

    IF(N2.NE.O.OR.L.NE.O) THENDO 60 II-MRt1,MRtN2tLDO 50 JJ-1,M-1

    50 IF(MPRTN(ZI,JJ).NE.IWORK(JJ)) GO TO 60IF(MPRTN(II,M).EQ.IWORK(M)) GO TO 80

    60 CONTINUE

    END ZFL-LtlDO 70 JJ~1,M

    70 MPRTN(MRtN2fL,JJ)sIWORK(JJ)

    80 CONTINUE

    END IF

    DO 90 II~1,MA90 MPRTN(II,1)~MPRTN(II,1)fl

    N1-M1

    N3~N2tL

    N2-MR-NI

    M1~MR

    MR-MRtN3

    99 CONTINUERETURN

    END

  • 16

    INTEGER FUNCTION INX(I,J)

    PICKS OUT THE APPROPRIATE ELEMENT OF A SYMMETRIC

    MATRIX STORED IN LOWER TRIANGULAR FORM

    IF(I.GE.J) THENINX~I'(I-1)~2tJ

    ELSEINXaJ'(J-1)~2fI

    END IFRETURN

    C

    CC

    10

    END

    REALtB FUNCTION FACT(N)

    CALCULATES N FACTORIAL

    FACT~1

    IF(N.LE.1) RETURN

    DO 10 I~2,N

    RZ~I

    FACT-FACT~RI

    RETURN

    END

    SUBROUTINE SEP(A,N,ONE,EPS,IFAIL)

    DECOMPOSES A(POSITIVE DEFINITE) INTO LL'

    (L LOWER TRIANGULAR) AND REPLACES A BY L

    IMPLICIT REAL'8(A-H,O-Z)

    DIMENSION A(')

    Z SQRT ( H)-D SQRT ( H)

    IFAIL-1

  • DO 20 I~1,NDn 2n J~r u

    IJ~INX(J,I)R~A(IJ)

    IF(I.NE.1) THENDO 10 K~1, I-1

    10 R~R-A(INX(I,K))'A(INX(J,KI)END IFIF(I.EQ.J) THENIF(R.LT.EPS) RETURND~ONE~ZSQRT(R)

    END IF20 A(IJ)~R'D

    IFAIL~ORETURN

    C

    CC

    END

    SUBROUTINE LOWINV(A,N,ZERO,ONE,EPS,IFAIL)

    INVERSION (IN PLACE) OF N X N LOWER TRIANGULAR MATRIX

    IMPLICIT REAL'8(A-H,O-Z)

    DIMENSION A(')ZABS(H)~DABS(H)

    IFAIL~1

    DO 20 I~1,NRsA(INX(I,I))IF(ZABS(R).LT.EPS) RETURND-ONE~R

    DO 20 J-1,I

    IJ-INX(I,J)

    IF(I.NE.J) THENSUM-ZERODO 10 K~J,I-1

  • 18

    10 SUM~SUMtA(INX(I,K))~A(INX(R,J))A(IJ)~-SUM~D

    ELSEA (IJ) ~D

    END IF

    20 CONTINUE

    IFAIL~O

    RETURN

    C

    C

    END

    SUHROUTINE MULT1(A,B,C,N,ZERO)

    FINDS C ~ B'AB (A-A', B LOWER TRIANGULAR)C ALL MATRICES N X NC

    IMPLICIT AEAL'8(A-H,O-Z)DIMENSION A(~),B(i),C(~)DO 20 I~1,NDO 20 J~1,I

    IJ~INX(I,J)

    SUM-ZERODO 10 K~I,NKI-INX(K,I)DO 10 L-J,N

    LJ~INX(L,J)KLaINX(K,L)

    10 SUM~SUMtB(KI)'A(KL)~H(LJ)20 C(IJ)~SUM

    RETURN

    END

  • Discumaion Paper Seriem, CentFR, Tilburg University, The Netherlands:(For previous papers please consult previous discussion papers.)No. Author(s)

    H901 Th. Ten Raa andP. Kop Jansen

    Title

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    8902 Th. Nijman and F. Palm

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    8904 F. van der Plceg

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    89G6 A.J. Markink andF. ven der Plceg

    8907 J. Osiewalski

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    8909 F. van der Plceg

    891G R. Gradus andA. de Zeeuw

    8911 A.P. Barten

    8912 K. Kamiye andA.J.J. Talman

    8913 G. van der Laan andA.J.J. Talman

    8914 J. Osiewalski andM.F.J. Steel

    8915 R.P. Gilles, P.H. Ruysand J. Shou

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    Title

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    Risk Aversion, Intertemporal Substitution andConsumption: The CARA-LQ ProblemInterdependent Preferences: An EconometricAnalysis

    Ownership Structure and Efficiency: AnIncentive Mechanism Approach

    On the Empirical Implementation of Some GameTheoretic Models of Household Labor SupplySignaling and Forward Induction in a MarketEntry Context

    Coherency end Regularity of Demand Systemswith Equality and Inequality Constraints

    Forecasting, Miaspecification and Unit Roots:The Case of AR(1) Versus ARMA(1,1)

    Wage Setting and Stabilization Policy in aGame with Renegotiation

    Alternative Approaches to Testing Non-NestedModela with Autocorrelated Disturbances: AnApplication to Models of U.S. Unemployment

    A Stochastic Analysis of an Input-OutputModel: Comment

    Keynesian and New Classical Models ofUnemployment Revisited

  • No. Author(s)

    9007 J. Osiewalski andM.F.J. Steel

    9008 O.W. imbens

    9009 c.w. Imbens

    9010 P. Deschampa

    9011 W. GUth andE. van Damme

    9012 A. Horaley andA. Wrobel

    9013 A. Horaley andA. Wrobel

    9014 A. Horsley andA. Wrobel

    9015 A. van den Elzen,G. van der Laan andD. Talman

    9016 P. Deschamps

    901~ B.J. Christensenand N.M. Kiefer

    9018 M. Verbeek andTh. Nijman

    9019 J.R. Magnus andB. Pesaran

    9020 A. Robson

    9021 J.R. Magnus andB. Pesaran

    Title

    Semi-Conjugate Prior Denaíties in Multi-variate t Regression Models

    Duration Models with Time-VaryingCcefficients

    An Efficient Method of Moments Estimatorfor Discrete Choice Models with Choice-BasedSampling

    Expectations and Intertemporal Separabilityin en Empirical Model of Consumption andInvestment under Uncertainty

    Gorby Games - A Qame Theoretic Analysis ofDisarmament Campaigns and the DefenseEfficiency-Hypothesis

    The Existence of an Equilibrium Densityfor Marginal Cost Prices, and the Solutionto the Shifting-Peak Problem

    The Closedness of the Free-Disposal Hullof a Production Set

    The Continuity of the Equilibrium PriceDenaity: The Case of Symmetric Joint Costs,and a Solution to the Shifting-PatternProblem

    An Adjustment Process for en ExchangeEconomy with Linear Production Technologies

    On Fractio..sl Demand Systems and BudgetShare Positivity

    The Exact Likelihood Function for anEmpirical Job Search Model

    Testing for Selectivity Bias in Panel DataModels

    Evaluation of Moments of Ratios of QuadraticForms in Normal Variables and RelatedStatistics

    Status, the Distribution of Wealth, Socialand Private Attitudes to Risk

    Evaluation of Moments of Quadratic Forms inNormal Variables

  • anY on~~~ ~nnn i F Tii R~ IRC~ THE NETHERLANDBibllotheek K. U. Brabant

    YII III~I~~NI~IY M lillll

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