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TRANSCRIPT
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No. 9021
EVALUATION OF MOMENTS OF QUADRATICFORMS IN NORMAL VARIABLES .
i~. -,~
by Jan R. Magnus '~ U. yzand Bahram Pesaran ~
April 1990
ISSN o924-7815
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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.
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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
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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
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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:
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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
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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.
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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
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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).
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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.
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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
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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
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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
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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
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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
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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
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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
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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
The Choice of Model in the Construction ofInput-Output Ccefficients Matrices
8902 Th. Nijman and F. Palm
8903 A. van Scest,I. Woittiez, A. Kapteyn
8904 F. van der Plceg
8905 Th. van de Klundert andA. van Schaik
89G6 A.J. Markink andF. ven der Plceg
8907 J. Osiewalski
8908 M.F.J. Steel
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
Generalized Least Squares Estimation ofLinear Models Containing Rational FutureExpectations
Labour Supply, Income Taxes and HoursRestrictions in The Netherlands
Capital Accumulation, Inflation and Long-Run Conflict in International Objectives
Unemployment Persiatence and Losa ofProductive Capacity: A Keynesian ApproachDynamic Policy Simulation of Linear Modelswith Rational Expectations of Future Events:A Computer PackagePosterior Denaities for Nonlinear Regressionwith Equicorrelated Errors
A Bayesian Analysis of Simultaneous EquationModels by Combining Recursive Analytical andNumerical Approaches
lwro Ess~ys on Political Economy(i) The Political Economy of Overvaluation(ii) Election Outcomes and the StockmarketCorporate Tax Rate Policy and Publicand Private Employment
Allais Characterisation of PreferenceStructures and the Structure of Demand
Simplicial Algorithm to Find Zero Pointsof a Function with Speciel Structure on aSimplotope
Price Rigiditiea and Rationing
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No. Author(s)
8916 A. Kapteyn, P. Kooremanand A. van Scest
891~ F. Canova
8918 F. van der Plceg
8919 W. Bossert andF. Stehling
892o F. van der Plceg
8921 D. Cenning
8922 C. Fershtman endA. Fishman
8923 M.B. Canzoneri andC.A. Rogers
8924 F. Groot, C. Withagenand A. de Zeeuw
8925 O.P. Attanasio andG. Weber
8926 N. Rankin
8927 Th. ven de Klundert
8928 C. Dang
8929 M.F.J. Steel andJ.F. Richard
893o F. ven der Plceg
Title
Quantity Rationing and Concavity in aFlexible Household Labor Supply Model
Seasonalities in Foreign Exchange Markets
Monetary Disinflation, Fiscal Expansion andthe Current Account in an InterdependentWorld
On the Uniqueneas of Cardinally InterpretedUtility Functions
Monetary Interdependence under AlternativeExchange-Rate RegimesBottlenecks and Persistent Unemployment:Why Do Booms End?
Price Cycles and Booms: Dynamic SearchEquilibrium
Is the European Community an Optimal CurrencyArea? Optimal Tax Smoothing versus the Costof Multiple Currencies
Theory of Natural Exhaustible Resources:The Cartel-Versus-Fringe Model Reconsidered
Consumption, Productivity Growth and theInterest Rate
Monetary and Fiscal Policy in a'Hartian'Model of Imperfect Competition
Reducing External Debt in a World withImperfect Asset and Imperfect CommoditySubstítutíon
The D t -Triangulation of Rn for SimplicielAlgorithms for Computing Solutions ofNonlinear Equationa
Bayesian Multivariate Exogeneity Analysis:An Application to a UK Money Demend Equation
Fiscal Aspects of Monetary Integration inEurope
8931 H.A. Keuzenkamp The Prehistory of Rational Expectations
-
No. Author(s)
8932 E. van Damme, R. Seltenand E. Winter
8933 H. Carlsson andE. van Demme
8934 H. Huizinga
8935 C. Dang anaD. Talman
8936 Th. Nijman andM. Verbeek
8937 A.P. Barten
8938 G. Marini
8939 w. cuth andE. van Damme
8940 G. Marini andP. Scaramozzino
8941 J.K. Dagsvik
8942 M.F.J. Steel
8943 A. Rcell
8944 C. Hsiao
8945 R.P. Gilles
8946 w.B. MacLeod andJ.M. Malcomson
8947 A. van Scest andA. Kapteyn
8948 P. Kooreman andB. Melenberg
Title
Alternating Bid Bargaining with a SmallestMoney Unit
Global Payoff Uncertainty and Risk Dominance
National Tax Policies towards Product-Innovating Multinational Enterprises
A New Triangulation of the Unit Simplex forComputing Economic Equilibria
The Nonresponse Bias in the Analysis of theDeterminants of Total Annuel Expendituresof Households Based on Panel Data
The Estimation of Mixed Demand Systems
Monetary Shocks and the Nominal Interest Rate
Equilibrium Selection in the Spence SignalingGame
Monopolistic Competition, Expected Inflationand Contract Length
The Generalized Extreme Value Random UtilityModel for Continuous Choice
Weak Exogeníty in Misspecified SequentialModelsDual Capacity Trading and the Quality of theMarket
Identification and Estimation of DichotomousLatent Variables Models Using Panel Data
Equilibrium in a Pure Exchange Economy withan Arbitrary Communication Structure
Efficient Specific Investments, IncompleteContracts, and the Role of Market Alterna-tives
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-
No. Author(s)
8949 C. Dang
8950 M. Cripps
8951 T. Wansbeek anaA. Kapteyn
Title
The D -Triangulation for SimplicialDefor~ation Algorithms for ComputingSolutions of Nonlinear Equations
Dealer Behaviour and Price Volatility inAsset MarketsSimple Estimators for Dynemic Panel DataModels with Errors in Variablea
8952 Y. Dai, G. van der Laan, A Simplicisl Algorithm for the NonlinearD. Talman and Stationary Point Problem on an UnboundedY. Yememoto Polyhedron
8953 F. van der Plceg
8954 A. Kapteyn,S. van de Geer,H. van de Stadt andT. Wansbeek
a955 L. Zou
8956 P.Kooreman andA. Kapteyn
8957 E. van Demme
9001 A. van Scest,P. Kooreman andA. Kapteyn
9002 J.R. Magnus andH. Pesaran
9003 J. Driffill andC. Schultz
9004 M. McAleer,M.H. Pesaran andA. Bera
9005 Th. ten Raa andM.F.J. Steel
9006 M. McAleer endC.R. McKenzie
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
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anY on~~~ ~nnn i F Tii R~ IRC~ THE NETHERLANDBibllotheek K. U. Brabant
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