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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

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

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

A Bayesian Analysis of Exogeneity in ModelsPooling Time-Series and Cross-Section Data

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8916 A. Kapteyn, P. Kooremanand A. van Scest

891~ F. Canova

8918 F. van der Plceg

8919 W. Bossert andF. Stehling

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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

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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

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Reducing External Debt in a World withImperfect Asset and Imperfect CommoditySubstítutíon

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8938 G. Marini

8939 w. cuth andE. van Damme

8940 G. Marini andP. Scaramozzino

8941 J.K. Dagsvik

8942 M.F.J. Steel

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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

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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

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a955 L. Zou

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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)

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

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Keynesian and New Classical Models ofUnemployment Revisited

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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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