Bock, M.E.

Judge, G.G.

Yancey, T.A.

English

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UNIVERSITY OFILLINOIS LIBRARYAT URBANA-CHAMPAIGNBOOKSTACKSCENTRAL CIRCULATION BOOKSTACKSThe person charging this material is re-sponsible for its renewal or its return tothe library from which it was borrowedon or before the Latest Date stampedbelow. You may be charged a minimumfee of $75.00 for each lost boolc.TiMft, mutHotloiv and undsriinlna of books ore reosonsfor dUclplinory action and may result In dismissal fromthe University.TO RENEW CALL TELEPHONE CENTER, 333-8400UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGNai:When renewing by phone, write new due date belowprevious due date. L162Digitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/weakmeansquareer08yancFaculty Working Papers«A WEAK MEAN SQUARE ERROR TESTFOR STOCHASTIC RESTRICTIONS IN REGRESSIONT. A. Yancey, M. E. Bock and G. G. JudgeCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignFACULTY WORKING PAPERSCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana - ChampaignItarch 19, 1971A WEAK MEAN SQUARE ERROR TESTFOR STOCHASTIC RESTRICTIONS IN REGRESSIONT. A. Yancey, M. E. Bock and G. G. JudgeNo. 8.«V?M :.A WEAK MEAN SQUARE ERROR TESTFOR STOCHASTIC RESTRICTIONS IN REGRESSIONT.A. Yancey, M.E. Bock and G.G. JudgeThe weak mean square error results of Wallace, relating to tests forlinear restrictions in regression, are extended to include tests for sto-chastic linear restrictions.1 . IntroductionIn a 1968 paper, Toro-Vizcarrondo and Wallace [6] showed that the non-central F distribution with a non-centrality parameter of one-half could beused to test the hypothesis that a linear equality restricted estimator, $,is better than an unrestricted least squares estimator, b, where betterrefers to the generalized mean square error (MSE) criterion. This meansfor the estimators 3 and b that MSE(^|) < MSE(£b) for every ^ j^ o. In acompanion paper, the same authors [8] present tables of critical points,review the test procedure and give some examples of how the test may beused. In a 1970 paper, Yancey, Judge and Bock [9] present a test for deter-mining the conditions under which stochastic linear prior information, whichmay be incorrect on the average, may improve the parameter estimates forthe linear hypothesis model over conventional sample information estimatesin the sense of having the same or smaller mean square errors for all esti-mates. In a 1971 paper in this Journal, Wallace [7] develops tests ofbetterness of the restricted estimator over the unrestricted least squaresestimator where betterness is defined in terms of average squared Euclidean-2-distance (weak mean square error (WMSE)), such that in estimation for thevector 6,(1.1) E(§-e3'(e-e) = tr (Egg + (Bias f) (Bias §)•) = i:MSE(e^).In a companion paper, Goodnight and Wallace [1] develop tables to be usedwith the weak mean square error tests for linear restrictions in regressionmodels.In this note, we extend the weak mean square error results of Wallace[7] to include stochastic linear restrictions and show that the test sta-tistic for the hypothesis(1.2) £(§-§)' (§-6) < E(b-§)'(b-§),where B is the stochastic restricted estimator and b is the unrestrictedleast squares estimator, has a non-central F distribution with a non-cen-trality parameter of J/2, where J is the number of stochastic restrictions.2. The Statistical ModelConsider the linear statistical modelC2.1) y = XB + u,(2.2)-'' E(u) = and E(uu') = a^I,where j^ is a (T x i) vector of observations, X is a (T x K) matrix of non-- Although we make use of the classical stochastic assumptions regard-ing the disturbances, the results of this paper are also valid for the moregeneral specification 0^6, where 9 is some symmetric positive definite matrix.-3-stochastic variables of rank K, u is a (T x 1) vector of unobservable randomnormal variables, I is a unit matrix of order T, and B is a (K x I) vectorof unknown parameters.Assume, in line with the work of Theil and Goldberger [5], Theil [4],Kakwani [3] and others, the following stochastic prior information about 3:(2.3) r = R6 + V,(2.4) E(v) = 6, E(vv') = a^J^^ and ECuv') = 0,where R is a (J x K) known matrix of rank J, r is a (J x i) vector of priorestimates of R6, v is a (J x 1) random vector of unobservable normal randomvariables with mean vector 6, representing the uncertainty about the possibly2biased prior information r, and o^ is a known non- singular matrix. The2 2ratio m = o /o is also assumed known. In the discussion to follow, wedefine Q = mJ^„.Under these specifications, the stochastic restricted estimator (Theiland Goldberger [5] and Theil [4]), which makes use of the Aitken estimatorto combine the sample and stochastic prior information, is(2.5) § = [or~^X'X + a'^R'n"-^R]~-^[a"^X'j' + a"^R'Q"-^:r]= [X'X + R'P."-^R]"-^[X'y + R'f2"-^r],and the variance- covariance of B is(2.6) E(3-EB)(3-EB)' = 0^[X'X + R'fl'-^R]"-^ = Zgg.- - --^pThe unrestricted least squares estimator, using only sample information,is(2.7) b = (X'X)'^X'y,where the variance-covariance of b is(2.8) ECb-Eb)(b-Eb)' = a^CX'X)""^ = Z, ^.- - -- ObIf we make use of the mean square error criterion, and the correspond-ing expected mean square error matrix of the estimator 3, then we may writethe risk matrix as(2.9) E(6-e)(e-B)' = MSE^s = oV-"- + W"-^Rf2'-^66'fi"-^R'W""^,-- -- tipwhereW = (X'X + R'J^"'^R).The difference between all linear combinations of the expected lossfor the stochastic restricted estimator B and the unrestricted least squaresestimator b is(2.10). A^ = rE(b-§)(b-§)'^ - rE(0-$)(0-§)'fc = rMSEj^^£ - i'MSEgg^= A' (X'X + R'J^~-^R)"-^R'i"i"-^[a^(R(X'X)"-^R' + «) - 66' ] [R^"'^ (X'X+ R'n"'^R)"-^]A,which must be greater than or equal to zero if the stochastically restrictedestimator is to be superior using the MSE criterion.In order to test the compatibility of the stochastic prior a.id sampleinformation using the mean square error criterion Yancey, Judge and Bock [9],following the reasoning of Toro-Vizcarrondo and Wallace [6], developed atest statistic which has a non-central F(J,T-K,A) distribution and showedthat every linear combination of the elements of the mean square error-5-matrix of § is as small or smaller than the same linear combinations of theelements of the mean square error matrix of b if and only if the non-cen-trality parameter A is < y ,5. The Weak Mean Square Error Criterion and TestAs noted by Wallace [7], to have the mean square error of every combin-ation of I better than the corresponding linear combination of b is quite astrong requirement. Wallace [7] suggests what he terms a weak mean squareerror (WMSE) criterion as a reasonable but weaker requirement. For vectorestimation, this criterion may be stated in general as(3.1) ^2 - E(e-e)'Ce-e) = tr (Egg + (Bias §) (Bias §)') = EMSE(6.),iwhere A_ is the expected squared Euclidean distance from 9 to G.The WMSE criterion for the stochastic linear restriction estimator maybe written as(3.2) WMSE = tr E(b-§)(b-§)' - tr E(§-§) (§-§) •= tr(X'X + R'rr-^R)"-^RJ^"-^[a^(R(X'X)"-^R' + fi) - 66' ] [J^"^R(X'X+ R'fi'V'"^].In order for the stochastic restricted estimator to be superior viathis criterion, equation (3.2) must be equal to or greater than zero.From (5.2), we can write(3.3) A2 = r(X'X+ R'fi"-^R)"-^R'n"-^[a^(R(X'X)~-^R' + fi) - 66 ' ]n"'^R(X'X+ R'ft'-^R)"'^^= £'A'BAJ?.,which is positive semi-definite for all non-zero (J x 1) vectors i if andonly if(3.4) B = [a^(R(X'X)"-^R' + fi) - 66']2/is positive semi -definite.—Hence,(3.5) tr B = tr[a^(R(X'X)"-^R' + ^) - 66'] > 0.The(3.6) tr[a^(R(X'X)"-^R' + f2) - 55']can be written as(3.7) tr[o-I - 66'(R(X'X)"-^R' + fi)"'^] (R(X' X)"-^R' + fi),where (R(X'X)~ R' + il) is positive definite. Since the characteristic rootsof a positive definite matrix are ail positive, from (3,5) the(3.8) tr[a^I - 66" (R(X'X)'-^R' + fi)~-^] (R(X'X)"-^R' + ft)is equal ton(3.9) I i>. r-y'^- >j^^j jCl) i =2/ _1 _1— This can be seen since A = (X'X + R'fi R) is positive definite byassumption, and we can let the s, a non-negative (J x 1) vector in thequadratic form, in s ' Bs > equal Ail. Given either s or i, we can findi = As or s = A'^a, Clearly, s = 0, a null vector. If and only if il = 0.Thus, C is positive semi-definite if and only if A'BA is positive semi-definite since V ABM = s'Bs > for all s and I not identically zero andA'AB.AK. = sBs = for some s and Z ^ 0. Furthermore, if a matrix is posi-tive semi-definite, its trace is non-negative (Graybill [2, p. 318]), andwe have tr A'BA > if and only if tr B > 0.-7-for some ordering of if). , where (}). are characteristic roots of (R(X'X) R' + fj)and ii. the characteristic roots of [a^I -. 66' (R(X'X)"-^R' + Q)""^] (Graybill[2, p. 233]).Since the <(). are all positive and since E^., ..<!>. > for all sets of-1 2positive definite and semi-definite matrices (RCX'X) R' + fi) and [o I- 66' (RCX'X) R' + f2) ] respectively, the 'i'-r--^ are all non-negative.Furthermore, the characteristic roots of positive semi-definite matrices2 -1 -1are non-negative, [a I - 66'(R(X'X) R' + fi) ] is positive semi-definite,and tr[a^l - 66' (R(X'X)'-^R' + ^)~^] > which implies(3.10) J0^ - 6'(R(X'X)"-'-R' ^ Qy^6 > 0,or(3.11) y > -iy 6'(R(X'X)"-^R' + fi)6 = X.'^ " 20^"Hence, if the non-centrality parameter A of the distribution of y is lessJthan or equal to j, the estimator 3 is better than b when judged by theweak MSE criterion. Similarly, X > ^ if and only if E(b-B)'Cb-B) <E(|-§) •(§-§).Given this result, one can test the hypothesis that the use jf sto-chastic linear restrictions improves the estimator in a weak MSE senseover its unrestricted least squares counterpart, by testing the hypothesisX < y against the alternative X > -j and by using the tables developed byGoodnight and Wallace [1], noting whether y > T^Cy.J.T-K) at the a testlevel.4. Concluding RemarksA test for stochastic restrictions in regression which recognizes thetrade-off between bias and variance has been developed. It should be notedin closing that if we remove the assumption that m, the ratio of the vari-ances of the prior restriction and sample errors, is known, statistics arecreated whose distributions are unknown and untabled. However, for thestatistical model of this paper, preliminary sampling experiments indicatethat the non-central F distribution is a reasonable approximation to theempirical distribution when m is replaced by an estims^te, m = ^o~/a . Fin-ally, let us note that as in the equality restricted estimator case, theproperties of the preliminary test estimator,which are based on the preliminary test about X, whereandfor^(0,c)^^^ " ° ^""^ ^(c,«)^^^ " "' "^^^^ Y > c.00/ fCY)dY = a,care yet to be determined. The authors are currently working on the samp-ling properties of this and related preliminary test estimators.University of Illinois-9-References[1] Goodnight, J. and T.D. Wallace (1971). "Operational Techniques andTables for Making Weak MSE Tests for Restrictions in Regression,"Econometrica,.[2] Graybill, F.A. (1969). Introduction to Matrices with Applications inStatistics,Wadsworth Publishing Company, Belmont, California,[3] Kakwani, N.C. (1968). "Note on the Unbiasedness of a Mixed RegressionEstimator," Econometrica,Vol. 36, pp. 610-11.[4] Theil, H. (1963). "On the Use of Incomplete Prior Information in Re-gression Analysis," Journal of the American Statistical Association,Vol. 58, pp. 401-14.[5] Theil, H. and A.S. Goldberger (1961). "On Pure and Mixed Estimationin Economics," International Economic Review, Vol. 2, pp. 65-78.[6] Toro-Vizcarrondo, C. and T.D. Wallace (1968). "A Test of the MeanSquare Error Criterion for Restrictions in Linear Regression," Journalof the American Statistical Association, Vol. 63, pp. 558-572.[7] Wallace, T.D. (1971). "Weaker Criteria and Tests for Linear Restric-tions in Regression," Econometrica, ,[8] Wallace, T.D. and C.E. Toro-Vizcarrondo (1969). "Tables for the MeanSquare Error Test for Exact Linear Restrictions in Regression,"Journal of the Ainerican Statistical Association,Vol. 64, pp. 1644-93.[9] Yancey, T.A., G.G. Judge and M.E. Bock (1970). "A Mean Square ErrorTest When Stochastic Restrictions are Used in Regression," WorkshopPaper No. 7001, Department of Economics, University of Illinois,SELECTIVE CONTROL OF INDEPENDENT ACTIVITIES:LINEAR PROGRAHiilNG OF HARKOVIAN DECISIOHStHirohlde HinomotoUniversity of IllinoisUrbana, Illinois

A weak mean square error test for stochastic restrictions in regression. BEBR No. 8

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