A Note on the Efficient Estimation of the Linear ... - UMD Econ

A Note on the Efficient Estimation of the Linear Expenditure System Author(s): John C. Ham Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 73, No. 361 (Mar., 1978), pp. 208210 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2286548 . Accessed: 13/03/2013 10:05 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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of A Note on the Efficient Estimation System the LinearExpenditure JOHNC. HAM*

n

An efficient methodof estimatingthe linearexpendituresystem A testforinsuring that likelihoodis introduced. (LES) by maximum maximization satisfythe parameter estimatesobtainedby numerical thenewmethod is given.In someexperiments first-order conditions method ofestimating theLES is comparedto themoreconventional lesscomputer timethan ofestimation andis foundto usesignificantly theconventional method.

rtHt=

rtyh -

5h(mt

rtyh -

E k==1

Pktyk)+

Eht ,

(l-b)

n

i=l1

Ei + 5h = 1 ' n

KEY WORDS: Commoditydemand; Labor supply; Linear expenditure system;Maximumlikelihoodestimation.

An importantarea of economic research is the estimation of systems of commodity demand and labor supply. One of the most widely used systems is the linear expendituresystem (LES). Maximum likelihood is the natural method of estimatingthe LES. Parks (1971) proposed two proceduresforestimatingthe LES by maximumlikelihood: a Gauss-Newtonroutineand a modificationof Stone's (1954) originalproposal. Abbott and Ashenfelter(1976) chose the quadratic hill-climbing method from Goldfeld and Quandt (1972, p. 5) to maximizenumericallythe LES likelihoodfunction,while Kiefer and MacKinnon (1976) used the DavidsonFletcher-Powell(DFP) algorithmof Powell (1971) for maximizationwhen estimatingthe LES. The users of numericalmaximizationface at least two seriousproblemsno matterwhichmethodofoptimization is chosen. First, using numerical optimizationrequires largeamountsof computertime. Second, it can oftenbe verydifficult to determineif the algorithmhas actually achieved a maximum or whetherit has simplystopped on a relatively flat section of the likelihood function. The purpose of this note is to show that both of these problemsof numericalmaximizationare easily overcome in the case of the LES. A computationallysimple procedureoffering substantialsavingsin the computertime necessaryto estimate the LES is introduced.Further,a conditions methodis givenforinsuringthatthefirst-order are satisfied. The LES may be written

+

it = i=l

Eht

(l.c) (ld)

whereEit,pit,rt, Ht, and mtare the expenditureson good i, the price of good i, the wage rate, the quantity of labor supplied, and nonlabor income, respectively,in periodt. The Eit consistofbroad expenditureclasses such as durables, clothing,food, and transportation.The r's and the 0's are the parameters to be estimated by maximumlikelihood. It is possible to offeran intuitiveexplanation of the allocation process described by (l.a) and (l.b). First, view each yi as the subsistencelevel of each commodity i and yh as the maximumnumberof hours an individual can physicallywork. Next, consider an individual with two resources: his time, which he can sell (supply) at constant wage rate rt, and his nonlabor income. If r is the total numberof hours duringeach period, the consumer starts with resources, or full income, equal to mt+ rtT. The consumer firstspends part of his full income on the subsistence quantities of each of the commodities, yi, and on the subsistence quantity of leisure, Tr - h. Afterhe has made his sublsistencepurchases, the consumer allocates a fraction5i of his remaining income for additional purchases of each commodity i and a fraction5h of his remainingincome for additional purchasesof leisure. Now consider the problem of estimatingfthe LES. From (l.d) it is clear that the n + 1 error terms elt, E2t, . . . eEnt, ht must have a singularcovariance matrix. Considerthe followingestimationprocedure:First,drop one of the equations in (1.a) and (l.b), say the labor supply equation. Next, assume that the n X 1 vector of errorterms Et = (Eit E2t, . . Ent)) is distributedidentically and independentlyover time as N(O, V), where V is a positive definiten X n matrix. Finally, form the likelihood functionand obtain parameter estimates by maximizingit with respectto the n X 1 vector 5 = (51, n)' and the (n + 1) X 1 vector y= (y1, 52, ... .,

Eit =

pityi

+ Pi(mnt+

rtYh -

E

k=1

n = 1, ...n,

pktyk) +

t =1.,T,

E-t

(1.a)

* JohnC. Ham is GraduateStudent,Department of Economics, PrincetonUniversity,Princeton,NJ 08540. This workwas supUniversity portedbytheCanda (tounciland a grantto thePrinceton EconomicsDepartmentfromthe Sloan Foundation.The author wouldliketo thankRichardE. QuandtforhelDfuldiscussions.

? Journalof the AmericanStatistical Association March 1978,Volume73, Number361

208

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Theonrvand Methods Sectinn

Ham: EfficientEstimationof the LES

209

Yh)', using (1.c) to obtain the maximum Y2) * * .Yn, likelihood estimate of 5h. Of course, one could repeat the estimationprocedure,replacingone ofthe commodity demand equations in (l.a) by the labor supply equation (l.b). Fortunately,Barten (1969, pp. 25-27) has proved the followingresult: The maximumlikelihoodestimates of the parametersof a commoditydemand systemwill be independentofthe equation droppedfromthe system. In the discussionthat followsthe labor supply equation will be dropped. Denote the maximumlikelihoodestimatesof 5 and y by 0 and ly.Parks (1971) has shown that given y, 0 is an ordinary least-squares (OLS) estimator, and that given a and V, A is a generalizedleast-squares (GLS) estimator.The techniques suggestedin this article are based on Parks' result. Since the expressionsforz and will be usefulin the followingdiscussion,they will now be presented.First, A

definethe scalarsyit = Et-

Zk

Y2t, ...,

PityTi and

Xt

mt

=

+

rtyh

and define the n X 1 vector yt = (ylt, Ynt)'. Then the OLS estimator0 may be written

PktTk,

(2)

0(y) = (X xt2)-lE Xtyt . t

t

Next, definethe scalar it = Eit-

vectorWt =

(Wit,

W2t,

....,

5mt

and the n X 1

let Zt be the )W?t)'. Further,

n X (n + 1) matrix with i, jth component equal to n f + 1, and 5irt for j = n + 1, (i5i)pjt, for j wherebijis the Kroneckerdelta. Then the GLS estimator may be written A (3) Z 'V-1w ) . (5) V) = (Z tlV-1Z t)-l( t t y

Now the usual methodof estimatingthe LES proceeds as follows: Form the log-likelihoodfunctionand differentiate it with respect to V. Next, solve the first-order conditionsfor V=

(yit -

-

i'Xt)(yjt - jxt

.

(4)

Then substitute(4) back into the log-likelihoodfunction to obtain the concentratedlikelihoodfunction L* (y,) )=

T k- - ln V

(5)

wherek is a constant.Finally, maximize (5) numerically withrespectto 5 and y. Maximizing (5) will be referred to as the direct method of estimation.However, using (2), one can substitutefor5 in termsof y, concentrating the likelihoodfunctioneven further,obtaining L* (y, 5()) wherenow 1 vi=

-

E

T = L**(y) = k --ln 2 (Yit

-

( -)(t)()jt

-j()t)

| V'I '

(6) *

(7)

p for and maximizing(6) instead of (5), By substituting the number of parameters estimated by numerical

methodsis reducedfrom2n + 1 to onlyn + 1. Maximizing (6) will be described as the constrainedmethod of estimation. Afteran earlierversionof this article had been completed,I discoveredthat Deaton (1975, Ch. 4) presentsa methodof estimatingthe LES whichis equivalent to the constrainedmethod just described. However, Deaton's work differsfrom the present study in two respects. First,he does not proposea methodforinsuringthat the first-order conditionsare satisfied;while such a method is givenin the followingparagraph.Second, Deaton does not give any indicationof how much computertime is actually saved by the constrained method, while the evidence presented here suggests that the use of the constrainedmethod results in a substantial saving in computertime. Now once and z have been estimated, one must insure that they really do satisfy the first-orderconditions. To insure that satisfiesthe first-order conditions, simplysubstitutez and V2into (3) and calculate the GLS estimate y= y( , v). If = a, y then y must the first-order conditions. If has been estimated satisfy then the constrained satisfies the first-order by method, 0 If the construction. direct method of conditions (2) by estimationhas been used, one must also substitute into (2) to obtain the OLS estimate = (i).A If 5 also equals , then 0 and must satisfythe first-order conditions. In order to gain an estimate of how much computer time the constrainedmethod would actually save, the systemof Abbott and Ashenfelter(1976) was reestimated fromseveral initialvalues of the parametersby both the A

A

A

A

A

Actual Computer Time (in Seconds) Necessary to Obtain Maximum Likelihood Estimates of the LES fromDifferentInitial Values of ya

Initial valueb of

v

(1) Constrained

(2) Directc

(3)

method

method

(1)1(2)

(.45, .45,.45,.45,.45,.45, .45,.45) (.55,.55,.55,.55,.55,.55,.55,.55) (.65,.65,.65,.65,.65,.65,.65,.65) (.75,.75,.75,.75,.75,.75,.75,.75) (.80,.80,.80,.80,.80,.80,.80,.80) (.85,.85,.85,.85,.85,.85,.85,.85) (.95,.95,.95,.95,.95,.95,.95,.95) (.15, .20, .25, .30, .35, .40, .45, .50) (.50,.45,.40,.35,.30,.25,.20,.15) (.90,.85,.80,.75,.70,.65,.60,.55) (.55,.60,.65,.70,.75,.80,.85,.90) (.95, .80,.80,.90,.70,.60,.90,.80)

7.9 6.5 6.5 6.9 6.3 8.0 7.2 7.9 9.1 7.3 6.3 6.9

23.4 19.4 24.2 24.8 15.0 19.3 21.8 19.3 21.7 21.7 15.7 20.5

.34 .34 .27 .28 .42 .41 .33 .41 .42 .34 .40 .34

Column averages

6.7

20.6

.36

aParameters were estimated for the following commodity groups: durables, food, clothing,other nondurables, housing services, transportationservices, and other services as well as for labor supply. The respective maximum likelihood estimates were y= (-.922, .699, .466, .651, .763, .510, .424, 2.357) and # = (.238, .163, .134, .0254, .0755, .0997, .142, .121). The data were annual US aggregate data from 1929-1967. All equations were estimated in first difference form. I have recently learned that the original data of Abbott and Ashenfelter (1976) contain some errors. Thus the results presented in this table should be used for computational comparisons only.

bAllquantities(exceptlaborsupply)were normalizedto 1 in 1929,thusthe choice of initialy's less than1. eThe initial,8 used in the directmethodwas the averageshare vectorfromAbbott and Ashenfelter (1976,Table3) equal to (.1188,.277,.1029,.1287,.1896,.0317,.1512).

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210

Journalof the American StatisticalAssociation, March1978

constrainedand directmethod,usingthe DFP numerical optimizationprocedureof Powell (1971). For each run, the two methods were started fromthe same initial y vector. When the direct method was used, the initial 5 vector was always set equal to the vector of average commodityshares. For both methods of estimationthe actual computertimeused fromthe timethe optimization algorithmwas startedto the time that it convergedwas calculated for each set of initial parameters.The computertimesgiven forthe constrainedmethodincludethe time necessary to calculate the optimal 3 from the optimal - using (2). The experimentswere carried out on the Princeton IBM 360/91 using a programfromthe GQOPT optimization package. The GQOPT package was writtenby James Ertel, Stephen Goldfeld,and Richard Quandt at Princeton University. The computer time spent on estimationwas calculated by a subroutinewrittenby the Princeton UniversityComputer Center. The subroutineis consideredaccurate to 1/30 of a second and uses a negligible amount of computer time itself. No problems of nonconvergenceof multiple maxima occurred. From the table we see that the constrained method used on the average only .36 of the computer time the directmethodused. While these resultscannot

be considereddefinitive,it would appear that the constrainedmethodis a significantly more efficient method of estimatingthe LES. [ReceivedDecember1976. RevisedJune 1977.] REFERENCES Abbott,M., and Ashenfelter, 0. (1976),"Labor Supply,Commodity Demand,and theAllocationofTime,"ReviewofEconomic Studies, 42, 389-411. Barten,A.P. (1969), "MaximumLikelihoodEstimationof a Complete Systemof Demand Equations," EuropeanEconomicReview,1, 7-73. Deaton, A. (1975), Modelsand Projections of Demandin Post-War Britain,London: Chapmanand Hall, Ltd. Goldfeld,S.M., and Quandt, R.E. (1972), NonlinearMethodsin Amsterdam: North-Holland PublishingCo. Econometrics, Kiefer,N.M., and MacKinnon,J.G. (1976),"Small SampleProperties of Demand System Estimates," in Studies in Nonlinear Estimation,eds. S.M. Goldfeldand R.E. Quandt, Cambridge, Mass.: BallingerPublishingCo. Parks,R.W. (1971),"MaximumLikelihoodEstimationoftheLinear Expenditure System,"JournaloftheAmericanStatistical Association,66, 900-903. Powell,R.J.D. (1971),"RecentAdvancesin Unconstrained Optimization,"Mathematical Programming, 1, 26-57. Stone, R. (1954), "Linear ExpenditureSystems and Demand to thePatternof BritishDemand,"The Analysis:An Application EconomicJournal,64, 511-527.

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