CBMR

76261985201

IINIINIIIII IIIIIIIIIIIIIIIIIII IIIIIII~InNVIIsubfaculteit der econometrie

RESEARCH MEMORANDUM

LBURG UNIVERSITY-PARTMENT OF ECONOMICS

stbus 90153 - 5000 LE Tilburgtherlarxis

FEW

201

~~~~. ~~.~.~~~` B;x,~~~~~ ~E~K .

~~i-u~.1~~~

Alternative specification tests for

Tobit and related models

rwPeter Kooreman~ r~`(S~~ )

December 1985 Tilburg UniversityDepartment of Econometrics

P.O. Box 90153

5000 LE Tilburg

The Netherlands

~) Helpful comments by Arie Kapteyn are gratefully acknowledged.

Table of contents page

Abstract1. Introduction 1

2. The Small-Hsiao specification test 2

3. An empirical illustration 4

References 6

Abstract

Recently Small and Hsiao (1985) proposed a specification testfor the multinomial logit model as an alternative for the Hausman speci-fication test. We show that Small and Hsiao's test procedure can also beapplied to Tobit and related models.

1

1. Introduction

Under the assumption of no misspecification, the parameters ofthe Tobit model can be consistently estimated by either maximizing theTobit likelihood function or by maximizing the truncated regressionlikelihood function. The latter estimator is ínefficient relative to theformer since it ignores the information contained in the zero-obaerva-tíons.

As has been set forth by Ruud (1984), the availability of thispair of estimators can be used to construct a Hausman test for misapeci-fication. A practical problem with the Hausman test is that it requiresthe inveraion of a matrix which is often nearly singular. As a conse-quence, the test result can be shaky due to computational inaccuracies.

In this note we show that the test procedure proposed by Small

and Hsiao (1985) within the context of the multinational logit model can

serve as an alternative which avoids the computational problems of the

Hausman test.

z

2. The Small-Hsiao specification test

Small and Hsiao are interested in testing the validity of the

multinomial logit model as a specification in discrete choíce analysis.

Suppose we observe a discrete choice from a choice set C for each obser-

vation. Let D be a subset of the orginal choice set C. If the multi-

nomial logit specification is correct (including its "independence of

irrelevant alternatives" property) the parameters can be conaistently

estimated either by maximizing the likelihood function over the complete

sample, or by maximízing the conditional likelihood function based on

probabilities conditional on choices within D.t) The latter estimator is

inefficient relative to the former since it ignores the information con-

tained in the observatíons for which a choice from D a C- D is ob-

served.Small and Hsiao propose the following 'paeudo likelihood ratio

test'. Divide the total sample randomly into two parts A and B of(asymptotically) equal sizes NA and NB. Let A~ and p~ be estimates forthe k vector of unknown parameters 6, obtained by maximizing the res-

pective loglikelihood functions LA and LB, and let

eóB 3 (~2)eó t (1-}rz)so (1)

Let OB be the estimate obtained by maximizing the conditíonal loglikelihood function over the NB observations of subsample B whosechoices belong to the restricted choice set D. Let

e - -2[LB(ooB) - LB(oB)] (2)

The authors prove that e is asymptotically XZ distributed with k degrees

of freedom.A common feature of this test and the Hausman test ís that they

do not require specifying an explicit alternative model. Unlike theHausman test statistic, the Small-Hsiao test statistic cannot assumenegative values.

1) Some parameters may not be identified from the restricted choice set.

3

Obviously, there is a strong similarity between testing themultinomial logit model and testing the Tobit model. In both cases theconsistent but inefficient estimator is based on a subsample where theendogenous variable only assumea valuea within a restricted range. Inthe proof of their proposition Small and Hsiao firat observe that

~ a (OB-O~B)'HB(OB)(OB-OÓB) (3)

ueing a Taylor Series expansion of LB around gB; HB is the nega:iveHessian of LB. Approximation (3)likelihood function LB. Next, it

B "B "ABtic variance of J N .(01-8~ )

sufficient to show that the asympto-holds for any twice differentiable logis

equals the

This part of the proof only requires thatasymptotic limit of (HB~NB)-1~the consistent but inefficient

eatimator is exclusively based on a part of the observations on whichthe efficient estimator is based. Hence, the Small-Hsiao test procedurecannot only be applied to the multinomial logit model but also to a muchwider range of models, particularly Tobit and related models.

4

3. An empirical illustration

In Kooreman and Kapteyn (1986) the followingswitching household labor supply model was estimated:

~kf m Bf(wm~wf.P.u) f ef

Rf a Rf} if 1Cf ~ T

Rm a gm(wm~wf.P~u) f Em

Rf ~ T} if Rf ~ T

Rm ~ gm(wm.wf.P.v) f Em

endogenoua

(4)

(5)

(6)

(7)

(8)

Rm and Rf denote male and female leisure, wm and wf are the male andfemale wage rate, respectively, p is the price of consumption and u isnon-labor family income; T is total time endowment. gm and gf are demandfunctions of the Almost Ideal Demand System and ef, em and Em are errorterms. The shadow wage wf is implicitly defined by

T ~ gf(wm.wf~P~V) (9)

Equation (4) determines whether the female partner worka in apaid job (Rf ~ T) or not (q,f ~ T) (all males are assianed to work). Equa-tions (5) and (6) describe the number of hours worked by the partners ina two earner household. Equation (8) describes the male labor supply ina household with a non-participating female.

An interesting hypothesie to test is whether this model iscapable of simultaneously describing theae different aspects of thehousehold's labor supply behavior. This seems to be an ideal applicationof the generalized Small-Hsiao specification test.

The model has been eatimated twice. First only data on houae-holds where both partners worked ín a paid job were employed (197 obaer-vations). Maximizing the appropriately conditioned likelihood function

5

yielded a consistent (but inefficient) estimator. Next, an efficientestimator was obtained by maximizing the complete líkelihood functionover the 197 two earner households plus 310 households where only themale partner worked in a paid job.

We first compared the two estimators using a Hauaman test. The

difference of the two relevant covariance matrices (cf. Ruud (1985), eq.

(2.9)), however, did not satisfy the requirement of being positive semi-

definite. The guaranteed positive semi-definite version was computed at

48.0 (Ruud, eq. (2.10)). The Small-Hsiao and the interchanged 3ma11-

Hsiao (obtained by reversing subsets A and B) test statistics are 46.5

and 53.8, respectively. So, in contrast to the empirical results repor-

ted by Small and Hsiao, the differences between the test statistics are

quite small. All statistics reject equality of the two sets of estimates

at any reasonable level of significance.l'2)

1) Under the null hypothesis all statistics follow a X~~~) distribution.

2) In finite samples, the two test statistics could lead to conflictinginferences; see Small and Hsiao for a discussion.

6

References

Kooreman, P. and A. Kapteyn (1986), "Eatimation of Rationed and Unra-

tioned Household Labour Supply Functions Using Flexible Functíonal

Forms", The Economic Journal, forthcoming.

Ruud, P. ( 1984), "Tests of Specification in Econometrics", EconometricReviewa, 3, pp. 211-276.

Small, K.A. and C. Hsiao (1985), "Multinomial Logit Specification

Tests", International Economic Review, 26, pp. 619-627.

i

IN 1984 REEDS VERSCHENEN

138 G.J. Cuypers, J.P.C. Kleijnen en J.W.M. van RooyenTesting the ;~[ean of an Asymetric Population:Four Procedures Evaluated

139 T. Wansbeek en A. KapteynEstimation in a linear model with serially correlated errors whenobservations are missing

140 A. Kapteyn, S. van de Geer, H, van de Stadt, T. WansbeekInterdependent preferences: an econometric analysis

141 W.J.H. van GroenendaalDiscrete and continuous univariate modelling

142 J.P.C. Kleijnen, P. Cremers, F. van BelleThe power of weighted and ordinary least squares with estimatedunequal variances in experimental design

143 J.P.C. KleijnenSuperefficientexperiments

estimation of power functione in simulation

144 P.A. Bekker, U.S.G. PollockIdentiEication of linear atochastic models with covariancerestrictions.

145 Max D. Merbis, Aart J. de ZeeuwFrom structural form to state-space form

146 T.M. Doup and A.J.J. TalmanA new variable dimension simplicial algorithm to find equilibria onthe product space of unit simplices.

147 G. van der Laan, A.J.J. Talman and L. Van der HeydenVariable dimension algorithms for unproper labellings.

148 G.J.C.Th, van SchijndelDynamic firm behaviour and fínancial leverage clienteles

149 M. Plattel, J. PeilThe ethíco-political and theoretical reconstruction of contemporaryeconomic doctrines

150 F.J.A.M. Hoes, C.W. VroomJapanese Businesa Policy: The Cash Flow Trianglean exercise in sociological demystification

151 T.M. D~R, G. van der Laan and A.J.J. TalmanThe (2 ~1-2)-ray algorithm: a new simplicial algorithm to computeeconomic equilibria

ii

IN 1984 REEDS VERSCHENEN (vervolg)

152 A.L. Hempenius, P.G.H. MulderTotal Mortality Analysis of the Rotterdam Sample of the Kaunas-Rotterdam Intervention Study (KRIS)

153 A. Kapteyn, P. KooremanA disaggregated analysis of the alloca[ion of time within thehousehold.

154 T. Wansbeek, A. KapteynStatistically and Computationally Effícient Estimation of theGravity Model.

155 P.F.P.M. NederstigtOver de kosten per ziekenhuisopname en levensduurmodellen

156 B.R. MeijboomAn input-output like corporate model including multipletechnologies and make-or-buy decisiona

157 P. Kooreman, A. KapteynEstimation of Rationed and Unrationed Household Labor SupplyFunctions Using Flexible Functional Forms

158 R. Heuts, J. van LieshoutAn implementation of an inventory model with stochastic lead time

159 P.A. BekkerComment on: Identification in the Linear Errors in Variables Model

16U P. MeysFuncties en vormen van de burgerlijke staatUver parlemen[arisme, corporatisme en autoritair etatisme

161 J.P.C. Kleijnen, H.M.M.T. Denís, R.M.G. KerckhoffsEfficient estimation of power functiona

162 H.L. TheunsThe emergence of research on third world tourism: 1945 to 1970;An introductory essay cum bibliography

163 F. Boekema, L. VerhoefDe "Grijze" sector zwart op witWerklozenprojecten en ondersteunende instanties in Nederland inkaart gebracht

164 G, van der Laan, A.J.J. Talman, L. Van der HeydenShortest paths for simplicial algorithms

165 J.H.F. SchilderinckInterregional structure of the European CommunityPart II:Interregional input-output tables of the European Com-

munity 1959, 1965, 1970 and 1975.

iii

IN (1984) REEDS VEKSCHENEN (vervolg)

166 P.J.F.G. MeulendijksAn exercise in welfare economics (I)

167 L. Elsner, M.H.C. PaardekooperOn measures of nonnormality of matrices.

iv

IN 1985 REEDS VERSCHENEN

168 T.M. Doup, A.J.J. TalmanA continuous deformation algorithm on the product space of unitsimplices

169 P.A. BekkerA note on the identification of restricted factor loading matrices

170 J.H.M. Donders, A.M. van NunenEconomische politiek i n een twee-aectoren-model

171 L.H.M. Bosch, W.A.M. de LangeShift work in health care

172 B.B, van der GenugtenAsymptotic Normality of Least Squares Eatimators in AutoregreasiveLinear Regreasion Models

173 R.J. de GroofGeisoleerde veraus gecoSrdineerde economische politiek in een twee-regiomodel

174 G. van der Laan, A.J.J. TalmanAdjustment processes for finding economic equilibria

175 B.R. MeijboomHorizontal mixed decomposition

176 F. van der Ploeg, A.J. de ZeeuwNon-cooperative strategies for dynamic policy games and the problemof time inconsietency: a comment

177 B.R. MeijboomA two-level planníng procedure with respect to make-or-buy deci-sions, including cost allocationa

178 N.J. de BeerVoorspelpreataties van het Centraal Planbureau in de periode 1953t~m 1980

178a N.J. de BeerBIJLAGEN bij Voorspelprestaties van het Centraal Planbureau in deperiode 1953 t~m 1980

179 R.J.M. Alessie, A. Kapteyn, W.H.J. de FreytasDe invloed van demografische factoren en inkomen op consumptieveuitgaven

180 P. Kooreman, A. KapteynEstimation of a game theoretic model of household labor supply

181 A.J. de Zeeuw, A.C. MeijdamOn Expectations, Information and Dynamic Game Equilibría

V

182 Cristina YennavajaPeriodization approaches of capitalist development.A critical survey

183 J.P.C. Kleijnen, G.L.J. Kloppenburg and F.L. MeeuwsenTesting the mean of an asymmetric population: Johnson's modified Ttest revisited

184 M.O. Nijkamp, A.M. van NunenFreia versus Vintaf, een analyse

185 A.H.M. GerardsHomomorphiams of graphs to odd cycles

186 P. Bekker, A. Kapteyn, T. WansbeekConsistent sets of estímates for regressions with correlated oruncorrelated measurement errors in arbitrary subsets of allvariables

187 P. Bekker, J. de LeeuwThe rank of reduced dispersion matrices

188 A.J. de Zeeuw, F. van der PloegConsistency of conjectures and reactions: a critique

189 E.N. KertzmanBelastingstructuur en privatisering

190 J.P.C. KleijnenSimulation with too many factors: review of random and group-screening designs

191 J.P.C. KleijnenA Scenario for Sequential Experimentation

192 A. DortmansDe loonvergelijkingAfwenteling van collectieve lasten door loontrekkers?

193 R. Heuts, J. van Lieshout, K. BakenThe quality of some approximation formulas in a continuous reviewinventory model

194 J.P.C. KleijnenAnalyzing símulation experiments with common random numbers

195 P.M. KortOptimal dynamic investment policy under financial restrictions andadjuetment costs

196 A.H, van den Elzen, G. van der Laan, A.J.J. TalmanAdjustment proceases for finding equilibria on the simplotope

vi

197 J.P.C. KleijnenVariance heterogeneity in experimental design

198 J.P.C. KleijnenSelecting random number seeds in practice

199 J.P.C. Kleijnenftegression analysis of simulation experiments: functional softwarespecífication

200 G. van der Laan and A.J.J. TalmanAn algorithm for the linear complementarity problem with upper andlower bounds

~ iiu i iiwP'uiiNNUii dïi iiii~~i~i~ i~iàu u i u