Tilburg University

Efficiency gains due to using missing data procedures in regression models

Nijman, T.E.; Palm, F.C.

Publication date:1986

Link to publication in Tilburg University Research Portal

Citation for published version (APA):Nijman, T. E., & Palm, F. C. (1986). Efficiency gains due to using missing data procedures in regression models.(Research memorandum / Tilburg University, Department of Economics; Vol. FEW 240). Unknown Publisher.

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EFFICIENCY GAINS DUE TO USINGMISSING DATA PROCEDURES

IN REGRESSION MODELS

Th.E. NijmanF.C. Palm

November 1986

EFFICIENCY GAINS DUE TO USING MISSING DATAPROCEDURES IN REGRESSION MODELS

Th.E. Nijmanwr

F.C. Palm

november 19f36

.

In the chapter on "Economic Data Issues" of the Handbook of Econometrics,

Griliches (1986) analyzes the asymptotic variance of an estimator of a

reqression coefficient using imputations for the missing regressor values

and he compares it with that of an estimation procedure based on the com-

plete observations only. His derivation of an expression for the relati-

ve efficiency is íncorrect. In this note, we qive the correct result and

show that the relative efficiency of three estimators desiqned to handle

incomplete samples depends on parameters that have a atraightforward sta-

tistical interpretation. In terms of a gain of asymptotíc efficiency,

the use of these estimators is equivalent to the observation of a percen-

tage of the values which are actually missing. This percentage depends on

three R2-measures only, which can be straiqhtforwardly computed in applied

work. Therefore it should be easy in practice to check whether it is

worthwhile to use a more elaborate estimator.

The authors thank Professor Z. Griliches for his comments on an earlierversion of this note.

t

rt

Department of Econometrics, Tilburq University, P.O.B. 90153,

5000 LE Tilburg, The Netherlands.

Department of Economics, University of Limburg, P.O.B. 616,6200 MD Maastricht, The Netherlands.

1

Griliches (1986) considers the following regression model

2yi - Rxi t yzi t ei . ei ~ IN(0,o ),

and

xi - dzi t vi , vi ~ IN(O,Q~),

where the regressors xi and zi are assumed to be independent of the

correspondinq disturbances ei and vi. Actually, the normality

assumption is not made by Griliches, but it will be required below

for maximum likelihood (ML) estimation and it does not affect the

results for the other estimators. The variables yi and zi are ob-

served for i- 1, ...~N1 t N2, whereas xi is observed for i- 1,

(1)

(2)

... N1 only.Besides the OLS estimator of the reqression of yi on xi and z1 for

i- 1, ... N1 only, denoted by ~a and ya, Griliches ( 1986) considers

an estimation procedure in which the missing xi's are replaced by

d z., where d is the OLS estimate of d in ( 2) usinq the first N1 ob-a i a -servations. The estimate yatb is subsequently computed by OIS on

yiwhere

i 5It

- Baxi - Yzi t wi.

t e Bv, f e.(~dwi - ei .i i iN1 and xi - dazi and Ai -

is straightforward to show yatb

Ai - 0 if

can be alternatively com-

- dasa)zi, with xi - xi and1 otherwise.

that

puted by OLS of yi on xi and zi

yi - Bxi t yzi t{ei t A1Bvi t Bei(d - da)zi}-

Contrary to what is stated by Griliches ( 1986), the contribution ofSAi(d - da)zi to the asymptotic variance of yatb is not negliqible

if plim N2 N-1 - À~ 0 for N-~ ~, with N~ N1 } N2.

As the three components of the disturbance c~ (4) are independent,

the large sample distributíon of yatb is given by

`IN (Yatb - Y) ~ N(O,V)a

(3)

(4)

with V~ plim N(X'X)-1{X'StX } X'WS2 E W'X} (X'X)-1, (5)

where X is the matrix of regressors in (4), W is a vector with typical2 2 2element Aizi, Sl is a diagonal matrix with typical element Q f 9is av,

and E is the asymptotic variance of da.

After some alqebra, we get2a

V - {(1 - r2 )-1 t ~(U-1 - 2)}.aQ z

2 xz

Nwhere QZ - plim N-1 E zi (assumed to exist), U a

o2(SZQV } Q2)-1 and1-1

r2 is the theoretical R2 of the regression (2) of x on z. The resultxz

in (6) has been obtained by Gouriéroux and Monfort [1981, expression

(11) on p. 583].

(6)

The relative efficiency of Yatb with respect to Yais

Avar(vNY )Eff(Y ) - -a{b - 1 t ~(U-1 - 2)(1 - r2 ). (7)atb Avar ( v N Ya) xz

Accordinq to (7), using imputed values as in (3) leads to a gain of

efficiency compared with using complete observations only if )t ~},1 - ~

which is more stringent than the condition U~ 2-~ given by

Griliches (1986). Both conditionsrrequire that the unpredictable part

of x from z is not too important relative to Q2, the overall noise

level of (1). 21 - r

As U - 2xz ~ ~ (8)

1 - ryz

where r2 and rz denote the theozetical Rz's of a reqression of y ony xz yz

respectively x and z and on z only, it is obvious that a sufficient con-

dition for an efficiency gain is ry Xz ~}, i.e. the predictible pazt of

y is small.

As noted by Griliches (1986) and others, an efflciency qain is assuzed

if (4) is estimated by a generalized least squares (GLS) method which

3

takes the correlation structure of the disturbance in (4) into account.Again, the term AiB(d - da)zi cannot be neqlected (see Palm and Nijman

(1982) and Nijman and Palm (1985)). Alternatively, the fully efficient

ML estimator can be computed, e.q. using the convenient reparametrisationsuggested by Gouriéroux and Monfort (1981). From their results, the re-

lative efficiency of the GLS and ML estimators with respect to that of Yacan be obtained

Eff(YGLS) -

1 - a)1(1 - rXZ)

and Eff(Y~) - 1- ay(1 - rXZ) - 2aU(1 - u)rxz'

(9)

(10)

The relative efficiency in (7), ( 9) and ( 10) only depends on the threemagnitudes ~, u and rXZ. Equation ( 9) indicates that in terms of a gainof asymptotic efficiency, the use of GLS is equivalent to the observationof 100 U(1 - rX2) B of the values of xi that are actually missing. Similarexpressions can be obtained from (7) and (10) for Yatb and Y~ respecti-vely. The values in Table 1 illustrate this result.

TABLE 1 Percentage of missing observations that are regained by theuse of missing data prcx:edures instead of the complete dataonly.

1- ryxzu - 21- ryz

2rXZGain in percentage points for- - -

' Yatb YGLS YML

.3 .2 -106 24 32

.3 .8 - 27 6 40

.6 .2 27 48 58

.6 .8 7 12 50

.9 .2 71 72 76

.9 .8 18 18 32

Note that a good fit in (2) yielding a"qood proxy" for the missingvalues of xi does not imply that a large part of the missing information

on x, can be recovered, because of the induced multicollinearity betweenixi and zi in (4). Especially, when rXZ is small, the efficiency gainobtained by usinq the appropriate estlmators can be substantial. Thevalue of u is crucial for the efficíency of Ya}b. The loss of efficiency

4

can be important when u ~}. This lose incr~ases ae rXZ decreasea.Finally, íf u is close to one, i.e. xi is not very important in ex-plaining y in equation (1), all three approaches which take intoaccount the incomplete data, yield about equally efficient estimators.

5

References

Gouriéroux, C., and A. Monfort (1981), "On the problem of missing data

in linear models", Review of Economic Studies, 48, 579-586.

Griliches, Z. (1986), "Economic data issues", in Z. Griliches and

M.D. Intriligator, eds, Handbook of Econometrics, North

Holland, Amsterdam, 1466-1514.

Nijman, Th.E., and F.C. Palm (1985), "Consistent estimation of a regression

model with incompletely observed exogenous variable",

Netherlands Central Bureau of Statistics, unpublished paper.

Palm, F.C., and Th.E. Nijman ( 1982), "Linear regression using both tem-

porally aggregated and temporally disaggregated data",

Journal of Econometrics, 19, 333-343.

i

IN 1985 REEllS VERSCHENEN

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

172 B.B. van der GenugtenAsymptotic Normality of Least Squares Estimators in AutoregressiveLinear Regression Models

173 R.J, de GroofGe3soleerde versus 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 inconsistency: a comment

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

178 N.J. de BeerVoorspelprestaties 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 ínkomen op consumptieveuitgaven

180 P. Kooreman, A. KapteynEstimatíon of a~;ame theoretic model of household labor supply

l81 A.J. de 'Lceuw, A.C. MeijdamOn Expectatíons, Information and Uynamic Game liquílibria

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183 J.P.C. Kleíjnen, 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. GerardsHomomorphisms of graphs to odd cycles

186 P. Bekker, A. Kapteyn, T. WansbeekConsistent sets of estimates 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 PloegConsístency of conjectures and reactíons: a critique

189 E.N. KertzmanBelastingstructuur en privatisering

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

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V

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M I ~ A ~ÍÍ~~~~I~~N~~~~ VII II

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