Nijman, T.E.

Palm, F.C.

English

Research Papers in Economics

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EFFICIENCY GAINS DUE TO USING
MISSING DATA PROCEDURES
IN REGRESSION MODELS
Th.E. Nijman
F.C. Palm
November 1986EFFICIENCY GAINS DUE TO USING MISSING DATA
PROCEDURES IN REGRESSION MODELS
Th.E. Nijman
wr
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 earlier
version 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
2
yi - 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
yi
where
i 5
It
- Baxi - Yzi t wi.
t e Bv, f e.(~d
wi - ei .i i i
N1 and xi - dazi and Ai -
is straightforward to show yatb
Ai - 0 if
can be alternatively com-
- dasa)zi, with xi - xi and
1 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 of
SAi(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 typical
2 2 2
element 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 get
2
a
V - {(1 - r2 )-1 t ~(U-1 - 2)}.
aQ
z
2 xz
N
where QZ - plim N-1 E zi (assumed to exist), U a o2(SZQV } Q2)-1
and
1-1
r2 is the theoretical R2 of the regression (2) of x on z. The result
xz
in (6) has been obtained by Gouriéroux and Monfort [1981, expression
(11) on p. 583].
(6)
The relative efficiency of Yatb
with respect to Ya is
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). 2
1 - r
As U - 2 xz ~ ~ (8)
1 - r
yz
where r2 and rz denote the theozetical Rz's of a reqression of y on
y 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 which3
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 reparametrisation
suggested by Gouriéroux and Monfort (1981). From their results, the re-
lative efficiency of the GLS and ML estimators with respect to that of Y
a
can 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 three
magnitudes ~, u and rXZ. Equation ( 9) indicates that in terms of a gain
of asymptotic efficiency, the use of GLS is equivalent to the observation
of 100 U(1 - rX2) B of the values of xi that are actually missing. Similar
expressions 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 the
use of missing data prcx:edures instead of the complete data
only.
1- ryxz
u - 2
1- ryz
2
rXZ
Gain 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 missing
values of xi does not imply that a large part of the missing information
on x, can be recovered, because of the induced multicollinearity between
i
xi and zi in (4). Especially, when rXZ is small, the efficiency gain
obtained by usinq the appropriate estlmators can be substantial. The
value of u is crucial for the efficíency of Ya}b. The loss of efficiency4
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 into
account 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
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Testing the mean of an asymmetric population: Johnson's modified T
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Aegrippen rondom "kwaliteit"MI~A~ÍÍ~~~~I~~N~~~~VIIII

A continuous deformation algorithm on the product space of unit simplices

A note on the identification of restricted factor loading matricea

A Scenario for Sequential Experimentation

A two-level planning procedure with respect to make-or-buy decisions, including cost allocations

Alessie and Arie Kapteyn Consumption, Savings and Demography

Alternative specífication tests for Tobit and related modelsiv IN

An algorithm for the linear complementarity problem with upper and lower bounds

An extension of Kónig's theorem to graphs with no odd-K4

Analyzing simulation experiments with common random numbers

Arthur van Soest Non-convex budget sets, institutional constraints and imposition of concavity in a flexibele household labor supply model.

Baken An inventory model: what is the influence of the shape of the lead time demand distributton?

Baken The quality of some approximation formulas in a continuous review inventory model

Belastingstructuur en privatisering

Consistent estimation of rational expectations models

Cristina Pennavaja Periodization approaches of capitalist development. A critical survey

de Beer BIJLAGEN bij Voorspelprestaties van het Centraal Planbureau in de periode 1953 t~m

de Freytas De invloed van demografische factoren en ínkomen op consumptieve uitgaven

de Gevel The theory of wage differentials: a correction

de Leeuw The rank of reduced dispersion matrices

De loonvergelijking Afwenteling van collectieve lasten door loontrekkera?

de Zeeuw Non-cooperative strategies for dynamic policy games and the problem of time inconsistency: a comment

Decision Support Systems ( DSS), en de kleren van de keizer ...

den Elzen and Dolf Talman A new strategy-adjustment process for computing a Nash equilibrium in a noncooperative more-person game

der Genugten Asymptotic Normality of Least Squares Estimators in Autoregressive Linear Regression Models

der Laan A price adjustment for an economy with a block-diagonal pattern

der Ploeg Consístency of conjectures and reactíons: a critique

Gerards Homomorphisms of graphs to odd cycles

Groenendaal Jackknifing estímated weighted least squares

Groof Ge3soleerde versus gecoSrdineerde economische politiek in een tweeregiomodel

Groof Internationale coSrdinatie van economische politiek in een tweeregio-twee-sectoren model.

Hempeníus The relation between dividends and profits

Huys en E.N. Kertzman Effectieve belastíngtarieven en kapitaalkosten

Jacobi-type algorithms for eigenvalues on vector- and parallel computervi

Kapteyn Estimatíon of a~;ame theoretic model of household labor supply

Kapteyn Estimation of the error components model with incomplete panels

Kapteyn Habit Formation and Interdependent Preferences in the Almost Ideal Demand System

Kooreman A Microeconometric Analysis of Vacation Behavior

Meijboom Horizontal mixed decomposition

Mulder On Superimposed Recurrent Cycles

Nagelkerke Labour Relations, Networks, Job-creatíon and Regional Development A view to the consequences of technological change

Nunen Economische politiek ín een twee-sectoren-model

On the problem of missing data in linear models&quot;,

Optimal dynamic investment policy under financial restrictions and adjustment costs

Peter Kooreman Comment on 'Microeconometric Demand Systems with Binding Non-Negativity Constraints: The Dual Approach'

Peters Price Inertia in a Macroeconomic Model of Monopolistíc Competition

Plasmans Maximum Likelihood Estimation of Econometric Models with Rational Expectations of Current Endogenous Variables

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Regression analysis of simulatíon experiments: functional software specification

Sameh's parallel eigenvalue algorithm revisited

Selecting random number seeds

Signed Graphs - Regular Matroids - Grafts

Simplicial algorithms for solving the non-linear complementarity problem on the simplotope

Simulation with too many factora: review of random and groupscreening designs

Standridge Experimental design and regression analysis in simulation: an FMS case study

Storcken Preferences revealed by the choice of friends

Talman A continuous deformation algorithm on the product space of unit simplices

Talman Adjustment processes for finding economic equilibria

Talman Adjustment processes for finding equílibria on the simplotopeiii

Testing Korteweg's rational expectations model for a small open economy

Testing the mean of an asymmetric population: Johnson's modified T test revisited

The 2-ray algorithm for solving equilibrium problems on the unit simplex

The firm's ínvestment policy under a concave adjustment cost function

van Groenendaal Regression analysis of factorial designs with sequential replicationV

van Lieshout A generalisation and aome properties of Markowitz' portfolio selection method

van Nunen Freia versus Vintaf,

Variance heterogeneity in experimental design

Víngerhoets Fabrication of copper and copper semis in developing countries. A review of evidence and opportunities.

Voorspelprestaties van het Centraal Planbureau in de periode

Wansbeek Consistent sets of estimates for regressions with correlated or uncorrelated measurement errors in arbitrary subsets of all variables

Yamamoto a globally convergent simplicial algorithm for stationary point problems on polytopes

Efficiency gains due to using missing data procedures in regression models

Efficiency gains due to using missing data procedures in regression models

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