Nijman, T.E.

Palm, F.C.

English

Research Papers in Economics

Q~:~~l~ Discussion
uuiiiiuuiiuuuiiiiiiiuimiiuuiiiumiNo. 8902
GENERALIZED LEAST SQUARES
ESTIMATION OF LINEAR MODELS
CONTAINING RATIONAL FUTURE
EXPECTATIONS
by Theo Nijman and Franz Palm
January, 1989GENERALIZED LEAST SQUARES ESTIMATION OF LINSAR MODELS
CONTAINING RATIONAL FUTURE EXPECTATIONS
Theo Nijman ~
Franz Palm ~~
ABSTRACT
We discuss the choice of approximations for unobserved expectations
underlying consistent eatimators ín linear RE models with future expec-
tations. We ahow how estimators whích are more efficient than the commonly
used GMM estimators can be obtained íf it ia asaumed that the future expec-
tation depends on a fíníte aumber of variables only. Numerical reaulta for
a aimple model illustrate the relative efficiency of various estimatora.
Dept. of Econometríca, Tílburg University, P.O. Box 90153, 5000 LE
Tilburg, The Netherlands.
~~ Dept. of Economics, University of Limburg, P.O. Box 616, 6200 MD
Maastrícht, The Netherlands.- 2 -
1. INTRODUCTION
Models with expectational variablea are widely used ia empirical econo-
metric research. Various estímators have been put forward to eatimate the
parametera in these models (aee e.g. Pesaran (1987)). Many of these esti-
matora belong to the general clasa of GMH eatimatora proposed by Hanaen
(1982). If a GMM estimator is used, the unobaerved expectationa are
approximated by the correapondíng realization following a auggestion of
McCallum (1976). For atatic models an alternatíve referred to as the
aubstitution approach by Hickena (1982) coneiets in fitting an auxiliary
regression and approxímating the unobserved expectation by the projection
from the auxiliary equatíon. In thia note we show how to approximate the
future expectation by the projection from an auxilíary regreseion and
obtain a generalized least squarea eatimator (GLS) that is at least ae
efficíent as the GMM eatimator based on future realizationa as proxiea í'or
the futute expectations, provided one ia wílling to asaume that the future
expectation depends on a fínite number of variablea only.
Although both the GLS estímator defíned ín this way and the G2~! estimators
are not fully efficient, in general they have many advantagea over the
efficíent maximum likelihood (ML) eatimator. For inatance, theq do not
requíre a fully specified model and are therefore expected to be more
robust with respect to apecification uncertainty than full information
methods. Moreover they are often computationally much more attractive than
ML estimation.
The paper ia organized as follows. In aection 2 we íntroduce the GLS eati-
mator and compare it with GMM eatimatora. In aectioa 3 numerical reaults
on the relatíve efficiency of the various estimatora illustrate the argu-
ment. Finally aection 4 contains aome concluding remarks.2. THE GLS ESTIMATOR
Consider the following linear model
Yt ' P E[Ytallltj t b'xt t Et, ft ~ IN(O,OÉ), (1)
P
xt ~ E ri xt-í t vt, vt ~ IN(0,~), (2)
í-1
assume that Et and vs are independent for all t and a, that vt ia indepen-
dent of xt-1, xt-2, ... and define IL -{Yt~ xt~ Yt-1~ xt-1~ ..-}~ Equation
(1) describes a standard RE model with future expectation while equation
(2) states that the k-dimenaíonal vector of exogenous variablea is
generated by a vector autoregressíon, possibly with reatrictions on the
parameter matríces ri. If xt ia atatíonary and ~p~ i 1 the model has a
unique stationary aolution which can be characterized by
P
Yttl ~ E~i xt-i t ut,
i-0
(3)
where ut is índependent of xt, xt-1, etc. and ~i is often a highly nonli-
near function of the structural parameters in (1) and (2). By conatruction
the error term ut satiafies
E utut-k ~ 0 if k~ 0; E
ut ~ ou '
E utEt-k - 0 if k~ 1; E utEt-1 ~ OE.
(4)
P
As (3) implies E[yttlllt) ' E~i xt-i, ML eatímatíon comea down to joint
i'G ~i
estimation of (2) andP
Yt -(P~YO ' s' )xt t P E V't xt-i t Et
i-1
(5)
imposing all the restríctions on the II's which can be computationally very
demanding. Even if these restrictions are neglected simultaneous eatima-
tion of ( 5) and ( 2) may not be computationally very attractive.
The class of GMM estímators is based on the substitution of (3) ínto (1)
which ylelds
Yt - P Yt,l t S~xt t Et - Put (6)
from whích p and b can be consistently estimated using IV methods because
xt, xi-1, ... are orthogonal to Et and ut. As Et - put is autocorrelated
(see (4)), the standard IV estímators can be improved upon by takíng into
account the properties of thia error term as proposed by Cumby, Huizinga
and Obstfeld (1983) and Hayashí and Sims (1983) fot the present línear
model and by Hansen (1982) for the general case. Note that all GMM eatima-
tors are based on the fact that xt-k (k 2 0) is orthogonal to yttl -
ElYtallItJ but do not use the restrictions on (3).
An alternative class of estimators atarts by accounting for the excluslon
reatrictíons on the regressíon ín (3) by writíng
Yttl ` zt4 t ut, (7)
where zt ís the vector of all elements of (xt, .. , xt- ) wíth nonzero
P
coefficient in (3) and 7f is a corresponding vector of parameters. As
E[yttllIt] - ztif, a natural approximation for the unobaerved expectationof yttl which can be used instead of the approximation yttl used in (6) is
given by z'nt, where n is the OLS estimator in (7). Thia approximation
yíelds
yt - P zin . b'xt t Et t Pzt(n-n). (e)
If p and 6 are estímated from (8) using OLS the resulting eatímator
obviously coincides with the IV estimator from (6) when zt is the vector of
instruments because of the two stage least squares ínterpretation of IV
estimators. The parameters in (8) however can alternatively be eatimated
by GLS which yields a more efficíent estímator. In obvious vector notation
(8) reads as
P
y-[z(z'z)-lz'yt I xl t E- P z(z'z)-lz'u.
b
(9)
In the appendix we present expressíons for the inverse of the covariance
matrix of E- p Z(Z'Z)-1Z'u which can be used to obtain the GLS estimator.
Of course unknown parameters in the covatiance matrix have to be replaced
by consistent estimates, but these can easíly be obtained and do not affect
the limiting dístríbution.
An ímportant point to note is that the GLS estimator ís at least as effi-
cient as any GMM estimator which is based on the orthogonality conditions
in (6) only. This ís true because the estímator proposed by Cumby,
Huizinga and Obstfeld ( 1983) can be interpreted as an estimator based on
premultiplication of (9) by Z' and therefore cannot be more efficient than
GLS on (9). Moreover ít is well known ( see e.g. Hansen ( 1982)) that a GMM
estimator based on (6) only, cannot be more effícient than the estimator
proposed by Cumby et.al. if the number of instruments used ín that eatima-- 6 -
tor tends to infinity. The GLS estimator can of course be mora efficient
than GMM estimators because the zero restrictíons ín (3) are exploited. A
GMM estimator whích simultaneously impoaes the orthogonality reatrictiona
in (3) and (6) in a bivariate model will probably be as efficient as the
GLS estimator, but thís estimator is no longer computationally
attractive.
3. NUMERICAL EXAMPLE
In order to illustrate the argument in the previous section we present
numerical results on the relatíve asymptotic efficiency of the varíous
estimators for the vety símple case where p- 2 and k- 1, with k being the
number of exogenous variables in (1). It can be easíly checked that in
this case
np - b(Y1 t PYZ) {1 - PY1 - P2yz~-1
nl - bYy {1 - PY1 - P2y2}-1 (l0)
and
ut ' Ettt t(P1rp . 6) vttl~
where lower case letters Y and 4 indicate that we consider a scalar case.
Using (6) it ís straightforward to evaluate the asymptotic variancea of the
estímators. Moreover ít can be shown that the relative efficiency does not
depend on all six parameters in the model but
E(yt-Et)ZIEyi only.
on p, yl, YZ and RZ -
In table 1 the asymptotic efficiency of four estimatora of p and b ia pre-
sented. The effícíency is defíned as the ratio of the large sample
variance of an estimator over that of the ML estimator. The fírst estima-
tor consídered is the IV estimator when xt and xt-1 are used as ínstrumentsin (6). It ís evident that increasíng the number of ínstruments doea not
change the asymptotic varíance of the IV estimator. The second eatímator
applied to (6) has been proposed by Cumby et.al. (1983), denoted by CHO,
with xt, xt-1 and xt-2 being the instruments. In thíe case increasing the
number of instruments would lower the asymptotic variance. Irreapective of
the number of instruments, in thia model, the CHO eatimator will be leea
efficient than the third estlmator that we conaíder whích has been proposed
by Hayashí and Sims (1983), HS. It is the efficient eatimator withín the
class of GMM estimators based on the orthogonality restrfctíone in (6)
only. Fínally in table 1 we present the relative effíciency of the GLS
estimator proposed in aectíon 2 when it ia a priori asaumed that p- 3. The
GLS eatímetor based on the more reatrictíve asswoption that p- 2(whích
coincides with the data generating procesa) ís in this apecial case fully
efficient and therefore the relative efficiency of this estímator ía not
presented ín the table.
Table 1. Relative efficíency of the maximum likelíhood estimator
compared with alternative estimators for p and b if k-1, p-2 and p- 0.9.
Re1.Eff. p Re1.Hff. ó
R2 Y1 Y2 IV CHO HS GLS IV CHO HS GLS
.5 1.2 -.35 1.87 1.45 1.32 1.19 2.01 1.53 1.37 1.22
.5 1.4 -.45 2.30 1.62 1.30 1.13 2.43 1.69 1.33 1.15
.5 1.5 -.56 2.29 1.63 1.34 1.17 2.45 1.71 1.39 1.19
.5 1.7 -.72 2.91 1.89 1.32 1.12 3.06 1.96 1.34 1.13
.9 1.2 -.35 1.23 1.18 1.18 1.17 1.23 1.18 1.17 1.16
.9 1.4 -.45 1.46 1.30 1.28 1.23 1.46 1.30 1.28 1.23
.9 1.5 -.56 1.35 1.24 1.22 1.20 1.33 1.22 1.21 1.19
.9 1.7 -.72 1.54 1.31 1.27 1.20 1.51 1.30 1.25 1.19- 8 -
efficíency between the varioua estimators can be conaíderable. Ia thie
example, the asymptotic variance of the CHO estímator with xt-2 included as
additíonal lnstrument ís subatantially smaller than that of the standard IV
estimator. Moreover it pays either to use more instruments in the com-
putation of the CHO estimator or to use the HS estimator. Finally it is
clear from table 1 the GLS estimator propoaed in section 2 can in turn be
substantially more efficient than the HS esiimator.
7. CONCLUSIONS
In this note we showed how the assumption that a future expectation dependa
on a fínite number of varíablea only can be exploited to increase the effi-
ciency of simple consistent estimators. In the example, the ML estimator
appeared to be only about 20 percent more effícient asymptoticallq than the
GLS estimator which is computationally more easy to ímplement. Thia
finding suggests that ín empirical wozk, it ie more appropriate to approxi-
mate the unobserved future expectatione by a conditional expectation based
on a finíte number of past observations and then apply GLS than to aubati-
tute the future observatíon and apply aome IV method. The result can be
extended to more general models where for example lagged eadogenous
variables are included in (2).Cumby, R.E., Huizinga, J. and M. Obstfeld ( 1983), "Two-step two-stage
least squares estimation in modela with rational expectatíons",
Journal of Econometrics, 21, p. 333-356.
Hansen, L.P. (1982), "Large sample properties of generalized methods of
moments estimators", Econometríca, 50, p. 1029-1054.
Hayashi, F. and C. Sims (1983), "Nearly effícient estimatíon of time series
models with predetermined, but not exogenous, inatrumenta",
Econometrica, 51, p. 783-798.
McCallum, B.T. (1976), "Rational expectatíons and the estimation of econo-
metric models : an alternatíve procedure", Econometrica, 17, p.
484-490.
Pesaran, M.H. (1987), "The Limits to Rational Expectations", Basíl Black-
well, Oxford.
Wickens, M.R. (1982), "The efficient estímation of econometric models
with rational expectationa", Review of Economic Studiea, 49,
p. 55-67.- lo -
APPENDIX : Details on the computation of the GLS eatímator.
In thia appendíx we eho~r hov to find a reighting matrix vrhich can be usad
to compute the GLS eatimator deacribed in aectíon 2. If (9) is re~rrittea
ae
Y- W 9 t v, (Al)
where W. [Z(Z'Z)-iZ'y~ ~X], 9' -(p,b') and vr - E- pZ(Z'Z)-iZ'u, the
problem is to fínd a matrix ~'i euch that
I~T (W'~-iW)-1 W'~-iM ' N(~~ Plim T(i1'i~-li1)-1)r (A2)
vhere T denotes the sample eize. A matzíx Si-1 Nhích is appropriate ín
thís respect ia tha invezae of the covariance matrix Si of v asauming that Z
is determiniatic,
Si - OÉIT - p0É {2(Z'Z)-iZ'itZ-i(Z'Z)-12'} t p20u Z(Z'Z)-iZ', (A3)
if unknown parameters are replaced by conaiatent eatímates. As S2 ia a
(TxT) matríx, ít ia
to defíne
S2 . S21 . S22S2-1St'
3 2
computationally attractive aot to invert S2 directly but
Sii - OÉIT - OÉOuZZ-1(Z'Z)-1Z~1
~Z - Z - p-iQEQuZZ-i
~9 - p-Zau2Z'Z
(A4)
and to uae the matrix inversíon lemma- li -
(~1 t~Zn31Q2)-i ~ S2ií - S2iiQZ(Si31 t SiZS2i1S22)-1S22S2ii (AS)
which only reyuires inversion of (kxk) dimensional matrices if the matríx
inverslon lemma ia applíed i n a similar way to lnvert (ii.Discussion Paper Series, CentER, Tilburg University, The Netherlands:
No. Author(s)
8801 Th. van de Klundert
and F. van der Ploeg
8802 J.R. Magnus and
B. Pesaran
8803 A.A. Weber
8804 F. van der Ploeg and
A . J . de 'I.eeuw
8805 M.F.J. Steel
8806 Th. Ten Raa and
E.N. Wolff
8807 F. van der Ploeg
8901 Th. Ten Raa end
P. Kop Jansen
8902 Th. Nijman and F. Palm
Title
Fiscal Policy and Finite Lives in Interde-
pendent Economies with Real end Nominal Wage
Rígidity
The Bias of Forecasts from a First-order
Autoregression
The Credibility of Monetary Policies, Policy-
makers' Reputation and the EMS-Hypothesis:
Empirical Evidence from 13 Countries
Perfect Equilibrium in a Model of Competitive
Arms Accumulation
Seemingly Unrelated Regression Equation
Systems under Diffuse Stochastic Prior
Information: A Recursive Analytical Approach
Secondary Products and the Measurement of
Productivity Growth
Monetary and Fiscal Policy in Interdependent
Economies with Capital Accumulation, Death
and Population Growth
The Choice of Model in the Construction of
Input-Output Coefficient Matrices
Generalized Least Squares Eatimation of
Linear Models Containing Rational Future
Expectationsonv nni~~ ~nnn i G TII RI IRr TH~ NFTHFRLANDS
Bibliotheek K. U. Brabant ~iui~u~~u~iw~r~~ri~did

Generalized least squares estimation of linear models containing rational future expectations

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