❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡
Pós✲●r❛❞✉❛çã♦
❡♠ ❊❝♦♥♦♠✐❛
❞❛ ❋✉♥❞❛çã♦
●❡t✉❧✐♦ ❱❛r❣❛s
◆
◦✸✸✸
■❙❙◆ ✵✶✵✹✲✽✾✶✵
❖♣t✐♠❛❧ ■❱ ❊st✐♠❛t✐♦♥ ♦❢ ❙②st❡♠s ❲✐t❤
❙t♦❝❤❛st✐❝ ❘❡❣r❡ss♦rs ❛♥❞ ❱❛r ❉✐st✉r❜❛♥❝❡s
❲✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❉②♥❛♠✐❝ ❙②st❡♠s
❈❛r❧♦s ▼❛rt✐♥s ❋✐❧❤♦✱ ❉❛✈✐❞ ▼✳ ▼❛♥❞②
❆❣♦st♦ ❞❡ ✶✾✾✽
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
▼❛rt✐♥s ❋✐❧❤♦✱ ❈❛r❧♦s
❖♣t✐♠❛❧ ■❱ ❊st✐♠❛t✐♦♥ ♦❢ ❙②st❡♠s ❲✐t❤ ❙t♦❝❤❛st✐❝ ❘❡❣r❡ss♦rs ❛♥❞ ❱❛r ❉✐st✉r❜❛♥❝❡s ❲✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❉②♥❛♠✐❝ ❙②st❡♠s✴ ❈❛r❧♦s ▼❛rt✐♥s ❋✐❧❤♦✱ ❉❛✈✐❞ ▼✳ ▼❛♥❞② ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✸✸✸✮
■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
OPT I M A L I V EST I M AT I ON OF SY ST EM S W I T H ST OCH A ST I C R EG R ESSOR S A N D VA R D I ST U R B A N C ES
W I T H A PP L I CAT I ON S T O D Y N A M I C SY ST EM S
Dav id M . M andy * and
Car l os M ar t ins-F il ho* *
* University of Missouri
** Oregon St at e University and Funda»c~ao Get ulio Vargas - FGV
February 14, 2000
A bst r ac t . T his paper considers t he general problem of Feasible Generalized L east Squares I nst rument al Variables (FGL S IV ) est imat ion using opt imal inst rument s. First we summarize t he su± cient condit ions for t he FGL S I V est imat or t o be asympt ot ically equivalent t o an opt imal GL S IV est imat or. T hen we specialize t o st at ionary dynamic syst ems wit h st at ionary VA R errors, and use t he su± cient condit ions t o derive new moment condit ions for t hese models. T hese moment condit ions produce useful I V s from t he lagged endogenous variables, despit e t he correlat ion bet ween errors and endogenous variables. T his use of t he informat ion cont ained in t he lagged endogenous variables expands t he class of I V est imat ors under considerat ion and t hereby pot ent ially improves bot h asympt ot ic and small-sample e± ciency of t he opt imal I V est imat or in t he class. Some M ont e Carlo experiment s compare t he new met hods wit h t hose of Hat anaka [1976]. For t he DGP used in t he M ont e Carlo experiment s, asympt ot ic e± ciency is st rict ly improved by t he new I V s, and experiment al small-sample e± ciency is improved as well.
Keywords. Dynamic Models, IV Est imat ion, VAR Errors.
JEL Classi¯ cation C30.
David M. Mandy Depart ment of Economics
118 Professional Building University of Missouri Columbia, MO 65211 USA Voice 573-882-1763; Fax 573-882-2697
e-mail Mandy@econ.missouri.edu
Carlos Mart ins-Filho Praia de Bot afogo, 190 - 11o. Andar Escola de Pos-Graduacao em Economia - EPGE
Funda»c~ao Get ulio Vargas - FGV Rio de Janeiro, RJ 22253-900
BRAZIL
Voice 55-21-536-9250; Fax 55-21-536-9450 e-mail cmart ins@fgv.br
* * Research support for C. M art ins-Filho from CNPq, PRONEX and t he Federal Universit y of Cear¶a, Brazil is grat efully acknowledged. We t hank A man Ullah, an anonymous referee and L ee A dkins for helpful comment s. T he aut hors ret ain responsibilit y for any remaining errors.
1. I nt r oduct ion
Mandy and Mart ins-Filho [1994 and 1997] provide su± cient condit ions for asympt ot ic equivalence of
feasible generalized least squares (FGLS) inst rument al variables (IV) and generalized least squares (GLS)
IV est imat ors. T here t he focus was on t he st ruct ure of t he error covariance mat rix, and due t o t he generality
of t he model lit t le at t ent ion was given t o inst rument design.
Here we not e t hat a minor rest at ement of our earlier condit ions is su± cient for asympt ot ic equivalence of
FGLS IV and GLS IV when t he inst rument s are opt imal. T his observat ion is new in t hat t he condit ions are
general, applying t o many familiar covariance st ruct ures, and t he est imat ors are opt imal in t he class of IV
est imat ors under considerat ion.
We t hen apply t he condit ions t o st at ionary dynamic syst ems wit h st at ionary VAR errors. T he su± cient
condit ions allow us t o expand t he class of IV est imat ors under considerat ion by ident ifying new moment
condit ions t hat enable use of (t ransformed) lagged endogenous variables as IVs, despit e t he presence of VAR
errors in t he dynamic syst em. T his raises t he prospect of bot h asympt ot ic and small sample e± ciency gains
relat ive t o t he IV est imat ors t hat have been considered t o dat e. For a part icular dat a generat ion process
(DGP), we show t hat a st rict improvement in asympt ot ic e± ciency is obt ained by using our new IVs, and
experiment al result s for t he DGP suggest t hat small-sample e± ciency is improved as well. T he result s
di®er from earlier lit erat ure, not ably Dhrymes and Taylor [1976] and Hat anaka [1976], in t hat t he e± ciency
propert ies do not depend on speci¯ c dist ribut ional assumpt ions but inst ead are at t ained in a class of IV
est imat ors.
Sect ion 2 discusses su± cient condit ions for FGLS IV est imat ors t o be opt imal IV est imat ors for some
given set of IVs. Sect ion 3 reviews est imat ion of error VAR(1) nuisance paramet ers t hat accommodat es
general st ochast ic regressors. Sect ion 4 examines dynamic models and proposes IVs t hat sat isfy t he su± cient
condit ions from Sect ion 2, t hereby proposing new opt imal IV est imat ors for dynamic models wit h VAR errors.
Sect ion 5 present s some Mont e Carlo comparisons of t he new e± cient IV est imat ors wit h each ot her and
2. Suf f icient Condit ions f or A sympt ot ic Equival ence of GL S I V and FGL S I V using Opt imal I nst r ument s
Consider t he linear regression model
y = x¯ + u; E (u) = 0; E (uu0) ´ - (µ): (1)
Here y is a T £ 1 st ochast ic observable vect or, x is a T £ K st ochast ic observable mat rix, ¯ is a K £ 1
nonst ochast ic unknown paramet er vect or t o be est imat ed, u is a T £ 1 st ochast ic unobservable error vect or,
and µ is a n £ 1 nonst ochast ic vect or of unknown nuisance paramet ers. When t he condit ional mean E (ujx)
is nonzero IV est imat ion is required. Whit e [1984, Chapt er VI I] est ablished t he asympt ot ic equivalence of
GLS IV and FGLS IV for some st andard forms of - (µ). Mandy and Mart ins-Filho [1994 and 1997] proposed
general condit ions on - (µ) t hat are su± cient for t his asympt ot ic equivalence, given by:
(A1.1) - (µ)¡ 1 has at most W < 1 dist inct nonzero element s for every T , denot ed g
w T(µ) for w = 1; : : : ; W .
T hat is, t here are T2¡ W element s t hat are eit her zero or duplicat es of ot her nonzero element s in
- (µ)¡ 1. For each w, g
w T(µ) converges uniformly as T ! 1 t o a real-valued funct ion gw(µ) on an open
set S cont aining t he t rue value of µ, at which gw is cont inuous.
(A1.2) T he number of nonzero element s in each column (and row) of - (µ)¡ 1is uniformly bounded by N < 1
as T ! 1 .
Corresponding t o a T £ ¹K ( ¹K ¸ K ) mat rix D of IVs are t he opt imal inst rument s
z(µ) ´ - (µ)¡ 1D (D0- (µ)¡ 1D )¡ 1D0- (µ)¡ 1x;
which are used t o form t he GLS IV est imat or ^¯ (µ) ´ (z(µ)0x)¡ 1z(µ)0y: St andard asympt ot ic behavior of
^
¯ (µ) and asympt ot ic equivalence of t he FGLS IV est imat or ^¯ ( ^µ) (for some consist ent est imat or ^µ) wit h ^¯ (µ)
require t hat t he IVs D possess some basic propert ies. From (A1), let Ii w T be t he index set of element s in
row i of - (µ)¡ 1 t hat are equal t o g
w, for w = 1; : : : ; W . In t his not at ion, t he IVs D are assumed t o sat isfy:
(A2.1) p1 TD
0- (µ)¡ 1u!d N (0; Q
D u) for some symmet ric ( ¹K £ ¹K ) mat rix QD u.
(A2.2) plim
T ! 1 1 TD
0- (µ)¡ 1D = Q
D D, a ¯ nit e nonsingular mat rix.
(A2.3) plim
T ! 1 1
(A2.4) Each IV Di h can be expressed as Di h = ¸0h´i h, where ¸h is a vect or of ¯ xed ¯ nit e dimension t hat is
Op(1) and const ant across i , and ´i h has uniformly bounded fourt h moment s and fourt h cross moment s
wit h xj q for i ; j = 1; 2; : : : ; h = 1; : : : ; ¹K ; and q = 1; : : : ; K .
(A2.5) PTi = 1P j 2 Ii w T Di huj = Op(T1=2) for h = 1; : : : ; ¹K and w = 1; : : : W .
Under (A1) and (A2) t here is a FGLS IV est imat or t hat is opt imal wit hin t he class of IV est imat ors t hat
are based on D , provided we have available a consist ent est imat or of t he nuisance paramet ers.
L emm a 1. Assume (A1) and (A2). If ^µ!p µ t henpT ( ^¯ ( ^µ) ¡ ^¯ (µ)) !p 0:
T he proof is an ext ension of Mandy and Mart ins-Filho [1994 and 1997] and is available on request from
t he aut hors. Condit ion (A1) applies t o VAR errors and cert ain forms of het erocedast ic (including random
coe± cient s) and panel errors, but does not apply t o moving average errors.
3. VA R Er r or s
Now we specialize t he model t o a syst em of T observat ions on G equat ions wit h VAR(1) errors,1 Y =
X B + U; where Y is a T £ G mat rix of endogenous variables, X is a T £ K mat rix of (possibly st ochast ic)
regressors, B is a K £ G mat rix of unknown paramet ers t hat incorporat es any exclusion rest rict ions, and U
is a T £ G mat rix of VAR(1) errors. Let t ing t he it h row and column of an arbit rary mat rix M be M i ¢and
M¢i, respect ively, we assume t hat :
(A3) Ut ¢= Ut ¡ 1¢R + Vt ¢for t = 0; § 1; § 2; : : : ; where
(A3.1) R is a G £ G mat rix of nuisance paramet ers wit h absolut e eigenvalues less t han one,
(A3.2) V0
t ¢» I I D (0; § ) for some symmet ric ¯ nit e posit ive de¯ nit e G £ G nuisance mat rix § ,
(A3.3) Vt ¢has ¯ nit e absolut e fourt h moment s, and
(A3.4) t here exist s a bound ¹B such t hat E (Xt ¡ h ;iXt jX¿¡ h ;iX¿j) · B and E (X¹ t ¡ h ;iX¿¡ h ;iUt `U¿`) · B ;¹
for every t; ¿ = 1; 2; : : : ; h = 0; 1; 2; : : : ; i ; j = 1; : : : ; K ; and ` = 1; : : : ; G.
T he aut ocovariance funct ion of Ut ¢is denot ed ¡ (h) = E (Ut ¡ h ¢0 Ut ¢) for h = 0; § 1; § 2; : : : . Assumpt ion (A3)
implies Ut ¢is bot h st rict ly st at ionary and covariance st at ionary (Anderson [1971], pp. 372-378). Hencefort h
1I t is unnecessary t o consider higher order error VA Rs since a VA R(p) process can always be writ t en as a VA R(1) process
we express t he error VAR as U = ¢ UR + ¢ ¢0V + (I
T ¡ ¢ ¢0)U for
¢ = 2 6 6 6 6 6 4 0 1 0
1 . .. . .. ... 1 0 3 7 7 7 7 7 5 :
In t his model, t he nuisance vect or µ of t he previous sect ion consist s of vec R and t he unique element s
of § . Given t hat E (UjX ) 6= 0, consist ent est imat ion of µ requires a T £ K inst rument mat rix C (possibly
di®erent from D ) which we assume sat is¯ es
(A4.1) t he eigenvalues of 1 TC
0X possess a uniform lower bound in probability, and
(A4.2) C0U = O
p(T1=2).
Est imat ion of t he nuisance paramet ers is st raight forward from (A3) and (A4).
L emm a 2. Assume (A3) and (A4), and let ~¡ (h) = 1 T
P T
t = h + 1U~t ¡ h ¢0 U~t ¢, where ~U = (IT ¡ X (C0X )¡ 1C0)Y:
T hen ~¡ (h) !p ¡ (h): In part icular, ~R ¡ R = Op(T¡ 1=2) and ~§ p
! § , where ~R = ~¡ (0)¡ 1¡ (1) and ~~ § =
~
¡ (0) ¡ ~R0¡ (1).~
T he st andard proof from, for example, Fuller [1976] must be modi¯ ed only slight ly t o accommodat e t he
nonzero condit ional mean. Det ails are available on request from t he aut hors. Not e t hat ~R converges fast er
t han op(1). T his fact is used in t he design of new IVs in Sect ion 4.
For checking assumpt ions (A1) and (A2) it is convenient t o st ack t he syst em by observat ion, t hereby
expressing t he model in t he not at ion of Sect ion 2 wit h dependent vect or y = vecY0, regressor mat rix
x = X - IG, paramet er vect or ¯ = vec B0, and error vect or u = vecU0. In t his not at ion t he VAR(1) inverse
error covariance is - (µ)¡ 1= P0(I
T - §¡ 1)P , where P is t he VAR(1) t ransformat ion mat rix given by
P = 2 6 6 6 6 6 6 4
A 0 : : : : : : 0 ¡ R0 I
G 0 : : : ...
0 ¡ R0 I
G . .. ...
..
. . .. ... ... 0 0 : : : 0 ¡ R0 I
G 3 7 7 7 7 7 7 5
= ((IT ¡ ¢ ¢0) - A) ¡ (¢ - R0) + (¢ ¢0- IG):
Here A is a lower t riangular mat rix such t hat AE (U0
derive an explicit expression for - (µ)¡ 1 t hat is useful for verifying (A1) and (A2):
- (µ)¡ 1= P0(IT - §¡ 1)P
= [(IT ¡ ¢ ¢0) - A0§¡ 1A] + [¢ ¢0- §¡ 1] ¡ [¢ - §¡ 1R0] ¡ [¢0- R§¡ 1] + [¢0¢ - R§¡ 1R0]
= 2
6 6 6 6 6 6 4
A0§¡ 1A + R§¡ 1R0 ¡ R§¡ 1 0 : : : 0
¡ §¡ 1R0 §¡ 1+ R§¡ 1R0 ¡ R§¡ 1 . .. ...
0 . . . . .. . .. 0
..
. . .. ¡ §¡ 1R0 §¡ 1+ R§¡ 1R0 ¡ R§¡ 1
0 : : : 0 ¡ §¡ 1R0 §¡ 1
3
7 7 7 7 7 7 5
: (2)
4. I V s f or Est imat ion of Dynamic Simul t aneous Equat ion M odel s w it h VA R Er r or s
T he availability of IVs is cont ext -dependent and can only be invest igat ed under some assumpt ion on t he
source of t he nonzero condit ional mean. Perhaps t he most obvious cont ext is a simult aneous equat ion model.
T he usual 2SLS and 3SLS IVs can be expect ed t o sat isfy (A2) under t radit ional assumpt ions when t here is
only cont emporaneous error correlat ion, an observat ion t hat only adds t o t radit ional result s for simult aneous
equat ion models by including t hem as special cases of well-behaved st ochast ic regressors. When errors sat isfy
(A3) t hree problems emerge. First , (A1) must be checked. As not ed in Mandy and Mart ins-Filho [1994],
inspect ion of (2) veri¯ es t hat (A1) is indeed sat is¯ ed by VAR errors. Second, we must derive IVs t hat
sat isfy (A2). T he usual 2SLS and 3SLS IVs su± ce if t here are no lagged endogenous variables,2 but if
Y = X B + U is a dynamic simult aneous equat ion model t hen, as is well-known from Wallis [1967 and 1972],
t he lagged endogenous variables as IVs int eract wit h t he VAR errors t o make t radit ional 2SLS and 3SLS
inconsist ent . We demonst rat e here t he ut ility of st at ing t he su± cient condit ions generally, in t he form of
(A1) and (A2), by using t hese condit ions t o develop new IVs t hat overcome t his problem. T he new IVs
are derived from lagged endogenous variables and t hereby provide a heret ofore unnot iced way of using some
informat ion from t he lagged endogenous variables for opt imal IV est imat ion of B . T hird, we must derive IVs
t hat sat isfy (A4). Following Wallis [1967 and 1972], t his is usually st raight forward provided t he st ruct ural
model cont ains exogenous variables.
Let t ing X = [ S Y0 Y¡ 1 ¢¢¢ Y¡ q] and B0= [ £0 B00 B10 ¢¢¢ Bq0], we have a dynamic simult
ane-ous equat ion model writ t en in normalized st ruct ural form Y = S£ +P qL = 0Y¡ LBL+ U; where subscript s on
2T his st at ement may appear at odds wit h t he ¯ nding by Turkingt on [1998] t hat when R is unknown a di®erent I V set must
t he endogenous mat rix Y indicat e lags, S is a T £ K0mat rix of variables not det ermined endogenously by t he
syst em under considerat ion (but perhaps st ochast ic), and £ and BL are K0£ G and G £ G unknown
param-et er mat rices, respect ively, t hat incorporat e normalizat ion and exclusion rest rict ions (so K = K0+ (q+ 1)G).
We assume t hroughout t hat all G equat ions are ident i¯ ed by t hese exclusion rest rict ions, t he st ochast ic
process St ¢is covariance st at ionary wit h St ¢independent of V¿¢for t · ¿, t he aut oregressive process in Yt ¢is
covariance st at ionary, and t hat (A3) holds. Since (A1) is sat is¯ ed by VAR errors and S (and lags t hereof )
can be used for (A4), t he cent ral issue is deriving IVs t hat sat isfy (A2).
4.1 A ssum pt ion (A 2) .
Usually t he IVs in a simult aneous equat ion model t ake t he form H - IG for some (T £ ^K ) mat rix H ,
where ¹K ´ K G. For example, H = [ S^ Y¡ 1 ¢¢¢ Y¡ q] yields t he usual 3SLS IVs. T his along wit h (2)
allows us t o st at e (A2) more simply for t he present model in t erms of H rat her t han D . To obt ain t he
simpli¯ cat ion not e ¯ rst from (2) t hat t here are ¯ ve dist inct nonzero G £ G blocks in - (µ)¡ 1, and t hat t he
upper left block appears only once and t herefore is of no consequence in (A2). Ignoring t his block, we can
subst it ut e D = H - IG, x = X - IG, (2) for - (µ)¡ 1, and P u = P vecU0= vecV0in (A2) t o rewrit e t he
assumpt ion as
(A2.1') A Cent ral Limit T heorem (CLT ) applies t o p1 T
¡
H0- §¡ 1¡ H0
¡ 1- R§¡ 1
¢ vec V0.
(A2.2') plim
T ! 1 1
TH0HL is a ¯ nit e nonsingular mat rix for lags L = ¡ 1; 0; 1.
(A2.3') plim
T ! 1 1 TH
0X
L is a ¯ nit e mat rix of full column rank3 for lags L = ¡ 1; 0; 1.
(A2.4') Each element of H can be expressed in t he speci¯ ed form.
(A2.5') H0U
L = Op(T1=2) for lags L = ¡ 1; 0; 1.
Assumpt ions (A2.1'){ (A2.4') hold for t he usual 3SLS IVs in dynamic models sat isfying t he assumpt ions
of t he present sect ion. T he problem wit h including lags of Y in t he IV set is t hat t hey violat e (A2.5'). So t he
cent ral problem in designing new IVs based on lags of Y is ¯ nding a t ransformat ion for t he lags t hat dest roys
t he asympt ot ic correlat ion between t hem and UL, t hereby int roducing addit ional ort hogonality condit ions
t hat can be exploit ed in IV est imat ion.
3W hen t he model is a simult aneous equat ion model (A 2.3' ) must be rest at ed t o re° ect t he ident ifying exclusion rest rict ions,
using a regressor mat rix obt ained from t he generic form X - IG by dropping t he known zeros in ¯ and t he corresponding columns of X - IG. T hen t he product in (A 2.3' ) is of full column rank since we have assumed t he model is ident i¯ ed by
4.2 Tr ansfor m i ng L agged Endogenous V ar i abl es.
Consider t he set © of all T £ T mat rices t hat can be writ t en as a ¯ nit e sumP i ri¢ti, where ri are real
numbers, ti are nonnegat ive int egers, and ¢0 ´ IT. T his set is a commut at ive ring whose addit ion and
mult iplicat ion operat ions are st andard mat rix addit ion and mult iplicat ion and whose mult iplicat ive ident ity
is IT. St acking t he error VAR (by equat ion) yields
[(IG - IT) ¡ (R0- ¢ )]vecU = (IG - ¢ ¢0)vecV + (IG- (IT ¡ ¢ ¢0))vecU; (3)
and A ´ (IG - IT) ¡ (R0- ¢ ) is a G £ G mat rix over ©. Alt hough A convert s vecU int o whit e noise,
(A2.5') requires t hat each column of U be asympt ot ically uncorrelat ed wit h t he IVs. So t he problem of
const ruct ing IVs from lags of Y can be st at ed as a problem of ¯ nding a t ransformat ion ¦ over © such t hat
¦ U is asympt ot ically uncorrelat ed wit h Y¡ L. T his is accomplished by solving (3) over © for U¢i, which
involves t he det erminant of A over © and t herefore includes ¢` for ` = 0; : : : ; G. To obt ain zero correlat ion,
we must ensure t hat t he t ime index on t he lags of Y are prior t o t he t ime index on Vt ¢. T hus for IVs ¦ 0Y¡ L
we require L > G.
To see t his formally let IG = IG - IT denot e t he G £ G ident ity mat rix over © and, in order t o avoid
confusing mat rix mult iplicat ion between mat rices over © wit h ordinary mat rix mult iplicat ion, use ¯ t o
denot e mat rix mult iplicat ion between mat rices over © and ¢ t o denot e mult iplicat ion between a scalar in
© (i.e., a T £ T mat rix) and a mat rix over ©. We cont inue t o denot e ordinary mat rix mult iplicat ion
by t he absence of an operat or. Since © is commut at ive, t he ordinary rules for mat rix inversion apply
t o mat rices over © (see Bourbaki [1989] I I x8.7). In part icular, (det A) ¢IG = A¤ ¯ A; where (det A)
and A¤ are t he det erminant and adjoint of A over ©, respect ively. It is st raight forward t o verify t hat
B ¯ C = B C for any conformable mat rices B and C over ©, and also t hat (det A) ¢IG = IG - (det A),
so IG - (det A) = A¤A: Ordinary mat rix premult iplicat ion of (3) by A¤ and t hen subst it ut ion of t he last
equality yields (IG- (det A))vecU = A¤[(IG- ¢ ¢0)vecV + (IG- (IT¡ ¢ ¢0))vecU]: In t erms of t he individual
columns of U t hat appear in (A2.5'), t his is
(det A)U¢i = [ K1i : : : KG i][(IG- ¢ ¢0)vecV + (IG- (IT ¡ ¢ ¢0))vec U]
=
G
X
j = 1
where Kj i is t he (j ; i ) cofact or of A over ©. Equat ion (4) expresses U¢i in t erms of only t he whit e noise V
(except for t he presence of (IT ¡ ¢ ¢0)U¢j, which only involves t he ¯ rst observat ion U1¢). Since t he coe± cient
(det A) t ransforms U¢i int o whit e noise, an est imat e of t his mat rix using ~R provides a t ransformat ion t o be
used on lags of Y t hat result s in no asympt ot ic correlat ion between t he IVs and U¢i. T his is exact ly what is
required by (A2.5').
T hus t he proposal is t o include ~¦ 0Y
¡ L in H for lags L > G, where ~¦ is (det A) wit h ~R replacing R.
T hese IVs can be calculat ed by not ing t hat each T £ T element of (IG- IT) ¡ ( ~R0- ¢ ) in © is eit her of
t he form IT ¡ ~R` `¢ or of t he form ¡ ~R` k¢ , so from t he de¯ nit ion of t he det erminant we may express t he
t ransformat ion mat rix as ~¦ = det ((IG- IT) ¡ ( ~R0- ¢ )) =
P G
` = 0(¡ 1)`r~`¢`; where ~r` is t he sum of all `t h
order principal minors of ~R0(and ~r
0´ 1 for not at ional convenience). Hence t he new IVs are
~ ¦ 0Y¡ L =
G
X
` = 0
(¡ 1)`r~`(¢0)`Y¡ L = G
X
` = 0
(¡ 1)`~r`Y` ¡ L: (5)
T hese IVs sat isfy (A2') for L > G.
T heor em . Assume (A3) and (A4). T hen H = ~¦ 0Y
¡ L sat is¯ es (A2') for L > G.
Proof. In t he Appendix.
T he t heorem shows t hat it is possible t o incur no asympt ot ic cost t o not knowing t he VAR error paramet ers
while st ill using some informat ion from t he lagged endogenous variables, t hereby expanding t he class of IVs
used in est imat ion and pot ent ially improving bot h asympt ot ic and small-sample e± ciency. Not e t hat t he
FGLS IV est imat ors considered in t his sect ion are possibly asympt ot ically e± cient relat ive t o t he est imat ors
in Dhrymes and Taylor [1976] or Hat anaka [1976], since t he e± ciency propert ies of t heir est imat ors rely on
more st ringent dist ribut ional assumpt ions t han t hose made herein.
Not e ¯ nally t hat , while P0y
¡ L or P0(Y¡ L - IG) have int uit ive appeal as IVs since P uh is whit e noise,
t hese IVs do not in fact sat isfy (A2.5'). For (A2.5'), D must embody a t ransformat ion of lagged endogenous
variables t hat result s in zero asympt ot ic correlat ion wit h each individual column of Uh. T he P t ransformat ion
does not meet t his requirement since P only t ransforms t he entire vector uh = vecUh0 int o whit e noise.
5. M ont e Car l o Compar isons
on various set s of IVs wit h each ot her and wit h Hat anaka's [1976] est imat ors in a dynamic simult aneous
equat ion model wit h VAR(1) errors. T he DGP is Ericsson's [1991] two-equat ion model wit h t he error
st ruct ure modi¯ ed t o be VAR(1), as in Hendry and Harrison [1974]. Hence
Yt 1= £11St 1+ Yt 2B02 1 + Yt ¡ 1;1B11 1 + Ut 1
Yt 2= £22St 2+ £32St 3+ £42St 4+ Yt 1B01 2 + Ut 2
Ut ¢= Ut ¡ 1¢R + Vt ¢;
whereVt ¢isgenerat ed ast hebivariat e mixed normal Vt ¢0= ¸tat+ (1¡ ¸t)bt.4 Here, ¸t is binomial wit h E (¸t) =
1
2; bt » N (0; ¥ ); at » N (0; 31¥); ¸t; at; and bt independent ; and ¥ =
· §
1 1
16 ¥12
¥12 161
¸ .
Following Ericsson and Hendry and Harrison St ¢ is generat ed by St ¢ = St ¡ 1¢¤0+ Et ¢; where Et ¢0 »
N I D (0; ª ), and t he following paramet er values are ¯ xed for all experiment s: ¤ = diagf 0:8; 0:7; 0:4; 0:2g,
£22= £32= £42= £11= 1, B01 2 = 0:3, and
ª = 2
6 4
0:25 0:237 0 0 0:237 0:25 0 0
0 0 0:49 0
0 0 0 0:49
3
7 5 :
We also use Ericsson's values for B02 1 2 f ¡ 0:5; 0:3g, B11 1 2 f ¡ 0:4; 0:2; 0:7g, and T 2 f 20; 40; 80g. T he values
chosen for t he error VAR paramet ers are R 2 ½·
:6 :6 :2 :2 ¸
; ·
:6 :2 :2 :6 ¸
; ·
:2 :6 :6 :2
¸ ¾
, t aken from Guilkey and
Schmidt [1973], and §112 f 0:25; 1; 4g, t aken from Hendry and Harrison [1974]. For each value of §11, ¥12
is set so t hat t he correlat ion between Vt 1 and Vt 2 is 0.5, as in Ericsson, and Vt ¢is designed so t hat §22 = 1
for all experiment s, also as in Ericsson.5
T hus, t here are 54 experiment s for each sample size T , and for each experiment we use Ericsson's speci¯
-cat ion of N = 80; 000=T repli-cat ions. Not e t hat Et ¢is independent of ¸t, at, and bt; all of t he \ primit ives" of
t he model have moment s of all orders; and St ¢, Yt ¢, and Ut ¢are st at ionary VAR processes for all paramet er
set s; so (A3) is clearly sat is¯ ed.
5.1 P seudor andom N umb er G ener at ion.
Generat ing mult ivariat e normal pseudorandom vect ors wit h a general covariance mat rix involves several
algorit hms. All univariat e normal random number generat ors ut ilize a uniform generat or and a t
ransfor-mat ion of t he simulat ed uniform random numbers t o produce simulat ed st andard normal random numbers.
4Normality of V
Mult ivariat e normal generat ors add an addit ional st ep by t ransforming t he simulat ed univariat e normal
random numbers int o simulat ed mult ivariat e normal random vect ors t hat possess t he speci¯ ed covariance
mat rix. Hence, t hree algorit hms are involved, and problems can arise at any st ep in t he process.
Not iceable improvement s in met hodology for all t hree of t hese st eps have appeared in t he st at ist ical
comput ing lit erat ure. Unfort unat ely, t hese advances have not all made t heir way int o t he economet ric
lit erat ure. We incorporat ed t hese advances by using Fushimi's [1990] generalized feedback shift regist er
recurrence formula t o produce uniform pseudorandom numbers t hat perform well in a series of t est s for
randomness, including a t est suggest ed by Marsaglia [1985]. Fushimi's algorit hm was implement ed wit h a
seed value of 1589 by t he IMSL V2.0 rout ine DRNUN. T his seed value was used successfully by Fushimi in
t est ing t he algorit hm.
Univariat e st andard normals were obt ained using t he algorit hm proposed by Kinderman and Ramage
[1976] as implement ed by t he IMSL V2.0 rout ine DRNNOA. T his algorit hm has been shown t o perform
bet t er t han t he t radit ional one based on Box and Muller [1958], at least when used wit h t radit ional mixed
congruent ial uniform generat ors.
T he met hod we used t o t ransform t he univariat e st andard normal pseudorandom numbers int o mult
i-variat e normal pseudorandom vect ors wit h a part icular covariance st ruct ure was t he t riangular fact orizat ion
met hod (Cholesky's) as implement ed by t he IMSL V2.0 rout ine DRNMVN. To obt ain t he Cholesky fact
or-izat ion of t he covariances mat rices, t he IMSL V2.0 rout ine DCHFAC is used. We suggest t hat fut ure Mont e
Carlo experiment ers consider using some of t hese improvement s in met hodology where appropriat e.
Following Ericsson, separat e Et ¢vect ors were generat ed for each observat ion of each replicat ion in each
experiment (i.e., t he St ¢vect ors are t ruly exogenous). However, t o reduce int erexperiment variability (Hendry
1984; Davidson and MacKinnon 1993, pp. 738-743) only enough at, bt, and ¸t were generat ed for one
experiment and t hen t hese were reused across experiment s (for variat ions in §11, t he same underlying
uniform deviat es were reused t o produce at and bt wit h t he speci¯ ed covariances). The ¸t were produced
from pseudorandom uniform numbers by set t ing ¸t = 1 if t he uniform variat e is less t han 12 and ¸t = 0
ot herwise.
Wit h t he simulat ed Vt ¢and Et ¢vect ors in place, GAUSS v3.2.13 was used t o generat e observat ions and
calculat e est imat ors. Following Ericsson, t he above VAR's in Ut ¢and St ¢, and t he st ruct ural model, were
used t o generat e T + 30 observat ions from init ial values U0¢, S0¢, and Y0¢set at t he uncondit ional means
of zero. T hen t he ¯ rst 30 observat ions were discarded in an e®ort t o obt ain st at ionarity. T his process uses
four t imes as many init ial values of Vt ¢when T = 20 t han when T = 80, since N varies from 4; 000 t o 1; 000.
For t he T = 40 and T = 80 experiment s we skipped t hrough t he dat abank t o maint ain alignment of t he Vt ¢
values, again reducing int erexperiment variability.
For est imat ion purposes it is more convenient t o st ack t he syst em by equat ion rat her t han by observat ion,
and we do so hencefort h. T he ¯ rst st ep in est imat ion is t o specify inst rument s C for generat ing ~R and ~§ .
We follow Wallis [1967 and 1972] in using S and lags t hereof as IVs in t he ¯ rst st ep. T hese IVs sat isfy (A4)
as long as S is t ruly exogenous because (A2.3') implies (A4.1), (A2.5') implies (A4.2), and (A2') is st andard
for exogenous S and lags t hereof.6 T his leads t o opt imal ¯ rst -st ep inst rument s C = M (M0M )¡ 1M0(I G- X )
t hat ignore t he nonspherical errors, from t he IVs M = IG - [ S S¡ 1 : : : S¡ L], where L is large enough
t o ident ify t he model. Since S alone ident i¯ es our Mont e Carlo DGP, t he ¯ rst nine est imat ors we consider
use M = I2- S as ¯ rst -st ep IVs t o obt ain inst rument s C for t he st ruct ural mat rix (including exclusion
rest rict ions) ·
Y¢2 Y¢¡ 1;1
Y¢1
¸
: T hese IVs sat isfy (A4) because St ¢ is independent of Ut ¢ in t he DGP
described above. The residuals from t his IV est imat ion were used t o calculat e ~R, ~P , ~§ , and ~¦ . Since adding
more IVs can only improve asympt ot ic e± ciency,7 t he second nine est imat ors we consider are t he same as
t he ¯ rst nine est imat ors in t he second st ep but use M = I2- [ S S¡ 1] in t he ¯ rst st ep. T his permit s us t o
st udy t he small-sample e®ect s of est imat ing t he error VAR paramet ers from an asympt ot ically more precise
est imat e of B .
T he ninet een est imat ors calculat ed for each replicat ion are summarized in t he t able below. In t he second
st ep, est imat ors 1-6 and 10-15 are FGLS IV est imat ors wit h all exclusion rest rict ions imposed. T heir IVs
are list ed in t he t able and all sat isfy (A2') by t he T heorem because, as discussed above, t he underlying St ¢
and Vt ¢processes sat isfy (A3) and t he ¯ rst -st ep IVs sat isfy (A4). Est imat ors 7-9 and 16-18 are t he t hree
6A s wit h (A 2.3' ), (A 4.1) must be rest at ed t o re° ect t he exclusion rest rict ions in a simult aneous equat ion model.
7A sympt ot ic e± ciency may be enhanced by including more I V s t han are needed for ident i¯ cat ion, alt hough t he small-sample
est imat ors proposed by Hat anaka. Finally, t o get a sense of t he small-sample loss associat ed wit h est imat ing
R and § we calculat ed t he GLS IV est imat or t hat uses t he t rue values of R and § , and t he same IVs as
est imat ors 6 and 15 (except t hat t he t rue value of ¦ is used) since t his is asympt ot ically t he most e± cient
set of IVs we consider.
Est imat or St ep 1 IVs (M ) St ep 2 IVs (H )
1 I2- S S
2 I2- S [ S ¦~0Y¢¡ 3;1]
3 I2- S [ S S¢¡ 1;1]
4 I2- S [ S S¢¡ 1;1 ¦~0Y¢¡ 3;1]
5 I2- S [ S S¡ 1]
6 I2- S [ S S¡ 1 ¦~0Y¢¡ 3;1]
7 I2- S Hat anaka met hod 1
8 I2- S Hat anaka met hod 2
9 I2- S Hat anaka met hod 3
10 I2- [ S S¡ 1] S
11 I2- [ S S¡ 1] [ S ¦~0Y¢¡ 3;1]
12 I2- [ S S¡ 1] [ S S¢¡ 1;1]
13 I2- [ S S¡ 1] [ S S¢¡ 1;1 ¦~0Y¢¡ 3;1]
14 I2- [ S S¡ 1] [ S S¡ 1]
15 I2- [ S S¡ 1] [ S S¡ 1 ¦~0Y¢¡ 3;1]
16 I2- [ S S¡ 1] Hat anaka met hod 1
17 I2- [ S S¡ 1] Hat anaka met hod 2
18 I2- [ S S¡ 1] Hat anaka met hod 3
19 True R and § [ S S¡ 1 ¦ 0Y¢¡ 3;1]
5.3 R esult s.
Since t hese are IV est imat ors wit h non-normal errors, small sample moment s may not exist . Hence we
compare t hem only by examining t heir asympt ot ic covariances and t heir empirical dist ribut ion funct ions.
Turning ¯ rst t o t he asympt ot ic covariances, larger set s of IVs can be expect ed t o yield a t rue variance
reduct ion in our DGP since all lags are t heoret ically relevant . To dat e, no opt imal set of IVs has been
derived for a model wit h bot h lagged endogenous variables and VAR errors, and as not ed in Sect ion 4 under
t hese condit ions t he 3SLS IVs do not even yield a feasible est imat or t hat is asympt ot ically equivalent t o
it s infeasible count erpart . Hence for t he model considered here we must direct ly calculat e t he asympt ot ic
covariances and t hen compare t hem.
T here are 54 di®erent paramet er set s considered in our experiment s and six di®erent set s of second-st ep
IVs. T he asympt ot ic covariance for IVs D is
"
E ·
1 TF
0- ¡ 1D
¸ · E
· 1 TD
0- ¡ 1D
¸ ¸¡ 1
E ·
1 TD
0- ¡ 1F
for any T , where
F = ·
S¢1 Y¢2 Y¢¡ 1;1
S¢2 S¢3 S¢4 Y¢1
¸
is t he regressor mat rix IG - X wit h columns dropped t o re° ect exclusion rest rict ions. T he asympt ot ic
covariance is calculat ed from t he aut ocovariance funct ion of t he [ Y S ] process viewed as one 6-dimensional
VAR(1), and di®ers for each paramet er and IV set , so t here are far t oo many comparisons t o present in det ail.
We calculat ed and st udied all 54 £ 6 versions, but indicat e here only some broad t rends.
T here are large gains in asympt ot ic e± ciency when t he IV set is ext ended beyond S, irrespect ive of
whet her t he addit ional IVs are S¢¡ 1;1, S¡ 1, or ~¦ 0Y¢¡ 3;1. T hese gains can reduce t he variances by as much as
several t housand percent for some element s of B and some paramet er set s. For example, when R = ·
:6 :2 :2 :6 ¸
,
§11= 4, B02 1 = 0:3, and B11 1 = ¡ 0:4 t he asympt ot ic variance of ^B11 1 improves from 165.4 t o 7.5 when t he
IV set expands from S t o [ S S¡ 1]. T he gains are generally largest on t he coe± cient s of t he endogenous and
lagged endogenous variables, and when t he t rue value of B11 1 is ¡ 0:4, but t hey vary widely across paramet ers.
Neit her S¢¡ 1;1nor S¡ 1dominat es ~¦ 0Y¢¡ 3;1for all paramet er set s and all element s of B , nor vice-versa. Adding
lagged values of S, even just S¢¡ 1;1, is bet t er t han adding ~¦ 0Y¢¡ 3;1alone for many paramet er set s but t here
are paramet er set s for which adding ~¦ 0Y
¢¡ 3;1 yields smaller variances for all element s of B t han adding S¡ 1.
Moreover, when [ S S¡ 1] is ext ended furt her t o include ~¦ 0Y¢¡ 3;1 t here somet imes remain large addit ional
e± ciency gains, wit h variances decreasing by several hundred percent in a few cases but usually improving
by less t han 5 percent . For t he experiment ment ioned above, adding ~¦ 0Y
¢¡ 3;1 t o t he IV set furt her reduces
t he asympt ot ic variance of ^B11 1 from 7.5 t o 1.8. T his veri¯ es t hat t he opt imality of t he [ S S¡ 1] IV set
shown by Turkingt on [1998] does not carry-over t o dynamic st ruct ural models, and demonst rat es t hat our
new moment condit ions indeed cont ribut e t o asympt ot ic e± ciency. Again, t he gains are usually largest on
t he coe± cient s of t he endogenous and lagged endogenous variables and when B11 1 = ¡ 0:4. We also st udied
adding furt her lags of S, up t o S¡ 3, and found furt her but less dramat ic e± ciency gains. Finally, we veri¯ ed
t hat t he usual 3SLS IVs are de¯ nit ely not asympt ot ically opt imal for t his model (our expression for t he
asympt ot ic covariance in t his case is only valid when R and § are known), but do yield smaller variances
t han [ S S¡ 1 ¦~0Y¢¡ 3;1] for some paramet er set s.
perfor-mance of t he est imat ors for t he endogenous and lagged endogenous coe± cient s B02 1 and B11 1. We begin
by ident ifying some general t endencies. First , con¯ rming t he asympt ot ic propert ies of t he est imat ors under
st udy, as T increases from 20 t o 40 and t o 80 we observe less disperse and bet t er cent ered est imat es for
all paramet ric speci¯ cat ions. T he e®ect s of increased sample size are st rong enough t o make t he est imat ors
increasingly similar in performance and t hus t he dist inct ions t hat can be made are much more pronounced
when T = 20. Second, t he larger ¯ rst -st ep IV set , I2- [ S S¡ 1], improves bot h dispersion and cent rality
of all est imat ors across all paramet ric speci¯ cat ions, but t he improvement is modest and is most not iceable
for small sample sizes. T his improved performance is less acut e and at t imes barely percept ible for t he
est imat ion met hods proposed by Hat anaka. T hird, increases in §11 produce much more disperse and worse
cent ered est imat es for all paramet ric speci¯ cat ions and all est imat ors, and t he noncent rality becomes severe
for §11= 4. Fourt h, t he speci¯ cat ion of R does not mat erially a®ect est imat or performance.
Among t he est imat ors proposed by Hat anaka, met hods 1 and 3 are superior t o met hod 2 overall in t hese
experiment s. Met hod 2 shows a marked t endency t o produce very disperse est imat es. In est imat ing B02 1,
met hod 1 overest imat es slight ly more oft en t han met hod 3 when t he t rue value is B02 1 = ¡ 0:5. When
B02 1 = 0:3 met hod 3 is much bet t er cent ered t han met hod 1 while t heir dispersions are comparable. Hence,
t hese experiment s suggest t hat Hat anaka's met hod 3 is preferable t o his ot her two met hods.
Among t he FGLS IV est imat ors 1-6, est imat or 6 displays t he t ight est dispersion in est imat ing bot h B02 1
and B11 1, suggest ing t hat increasing t he number of IVs, including t he new IVs derived here, brings bene¯ t s
even in small samples. Int erest ingly, est imat ors 2 and 3, which use t he same number of IVs and di®er only
in t he use of a t ransformed lagged endogenous IV rat her t han a lagged exogenous IV, have almost ident ical
dispersions across experiment s. T his is mildly di®erent from t he asympt ot ics, which showed modest ly smaller
variances for est imat or 2 for most , but not all, paramet er set s. T he marginal gains from adding more IVs
decrease wit h t he number of IVs used. T hus, est imat ors 2 and 3 are signi¯ cant ly less disperse t han est imat or
1 but est imat or 6 is only slight ly less disperse t han est imat or 5.
T he e®ect on cent ral t endency of using more IVs is much less de¯ nit ive. Since all t he FGLS IV est imat ors
are asympt ot ically pT -normal cent ered at t he t rue value, t his is consist ent wit h t heory. All feasible est
wit hin any paramet er set as t he number of IVs increases.
For a ¯ xed value of B02 1, dispersion and cent ral t endency of t he FGLS IV est imat ors of B02 1 are not
sensit ive t o t he value of B11 1, but dispersion is usually slight ly smaller when B02 1 = 0:3. In cont rast ,
recall t hat t he asympt ot ic variances are improved more subst ant ially by adding IVs when B11 1 = ¡ 0:4, so
asympt ot ically t he dispersion of at least some of t he est imat ors is indeed sensit ive t o t he value of B11 1.
Int erest ingly, t here is considerable improvement in bot h t he dispersion and cent ral t endency of all FGLS
IV est imat ors of B11 1 when B02 1 increases t o 0:3. However, t heir relat ive performance is una®ect ed.
Given t hese general observat ions, it seems most useful t o focus on Hat anaka's met hod 3 (est imat or 18), t he
FGLS IV est imat or using a large IV set (est imat or 15), and t he infeasible est imat or 19 for a closer comparison.
Panels 1 and 2 show cumulat ive frequencies for t hese t hree est imat ors of B02 1 and B11 1, respect ively, for t he
following six represent at ive experiment s:
All Experiment s:
T = 20; R = ·
0:6 0:6 0:2 0:2 ¸
; §11= 0:25;
8 > > > > > > > <
> > > > > > > :
Experiment B02 1 B11 1
1 ¡ 0:5 ¡ 0:4 2 ¡ 0:5 0:2 3 ¡ 0:5 0:7 4 0:3 ¡ 0:4
5 0:3 0:2
6 0:3 0:7:
Panel 1 shows t hat all t hree est imat ors usually overest imat e B02 1, but t his t endency is mild for t he
infeasible est imat or 19, followed by Hat anaka's est imat or 18 and t hen t he FGLS IV est imat or 15. Likewise,
t he infeasible est imat or is most t ight ly dist ribut ed, followed by Hat anaka's est imat or and t hen t he FGLS
IV est imat or. Hence ignorance of R and § is indeed cost ly is small samples, and Hat anaka's met hod 3
performed best among t he feasible est imat ors of B02 1 in t hese experiment s.
Panel 2 shows t hat t hese conclusions do not apply t o est imat ion of B11 1. T he infeasible est imat or t ends
t o underest imat e when B11 1 = ¡ 0:5, and t his t endency becomes more pronounced as B02 1 increases, so t hat
t he feasible est imat ors are act ually bet t er cent ered t han t he infeasible est imat or in experiment s 2 and 3.
T he infeasible est imat or cont inues t o display smaller dispersion in most experiment s, but surprisingly does
not dominat e as it does in est imat ing B02 1 because in experiment 6 of Panel 2 bot h feasible est imat ors are
more t ight ly dist ribut ed t han t he infeasible est imat or. Also unlike Panel 1, Hat anaka's est imat or does not
comparable or bet t er cent ering in experiment s 2, 3, 5, and 6.
T he qualit at ive and comparison conclusions derived from t he analysis of Panels 1 and 2 are essent ially
unalt ered for T = 40 and T = 80. However, two observat ions are in order. First , as T grows t he est imat ors
become bet t er cent ered and less disperse t o t he point t hat drawing dist inct ions at T = 80 becomes rat her
di± cult for some paramet ers. Second, t he posit ive e®ect s of increased sample size are much more pronounced
on t he feasible est imat ors t han on t he infeasible est imat or 19. A result t o be expect ed, as t he increased sample
size bene¯ t s t he est imat ion of t he nuisance, as well as t he paramet ers of int erest .8
In summary, for t he sample sizes and paramet er con¯ gurat ions considered in our Mont e Carlo st udy,
neit her t he best FGLS IV nor t he best Hat anaka met hod emerges as a clear best choice. In fact , t hese
est imat ors perform rat her similarly and become more similar as t he sample size increases. T he experiment s
show t hat int roducing addit ional IVs when const ruct ing FGLS IV est imat ors, such as t hose proposed in
Sect ion 4 above, is bene¯ cial bot h asympt ot ically and in small samples. More IVs in t he ¯ rst st ep is
bene¯ cial as well. T he bene¯ t s of addit ional IVs diminish, however, as t he number of IVs expands. Finally,
it is somewhat dist urbing t hat t here is a marked t endency of all est imat ors t o overest imat e B02 1. Cent ering
is bet t er in t he est imat ion of B11 1, but t here is st ill a t endency for t he feasible est imat ors t o overest imat e.
A ppendix: Pr oof of T heor em
(A2.1') Let ¦ = (det A) (i.e., wit h t he t rue R rat her t han ~R). T he count erpart t o (5) based on ¦ is
¦ 0Y¡ L = G
X
` = 0
(¡ 1)`r`Y` ¡ L: (P1)
Suppose moment arily t hat H is given by (P1) rat her t han t he feasible version of (P1) based on ~¦ . Not e t hat
(H0- §¡ 1¡ H¡ 10 - R§¡ 1)vecV0=
2
6 4
P T t = 1
¡
Ht 1§¡ 1¡ Ht ¡ 1;1R§¡ 1
¢ (Vt ¢)0
.. . PT
t = 1
³
Ht ^K§¡ 1¡ H
t ¡ 1; ^KR§¡ 1
´ (Vt ¢)0
3
7
5 : (P2)
Hence, using t he Cram¶er-Wold Device, we need only apply a CLT t o
1 p
T¸
0(H0- §¡ 1¡ H0
¡ 1- R§¡ 1)vecV0=
1 p
T
T
X
t = 1
2
4
^ K
X
i = 1
¸0 i
¡
Ht i§¡ 1¡ Ht ¡ 1;iR§¡ 1
¢ (Vt ¢)0
3
5 ; (P3)
where ¸0= ( ¸0
1 : : : ¸0K^ ) is an arbit rary nonst ochast ic vect or composed of (G £ 1) subvect ors ¸i. Su± cient
condit ions for t he m-dependent CLT t o apply t o (P3) for t he special case of m = 0 are (Schmidt 1976, p.
258):
(CLT 1) EhP Tt = 1¸0
i(Ht i§¡ 1¡ Ht ¡ 1;iR§¡ 1)(Vt ¢)
i = 0.
(CLT 2) E · ¯
¯
¯P Tt = 1¸0i(Ht i§¡ 1¡ Ht ¡ 1;iR§¡ 1)(Vt ¢)
¯ ¯ ¯
3¸
is bounded across t.
(CLT 3) ¾2´ lim N ! 1 N1
PN
n = 1At + n exist s and is independent of t, where
At = E
2 6 4 0 @ ^ K X
i = 1
¸0i(Ht i§¡ 1¡ Ht ¡ 1;iR§¡ 1)(Vt ¢)0
1
A
23
7 5 :
(CLT 1) is immediat e since t he t ime index on row t of each lag of Y in (P1) is less t han t (since L > G),
making t he lag of Y independent of Vt ¢. (CLT 2) follows from (A3) and (A2.4'). (CLT 3) holds because
At = ^ K
X
i = 1 ^ K
X
j = 1
¸0i
£
¨ i j(0)(§¡ 1+ R§¡ 1R0) ¡ ¨ i j(1)(R§¡ 1+ §¡ 1R0)
¤ ¸j
for every t, where ¨ i j(h) = E [Ht iHt ¡ h ;j] is independent of t by covariance st at ionarity of Yt ¢, and we have
again made use of independence between Ht ¢or Ht ¡ 1;¢and Vt ¢for t he lags t hat are present in At. T hus
(A2.1') holds when t he IVs are based on ¦ rat her t han ~¦ .
Now we show t hat (P3) converges in probability t o t he analogous expression based on ~¦ , so t hat using ~¦
does not a®ect t he asympt ot ic dist ribut ion. From (P3), t he di®erence between using ~¦ and ¦ in (A2.1') is
1 p
T
T
X
t = 1 G
X
i = 1
¸0i
h
( ~¦ ¢t¡ ¦ ¢t)0Y¢¡ L ;i§¡ 1¡ ( ~¦ ¢;t ¡ 1¡ ¦ ¢;t ¡ 1)0Y¢¡ L ;iR§¡ 1
i
(Vt ¢)0: (P4)
From (5),
h ~
¦ ¢;t ¡ h¡ ¦ ¢;t ¡ h
i0
Y¢¡ L ;i = G
X
` = 1
(¡ 1)`(~r` ¡ r`)
¡
(¢`)¢;t ¡ h
¢0
Y¢¡ L ;i
=
G
X
` = 1
(¡ 1)`(~r` ¡ r`)Yt ¡ L ¡ h + ` ;i: (P5)
Since t ¡ L ¡ h + ` < t when L > G, ` · G, and h ¸ 0, by (A3) we can apply Chebyshev's Inequality t o show
1 p
T
T
X
t = 1
Yt ¡ L + ` ;i
¡
¸0i§¡ 1(Vt ¢)0
¢
= Op(1) and
1 p
T
T
X
t = 1
Yt ¡ L ¡ 1+ ` ;i
¡
¸0iR§¡ 1(Vt ¢)0
¢
= Op(1): (P6)
Subst it ut ing (P5) and (P6) int o (P4) yields
G
X
i = 1 G
X
` = 1
Since ~r` p
! r` by Lemma 2, we have (P4) p
! 0.
(A2.2') From (5), Hh =
P G
` = 0(¡ 1)`r~`Y` ¡ L + h. So
1 TH
0H
h = 1
T
G
X
` = 0 G
X
i = 0
(¡ 1)` + 1r~`r~iY` ¡ L0 Yi ¡ L + h: (P7)
Since ~r` p
! r` by Lemma 2 and t he fourt h moment s of t he Yt ¢process exist by (A3.4), (P7) converges t o
G
X
` = 0 G
X
i = 0
(¡ 1)` + 1r`riE (Y` ¡ L0 Yi ¡ L + h) for h = ¡ 1; 0; 1:
(A2.3') T he convergence argument is ident ical t o (A2.2').
(A2.4') From (5),
Ht h = G
X
` = 0
(¡ 1)`~r`Yt + ` ¡ L ;h:
Since (¡ 1)`r~
` is independent of t and is Op(1) by Lemma 2, and t he fourt h moment s of Yt + ` ¡ L ;h have t he
speci¯ ed property by (A3.4), Ht h t akes t he form required by (A2.4').
(A2.5') From (5),
H0U h =
G
X
` = 1
(¡ 1)`(~r
` ¡ r`)Y` ¡ L0 Uh+ Y¡ L0 ¦ Uh: (P8)
By Lemma 2, t he sum is Op(1) if Y` ¡ L0 Uh = Op(T ) (not e t he import ance of ~R converging fast er t han op(1)).
A rout ine applicat ion of Chebyshev's Inequality est ablishes t his because t he moment s involved are bounded
by (A3.4). From (4),
Kj i = (¡ 1)j + i G¡ 1X
` = 0
(¡ 1)`r
` j i¢`; (P9)
where r` j i is t he sum of all `t h order principal minors of t he submat rix of R0obt ained by delet ing row j and
column i (and r0j i ´ 1 for not at ional convenience). Subst it ut ing (P9) int o (4) yields
Y0
¡ L¦ U¢+ h ;i = Y¡ L0 (det A)U¢+ h ;i = G
X
j = 1
(¡ 1)j + i G ¡ 1X
` = 0
(¡ 1)`r
` j iY¡ L0 ¢`[¢ ¢0V¢+ h ;j + (IT ¡ ¢ ¢0)U¢+ h ;j]: (P10)
Since t he only nonzero element of (IT¡ ¢ ¢0)U¢+ h ;j is U1+ h ;j, t his t erm is clearly Op(T1=2). For t he remaining
t erm, not e t hat
1 p
TY
0
¡ L¢`¢ ¢0V¢+ h ;j =
1 p
TY
0 ` ¡ LV¢+ h ;j
except for t he ¯ rst row of V . As shown above, t his sat is¯ es t he condit ions for t he m-dependent CLT (and
t hus is Op(1)) since t he index on row t of each lag of Y is less t han t + h when ` · G ¡ 1, L > G, and
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