Simultaneous Equations Model
11.1 The Inconsistency of OLS. The OLS estimator from Eq. (11.14) yields 8ols =
E Ptqt/ E Pt2 where pt = Pt — l3 and qt = Qt — Q. Substituting qt = 8pt C
t=i t=i
TT
(u2t — U2) from (11.14) we get 8ois = 8 C E Pt(u2t — N/Y, P2. Using (11.18),
t=i t=i
T
we get plim £ Pt(U2t — U2)/T = (012 — 022)/(8 — ") where Oij = cov. Uit, Ujt)
t=1
for i, j = 1,2 and t = 1,2,.., T. Using (11.20) we get
Plim 8ois = 8 C [(CT12 — 022)/(8 — ")]/[(011 C CT22 — 2ст12)/(8 — ")2] = 8 C (o12 — 022)(8 — ")/(o11 C 022 — 2°12).
11.2 When Is the IVEstimator Consistent?
a. ForEq.(11.30)y1 = a^y2 C ‘1зУз C "11X1 C "12X2 C U1.Whenweregress
T
У2 on X1, X2 and X3 to get У2 = у2 C V2, the residuals satisfy E y2tV2t = 0
t=1
TTT
and 22 V2tXu = 22 V2tX2t = 22 V2tX3t = 0.
t=1 t=1 t=1
Similarly, when we regress y3 on X1, X2 and X4 to get y3 = y3 C
T T T
V3, the residuals satisfy E y3t83t = 0 and E V3tXu = E ^3tX2t =
t=1 t=1 t=1
T
E v3tX4t = 0. In the second stage regression y1 = a^y2 C ’13y3 C
t=1
"11X1 C "12X2 C £1 where Є1 = ’12 (y2 — 382) C ’13(y3 — 383) C u =
T
‘12882 C’13883Cu1. For this to yield consistent estimates, we needE 2t©1t =
t=1
Ey^m E y3t©1t = E y3^; £ X1t£1t = E X1tu1t and E X2t£1t =
t=1 t=1 t=1 t=1 t=1 t=1
E X2tu1t. But E y 2t©1t = ’12 E 882ty2t C ’13 E 88 3t:8 2t C E y2tu1t with
t=1 t=1 t=1 t=1 t=1
T
E V2ty2t = 0 because 882 are the residuals and y2 are the Predicted val
t=1
T
ues from the same regression. However, 883ty2t is not necessarily zero
t=1
B. H. Baltagi, Solutions Manual for Econometrics, SPringer Texts in Business and Economics, DOI 10.1007/9783642545481—11, © SPringerVerlag Berlin Heidelberg 2015
because V3 is only orthogonal to Xi, X2 and X4, while У2 is a perfect linear
T
combination of Xi, X2 and X3. Hence, Y $3tX3t is not necessarily zero,
t=i
T
which makes Y v3ty2t not necessarily zero.
t=i
T T T T
Similarly, y3t©it = ai2 £ V2ty3t C ‘nY v3ty3t C Y y3tuit with
t=i t=i t=i t=i
TT
Y v3ty3t = 0 and Y V2ty^3t not necessarily zero. This is because V2t is
t=i t=i
only orthogonal to Xi, X2 and X3, while y3 is a perfect linear combina
T
tion of Xi, X2 and X4. Hence, Y v2tX4t is not necessarily zero, which
t=i
T
makes Y V2ty3t not necessarily zero. Since both Xi and X2 are included
t=i
TT
in the first stage regressions, we have J]Xit£it = Y XitUit using the
t=i t=i
T T T T
fact that XitV2t = Y XitV3t = 0. Also, X2teit = Y X2tuit since
t=i t=i t=i t=i
TT
Y X2tV2t = Y X2tV3t = 0.
t=i t=i
b. The first stage regressions regress y2 and y3 on X2, X3 and X4 to get
T
y2 = y2 C V2 and y3 = y3 C V3 with the residuals satisfying Y V2ty2t =
t=i
Y V2tX2t = J2 V2tX3t = J2 V2tX4t = 0 and Y V3ty3t = J2 V3tX2t =
t=i t=i t=i t=i t=i
TT
v3tX3t = V v3tX4t = 0. In the second stage regression yi = ai2y2 C
t=i t=i
‘i3y3 C PiiXi C "i2X2 C £i where ei = ап(у2 – У2) C ai3.y3 – У3) C ui = ai2v2 C ai3v3 C ui. For this to yield consistent estimates, we need p y2t©it = p y2tuit; p y3teit = p y3tuit; p X^it = p Xituit and
t=i t=i t=i t=i t=i t=i
P X2t©it = P X2tuit.
t=i t=i
The first two equalities are satisfied because
T T T T
y2tv2t = y2tv3t = 0 and y3tv2t = y3tv3t = 0
t=i t=i t=i t=i
since the same set of X’s are included in both first stage regressions.
T T T T
Y X2t©it = Y X2tuit because X2tV2t = Y X2tV3t = 0 since X2 is
t=i t=i t=i t=i
included in both firststage regressions. However, X1te1t ф X1tu1t
t=i t=i
TT
since y^X1tv2t ф 0 and E XitV3t ф 0
t=i t=i
because Xi was not included in the first stage regressions.
11.3 JustIdentification and Simple IV. If Eq. (11.34) is justidentified, then X2 is of the same dimension as Yi, i. e., both are Txgi. Hence, Zi is of the same dimension as X, both of dimension Tx (g1 + k1). Therefore, X0Z1 is a square nonsingular matrix of dimension (g1 + k1). Hence, (ZjX) 1 exists. But from (11.36), we know that
•i,2SLS = (ZjPxZi)1 ZjPxyi = [Z1X(X, X)1X, Z1]“1 Z1X(X, X)1X, y1
= (X, Z1)1(X, X) (z;x)1 (z;x) (x0x)1x0y1 = (x0ZO1x0y1
as required. This is exactly the IV estimator with W = X given in (11.41). It is important to emphasize that this is only feasible if X0Z1 is square and nonsingular.
11.4 2SLS Can Be Obtained as GLS. Premultiplying (11.34) by X0 we get X0y1 = X, Z181 + X0u1 with X0u1 having mean zero and var(X0u1) = o11X0X since var(u1) = o11 IT. Performing GLS on this transformed equation yields
•i, gls = [Z01X(anX0X)1X0Z1]1Z01X(an X0X)1X0y1 = (Z^Zi)^0^
as required.
11.5 The Equivalence of 3SLS and 2SLS.
a. From (11.46), (i) if X is diagonal then X1 is diagonal and X1 <S> PX is blockdiagonal with the ith block consisting of PX/(tii. Also, Z is block – diagonal, therefore, fZ0[X1 <S> PX]Z}1 is blockdiagonal with the ith block consisting of (jii (Z0PXZ^ 1. In other words,
/Zi 0 .. o 0 Z2 .. 0 
y0 0 .. Zoy
/ZiPXZi/CT11 0 .. 0
0 Z2 Px Z2 /f^22 .. 0
у 0 0 .. Z0PxZo/6ooy
VZ0Pxyo/ООО у
Hence, from (11.46) 83SLS = {Z0[C 1 <8> Px]Z} 1{Z0[^ 1 (g> Px]y}
(ii) If every equation in the system is justidentified, then ZiX is square and nonsingular for i = 1,2, ..,G. In problem 11.3, we saw that for say the first equation, both Z1 and X have the same dimension Tx(g1 + k1). Hence, Z1X is square and of dimension (g1 + k1). This holds for every
equation of the justidentified system. In fact, problem 11.3 proved that 8il2SLS = (X, Zi)_1 X0yifori = 1,2,.., G. Also, from (11.44), we get
83SLS = {diag [Z0X] [t1 ® (X0X)1] diag [X0Zi]}1
{diag [Z0X] [£1 ® (X0X)1] (Ig ® X0)y}.
But each Zi0X is square and nonsingular, hence
83SLS = [diag (X’Zi)1 (£ ® (X0X)) diag (Z0X)1
{diag (Z0X) [e1 ® (X0X)1] (Ig ® X0)y}.
which upon cancellation yields
O3SLS = diag(X0Zi)1 (Ig ® X0)y
/(X, Z1)1 0 .. 0 1 0 X’yA
b. Premultiplying (11.43) by (IG <g> PX) yields y* = Z*8 + u* where y* = (IG <8>PX )y, Z* = (IG <S>PX )Zandu* = (IG <g>PX)u. OLS on this transformed model yields
Sols = (Z*0Z*)1Z*0y* = [Z0(Ig ® Px)Z]1 [Z0(Ig ® PX)y]
v 0 0 
.. (ZG 

0 (z;PxZ1)1 ZjPXy1 1 
*01,2SLS^ 

(z2PxZ2)1 Z2PXy2 
= 
8 2,2SLS 
V(ZGPxZg)1 ZGPXyGyi 
V8 G,2SLS / 
which is the 2SLS estimator of each equation in (11.43). Note that u* has mean zero and var(u*) = £ <S> PX since var(u) = £ <S> IT. The generalized inverse of var(u*) is £1 <S> PX. Hence, GLS on this transformed model yields
8gls = [Z*0(£1 ® Px)Z*]1Z*0(£1 ® Px)y*
= [Z0(Ig <8> Px)(£ 1 <S> Px)(Ig <S> Px)Z] 1Z0(Ig <8> Px)(£ 1 <S> Px) x (Ig <S> Px)y
= [Z0(£1 ® Px)Z]1Z0[£1 ® Px]y.
Using the Milliken and Albohali condition for the equivalence of OLS and GLS given in Eq. (9.7) of Chap. 9, we can obtain a necessary and sufficient condition for 2SLS to be equivalent to 3SLS. After all, the last two estimators are respectively OLS and GLS on the transformed * system. This condition gives Z*0(£1 <S> PX)PZ* = 0 since Z* is the matrix of regressors and £ <S> PX is the corresponding variancecovariance matrix. Note that Z* = diag[PXZ0] = diag[Z0] or a blockdiagonal matrix with the ith block being the matrix of regressors of the secondstage regression of 2SLS. Also, PZ* = diag[PZ. ] andPZ* = diag[Pz]. Hence, the above condition reduces to оijZ0PZj = 0 for i ф j and i, j = 1,2,.., G.
If £ is diagonal, this automatically satisfies this condition since cry = оij=0 for i ф j. Also, if each equation in the system is justidentified, then Z0X and X0Zj are square and nonsingular. Hence,
I0Z (z0j 1 Zj = Z0X(X0X)1X0Zj ZjX(X0X)1X0Zj 1
Z0X(X0X)1X0 = Z0X(X0X)1X0 = Z0PX = Z0
after some cancellations. Hence, Z0P = Z0 — Z0P = Z0 — Z0 = 0, under
0 Z 0 0 Z 0 0
justidentification of each equation in the system.
11.6 a. Writing this system of two equations in the matrix form given by (A.1) we
(?)
b. There are two zero restrictions on Г. These restrictions state that Wt and Lt do not appear in the demand equation. Therefore, the first equation is overidentified by the order condition of identification. There are two excluded exogenous variables and only one right hand side included endogenous variable. However, the supply equation is not identified by the order condition of identification. There are no excluded exogenous variables, but there is one right hand side included endogenous variable.
c. The transformed matrix FB should satisfy the following normalization restrictions: fll + fl2 = 1 and f2i + f22 = 1.
These are the same normalization restrictions given in (A.6). Also, Fr must satisfy the following zero restrictions:
f110 – f12e = 0 and f110 – f12f = 0.
If either e or f are different from zero, so that at least Wt or Lt appear in the supply equation, then f12 = 0. This means that f11 + 0 = 1 or f11 = 1, from the first normalization equation. Hence, the first row of F is indeed the first row of an identity matrix and the demand equation is identified. However, there are no zero restrictions on the second equation and in the absence of any additional restrictions f21 ф 0 and the second row of F is not necessarily the second row of an identity matrix. Hence, the supply equation is not identified.
11.7 a. In this case,
1 
b 
; Г = 
a – c 
d 
0 0 
1 
f 
e 0 
0 
g – h 

1,Yt, At 
Wt, Lt] 
and u( 
= (u1t 
,u2t). 
B 
I yt = 
b. There are four zero restrictions on Г. These restrictions state that Yt and At are absent from the supply equation whereas Wt and Lt are absent from the demand equation. Therefore, both equations are overidentified. For each equation, there is one right hand sided included endogenous variable and two excluded exogenous variables.
c. The transformed matrix FB should satisfy the following normalization restrictions:
fll + fl2 = 1 and f2i + f22 = 1.
Also, Fr must satisfy the following zero restrictions:
—f21 c C f220 = 0 – f21 d + f220 = 0.
and
fii0 — f12g = 0 fii0 — f12h = 0.
If either c or d are different from zero, so that at least Yt or At appear in the demand equation, then f2i = 0 and from the second normalization equation we deduce that f22 = i. Hence, the second row of F is indeed the second row of an identity matrix and the supply equation is identified. Similarly, if either g or h are different from zero, so that at least Wt or Lt appear in the supply equation, then fi2 = 0 and from the first normalization equation we deduce that fii = i. Hence, the first row of F is indeed the first row of an identity matrix and the demand equation is identified.
11.8 a. From example (A.1) and Eq. (A.5), we get
i b —a —c 0
i —e —d 0 —f
Therefore, ® for the first equation consists of only one restriction, namely that Wt is not in that equation, or yi3 = 0. This makes ф0 = (0,0,0,0, i), since
‘ф = 0 gives уїз = 0. Therefore,
**=0=(.”,).
This is of rank one as long as f ф 0. Similarly, for * the second equation consists of only one restriction, namely that Yt is not in that equation, or У22 = 0. This makes * = (0,0,0,1,0/, since al* = 0 gives у22 = 0. ‘У 12 I
/(e + b).
This can be easily verified by solving the two equations in (A.3) and (A.4) for Pt and Qt in terms of the constant, Yt and Wt.
From part (a) and Eq. (A.17), we can show that the parameters of the first structural equation can be obtained from al [W, *] = 0 where ‘1 = ("11, "12, "її, у12, уїз) represents the parameters of the first structural equation. W0 = [П, I3] and *0 = (0,0,0,0,1) as seen in part (a).
"11^11 + "12^21 + У11 = 0 "11^12 + "12^22 + У12 = 0 "11^13 + "12^23 + У13 = 0 у1з = 0
If we normalize by setting "11 = 1, we get "12 = —л13/л23 also
Л13 л22л13 — л12л23
У12 = —12 H————— Л22 = ——————————— and
Л23 Л23
Л13 ТС21Л13 — ли Л23
У11 = — л11 H————— л21 = ———————————— .
л23 л23
One can easily verify from the reduced form parameters that "12 = – Л13/Н23 = – bf/ — f = b.
Similarly, "12 = (л 22 л 13 – л 12 л 23)/ Л23 = (cbf + ecf)/ – f(e + b) = – c. Finally, "її = (л 2i л із – лїї л23)/ л 23 = (abf + aef)/ – f(e + b) = – a. Similarly, for the second structural equation, we get,
л 11 
12 
13 
0 

л 21 
22 
23 
0 

("21, "22, "2b "22, "23) 
61 
0 
0 
0 
= (0, 0, 0, 0, 0) 
0 
1 
0 
1 

0 
0 
1 
0 

which can be rewritten 
as 

"21 л 11 + "22 л 21 + "21 
=0 

"21 л 12 + "22 л 22 + "22 
=0 

"21 л 13 + "22 л 23 + "23 
=0 

"22 
= 0. 
If we normalize by setting "21 = 1, we get "22 = — л 12/ л 22 also
л 12 л 23л 12 – л 13л 22 ,
"23 = – л13 Н————— л 23 = ——————————— and
22 22
л 12 л12 л 21 – л 11 л 22
"21 = — л11 Н————— л 21 = ———————————– .
22 22
One can easily verify from the reduced form parameters that "22 = – л 12/ л 22 = – ec/c = – e. Similarly,
"23 = (л 23л 13 – л 13л 22)/ л 22 = (ecf – bdf)/c(e + b) = – f. Finally, "21 = (л 12 л 21 – лпл 22)/ л 22 = (dec – bdc)/c(e + b) = – d.
11.9 For the simultaneous model in problem 11.6, the reduced form parameters
are given
11 
12 
13 
ab C bc 
be 
bf 

21 
22 
23 
a – c 
e 
f 
by 
/(b C d) 
This can be easily verified by solving for Pt and Qt in terms of the constant, Wt and Lt. From the solution to 11.6, we get
A = [B, Г] =
For the first structural equation,
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
Ф = 
with ‘ ф 
= 0 yielding 
"12 = 
"13 = 
= 0. Therefore, (A.17) reduces to 

т 11 
12 
т 13 
0 
0 

т 21 
22 
23 
0 
0 

("11, "12 
"11, "12, "13) 
1 
0 
0 
0 
0 
= (0, 0, 0, 0, 0) 
60 
1 
0 
1 
0 

0 
0 
1 
0 
1 
This can be rewritten as
"іітс її + Р12Л 21 + "її = 0 "11 те 12 + Р12Л 22 + "12 = 0 "її те 13 + Р12Л 23 + "13 = 0 "12 = 0 "13 = 0
If we normalize by setting "11 = 1,weget"12 = — тт13/тт23; "12 = — т 12/ тт22.
Also, "11 = — ти + (^12/^22)^21 =
(т 13/ тт23)тт21 = —21—13——————– 11—23 one can easily verify from the reduced form
23
parameters that
"12 = – т 13/T23 = – bf/ — f = b. Also "12 = — т 12/т22 = —be/ — e = b. Similarly,
Also,
n 21n 13 — n 11n 23 abf — bcf + adf + bcf a(bf + df)
Y11 n 23 —f(b + d) —(bf + df) a
For the second structural equation there are no zero restrictions. Therefore, (A.15) reduces to
(0,0,0,0,0)
This can be rewritten as
"21n 11 + "22n 21 + "21 = 0 "21n 12 + "22 n 22 + "22 = 0
"21 n 13 + "22 n 23 + "23 = 0
Three equations in five unknowns. When we normalize by setting "21 = 1, we still get three equations in four unknowns for which there is no unique solution.
11.10 JustIdentified Model.
a. The generalized instrumental variable estimator for 81 based on W is given below (11.41)
8Uv = (Z^wZ!)1 ZiPWy!
Z1W (w0w) 1 W0 Z1
= (W0Z1)—!(W0W) (ZiW 1 (ZiW (W0W)—!W—1y1 = (W0 Z1)—1W0y1
since W0Z1 is square and nonsingular under justidentification. This is exactly the expression for the simple instrumental variable estimator for 81 given in (11.38).
b. Let the minimized value of the criterion function be
Q = (yi — Zi8i, iv),Pw(Yi — Z18 i, iv).
Substituting the expression for 81jv and expanding terms, we get
Q = yiPWy1 — yiW (ZiW)1 ZiPWy1 — yiPWZ1(W, Zi)1W, yi + y1W (Z1W)1 Z1 PwZ 1 ( W0Z1)1 W0y 1 Q = y,1PWy1 — y1W (Z1W)1 Z,1W(W, W)1 W0y1 — y’^^W/^W^ (W,1Z1)1 W0y1 + y1W (Z1W)1 Z,1W(W/W)1W0Z1(W, Z1)1W, y1 Q = y1Pwy1 — y1Pwy1 — y1Pwy1 + y1Pwy1 = 0
as required.
c. Let Z1 = PWZ1 be the set of second stage regressors obtained from regressing each variable in Z1 on the set of instrumental variables W. For the justidentified case
PZ1 = PpwZ1 = Z1 (Z1Z1) Z1 = PwZ1 (Z1PWZ1)1 Z1Pw
= W(W0W)1W0 Z1 [Z,1W(W, W)1 W0 Z1]1 Z1W(W0W)1W0 = W(W0W)1W0Z1(W0Z1)1(W0W) (Z,1W)1 Z,1W(W0W)1W0 = W(W0W)1W0 = PW.
Hence, the residual sum of squares of the second stage regression for the justidentified model, yields y1P£ y1 = y1PWy1 since P% = PW. This is exactly the residual sum of squares from regression y1 on the matrix of instruments W.
11.11 The More Valid Instruments the Better. If the set of instruments W1 is spanned by the space of the set of instruments W2, then PW2Wi = W1. The corresponding two instrumental variables estimators are
with asymptotic covariance matrices cm plim (Z,1PwiZ1/T)_1 fori = 1,2.
The difference between the asymptotic covariance matrix of 81,w1 and that of 8i, w2 is positive semidefinite if
is positive semidefinite. This holds, if Z,1PW2Z1 — Z,1PW1Z1 is positive semidefinite. To show this, we prove that PW2 — PW1 is idempotent.
(Pw2 — Pw1 )(Pw2 — Pw1 ) = Pw2 — Pw2Pw1 — Pw1Pw2 + Pw1
= PW2 — PW1 — PW1 + PW1 = PW2 — PW1.
This uses the fact that PW2PW1 = PW2W1 (W’W1) 1 W’ = W1 (W’W1) 1 W1 = Pw1, since PW2W1 = W1.
11.12 Testing for OverIdentification. The main point here is that W spans the same space as [Z 1,W*] and that both are of full rank ‘. In fact, W* is a subset of instruments W, of dimension (‘ — k1 — g1), that are linearly independent of Z1 = PwZ1.
a. Using instruments W, the second stage regression on y1 = Z181+W*y+u1 regresses y1 on PW[Z1,W*] = [Z 1,W*] since Z 1 = PWZ1 and W* = PWW*. But the matrix of instruments W is of the same dimension as the matrix of regressors [Z 1,W*]. Hence, this equation is justidentified, and from problem 11.10, we deduce that the residual sum of squares of this second stage regression is exactly the residual sum of squares obtained by regressing y1 on the matrix of instruments W, i. e.,
URSS* = yjPwy1 = yiy1 — yiPwy1.
b. Using instruments W, the second stage regression on y1 = Z181 + u1
regresses y1 on Z 1 = PWZ1. Hence, the RRSS* = y,1P21 y1 = y1y1 — y,1Pz 1 y1 where P^ = I — 1 and P^ = Z1 (^Z1) Z1 =
PWZ1 (Z,1PWZ1) 1 Z’jPw. Therefore, using the results in part (a) we get RRSS* – URSS* = y,1PWy1 – ylPZiy1 as given in (11.49).
c. Hausman (1983) proposed regressing the 2SLS residuals (y1 — Z1812SLS) on the set of instruments W and obtaining nR^ where R is the uncentered R2 of this regression. Note that the regression sum of squares yields
(y1 — Z1Q1,2sls )0Pw(y1 — Z1§1,2SLs)
= y1PWy1 — y,1PWZ1(Z,1PWZ1)_1Z,1PWy1 — y,1PWZ1(Z,1PWZ1)_1Z,1PWy1 + y,1PWZ1(Z1PWZ1)1Z,1PWZ1 (z;PwZ1)“1 Z1PWy1 = y1Pwy1 — y1PwZ1 (Z^PwZ1) 1 Z1Pwy1 = y,1PWy1 — yiPZl y1 = RRSS* — URSS*
as given in part (b). The total sum of squares of this regression, uncentered, is the 2SLS residuals sum of squares given by
(y1 — Z1Q1,2sls )0(y1 — Z1Q1,2SLs) = T5n
where an is given in (11.50). Hence, the test statistic
RRSS* — URSS* Regression Sum of Squares
&П Uncentered Total Sum of Squares / T
= T(uncentered R2) = T R2 as required.
This is asymptotically distributed as x2 with ‘ — (g1 + k1) degrees of freedom. Large values of this test statistic reject the null hypothesis.
d. The GNR for the unrestricted model given in (11.47) computes the residuals from the restricted model under Ho; " = 0. These are the 2SLS residuals (y1 — Z1q12SLS) based on the set of instruments W. Next, one differentiates the model with respect to its parameters and evaluates these derivatives at the restricted estimates. For instrumental variables W one has to premultiply the right hand side of the GNR by PW, this yields (yi – Zi81,2sls) = PWZ1b1 + PWW*b2 + residuals.
But Z1 = PWZ1 and PWW* = W* since W* is a subset of W. Hence, the GNR becomes (y1 — Z181,2SLS) = Z 1b1 + W*b2 + residuals. However, [Z1, W*] spans the same space as W, see part (a). Hence, this GNR regresses 2SLS residuals on the matrix of instruments W and computes TR where Ru is the uncentered R2 of this regression. This is Hausman’s (1983) test statistic derived in part (c).
11.15 a. The artificial regression in (11.55) is given by
y 1 = Z181 + (Y1 — Y1)^ + residuals
This can be rewritten as y1 = Z181 + IіWY1q + residuals where we made use of the fact that Y1 = PWY1 with Y1 — Y1 = Y1 — PWY1 = IіWY1. Note that to residual out the matrix of regressors PWY1, we need
P pwy1 = I — PwY1(Y1PwY1)1Y1Pw
so that, by the FWL Theorem, the OLS estimate of 81 can also be obtained from the following regression
P PwY1 y1 = P PwY1 Z181 + residuals.
Note that
Ppwy1 Z1 = Z1 — PwY1(Y1PwY1)1Y1 PwZ1
with PWZ1 = [PWY1,0] and PWX1 = 0 since X1 is part of the instruments in W. Hence, PpwY1 Z1 = [Y1,X1] — [PwY1,0] = [PWY1,X1] = [Y1 ,X1] = Pw[Y1,X1] = PwZi = Z1 In this case,
8l, ols = (Z1PpwY1Z1)1 Z1PPWY1 y1 = (Z1Z1) ^ 1y1
= (Z1PwZ1) 1 Z1Pwy1 = 8 1,iv as required.
b. The FWL Theorem also states that the residuals from the regressions in part (a) are the same. Hence, their residuals sum of squares are the same. The last regression computes an estimate of the var(81,ojs) as s2 (Z^PwZ^ 1 where s2 is the residual sum of squares divided by the degrees of freedom of the regression. In (11.55), this is the MSE of that regression, since it has the same residuals sum of squares. Note that when r ф 0 in (11.55), IV estimation is necessary and s11 underestimates o11 and will have to be replaced by
(y1 — Z181,Iy),(y1 — Z1§1,iv)/T.
11.16 Recursive Systems.
a. The order condition of identification yields one excluded exogenous variable (x3) from the first equation and no right hand side endogenous variable i. e., k2 = 1 andg1 = 0. Therefore, the first equation is overidentified with degree of overidentification equal to one. The second equation has two excluded exogenous variables (x1 and x2) and one right hand side endogenous variable (y1) so that k2 = 2 and g1 = 1. Hence, the second equation is overidentified of degree one. The rank condition of identification can be based on
"12 0
0 "23_
(x1t, x2t, x3t). For the first equation, the set of
where ‘ is the first row of A. For the second equation, the set of zero restrictions yield
where a2 is the second row of A. Hence, for the first equation
which is of rank 1 in general. Hence, the first equation is identified by the rank condition of identification. Similarly, for the second equation
"11 "12 0 0
which is of rank 1 as long as either "11 or "12 are different from zero. This ensures that either x1 or x2 appear in the first equation to identify the second equation.
b. The first equation is already in reduced form format y1t — —"lix1t – "I2x2t + u1t
y1t is a function of x1t, x2t and only u1t, not u2t. For the second equation, one can substitute for y1t to get
y2t — "23x3t — "21 (—"11 x1t — "I2x2t + u1t) + u2t
— "21"11x1t + "21" 12x2t – "23x3t + u2t – "21 u1t
so that y2t is a function of x1t, x2t, x3t and the error is a linear combination of u1t and u2t.
c. OLS on the first structural equation is also OLS on the first reduced form equation with only exogenous variables on the right hand side. Since x1 and x2 are not correlated with u1t, this yields consistent estimates for "11
and y12. OLS on the second equation regresses y2t on y1t and x3t. Since x3t is not correlated with u2t, the only possible correlation is from y1t and u2t. However, from the first reduced form equation y1t is only a function of exogenous variables and u1t. Since u1t and u2t are not correlated because £ is diagonal for a recursive system, OLS estimates of "21 and y23 are consistent.
d. Assuming that ut ~ N(0, £) where u( = (u1t, u2t), this exercise shows that OLS is MLE for this recursive system. Conditional on the set of x’s, the likelihood function is given by
L(B, Г, £) = (2 Tt)T/2BT£T/2 exp(2 £u0£1utj so that
1 T
logL = (T/2) log2 к + Tlog B – (T/2) log £ – ^ ^u0£ 1ut.
2 t=1
Since B is triangular, B = 1 so that log B = 0. Therefore, the only place
T
in logL where B and Г parameters appear is in u0£1ut. Maximizing
t=1
T
logL with respect to B and Г is equivalent to minimizing ^ u0£1ut with
t=1
respect to B and Г. But £ is diagonal, so £1 is diagonal and
T T T
£u0£4 = J2 u2t/CT11 + £ u2‘/CT22.
t=1 t=1 t=1
The partial derivatives of logL with respect to the coefficients of the first
T
structural equation are simply the partial derivatives of J^u2t/on with
t=1
respect to those coefficients. Setting these partial derivatives equal to zero yields the OLS estimates of the first structural equation. Similarly, the partial derivatives of logL with respects to the coefficients of the second
T
structural equation are simply the derivatives of u22t/o22 with respect to
t=1
those coefficients. Setting these partial derivatives equal to zero yields the OLS estimates of the second structural equation. Hence, MLE is equivalent to OLS for a recursive system.
11.17 Hausman s Specification Test: 2SLS Versus 3SLS. This is based on Baltagi (1989).
a. The two equations model considered is
yi = Z181 + ui and y2 = Z282 + u2 (1)
where yi and y2 are endogenous; Z1 = [y2,x1,x2], Z2 = [y1,x3]; and x1, x2 and x3 are exogenous (the y’s and x’s are Tx1 vectors). As noted in the problem, 8 = 2SLS, 8 = 3SLS, and the corresponding residuals are denoted by u and u. = (a, "1, "2) and 82 = (у, "3) with ay ф 1,so that
the system is complete and we can solve for the reduced form.
Let X = [x1,x2,x3] and PX = X(X0X)_1X0, the 3SLS equations for model (1) are given by
Z0 (Ё_1 <S> Px) ZQ = Z0 (Ё_1 <S> Px) y (2)
where Z = diag[Zi], y0 = (y1,y2), and Ё_1 = [(8ij] is the inverse of the estimated variancecovariance matrix obtained from 2SLS residuals. Equation (2) can also be written as
Z (Ё_1 <S> Px) u = 0, (3)
or more explicitly as
511Z,1Px^ 1 +1812Z11Pxu2 = 0 (4)
512Z2Pxu 1 C 522Z2Pxu2 = 0 (5)
Using the fact that Z1X is a square nonsingular matrix, one can premultiply (4) by (X0X) (Z01X)“1 to get
511X0ti 1 C cr12X0ti2 = 0. (6)
b. Writing Eq. (5) and (6) in terms of Q1 and Q2, one gets
512 (Z2PxZ1) 81 C 522 (Z2PxZ2) 82 = 512 (Z2PxyO C (822 (Z2Pxy2) (7)
and
511X, ZiQi C &12X0Z28l2 = 511X, yi C 812X0y2. (8)
Premultiply (8) by 512Z’2X(X’X)~1 and subtract it from 511 times (7). This eliminates 81, and solves for 82:
82 = (Z2PxZ2)_1 Z2PXy2 = 82. (9)
Therefore, the 3SLS estimator of the overidentified equation is equal to its 2SLS counterpart.
Substituting (9) into (8), and rearranging terms, one gets 511X, Z1Q1 = 511X0y1 +512X0u2.
Premultiplying by Z,1X(X, X)_1/511, and solving for 81, one gets Q1 = (Z’^xZ!)1 Z1Pxy1 C (512/511) (ZjPxZi)"1 ZjPxu. (10)
Using the fact that 512 = — 512/£, and 511 = 522/£, (10) becomes 81 = 81 — (512 /522) (Z’iPxZi)"1 Z’iPxu2. 01
Therefore, the 3SLS estimator of the justidentified equation differs from its 2SLS (or indirect least squares) counterpart by a linear combination of the 2SLS (or 3SLS) residuals of the overidentified equation; see Theil (1971).
c. A Hausmantype test based on and is given by
m = (Qi —81)0 [v (81) — v (81)] 1 (8i — Qi)
where V(8′) = 5n (Z’PxZi) 1
and V(8i) = (1/5”) (Z’PxZi)1 C (^/o^) (Z’PxZi)"1
(Z’iPXZ2) (Z2PxZ2)1 (Z2PxZi) (Z’PxZi)"1 .
The latter term is obtained by using the partitioned inverse of Z0(£"’ <S>
PX)Z. Using (11), the Hausman test statistic becomes m = (5i22/5222) (u2PxZi) [(5ii — 1/5n) (Z’iPxZi)
— (5i22/522) (Z’iPxZ2) (Z2PxZ2)1 (Z2PxZi)]1 (Z’iPxu2) .
However, (on — 1/(7n) = (@12І®22); this enables us to write
where Zi = PXZi is the matrix of second stage regressors of 2SLS, for i = 1,2. This statistic is asymptotically distributed as /J, and can be given the following interpretation:
Claim: m = TR2, where R2 is the Rsquared of the regression of u2 on the set of second stage regressors of both equations, Z1 and Z2.
Proof. This result follows immediately using the fact that cr22 =
u2u2/T = total sums of squares/T, and the fact that the regression sum of squares is given by
where the last equality follows from partitioned inverse and the fact that
Z 2u 2 = o.
11.18 a. Using the order condition of identification, the first equation has two excluded exogenous variables x2 and x3 and only one right hand sided included endogenous variable y2. Hence,
k2 = 2 > g1 = 1
with yt = (y1t, y2t) and xt = (x1t, x2t, x3t). The first structural equation has the following zero restrictions:
where ‘і is the first row of A. Similarly, the second structural equation has the following zero restriction:
where a2 is the second row of A. Hence, A¥1 = which has
"22 "2з
rank one provided either "22 or "23 is different from zero. Similarly, A¥2 = /"11
identified by the rank condition of identification. b. For the first equation, the OLS normal equations are given by
(ЗДІ “12 = Z1y1
V"" 1V ols
where Z1 = [y2,x1]. Hence,
obtained from the matrices of crossproducts provided by problem 11.18.
Hence, the OLS normal equations are
8 1^/p 12 10 20j "и
solving for the OLS estimators, we get
OLS on the second structural equation can be deduced in a similar fashion.
c. For the first equation, the 2SLS normal equations are given by
where Z’ = [y2,x1] andX = [x1,x2,x3]. Hence, Z’X =
with
Therefore, the 2SLS normal equations are
2SLS on the second structural equation can be deduced in a similar fashion.
d. The first equation is overidentified and the reduced form parameter estimates will give more than one solution to the structural form parameters.
However, the second equation is justidentified. Hence, 2SLS reduces to indirect least squares which in this case solves uniquely for the structural parameters from the reduced form parameters. ILS on the second equation yields
" 21
"22 = (X, Zi)1X, y2
V23 ILS
where Z1 = [y1,x2,x3] andX = [x1,x2,x3]. Therefore,
and
‘x’^ 
10 

X0y2 = 
x2y2 x03y2 
= 
20C 30 
so that 
" 21 
5 
0 
0 
1 
10 

" 22 
= 
40 
20 
0 
20C 

" 23 
ILS 
20 
0 
10 
30 
Note that X0Z1 for the first equation is of dimension 3 x 2 and is not even square.
11.19 Supply and Demand Equations for Traded Money. This is based on Laffer (1970).
b. OLS Estimation
SYSLIN Procedure
Ordinary Least Squares Estimation
Model: SUPPLY Dependent variable: LNTM_P
Analysis of Variance

Root MSE 0.03068 RSquare 0.9442
DepMean 5.48051 Adj RSQ 0.9380
C. V. 0.55982
Parameter Estimates

OLS ESTIMATION OF MONEY DEMAND FUNCTION
SYSLIN Procedure
Ordinary Least Squares Estimation
Model: DEMAND Dependent variable: LNTM. P
Analysis of Variance 

Sum of 
Mean 

Source 
DF 
Squares 
Square 
F Value 
Prob>F 
Model 
4 
0.29889 
0.07472 
245.207 
0.0001 
Error 
16 
0.00488 
0.00030 

C Total 
20 
0.30377 

Root MSE 0.01746 RSquare 
0.9839 

Dep 
Mean 5.48051 Adj RSQ 
0.9799 

C. V. 
0.31852 

Parameter Estimates 

Parameter 
Standard 
T for HO: 

Variable 
DF 
Estimate 
Error Parameter=0 
Prob > T 

INTERCEP 
1 
3.055597 
0.647464 
4.719 
0.0002 
LNY_P 
1 
0.618770 
0.055422 
11.165 
0.0001 
LNI 
1 
0.015724 
0.021951 
0.716 
0.4841 
LNS1 
1 
0.305535 
0.101837 
3.000 
0.0085 
LNS2 
1 
0.147360 
0.202409 
0.728 
0.4771 
c. 2SLS Estimation
SYSLIN Procedure
TwoStage Least Squares Estimation
Model: SUPPLY Dependent variable: LNTM. P
Analysis of Variance

2SLS ESTIMATION OF MONEY DEMAND FUNCTION
SYSLIN Procedure
TwoStage Least Squares Estimation
Model: DEMAND
Dependent variable: LNTM. P
Analysis of Variance
Sum of 
Mean 

Source 
DF 
Squares 
Square 
F Value 
Prob>F 
Model 
4 
0.30010 
0.07503 
154.837 
0.0001 
Error 
16 
0.00775 
0.00048 

C Total 
20 
0.30377 
Root MSE 0.02201 RSquare 0.9748
Dep Mean 5.48051 Adj RSQ 0.9685
C. V. 0.40165
Parameter Estimates

The total number of instruments equals the number of parameters in the equation. The equation is just identified, and the test for over identification is not computed.
d. 3 SLS Estimation
SYSLIN Procedure
ThreeStage Least Squares Estimation
Cross Model Covariance
Sigma SUPPLY DEMAND
SUPPLY 0.0009879044 0.000199017
DEMAND 0.000199017 0.000484546
Cross Model Covariance
Corr SUPPLY DEMAND
SUPPLY 1 0.287650003
DEMAND 0.287650003 1
Cross Model Inverse Correlation

System Weighted MSE: 0.53916 with 34 degrees of freedom. System Weighted RSquare: 0.9858 Model: SUPPLY Dependent variable: LNTM. P
3SLS ESTIMATION OF MODEY SUPPLY AND DEMAND FUNCTION
SYSLIN Procedure
ThreeStage Least Squares Estimation
Parameter Estimates
Variable 
DF 
Parameter Estimate 
Standard Error 
T for HO: Parameter=0 
Prob > T 
INTERCEP 
1 
3.741721 
0.280279 
13.350 
0.0001 
LNRM. P 
1 
0.524645 
0.092504 
5.672 
0.0001 
LNI 
1 
0.177757 
0.013347 
13.318 
0.0001 
Model: DEMAND
Dependent variable: LNTM. P
Parameter Estimates
Variable 
DF 
Parameter Estimate 
Standard Error 
T for HO: Parameter=0 
Prob > T 
INTERCEP 
1 
1.887370 
1.186847 
1.590 
0.1313 
LNY_P 
1 
0.764657 
0.110168 
6.941 
0.0001 
LNI 
1 
0.077564 
0.048797 
1.590 
0.1315 
LNS1 
1 
0.234061 
0.141070 
1.659 
0.1165 
LNS2 
1 
0.126953 
0.250346 
0.507 
0.6190 
e. HAUSMANTEST
Hausman Test
2SLS vs OLS for SUPPLY Equation
HAUSMAN
6.7586205
2SLS vs OLS for DEMAND Equation
HAUSMAN
2.3088302
SAS PROGRAM
Data Laffer1; Input TM RM Y S2 i S1 P; Cards;
Data Laffer; Set Laffer1;
LNTM_P=LOG(TM/P);
LNRM_P=LOG(RM/P);
LNi=LOG(i);
LNY_P=LOG(Y/P);
LNS1=LOG(S1);
LNS2=LOG(S2);
T=21;
Proc syslin data=Lafferoutest=ols_s outcov;
SUPPLY: MODEL LNTM_P=LNRM_P LNi;
TITLE ‘OLS ESTIMATION OF MONEY SUPPLY FUNCTION’;
Proc syslin data=Lafferoutest=ols_s outcov;
DEMAND: MODEL LNTM_P=LNY_P LNi LNS1 LNS2;
TITLE ‘OLS ESTIMATION OF MONEY DEMAND FUNCTION’;
Proc syslin 2SLS data=Lafferoutest=TSLS_s outcov;
ENDO LNTMP LNi;
INST LNRM_P LNY_P LNS1 LNS2;
SUPPLY: MODEL LNTM_P=LNRM_P LNi/overid;
TITLE ‘2SLS ESTIMATION OF MONEY SUPPLY FUNCTION’;
Proc syslin 2SLS data=Lafferoutest=TSLS_d outcov;
ENDO LNTM_P LNi;
INST LNRM_P LNY_P LNS1 LNS2;
DEMAND: MODEL LNTM_P=LNY_P LNi LNS1 LNS2/overid; TITLE ‘2SLS ESTIMATION OF MONEY DEMAND FUNCTION’;
Proc SYSLIN 3SLS data=Lafferoutest=S_3SLS outcov; ENDO LNTMP LNi;
INST LNRM_P LNY_P LNS1 LNS2;
SUPPLY: MODEL LNTM_P=LNRM_P LNi;
DEMAND: MODEL LNTM_P=LNY_P LNi LNS1 LNS2; TITLE ‘3SLS ESTIMATION’;
RUN;
PROCIML;
TITLE ‘HAUSMAN TEST’;
use ols_s; read all var fintercep lnrm_p ini) into olsl;
olsbLs=ols1[1,]; ols_v_s=ols1[2:4,];
use ols_d; read all var f intercep lny_p lni lnsl lns2) into
ols2;
olsbLd=ols2[1,]; ols_v_d=ols2[2:6,];
use tsls_s; read all var f intercep lnrm_p lni} into tsls1;
tslsbts=tsls1[13,]; tsls_v_s=tsls1[14:16,];
use tsls_d; read all var f intercep lny_p lni lns1 lns2} into
tsls2;
tslsbtd=tsls2[13,]; tsls_v_d=tsls2[14:18,];
d=tslsbts ‘olsbt_s‘; varcov=tsls_v_sols_v_s;
Hausman=d ‘*inv(varcov)*d;
print ‘Hauman Test’,, ‘2SLS vs OLS for SUPPLY Equation’,,
Hausman;
d=tslsbt_d ‘olsbt_d‘; varcov=tsls_v_dols_v_d;
Hausman=d ‘*inv(varcov)*d;
print ‘2SLS vs OLS for DEMAND Equation’,, Hausman;
11.20 a. Writing this system of two equations in matrix form, as described in (A.1) we get xt = [1,Xt] and u0 = (u1t, u2t). There are no zero restrictions for the demand equation and only one zero restriction for the supply equation (Xt does not appear in the supply equation). Therefore, the demand equation is not identified by the order condition since the number of excluded exogenous variables k2 = 0 is less than the number of right hand side included endogenous variables g1 = 1 . 
Similarly, the supply equation is justidentified by the order condition since k2 = gi = 1.
By the rank condition, only the supply equation need be checked for identification. In this case, ф0 = (0,1/ since y22 = 0. Therefore,
Аф = Р’гіф = ("22) = (?).
This is of rank one as long as a2 ф 0, i. e., as long as Xt is present in the demand equation. One can also premultiply this system of two equations by
F = fi1 fi2 f21 f22
the transformed matrix FB should satisfy the following normalization restrictions: f11 + f12 = 1 and f21 + f22 = 1.
Also, Fr must satisfy the following zero restriction:
—«2f21 + 0 f22 = 0.
If a2 ф 0, then f21 = 0 which also gives from the second normalization equation that f22 = 1. Hence, the second row of F is indeed the second row of an identity matrix and the supply equation is identified. However, for the demand equation, the only restriction is the normalization restriction f11 + f12 = 1 which is not enough for identification.
„ T T
b. The OLS estimator of the supply equation yields " 1ols = J2 ptqt/ J2 p2
where pt = Pt — l5 and qt = Qt — Q. Substituting qt = "1 pt + (u2t — u2/
TT
we get " 1,ols = "1 + J2 pt(u2t — 7)/ E pj2.
« "o ‘2
The reduced form equation for Pt yields Pt = —————————— —————— Xt +
uit — u2t «1 + "1
T
Defining mxx = plim J2 x2/T, we get
b. Regress yi and y3 on all the X’s including the constant, i. e., [1,Xi, X2,X5]. Get y i andy 3 and regress y2 on a constant, y 1, y 3 andX2 in the secondstep regression.
c. For the first equation, regressing y2 on X2 and X3 yields
y2 — ft21 C ft22X2 + O23X3 C ‘V2 — y2 + ‘O2 T T T
with £ v2t — J2 v2tX2t — J2 v2tX3t — 0 by the property of least squares.
t=i t=i t=i
Replacing y2 by y2 in Eq. (1) yields yi — ai C "2 (y2 C v^2) C "iXi C ui — ai C "2y2 C "iXi C ("2v2 C ui) .
T
In this case, y2tv2t — 0 because the predictors and residuals of the same
t=i
T
regression are uncorrelated. Also, y2tu1t — 0 since y2t is a linear com
t=i
T
bination of X2 and X3 and the latter are exogenous. However, X1tv2t
t=i
is not necessarily zero since Xi was not included in the first stage regression. Hence, this estimation method does not necessarily lead to consistent estimates.
d. The test for overidentification for Eq. (1) can be obtained from the Ftest given in Eq. (11.48) with RRSS* obtained from the second stage regression residual sum of squares of 2SLS run on Eq. (1), i. e., the regression of yi on [i,^2,Xi] where y2 is obtained from the regression of y2 on [i, X1, X2.., X5]. The URSS* is obtained from the residual sum of squares of the regression of yi on all the set of instruments, i. e., [i, X1,.., X5]. The URSS is the 2SLS residual sum of squares of Eq. (1) as reported from any 2SLS package. This is obtained as
2
Note that this differs from RRSS* in that y2t and not y2t is used in the computation of the residuals. ‘, the number of instruments is 6 and ki — 2 while gi — i. Hence, the numerator degrees of freedom of the Ftest is
‘ — (gi + ki) = 3 whereas the denominator degrees of freedom of the Ftest isT — ‘ = T — 6. This is equivalent to running 2SLS residuals on the matrix of all predetermined variables [1 ,X1,.., X5] and computing T times the uncentered R2 of this regression. This is asymptotically distributed as X2 with three degrees of freedom. Large values of this statistic reject the overidentification restrictions.
11.23 Identification and Estimation of a Simple TwoEquation Model. This solution is based upon Singh and Bhat (1988).
®12
®12 m2
The structural and reduced form parameters are related as follows:
1 —p" 
Я11 
a 

—1 1 
Я12 
у 
or
‘ = Я11 — "^12 У = Я12 — Я11
We assume that the reducedform parameters л11, я12, m2, m12 and m2 have been estimated. The identification problem is one of recovering a, ", ", oj2 and o12 and o from the reducedform parameters.
From the second equation, it is clear that у is identified. We now examine the relation between £ and fi given by £ = BfiB0 where B = ^ — . Thus, we have
of = m1 — 2Pm12 + P2m2,
012 = – m2 + (1 + P)mi2 — Pm^ o2 = m2 — 2m12 C m.
Knowledge of mi, m12, and m2 is enough to recover o. Hence, o is identified. Note, however, that o2 and o12 depend on ". Also, a is dependent on ". Hence, given any " = "*, we can solve for a, oj2 and o12. Thus, in principle, we have an infinity of solutions as the choice of " varies. Identification could have been studied using rank and order conditions. However, in a simple framework as this, it is easier to check identification directly. Identification can be reached by imposing an extra restriction f (", a, oj2, o12) = 0, which is independent of the three equations we have for of, o12 and a. For example, " = 0 gives a recursive structure. We now study the particular case where f (", a, o2, o12) = o12 = 0.
b. Identification with oj2 = 0: When o12 = 0, one can solve immediately for P from the o12 equation:
P = m12 — mf
m22 — m12
Therefore, P can be recovered from m^ m12, and m2. Given ", all the other parameters (of, a) can be recovered as discussed above.
c. OLS estimation of ft when aj2 = 0: The OLS estimator of " is
T T
л E (yt1— y 1 )(yt2— y2) E (ut1 — u1 )(yt1— y 1)
P ols = V = P C————— V
T
E(ut1 — u1)(ut2 — ЇЇ2)
t=1_____________________
T
E(yt2 — y2)2
t=1
The second term is nonzero in probability limit as yti and uti are correlated from the first structural equation. Hence "ols is not consistent.
d. From the second structural equation, we have yt2 — yt1 = у + ut2, which means that (yt2 — y2) — (yt1 — y 1) = ut2 — u2, or zt = ut2 — u2. When
an = 0, this zt is clearly uncorrelated with ut1 and by definition correlated withyt2. Therefore, zt = (yt2 —y2) — (yt1 —y1) maybe taken as instruments, and thus
E zt(yt1 — yi) E(yt1 — yi)[(yt2 — y2) — (yt1 — yi)]
" t=i t=i
"IV = —————————– = ——————————————————————–
Ё zt(yt2 — y2) Ё(yt2 — y2)[(yt2 — y2) — (yt1 — yi)]
t=i t=i
obtained in part (d). The OLS estimators of the reducedform parameters are consistent. Therefore, "ILS which is a continuous tranform of those estimators is also consistent. It is however, useful to show this consistency directly since it brings out the role of the identifying restriction.
From the structural equations, we have
" i — p" 
yt1 — yi 
ut1 — ui 

—i i 
yt2 — y2 
ut2 — Й2 
Using the above relation and with some algebra, it can be shown that for
" Ф i:
T T
E(Ul – Ul)(Ut2 – Ui) + " E(^2 – П2)2
P ILS = T—————————————– T since,
(utl – ui)(ut2 – u2) + (ut2 – u2)2
TT plimT £(ut! – TN1)(ut2 – 1N2) = 012, and plimT £(ut2 – u)2 = of.
Therefore, plim "ILS = (о12 + bof) / (o12 C of) . The restriction o12 = 0 implies plim pils = " which proves consistency. It is clear that"ILS can be interpreted as a methodofmoments estimator.
11.24 Errors in Measurement and the Wald (1940) Estimator. This is based on Farebrother (1987).
a. The simple IV estimator of " with instrument z is given by "IV =
n n n
E ziyi zixi. But E ziyi = sum of the yi’s from the second sam
i=1 i=1 i=1
ple minus the sum of the yi’s from the first sample. If n is even, then
nn
Eziyi = (y2 – y 1). Similarly, £ziXi = (x2 – x 1), so that "iv =
i=1 i=1
(y2 – y1)/(x2 – X0 = "W.
b. Let p2 = of/ (^u C of) and x2 = of/ (o2 + of) then Wi = p2Xi* – x2u
has mean E(wi) = 0 since E (xi*) = E(ui) = 0. Also, var(wD = (P2)2°* + (£2)2ou2 = o*ou= (o* + ou).
Since wi is a linear combination of Normal random variables, it is also Normal.
E(xiwi) = p2E (xix*) – x2E(xiui) = p2of – x2o2 = 0.
c. Now £2xi C wi = X2 (xi* C ui) C p2xi* – £2ui = (x2 C p2)xi* = xi* since
X2 C p2 = 1.
Using this result and the fact that yi = "xi* C £i, we get
nn
PW = ^2 *yi/l>i
i=1 i=1
(n n n n
zixi + ^i/ ^ zixi
i=i i=i i=i i=i
so that E("W/x1, ..,xn) = "x2. This follows because wi and ei are independent of xi and therefore independent of zi which is a function of xi. Similarly, "ols = p xiyi p xi2 = "x2 + " p wixi p xi2 +
i=i i=i i=i i=i
nn
Exi£i xi2 so that E("ols/x1, ..,xn) = "x2 since wi and ei are inde
i=i i=i
pendent of xi. Therefore, the bias of "ols = bias of "W = ("x2 — ") = _PCT,2 / (CT* + CT,2) for all choices of zi which are a function of xi.
11.25 Comparison oftratios. This is based onFarebrother(1991). LetZ1 = [y2,X], then by the FWLTheorem, OLS on the first equation is equivalent to that on P Xyi = рхУ2а+р xui. This yields a ols = (у2РхУ2) 1 y^j5 Xyi and the residual sum of squares are equivalent
yiPzyi = yiPxyi — yiPxy2 (y2pxy^ i y2pxyi so that var(aols) = s^ (y213Xy2) i where s^ = yiPzyi/(T — K). The tratio for
a = 0 is
= (T — к)1/2(у2РхУі!) [(yiРхУі)(у2РхУ2) — (yipxy2)ri/2 .
This expression is unchanged if the roles of yi and y2 are reversed so that the tratio for Hb у = 0 in the second equation is the same as that for Ho; a = 0 in the first equation.
Foryi andy2 jointly Normal with correlation coefficient p, Farebrother(1991) shows that the above two tests correspond to a test for p = 0 in the bivariate Normal model so that it makes sense that the two statistics are identical.
11.28 This gives the backup runs for Crime Data for 1987 for Tables 11.2 to 11.6
. ivreg lcrmrte (lprbarr lpolpc= ltaxpc lmix) lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban if
year==87
Instrumental variables (2SLS) regression
Number of obs= 90
F(20,69) = 17.35
Prob > F =0.0000 Rsquared =0.8446 Adj Rsquared =0.7996 Root MSE =.24568
lcrmrte 
Coef. 
Std. Err. 
t 
P>t 
[95% Conf. Interval] 

lprbarr 
.4393081 
.2267579 
1.94 
0.057 
.8916777 
.0130615 
lpolpc 
.5136133 
.1976888 
2.60 
0.011 
.1192349 
.9079918 
lprbconv 
.2713278 
.0847024 
3.20 
0.002 
.4403044 
.1023512 
lprbpris 
.0278416 
.1283276 
0.22 
0.829 
.2838482 
.2281651 
lavgsen 
.280122 
.1387228 
2.02 
0.047 
.5568663 
.0033776 
ldensity 
.3273521 
.0893292 
3.66 
0.000 
.1491452 
.505559 
lwcon 
.3456183 
.2419206 
1.43 
0.158 
.137 
.8282366 
lwtuc 
.1773533 
.1718849 
1.03 
0.306 
.1655477 
.5202542 
lwtrd 
.212578 
.3239984 
0.66 
0.514 
.433781 
.8589371 
lwfir 
.3540903 
.2612516 
1.36 
0.180 
.8752731 
.1670925 
lwser 
.2911556 
.1122454 
2.59 
0.012 
.5150789 
.0672322 
lwmfg 
.0642196 
.1644108 
0.39 
0.697 
.263771 
.3922102 
lwfed 
.2974661 
.3425026 
0.87 
0.388 
.3858079 
.9807402 
lwsta 
.0037846 
.3102383 
0.01 
0.990 
.615124 
.6226931 
lwloc 
.4336541 
.5166733 
0.84 
0.404 
1.464389 
.597081 
lpctymle 
.0095115 
.1869867 
0.05 
0.960 
.3635166 
.3825397 
lpctmin 
.2285766 
.0543079 
4.21 
0.000 
.1202354 
.3369179 
west 
.0952899 
.1301449 
0.73 
0.467 
.3549219 
.1643422 
central 
.1792662 
.0762815 
2.35 
0.022 
.3314437 
.0270888 
urban 
.1139416 
.143354 
0.79 
0.429 
.3999251 
.1720419 
_cons 
1.159015 
3.898202 
0.30 
0.767 
8.935716 
6.617686 
Instrumented: lprbarr lpolpc
Instruments: lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg
lwfed lwsta lwloc lpctymle lpctmin west central urban ltaxpc lmix
. estimates store b2sls
. reg lcrmrte lprbarr lprbconv lprbpris lavgsen lpolpc ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban if year==87
Source 
SS 
df 
MS 
Number of obs 
= 
90 
F(20,69) 
= 
19.71 

Model 
22.8072483 
20 
1.14036241 
Prob > F 
= 
0.0000 
Residual 
3.99245334 
69 
.057861643 
Rsquared 
= 
0.8510 
Adj Rsquared 
= 
0.8078 

Total 
26.7997016 
89 
.301120243 
Root MSE 
= 
.24054 
lcrmrte 
Coef. 
Std. Err. 
t 
P>t 
[95% Conf. Interval] 

lprbarr 
.4522907 
.0816261 
5.54 
0.000 
.6151303 
.2894511 
lprbconv 
.3003044 
.0600259 
5.00 
0.000 
.4200527 
.180556 
lprbpris 
.0340435 
.1251096 
0.27 
0.786 
.2836303 
.2155433 
lavgsen 
.2134467 
.1167513 
1.83 
0.072 
.4463592 
.0194659 
lpolpc 
.3610463 
.0909534 
3.97 
0.000 
.1795993 
.5424934 
ldensity 
.3149706 
.0698265 
4.51 
0.000 
.1756705 
.4542707 
lwcon 
.2727634 
.2198714 
1.24 
0.219 
.165868 
.7113949 
lwtuc 
.1603777 
.1666014 
0.96 
0.339 
.171983 
.4927385 
lwtrd 
.1325719 
.3005086 
0.44 
0.660 
.4669263 
.7320702 
lwfir 
.3205858 
.251185 
1.28 
0.206 
.8216861 
.1805146 
lwser 
.2694193 
.1039842 
2.59 
0.012 
.4768622 
.0619765 
lwmfg 
.1029571 
.1524804 
0.68 
0.502 
.2012331 
.4071472 
lwfed 
.3856593 
.3215442 
1.20 
0.234 
.2558039 
1.027123 
lwsta 
.078239 
.2701264 
0.29 
0.773 
.6171264 
.4606485 
lwloc 
.1774064 
.4251793 
0.42 
0.678 
1.025616 
.670803 
lpctymle 
.0326912 
.1580377 
0.21 
0.837 
.2825855 
.3479678 
lpctmin 
.2245975 
.0519005 
4.33 
0.000 
.1210589 
.3281361 
west 
.087998 
.1243235 
0.71 
0.481 
.3360167 
.1600207 
central 
.1771378 
.0739535 
2.40 
0.019 
.3246709 
.0296046 
urban 
.0896129 
.1375084 
0.65 
0.517 
.3639347 
.184709 
_cons 
3.395919 
3.020674 
1.12 
0.265 
9.421998 
2.630159 
. estimates store bols. hausman b2sls bols 
Coefficients

b = consistent under Ho and Ha; obtained from ivreg B = inconsistent under Ha, efficient under Ho; obtained from regress
Test: Ho: difference in coefficients not systematic chi2(20) = (bB)'[(V_bV_B)~ (1)](bB)
= 0.87
Prob > chi2 = 1.0000
. ivreg Icrmrte (Iprbarr lpolpc= Itaxpc Imix) Iprbconv Iprbpris lavgsen Idensity Iwcon Iwtuc Iwtrd Iwfir Iwser Iwmfg Iwfed Iwsta IwIoc IpctymIe Ipctmin west centraI urban if year==87, first
Firststage regressions
90
3.11
0.0002
0.4742
0.3218
.33174
Iprbarr 
Coef. 
Std. Err. 
t 
P>t 
[95% Conf. IntervaI] 

Iprbconv 
.1946392 
.0877581 
2.22 
0.030 
.3697119 
.0195665 
Iprbpris 
.0240173 
.1732583 
0.14 
0.890 
.3696581 
.3216236 
Iavgsen 
.1565061 
.1527134 
1.02 
0.309 
.1481488 
.4611611 
Idensity 
.2211654 
.0941026 
2.35 
0.022 
.408895 
.0334357 
Iwcon 
.2024569 
.3020226 
0.67 
0.505 
.8049755 
.4000616 
Iwtuc 
.0461931 
.230479 
0.20 
0.842 
.5059861 
.4135999 
Iwtrd 
.0494793 
.4105612 
0.12 
0.904 
.769568 
.8685266 
Iwfir 
.050559 
.3507405 
0.14 
0.886 
.6491492 
.7502671 
Iwser 
.0551851 
.1500094 
0.37 
0.714 
.2440754 
.3544456 
Iwmfg 
.0550689 
.2138375 
0.26 
0.798 
.3715252 
.481663 
Iwfed 
.2622408 
.4454479 
0.59 
0.558 
.6264035 
1.150885 
Iwsta 
.4843599 
.3749414 
1.29 
0.201 
1.232347 
.2636277 
IwIoc 
.7739819 
.5511607 
1.40 
0.165 
.3255536 
1.873517 
IpctymIe 
.3373594 
.2203286 
1.53 
0.130 
.776903 
.1021842 
Ipctmin 
.0096724 
.0729716 
0.13 
0.895 
.1552467 
.1359019 
west 
.0701236 
.1756211 
0.40 
0.691 
.280231 
.4204782 
centraI 
.0112086 
.1034557 
0.11 
0.914 
.1951798 
.217597 
lcrmrte 
Coef. 
Std. Err. 
t 
P>t 
[95% Conf. Interval] 

lprbarr 
.4393081 
.2267579 
1.94 
0.057 
.8916777 
.0130615 
lpolpc 
.5136133 
.1976888 
2.60 
0.011 
.1192349 
.9079918 
lprbconv 
.2713278 
.0847024 
3.20 
0.002 
.4403044 
.1023512 
lprbpris 
.0278416 
.1283276 
0.22 
0.829 
.2838482 
.2281651 
lavgsen 
.280122 
.1387228 
2.02 
0.047 
.5568663 
.0033776 
ldensity 
.3273521 
.0893292 
3.66 
0.000 
.1491452 
.505559 
lwcon 
.3456183 
.2419206 
1.43 
0.158 
.137 
.8282366 
if year==87 

Source 
1 SS 
df 
MS 
Number of obs 
= 90 

= 4.42 

F(20,69) 

Model 
6.99830344 
20 
.349915172 Prob > F 
= 0.0000 

Residual 
5.46683312 
69 
.079229465 Rsquared 
= 0.5614 

= 0.4343 

Total 
1 12.4651366 
89 
.140057714 Root MSE 
= .28148 

lpolpc 
Coef. 
Std. Err. 
t 
P>t 
[95% Conf. Interval] 

lmix 
.2177256 
.0733414 
2.97 
0.004 
.0714135 
.3640378 
ltaxpc 
.5601989 
.1489398 
3.76 
0.000 
.2630721 
.8573258 
lprbconv 
.0037744 
.0744622 
0.05 
0.960 
.1447736 
.1523223 
lprbpris 
.0487064 
.1470085 
0.33 
0.741 
.3419802 
.2445675 
lavgsen 
.3958972 
.1295763 
3.06 
0.003 
.1373996 
.6543948 
ldensity 
.0201292 
.0798454 
0.25 
0.802 
.1391581 
.1794165 
lwcon 
.5368469 
.2562641 
2.09 
0.040 
1.04808 
.025614 
lwtuc 
.0216638 
.1955598 
0.11 
0.912 
.411795 
.3684674 
lwtrd 
.4207274 
.3483584 
1.21 
0.231 
1.115683 
.2742286 
lwfir 
.0001257 
.2976009 
0.00 
1.000 
.5935718 
.5938232 
lwser 
.0973089 
.1272819 
0.76 
0.447 
.1566116 
.3512293 
lwmfg 
.1710295 
.1814396 
0.94 
0.349 
.1909327 
.5329916 
lwfed 
.8555422 
.3779595 
2.26 
0.027 
.1015338 
1.609551 
lwsta 
.1118764 
.3181352 
0.35 
0.726 
.7465387 
.5227859 
lwloc 
1.375102 
.4676561 
2.94 
0.004 
.4421535 
2.30805 
lpctymle 
.4186939 
.1869473 
2.24 
0.028 
.0457442 
.7916436 
lpctmin 
.0517966 
.0619159 
0.84 
0.406 
.1753154 
.0717222 
west 
.1458865 
.1490133 
0.98 
0.331 
.151387 
.4431599 
central 
.0477227 
.0877814 
0.54 
0.588 
.1273964 
.2228419 
urban 
.1192027 
.1719407 
0.69 
0.490 
.4622151 
.2238097 
_cons 
16.33148 
3.221824 
5.07 
0.000 
22.75884 
9.904113 
The test that the additional instruments are jointly significant in the first stage regression for Ipolpc is performed as follows:
. test lmix=ltaxpc=0
(1) lmix – ltaxpc = 0
(2) lmix = 0
F(2,69) = 10.56 Prob > F = 0.0001
. ivregress 2sls Icrmrte (Iprbarr lpolpc= Itaxpc Imix) Iprbconv Iprbpris lavgsen Idensity Iwcon Iwtuc Iwtrd Iwfir Iwser Iwmfg Iwfed Iwsta Iwloc Ipctymle Ipctmin west central urban if year==87
InstrumentaI variabIes (2SLS) regression 
Number of obs 
= 90 
WaId chi2(20) 
= 452.49 

Prob > chi2 
= 0.0000 

Rsquared 
= 0.8446 

Root MSE 
= .21511 
Icrmrte 
Coef. 
Std. Err. 
z 
P>z 
[95% Conf. IntervaI] 

Iprbarr 
.4393081 
.1985481 
2.21 
0.027 
.8284552 
.050161 
IpoIpc 
.5136133 
.1730954 
2.97 
0.003 
.1743526 
.852874 
Iprbconv 
.2713278 
.074165 
3.66 
0.000 
.4166885 
.1259672 
Iprbpris 
.0278416 
.112363 
0.25 
0.804 
.2480691 
.1923859 
Iavgsen 
.280122 
.121465 
2.31 
0.021 
.5181889 
.042055 
Idensity 
.3273521 
.0782162 
4.19 
0.000 
.1740512 
.4806531 
Iwcon 
.3456183 
.2118244 
1.63 
0.103 
.06955 
.7607865 
Iwtuc 
.1773533 
.1505016 
1.18 
0.239 
.1176244 
.4723309 
Iwtrd 
.212578 
.2836914 
0.75 
0.454 
.3434468 
.7686029 
Iwfir 
.3540903 
.2287506 
1.55 
0.122 
.8024333 
.0942527 
Iwser 
.2911556 
.0982815 
2.96 
0.003 
.4837837 
.0985274 
Iwmfg 
.0642196 
.1439573 
0.45 
0.656 
.2179316 
.3463707 
Iwfed 
.2974661 
.2998936 
0.99 
0.321 
.2903145 
.8852468 
Iwsta 
.0037846 
.2716432 
0.01 
0.989 
.5286262 
.5361954 
Iwtuc 
.1773533 
.1718849 
1.03 
0.306 
.1655477 
.5202542 
Iwtrd 
.212578 
.3239984 
0.66 
0.514 
.433781 
.8589371 
Iwfir 
.3540903 
.2612516 
1.36 
0.180 
.8752731 
.1670925 
Iwser 
.2911556 
.1122454 
2.59 
0.012 
.5150789 
.0672322 
Iwmfg 
.0642196 
.1644108 
0.39 
0.697 
.263771 
.3922102 
Iwfed 
.2974661 
.3425026 
0.87 
0.388 
.3858079 
.9807402 
Iwsta 
.0037846 
.3102383 
0.01 
0.990 
.615124 
.6226931 
IwIoc 
.4336541 
.5166733 
0.84 
0.404 
1.464389 
.597081 
IpctymIe 
.0095115 
.1869867 
0.05 
0.960 
.3635166 
.3825397 
Ipctmin 
.2285766 
.0543079 
4.21 
0.000 
.1202354 
.3369179 
west 
.0952899 
.1301449 
0.73 
0.467 
.3549219 
.1643422 
centraI 
.1792662 
.0762815 
2.35 
0.022 
.3314437 
.0270888 
urban 
.1139416 
.143354 
0.79 
0.429 
.3999251 
.1720419 
_cons 
1.159015 
3.898202 
0.30 
0.767 
8.935716 
6.617686 
Instrumented: Iprbarr Ipolpc
Instruments: Iprbconv Iprbpris lavgsen Idensity Iwcon Iwtuc Iwtrd Iwfir Iwser Iwmfg
Iwfed Iwsta Iwloc Ipctymle Ipctmin west central urban Itaxpc Imix
. reg Iprbarr Imix Itaxpc Iprbconv Iprbpris lavgsen Idensity Iwcon Iwtuc Iwtrd Iwfir Iwser Iwmfg Iwfed Iwsta Iwloc Ipctymle Ipctmin west central urban if year==87
Source 
1 SS 
df 
MS 
Number of obs F(20, 69) 
= 90 = 3.11 
Model 

6.84874028 
20 
.342437014 
Prob > F 
= 0.0002 

Residual 
7.59345096 
69 
.110050014 
Rsquared Adj Rsquared 
= 0.4742 = 0.3218 
Total 
1 14.4421912 
89 
.162271812 
Root MSE 
= .33174 
lprbarr 
Coef. 
Std. Err. 
t 
P>t [95% Conf. Interval] 

lmix 
.2682143 
.0864373 
3.10 
0.003 .0957766 
.4406519 
ltaxpc 
.1938134 
.1755345 
1.10 
0.273 .5439952 
.1563684 
lprbconv 
.1946392 
.0877581 
2.22 
0.030 .3697119 
.0195665 
lprbpris 
.0240173 
.1732583 
0.14 
0.890 .3696581 
.3216236 
lavgsen 
.1565061 
.1527134 
1.02 
0.309 .1481488 
.4611611 
ldensity 
.2211654 
.0941026 
2.35 
0.022 .408895 
.0334357 
lwcon 
.2024569 
.3020226 
0.67 
0.505 .8049755 
.4000616 
lwtuc 
.0461931 
.230479 
0.20 
0.842 .5059861 
.4135999 
lwtrd 
.0494793 
.4105612 
0.12 
0.904 .769568 
.8685266 
lwfir 
.050559 
.3507405 
0.14 
0.886 .6491492 
.7502671 
lwser 
.0551851 
.1500094 
0.37 
0.714 .2440754 
.3544456 
lwmfg 
.0550689 
.2138375 
0.26 
0.798 .3715252 
.481663 
lwfed 
.2622408 
.4454479 
0.59 
0.558 .6264035 
1.150885 
lwsta 
.4843599 
.3749414 
1.29 
0.201 1.232347 
.2636277 
lwloc 
.7739819 
.5511607 
1.40 
0.165 .3255536 
1.873517 
lpctymle 
.3373594 
.2203286 
1.53 
0.130 .776903 
.1021842 
lpctmin 
.0096724 
.0729716 
0.13 
0.895 .1552467 
.1359019 
west 
.0701236 
.1756211 
0.40 
0.691 .280231 
.4204782 
central 
.0112086 
.1034557 
0.11 
0.914 .1951798 
.217597 
urban 
.0150372 
.2026425 
0.07 
0.941 .4192979 
.3892234 
_cons 
4.319234 
3.797113 
1.14 
0.259 11.89427 
3.255799 
Then we can test that the additional instruments are jointly significant in the first stage regression for Iprbarr as follows:
. test lmix=ltaxpc=0
(1) lmix – ltaxpc = 0
(2) lmix = 0
F(2, 69) = 5.78
Prob > F = 0.0048
For lpolpc, the first stage regression is given by
. reg lpolpc lmix ltaxpc lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban
lwloc 
.4336541 
.4523966 
0.96 
0.338 
1.320335 
.453027 
lpctymle 
.0095115 
.1637246 
0.06 
0.954 
.3113827 
.3304058 
lpctmin 
.2285766 
.0475517 
4.81 
0.000 
.135377 
.3217763 
west 
.0952899 
.1139543 
0.84 
0.403 
.3186361 
.1280564 
central 
.1792662 
.0667917 
2.68 
0.007 
.3101756 
.0483569 
urban .1139416 .1255201 0.91 0.364 .3599564 .1320732
.cons 1.159015 3.413247 0.34 0.734 7.848855 5.530825
Instrumented: Iprbarr Ipolpc
Instruments: lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed
lwsta lwloc lpctymle lpctmin west central urban ltaxpc lmix
. estat firststage
Shea’s partial Rsquared
Shea’s 
Shea’s 

Variable 
Partial Rsq. 
Adj. Partial Rsq. 
lprbarr 
0.1352 
0.0996 
lpolpc 
0.2208 
0.0093 
Minimum eigenvalue statistic = 5.31166
Critical Values # of endogenous regressors: 2
Ho: Instruments are weak # of excluded instruments: 2
5% 
10% 
20% 
30% 

2SLS relative bias 
(not available) 

10% 
15% 
20% 
25% 

2SLS Size of nominal 5% Wald test 
7.03 
4.58 
3.95 
3.63 
LIML Size of nominal 5% Wald test 
7.03 
4.58 
3.95 
3.63 
One can run the same IV regressions for other years, but this is not done here to save space. 
11.30 Growth and Inequality Reconsidered
a. The 3SLS results are replicated using Stata below:
reg3 (Growth: dly = yrt gov m2y inf swo dtot f_pcy d80 d90) (Inequality: gih = yrt m2y civ mlg mlgldc), exog(commod f. civ f_dem Ldtot f. flit f. gov Lint f_m2y f_swo
f. yrt pop urb lex lfr marea oil legor_fr legor_ge legor_mx legor_ sc legor_uk) endog (yrt gov m2y inf swo civ mlg mlgldc)
Threestage leastsquares regression
Equation 
Obs 
Parms 
RMSE 
“Rsq” 
chi2 
P 
Growth 
119 
9 
2.34138 
0.3905 
65.55 
0.0000 
Inequality 
119 
5 
7.032975 
0.4368 
94.28 
0.0000 
Coef. 
Std. Err. 
z 
P>z 
[95% Conf. Interval] 

Growth 

yrt 
.0280625 
.1827206 
0.15 
0.878 
.3861882 
.3300632 
gov 
.0533221 
.0447711 
1.19 
0.234 
.1410718 
.0344276 
m2y 
.0085368 
.0199759 
0.43 
0.669 
.0306152 
.0476889 
inf 
.0008174 
.0025729 
0.32 
0.751 
.0058602 
.0042254 
swo 
4.162776 
.9499015 
4.38 
0.000 
2.301003 
6.024548 
dtot 
26.03736 
23.05123 
1.13 
0.259 
19.14221 
71.21694 
fpcy 
1.38017 
.5488437 
2.51 
0.012 
2.455884 
.3044564 
d80 
1.560392 
.545112 
2.86 
0.004 
2.628792 
.4919922 
d90 
3.413661 
.6539689 
5.22 
0.000 
4.695417 
2.131906 
_cons 
13.00837 
3.968276 
3.28 
0.001 
5.230693 
20.78605 
Inequality 

yrt 
1.244464 
.4153602 
3.00 
0.003 
2.058555 
.4303731 
m2y 
.120124 
.0581515 
2.07 
0.039 
.2340989 
.0061492 
civ 
.2531189 
.7277433 
0.35 
0.728 
1.173232 
1.67947 
mlg 
.292672 
.0873336 
3.35 
0.001 
.1215012 
.4638428 
mlgldc 
.0547843 
.0576727 
0.95 
0.342 
.1678207 
.0582522 
_cons 
33.13231 
5.517136 
6.01 
0.000 
22.31893 
43.9457 
Endogenous variables: dly gih yrt gov m2y inf swo civ mlg mlgldc Exogenous variables: dtot f_pcy d80 d90 commod Lciv Ldem Ldtot f_flit Lgov Linf Lm2y Lswo Lyrt pop urb lex lfr marea oil legor_fr legor_ge legor_mx legor_sc legor_uk
b. The 3sls results of the respecified equations are replicated using Stata below:
. reg3 (Growth: dly = gih yrt gov m2y inf swo dtot f_pcy d80 d90) (Inequality: gih = dly yrt m2y civ mlg mlgldc), exog(f_civ Ldem Ldtot Lflit Lgov Linf f_m2y Lswo Lyrt pop urb lex lfr marea commod oil legor_fr legor_ge legor_mx legor_sc legon. uk) endog(yrt gov m2y inf swo civ mlg mlgldc)
Threestage leastsquares regression
Equation 
Obs 
Parms 
RMSE 
“Rsq” 
chi2 
P 
Growth 
119 
10 
2.366525 
0.3773 
65.26 
0.0000 
Inequality 
119 
6 
6.969627 
0.4469 
96.06 
0.0000 
Coef. 
Std. Err. 
z 
P>z 
[95% Conf. Interval] 

———————– Growth 

gih 
.0181063 
.0452471 
0.40 
0.689 
.106789 
.0705763 
yrt 
.0642514 
.1971126 
0.33 
0.744 
.4505851 
.3220822 
gov 
.0559828 
.0457033 
1.22 
0.221 
.1455597 
.0335941 
m2y 
.0084914 
.0201616 
0.42 
0.674 
.0310246 
.0480074 
inf 
.0002947 
.0031579 
0.09 
0.926 
.0058948 
.0064841 
swo 
4.148469 
.9771934 
4.25 
0.000 
2.233205 
6.063733 
dtot 
28.56603 
23.60078 
1.21 
0.226 
17.69064 
74.8227 
fpcy 
1.35157 
.5565555 
2.43 
0.015 
2.442399 
.2607418 
d80 
1.581281 
.5498957 
2.88 
0.004 
2.659057 
.503505 
d90 
3.392124 
.6624323 
5.12 
0.000 
4.690467 
2.09378 
cons 
13.67011 
4.382862 
3.12 
0.002 
5.079854 
22.26036 
Inequality 

dly 
.2642484 
.3586634 
0.74 
0.461 
.438719 
.9672158 
yrt 
1.221263 
.4122647 
2.96 
0.003 
2.029286 
.4132386 
m2y 
.115859 
.0577899 
2.00 
0.045 
.2291251 
.0025929 
civ 
.1510064 
.7449721 
0.20 
0.839 
1.309112 
1.611125 
mlg 
.2951145 
.0870164 
3.39 
0.001 
.1245654 
.4656635 
mlgldc 
.0451182 
.0581734 
0.78 
0.438 
.159136 
.0688996 
_cons 
32.06252 
5.611046 
5.71 
0.000 
21.06507 
43.05997 
Endogenous variables: dly gih yrt gov m2y inf swo civ mlg mlgldc Exogenous variables: dtot f_pcy d80 d90 f_civ Ldem Ldtot Lflit Lgov Linf Lm2y Lswo Lyrt pop urb lex lfr marea commod oil legorJr legor_ge legor_mx legor_sc legor_uk
11.31 Married Women Labor Supply
b. The following Stata output replicates Equation 2 of Table IV of Mroz (1987, p. 770) which runs 2SLS using the following instrumental variables:
. local B unem city motheduc fatheduc. local E exper expersq. local control age educ
. ivregress 2sls hours (lwage= ‘B’ ‘E’) nwifeinc kidslt6 kidsge6 ‘control’, vce(robust)
Instrumental variables (2SLS) regression Number of obs = 428
Wald chi2(6) = 18.67
Prob > chi2 = 0.0048
Rsquared = .
Root MSE = 1141.3
hours 
Coef. 
Robust Std. Err. 
z 
P>z 
[95% Conf. Interval] 

lwage 
1261.574 
460.7772 
2.74 
0.006 
358.4673 
2164.681 
nwifeinc 
8.341979 
4.586405 
1.82 
0.069 
17.33117 
.6472101 
kidslt6 
234.7069 
182.201 
1.29 
0.198 
591.8143 
122.4005 
kidsge6 
59.78876 
49.04792 
1.22 
0.223 
155.9209 
36.34341 
age 
10.23035 
9.264046 
1.10 
0.269 
28.38755 
7.926846 
educ 
147.895 
52.71884 
2.81 
0.005 
251.222 
44.56794 
_cons 
2374.632 
531.9775 
4.46 
0.000 
1331.976 
3417.289 
Instrumented: Iwage
Instruments: nwifeinc kidslt6 kidsge6 age educ unem city motheduc fatheduc exper expersq
estat overid
Test of overidentifying restrictions: Score chi2(5) = 5.9522 (p = 0.3109)
estat firststage, forcenonrobust all
Firststage regression summary statistics
Adjusted 
Partial 
Robust 

Variable 
Rsq. 
Rsq. 
Rsq. 
F(6,416) 
Prob > F 
lwage 
0.1719 
0.1500 
0.0468 
2.58071 
0.0182 
Shea’s partial Rsquared
Shea’s 
Shea’s 

Variable 
Partial Rsq. Adj. Partial Rsq. 

lwage 
0.0468 
0.0240 
Minimum eigenvalue statistic = 3.40543
Critical Values # of endogenous regressors: 1
Ho: Instruments are weak # of excluded instruments: 6
5% 
10% 
20% 
30% 

2SLS relative bias 
19.28 
11.12 
6.76 
5.15 
10% 
15% 
20% 
25% 

2SLS Size of nominal 5% Wald test 
29.18 
16.23 
11.72 
9.38 
LIML Size of nominal 5% Wald test 
4.45 
3.34 
2.87 
2.61 
Similarly, the following Stata output replicates Equation 3 of Table IV of Mroz(1987, p. 770) which runs 2SLS using the following instrumental variables: . local F2 age2 educ2 age_educ
. ivregress 2sls hours (lwage= ‘B’ ‘E’ ‘F2’) nwifeinc kidslt6 kidsge6 ‘control’, vce(robust)
Instrumental variables (2SLS) regression 
Number of obs 
= 428 
Wald chi2(6) 
= 24.56 

Prob > chi2 
= 0.0004 

Rsquared 
=. 

Root MSE 
= 942.03 
hours 
Coef. 
Robust Std. Err. 
z 
P>z 
[95% Conf. Interval] 

lwage 
831.2505 
312.1889 
2.66 
0.008 
219.3715 
1443.129 
nwifeinc 
6.963789 
3.814549 
1.83 
0.068 
14.44017 
.5125885 
kidslt6 
270.9764 
154.5197 
1.75 
0.079 
573.8294 
31.87667 
kidsge6 
78.37192 
39.257 
2.00 
0.046 
155.3142 
1.429619 
age 
9.38908 
7.589533 
1.24 
0.216 
24.26429 
5.486132 
educ 
102.9946 
36.35213 
2.83 
0.005 
174.2435 
31.74576 
_cons 
2287.175 
432.9381 
5.28 
0.000 
1438.632 
3135.718 
Instrumented: lwage
Instruments: nwifeinc kidslt6 kidsge6 age educ unem city motheduc fatheduc exper expersq age2 educ2 age_educ
. estat overid
Test of overidentifying restrictions: Score chi2(8) = 19.8895 (p = 0.0108)
. estat firststage, forcenonrobust all Firststage regression summary statistics
Adjusted 
Partial 
Robust 

Variable 
Rsq. 
Rsq. 
Rsq. 
F(6,416) 
Prob > F 
lwage 
0.1846 
0.1570 
0.0614 
2.3555 
0.0133 
Shea’s partial Rsquared 
Shea’s 
Shea’s 

Variable 
Partial Rsq. Adj. PartialRsq. 

lwage 
0.0614 
0.0320 
Minimum eigenvalue statistic = 3.0035
Critical Values # of endogenous regressors: 1
Ho: Instruments are weak # of excluded instruments: 9
5% 
10% 
20% 
30% 

2SLS relative bias 
20.53 
11.46 
6.65 
4.92 
10% 
15% 
20% 
25% 

2SLS Size of nominal 5% Wald test 
36.19 
19.71 
14.01 
11.07 
LIML Size of nominal 5% Wald test 
3.81 
2.93 
2.54 
2.32 
Similarly, one can generate the rest of the regressions in Table IV and their related diagnostics. 
References
Baltagi, B. H. (1989), “A Hausman Specification Test in a Simultaneous Equations Model,”Econometric Theory, Solution 88.3.5, 5: 453467.
Farebrother, R. W. (1987), “The Exact Bias of Wald’s Estimation,” Econometric Theory, Solution 85.3.1, 3: 162.
Farebrother, R. W. (1991), “Comparison of tRatios,” Econometric Theory, Solution 90.1.4,7: 145146.
Singh, N. and A N. Bhat (1988), “Identification and Estimation of a Simple Two – Equation Model,” Econometric Theory, Solution 87.3.3, 4: 542545.
Theil, H. (1971), Principles of Econometrics (Wiley: New York).
Wald, A. (1940), “Fitting of Straight Lines if Both Variables are Subject to Error,” Annals of Mathematical Statistics, 11: 284300.
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