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An asymptotic variance inequality for instrumental variable estimators signaling proportional bias increases
Economics Letters 112 (2011) 53–55
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An asymptotic variance inequality for instrumental variable estimators signaling proportional bias increases Yun-Yeong Kim ∗ Department of International Trade, Dankook University, Yong-In, 448-701, Republic of Korea
article
info
Article history: Received 4 May 2007 Received in revised form 9 March 2011 Accepted 18 March 2011 Available online 2 April 2011
abstract An asymptotic variance inequality for instrumental variable (IV) estimators is proposed, which suggests a critical variance that signals proportional increases in the bias of IV estimators through the augmentation of a set of instruments. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C3 Keywords: IV estimator Asymptotic variance Bias
1. Introduction
Assumption 2.1. We assume the following:
Buse (1992) showed that a decrease in the bias of IV estimators is possible through a change in the variance after the augmentation of a set of instruments, supplementing Phillips (1980). The paper suggests a critical variance that signals proportional increases in the bias of IV estimators. 2. Asymptotic variance inequality
(1)
where (zt ) are ℓ-dimensional endogenous variables and (ut ) is the mean-zero error term. We denote by T the sample size and define y = (y1 , . . . , yT )′ ; Z = (z1 , . . . , zT )′ ; and u = (u1 , . . . , uT )′ . Suppose that there are m instrumental variables (xt ) , m ≥ ℓ, and let X = (x1 , . . . , xT )′ . Then the optimal IV estimator using X as the instrument matrix is given by
−1 ′ δˆX = Z ′ PX Z Z PX y , ′ −1 ′ where PX = X X X X is the projection onto the range of X .
p
Tel.: +82 31 8005 3402; fax: +82 31 8021 7210. E-mail address: [email protected].
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p
p
(d) T −1 X ′ Z → Mxz , which is of rank ℓ a.s. as T → ∞. Then we can derive the standard asymptotic distribution of the IV estimator.
√ T δˆ X − δ →d N 0, avar δˆ X
(2)
as T → ∞, where σu2 denotes the variance of ut and avar δˆ X −1 Mxz σu Mxz′ Mxx
2
−1
=
.
We now define
wt = zt − σzu /σu2 ut and let W = (w1 , . . . , wT )′ . Note that (wt ) is the residual from the projection of (zt ) onto (ut ). Under Assumption 2.1, Mww = plimT →∞ W ′ W /T exists and is given by ′
Mww = Mzz −
∗
p
(c) T −1 Z ′ u → σzu ; T −1 X ′ X → Mxx > 0; and T −1 Z ′ Z → Mzz > 0.
Proposition 2.2. Under Assumption 2.1, we have
For (yt ), we consider the regression model yt = zt′ δ + ut
(a) x′t , ut is an independent and identically distributed sequence. (b) xt is independent of us .
σzu σzu . σu2
(3)
The following theorem gives the inequality for the asymptotic variance of IV estimators.
54
Y.-Y. Kim / Economics Letters 112 (2011) 53–55
Then the bias of order T −1 in Buse (1992, Equation 18), can be given as, conditional on X(i) ,
Theorem 2.3. Let Assumption 2.1 hold. Then we have −1 avar(δˆ X ) ≥ σu2 Mww
(4)
for any choice of instruments (xt ). p
Proof. Let Mxw be defined similarly to Mxz . Because T −1 X ′ u → 0, we have Mxw = Mxz , and therefore, ′
′
Mxz Mxx Mxz = Mxw Mxx Mxw . −1
−1
(5)
Moreover, we have ′
Mxw Mxx Mxw ≤ Mww
(6)
because ′
−1 T −1 W ′ Px W → Mxw Mxx Mxw
p
and T −1 W ′ W → Mww
ar (L2 , L1 ) = a (L2 ) /a (L1 ) , where Q(i) =
Z˜(′i) PX(i) Z˜(i)
Given (2), the stated result now follows immediately from (5) and (6). If the IV estimator satisfies the lower bound of the equality in (4), it becomes the best linear consistent estimator (BLCE). The necessary and sufficient condition for the IV estimator to be the BLCE is given as follows: Lemma 2.4. Let Assumption 2.1 hold. Then δˆ X becomes the BLCE if and only if zt = Φ xt + σzu /σu2 ut ,
(7)
where Φ is a nonrandom coefficient of conformable dimensions.
Then the asymptotic ratio ar (L2 , L1 ) increases after δˆ X(1) attains the lower bound in (4) as follows:
T →∞
Proof. The claimed result holds because
(L2 − 1)2 σzu′ σu4 plim T 2 Q(22) σzu T →∞ plim ar (L2 , L1 ) = T →∞ 2 ′ 2 4 2 (L1 − 1) σzu σu plim T Q(1) σzu T →∞ 2 ′ (L2 − 1) σzu plim V2(2) σzu T →∞ = (L1 − 1)2 σzu′ plim V2(1) σzu T →∞
Proof. The sufficiency is obvious. To show the necessity, we suppose that the representation (7) is not allowed. Then we may write −1 zt = Mxz Mxx
xt + σzu /σu2 ut + υt
T
p
X V → 0;
T
−1
′
p
V u → 0;
and T
−1
p
V V → Mυυ > 0, ′
L2 − 1 L1 − 1
>1
(9)
which implies that δˆ X is not the BLCE as expected.
(a) the second equality in (9) holds because
σ plim TQ(i) 2 u
T →∞
T →∞
′ −1 Mww = Mxz Mxx Mxz + Mυυ ,
−1 −1 ′ −1 = σu2 plim T −1 Z ′ X(i) T −1 X(′i) X(i) T X(i) Z T →∞
= plim V(i) T →∞
Example 2.5. Consider the simultaneous equation model
through plimT →∞ T −1 Z˜(′i) X(i) = plimT →∞ T −1 Z ′ X(i) , where T −1 X(′i)
rt = ct + it ct = α + β rt + ut
(8)
where rt , ct , and it are income, consumption, and investment, respectively. Note that the IV estimator of Eq. (8) uses instruments (1, it )′ , and is the BLCE because
α 1 1 + it + ut 1−β 1−β 1−β
from Lemma 2.4.
=
2
−1 −1 −1 ′ = σu2 plim T −1 Z˜(′i) X(i) T −1 X(′i) X(i) T X(i) Z˜(i)
with V defined similarly to (υt ) as X and u. Therefore, we have
rt =
from L2 > L1 and because of the following:
by the projection of (zt ) onto (xt ) and (ut ), where (υt ) is a residual satisfying −1 ′
and a (Li ) = E (δˆ X(i) − δ|X(i) )′ E (δˆ X(i)
plim ar (L2 , L1 ) > 1.
W ′ Px W ≤ W ′ W .
′
−1
Theorem 3.1. Suppose Assumption 2.1 holds and plimT →∞ V(1) = −1 −1 σu2 Mww > 0, where V(i) = σu2 T Z ′ PX(i) Z . Then we have
as T → ∞ and, for all T ,
= (Li − 1) Q(i) σzu ,
with the ratio of aggregate biases as
− δ|X(i) ).
−1
p
E δˆ X(i) − δ|X(i)
p
N(i) → 0 from the assumption. (b) the third equality in (9) holds because −1 plim V(2) = σu2 Mww
T →∞
through the following: (i) −1 plim V(2) ≥ σu2 Mww = plim V(1)
T →∞
T →∞
3. Proportional bias increases
from Theorem 2.3, and (ii)
Let X(i) be a set of instrumental variables for i = 1, 2 such that L2 > L1 and X(1) ⊂ X(2) , where the number of instrument is mi = Li + ℓ. Let the endogenous variable in Eq. (1) be
−1 −1 σu2 Mww = plim V(1) = σu2 plim T Z ′ PX(1) Z
Z = Z˜(i) + N(i) ,
T →∞
T →∞
−1 ≥ σu2 plim T Z ′ PX(2) Z = plim V(2) T →∞
T →∞
p
where Z˜(i) = X(i) Πi with T −1 X(′i) N(i) → 0 for a conformable coefficient Πi .
through Z ′ PX(1) Z
−1
≥ Z ′ PX(2) Z
−1
from X(1) ⊂ X(2) .
Y.-Y. Kim / Economics Letters 112 (2011) 53–55
To check the closeness of the IV estimator to the BLCE, we suggest asymptotic relative efficiency (ARE),1 which indicates the magnitude of the asymptotic variance relative to the minimum variance bound:
ARE c ′ δˆ X
55
Table A.1
−1 c ′ Mww c
=
′ M −1 M c ′ Mxz xz xx
−1
c
for any nonzero ℓ-dimensional vector c. Note that the higher the ARE, the closer the IV estimator is to the BLCE, where 0 ≤ ARE ≤ 1.
−1 ˆ ˆ xz′ M ˆ xx Then the ARE of c ′ δˆ X is estimated by c ′ M Mxz
−1
−1 ˆ ww c /c ′ M c consistently under Assumption 2.1, where
ˆ xx = M
X ′X T
,
X ′Z
ˆ xz = M
T
,
and
σˆ zu σˆ zu , σˆ u2 ′
ˆ ww = M ˆ zz − M with
σˆ u2 =
uˆ ′ uˆ
and σˆ zu =
T
Z ′ uˆ T
,
where uˆ is the IV residual. Example 3.2. Note that zT′ +1 avar(δˆ X )zT +1 gives the forecast error variance of yT +1 and the above ARE becomes predictive relative efficiency (PRE) as −1 zT′ +1 Mww zT +1 ′ M −1 M zT′ +1 Mxz xz xx
−1
z T +1
with c = zT +1 . Kim and Park (1998) computed PRE for Klein’s model and demonstrated that PRE quickly approaches one as the number of instruments increases. 4. Simulation
Acknowledgments
For the simulation, we consider the following limited information model: yt = β zt + ut , zt = π1
5 −
xit + π2
i=1
10 −
xjt + γ ut ,
j =6
where β = 1; γ = 0.5; π1 = 0.1, . . . , 0.9; and π2 = 0.01, respectively. Here the data-generating process is d
(x1t , x2t , . . . , x10t , ut )′ ∼ i.i.d. n(0, I11 ), where I11 denotes an identity matrix of 11-dimensions. Define IV estimators with instruments IVj = x1t , x2t , . . . , xjt . Note IV1 , IV2 , . . . , IV4 are not for BLCEs, whereas IV5 , IV6 , . . . , IV10 are almost for BLCEs from Lemma 2.4.
1 The inequality (4) implies 1 − R2zu ≥ R2xz −1 because σu2 Mww = σu2 / Mzz 1 − R2zu
Mxx Mxz )
−1
= σ / 2 u
Mzz R2xz
I am grateful to Joon Y. Park and the anonymous referee for their invaluable comments. Appendix. Simulation results See Table A.1.
t = 1, 2, . . . , 100,
−1
The simulation results (the Appendix), which were based on 10,000 repetitions, are consistent with theoretical expectations. The median bias decreased after the addition of an instrument in lined rectangular areas. However, after closely approaching the variance bound, the bias increased proportionally with no reduction in the RMSE. Finally, the AREs well reflected this increase in the bias.
, where
R2ab
′ from the decomposition (3) and σu2 (Mxz
denotes the correlation coefficient of the
zero-mean univariate variables a and b. Note the IV estimator (even one whose R2xz is small) may be close to the BLCE if 1 − R2zu is small.
References Buse, A., 1992. The bias of instrumental variable estimators. Econometrica 60, 173–180. Kim, Yun-Yeong, Park, Joon Y., 1998. The asymptotic variance bound for instrumental variables estimators. Institute of Economic Research Discussion Paper 10. Seoul National University. Phillips, P.C.B., 1980. The exact distribution of instrumental variable estimators in an equation containing n + 1 endogenous variables. Econometrica 48, 861–878.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An asymptotic variance inequality for instrumental variable estimators signaling proportional bias increases Yun-Yeong Kim ∗ Department of International Trade, Dankook University, Yong-In, 448-701, Republic of Korea
article
info
Article history: Received 4 May 2007 Received in revised form 9 March 2011 Accepted 18 March 2011 Available online 2 April 2011
abstract An asymptotic variance inequality for instrumental variable (IV) estimators is proposed, which suggests a critical variance that signals proportional increases in the bias of IV estimators through the augmentation of a set of instruments. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C3 Keywords: IV estimator Asymptotic variance Bias
1. Introduction
Assumption 2.1. We assume the following:
Buse (1992) showed that a decrease in the bias of IV estimators is possible through a change in the variance after the augmentation of a set of instruments, supplementing Phillips (1980). The paper suggests a critical variance that signals proportional increases in the bias of IV estimators. 2. Asymptotic variance inequality
(1)
where (zt ) are ℓ-dimensional endogenous variables and (ut ) is the mean-zero error term. We denote by T the sample size and define y = (y1 , . . . , yT )′ ; Z = (z1 , . . . , zT )′ ; and u = (u1 , . . . , uT )′ . Suppose that there are m instrumental variables (xt ) , m ≥ ℓ, and let X = (x1 , . . . , xT )′ . Then the optimal IV estimator using X as the instrument matrix is given by
−1 ′ δˆX = Z ′ PX Z Z PX y , ′ −1 ′ where PX = X X X X is the projection onto the range of X .
p
Tel.: +82 31 8005 3402; fax: +82 31 8021 7210. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.03.001
p
p
(d) T −1 X ′ Z → Mxz , which is of rank ℓ a.s. as T → ∞. Then we can derive the standard asymptotic distribution of the IV estimator.
√ T δˆ X − δ →d N 0, avar δˆ X
(2)
as T → ∞, where σu2 denotes the variance of ut and avar δˆ X −1 Mxz σu Mxz′ Mxx
2
−1
=
.
We now define
wt = zt − σzu /σu2 ut and let W = (w1 , . . . , wT )′ . Note that (wt ) is the residual from the projection of (zt ) onto (ut ). Under Assumption 2.1, Mww = plimT →∞ W ′ W /T exists and is given by ′
Mww = Mzz −
∗
p
(c) T −1 Z ′ u → σzu ; T −1 X ′ X → Mxx > 0; and T −1 Z ′ Z → Mzz > 0.
Proposition 2.2. Under Assumption 2.1, we have
For (yt ), we consider the regression model yt = zt′ δ + ut
(a) x′t , ut is an independent and identically distributed sequence. (b) xt is independent of us .
σzu σzu . σu2
(3)
The following theorem gives the inequality for the asymptotic variance of IV estimators.
54
Y.-Y. Kim / Economics Letters 112 (2011) 53–55
Then the bias of order T −1 in Buse (1992, Equation 18), can be given as, conditional on X(i) ,
Theorem 2.3. Let Assumption 2.1 hold. Then we have −1 avar(δˆ X ) ≥ σu2 Mww
(4)
for any choice of instruments (xt ). p
Proof. Let Mxw be defined similarly to Mxz . Because T −1 X ′ u → 0, we have Mxw = Mxz , and therefore, ′
′
Mxz Mxx Mxz = Mxw Mxx Mxw . −1
−1
(5)
Moreover, we have ′
Mxw Mxx Mxw ≤ Mww
(6)
because ′
−1 T −1 W ′ Px W → Mxw Mxx Mxw
p
and T −1 W ′ W → Mww
ar (L2 , L1 ) = a (L2 ) /a (L1 ) , where Q(i) =
Z˜(′i) PX(i) Z˜(i)
Given (2), the stated result now follows immediately from (5) and (6). If the IV estimator satisfies the lower bound of the equality in (4), it becomes the best linear consistent estimator (BLCE). The necessary and sufficient condition for the IV estimator to be the BLCE is given as follows: Lemma 2.4. Let Assumption 2.1 hold. Then δˆ X becomes the BLCE if and only if zt = Φ xt + σzu /σu2 ut ,
(7)
where Φ is a nonrandom coefficient of conformable dimensions.
Then the asymptotic ratio ar (L2 , L1 ) increases after δˆ X(1) attains the lower bound in (4) as follows:
T →∞
Proof. The claimed result holds because
(L2 − 1)2 σzu′ σu4 plim T 2 Q(22) σzu T →∞ plim ar (L2 , L1 ) = T →∞ 2 ′ 2 4 2 (L1 − 1) σzu σu plim T Q(1) σzu T →∞ 2 ′ (L2 − 1) σzu plim V2(2) σzu T →∞ = (L1 − 1)2 σzu′ plim V2(1) σzu T →∞
Proof. The sufficiency is obvious. To show the necessity, we suppose that the representation (7) is not allowed. Then we may write −1 zt = Mxz Mxx
xt + σzu /σu2 ut + υt
T
p
X V → 0;
T
−1
′
p
V u → 0;
and T
−1
p
V V → Mυυ > 0, ′
L2 − 1 L1 − 1
>1
(9)
which implies that δˆ X is not the BLCE as expected.
(a) the second equality in (9) holds because
σ plim TQ(i) 2 u
T →∞
T →∞
′ −1 Mww = Mxz Mxx Mxz + Mυυ ,
−1 −1 ′ −1 = σu2 plim T −1 Z ′ X(i) T −1 X(′i) X(i) T X(i) Z T →∞
= plim V(i) T →∞
Example 2.5. Consider the simultaneous equation model
through plimT →∞ T −1 Z˜(′i) X(i) = plimT →∞ T −1 Z ′ X(i) , where T −1 X(′i)
rt = ct + it ct = α + β rt + ut
(8)
where rt , ct , and it are income, consumption, and investment, respectively. Note that the IV estimator of Eq. (8) uses instruments (1, it )′ , and is the BLCE because
α 1 1 + it + ut 1−β 1−β 1−β
from Lemma 2.4.
=
2
−1 −1 −1 ′ = σu2 plim T −1 Z˜(′i) X(i) T −1 X(′i) X(i) T X(i) Z˜(i)
with V defined similarly to (υt ) as X and u. Therefore, we have
rt =
from L2 > L1 and because of the following:
by the projection of (zt ) onto (xt ) and (ut ), where (υt ) is a residual satisfying −1 ′
and a (Li ) = E (δˆ X(i) − δ|X(i) )′ E (δˆ X(i)
plim ar (L2 , L1 ) > 1.
W ′ Px W ≤ W ′ W .
′
−1
Theorem 3.1. Suppose Assumption 2.1 holds and plimT →∞ V(1) = −1 −1 σu2 Mww > 0, where V(i) = σu2 T Z ′ PX(i) Z . Then we have
as T → ∞ and, for all T ,
= (Li − 1) Q(i) σzu ,
with the ratio of aggregate biases as
− δ|X(i) ).
−1
p
E δˆ X(i) − δ|X(i)
p
N(i) → 0 from the assumption. (b) the third equality in (9) holds because −1 plim V(2) = σu2 Mww
T →∞
through the following: (i) −1 plim V(2) ≥ σu2 Mww = plim V(1)
T →∞
T →∞
3. Proportional bias increases
from Theorem 2.3, and (ii)
Let X(i) be a set of instrumental variables for i = 1, 2 such that L2 > L1 and X(1) ⊂ X(2) , where the number of instrument is mi = Li + ℓ. Let the endogenous variable in Eq. (1) be
−1 −1 σu2 Mww = plim V(1) = σu2 plim T Z ′ PX(1) Z
Z = Z˜(i) + N(i) ,
T →∞
T →∞
−1 ≥ σu2 plim T Z ′ PX(2) Z = plim V(2) T →∞
T →∞
p
where Z˜(i) = X(i) Πi with T −1 X(′i) N(i) → 0 for a conformable coefficient Πi .
through Z ′ PX(1) Z
−1
≥ Z ′ PX(2) Z
−1
from X(1) ⊂ X(2) .
Y.-Y. Kim / Economics Letters 112 (2011) 53–55
To check the closeness of the IV estimator to the BLCE, we suggest asymptotic relative efficiency (ARE),1 which indicates the magnitude of the asymptotic variance relative to the minimum variance bound:
ARE c ′ δˆ X
55
Table A.1
−1 c ′ Mww c
=
′ M −1 M c ′ Mxz xz xx
−1
c
for any nonzero ℓ-dimensional vector c. Note that the higher the ARE, the closer the IV estimator is to the BLCE, where 0 ≤ ARE ≤ 1.
−1 ˆ ˆ xz′ M ˆ xx Then the ARE of c ′ δˆ X is estimated by c ′ M Mxz
−1
−1 ˆ ww c /c ′ M c consistently under Assumption 2.1, where
ˆ xx = M
X ′X T
,
X ′Z
ˆ xz = M
T
,
and
σˆ zu σˆ zu , σˆ u2 ′
ˆ ww = M ˆ zz − M with
σˆ u2 =
uˆ ′ uˆ
and σˆ zu =
T
Z ′ uˆ T
,
where uˆ is the IV residual. Example 3.2. Note that zT′ +1 avar(δˆ X )zT +1 gives the forecast error variance of yT +1 and the above ARE becomes predictive relative efficiency (PRE) as −1 zT′ +1 Mww zT +1 ′ M −1 M zT′ +1 Mxz xz xx
−1
z T +1
with c = zT +1 . Kim and Park (1998) computed PRE for Klein’s model and demonstrated that PRE quickly approaches one as the number of instruments increases. 4. Simulation
Acknowledgments
For the simulation, we consider the following limited information model: yt = β zt + ut , zt = π1
5 −
xit + π2
i=1
10 −
xjt + γ ut ,
j =6
where β = 1; γ = 0.5; π1 = 0.1, . . . , 0.9; and π2 = 0.01, respectively. Here the data-generating process is d
(x1t , x2t , . . . , x10t , ut )′ ∼ i.i.d. n(0, I11 ), where I11 denotes an identity matrix of 11-dimensions. Define IV estimators with instruments IVj = x1t , x2t , . . . , xjt . Note IV1 , IV2 , . . . , IV4 are not for BLCEs, whereas IV5 , IV6 , . . . , IV10 are almost for BLCEs from Lemma 2.4.
1 The inequality (4) implies 1 − R2zu ≥ R2xz −1 because σu2 Mww = σu2 / Mzz 1 − R2zu
Mxx Mxz )
−1
= σ / 2 u
Mzz R2xz
I am grateful to Joon Y. Park and the anonymous referee for their invaluable comments. Appendix. Simulation results See Table A.1.
t = 1, 2, . . . , 100,
−1
The simulation results (the Appendix), which were based on 10,000 repetitions, are consistent with theoretical expectations. The median bias decreased after the addition of an instrument in lined rectangular areas. However, after closely approaching the variance bound, the bias increased proportionally with no reduction in the RMSE. Finally, the AREs well reflected this increase in the bias.
, where
R2ab
′ from the decomposition (3) and σu2 (Mxz
denotes the correlation coefficient of the
zero-mean univariate variables a and b. Note the IV estimator (even one whose R2xz is small) may be close to the BLCE if 1 − R2zu is small.
References Buse, A., 1992. The bias of instrumental variable estimators. Econometrica 60, 173–180. Kim, Yun-Yeong, Park, Joon Y., 1998. The asymptotic variance bound for instrumental variables estimators. Institute of Economic Research Discussion Paper 10. Seoul National University. Phillips, P.C.B., 1980. The exact distribution of instrumental variable estimators in an equation containing n + 1 endogenous variables. Econometrica 48, 861–878.