Economics Letters 127 (2015) 89–92
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Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The true limit distributions of the Anderson–Hsiao IV estimators in panel autoregression✩ Peter C.B. Phillips a,b,c,d,∗ , Chirok Han e a
Yale University, United States
b
University of Auckland, New Zealand
c
Singapore Management University, Singapore
d
University of Southampton, United Kingdom
e
Korea University, Republic of Korea
article
info
Article history: Received 17 September 2014 Accepted 26 November 2014 Available online 29 December 2014 JEL classification: C230 C360
abstract This note derives the correct limit distributions of the Anderson–Hsiao (1981) levels and differences instrumental variable estimators, provides comparisons showing that the levels IV estimator has uniformly smaller variance asymptotically as the cross section (n) and time series (T ) sample sizes tend to infinity, and compares these results with those of the first difference least squares (FDLS) estimator. © 2014 Elsevier B.V. All rights reserved.
Keywords: Dynamic panel IV estimation Levels and difference instruments
1. Introduction In pioneering work on dynamic panel models, Anderson and Hsiao (1981, AH hereafter) developed two consistent instrumental variable (IV) estimators for the common slope coefficient in first order panel autoregression. These estimators used lagged levels and lagged differences as instruments and they form the core of much later work on GMM approaches to inference in dynamic panels. This note corrects the AH limit theory and provides an interesting asymptotic equivalence between their levels IV estimator and the first difference least squares estimator of Phillips and Han (2008) and Han and Phillips (2010). The levels IV estimator is shown to have asymptotically uniformly smaller variance than the difference IV estimator when (n, T ) → ∞. For fixed T , the levels estimator is also more efficient except when T is very small. The present work deals with the stationary case. See
Han and Phillips (2013), Moon and Phillips (2004) and the references therein for recent work on the panel unit root case. 2. Asymptotic distributions of IV estimators For the simple panel dynamic model yit = αi + β yit −1 + uit , after eliminating the nuisance fixed effects by first-differencing the equation, AH (1981, Section 8) propose using lagged variables in levels or differences as potential instrumental variables. The resulting levels and difference IV estimators are n T
βˆ l =
i=1 t =2 n T i=1 t =2
n T
yit −2 1yit
yit −2 1yit −1
and
βˆ d =
1yit −2 1yit
i=1 t =3 n T
,
1yit −2 1yit −1
i=1 t =3
where 1yit = yit − yit −1 . We provide the correct asymptotics for these two estimators under stationarity. ✩ Phillips acknowledges support from the NSF under Grant No. SES 12-58258. Research by Han was supported by the Korean Government (2014S1A2A2027803) and Korea University (K1421311). ∗ Corresponding author at: Yale University, United States. E-mail address:
[email protected] (P.C.B. Phillips).
http://dx.doi.org/10.1016/j.econlet.2014.11.030 0165-1765/© 2014 Elsevier B.V. All rights reserved.
Levels IV Let yit = αi + β yit −1 + uit (i =1, . . . , n; t = 1, . . . , T) with αi j 2 |β| < 1, uit ∼iid 0, σ 2 , yi0 = 1−β + ∞ j=0 ρ ui,−j , and αi ∼iid 0, σα
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P.C.B. Phillips, C. Han / Economics Letters 127 (2015) 89–92
and the denominator DlnT → a.s. − T1 1σ+β , we have 2
independent of uit . The levels IV estimator βˆ l satisfies √1
n T
n
√
n(βˆ l − β) = 1 n
yit −2 1uit
i=1 t =2
n T
l NnT DlnT
=:
yit −2 1yit −1
√
.
α
∞
j=0
σ2 σα2 + , 1 − β2 (1 − β)2 σα2 β | j| σ 2 E yit yit −j = + . 1 − β2 (1 − β)2 Hence, as n → ∞, n T 1 yit −2 1yit −1 DlnT =
(2)
→
a.s. E
T
yit −2 1yit −1
=
t =2
T
E (yit −2 yit −1 ) − E y2it −2
t =2
= T1 E (yit −1 yit −2 ) − E
T1 (β − 1) σ 2
y2it −2
T1 σ 2
, (3) 1+β (1 − β) where Tk = T − k. By partial summation ∆(yit −2 uit ) = yit −2 1uit + (1yit −2 )uit −1 for t ≥ 3 and we have T T yit −2 1uit = yiT −2 uiT − yi0 ui2 − (1yit −2 )uit −1 , =
2
=−
t =3
t =3
yi0 1ui2 = yi0 ui2 − yi0 ui1 , and adding gives T
yit −2 1uit = yiT −2 uiT − yi0 ui1 −
t =2
T (1yit −2 )uit −1 .
(4)
E(1yit )2 = 2E(y2it ) − 2E(yit yit −1 ) =
2σ 2 (1 − β)
1 − β2
=
2σ 2 1+β
we have
E
T
(6)
βˆ fd =
yit −2 1yit + w1 + w2
i=1 t =2 n T
,
(7)
yit −2 1yit −1 + w1
i =1 t =2
n (y2i0 − y2iT −1 ) and w2 = i=1 (yi0 1yi1 − yiT −1 1yiT ). The stationarity requirement for βˆ fd affects w1 and w2 . When E (y2it ) is stable, the terms w1 and w2 are dominated by the
yit −2 1uit
t =2
= E yiT −2 uiT − yi0 ui1 −
that relates to βˆ l with end corrections, viz.,
where w1 =
2
σ 2β 1−β 2
which is the same limit distribution in the stationary case as the GMM estimator in Han and Phillips (2010, HP). Importantly, in (6) there is no dependence in the limit variance on σα2 . Also, note that there is a discontinuity in the limit theory as β → 1 because in that √ case βˆ l is only T consistent and has a limit Cauchy distribution (Phillips, 2014). The expression in (5) differs from AH’s result, (8.4). The error in AH seems to arise because yit −2 1uit is mistaken as a martingale difference sequence and the asymptotic variance actually involves cross product terms and a (finite T ) long run variance. The above demonstration simply avoids this calculation by using partial T summation to put the sum t =2 yit −2 1uit into a more convenient form. The equivalence of the AH and HP estimators for large T is unexpected, because the AH estimator is derived under weaker orthogonality conditions E(yit −2 1uit ) = 0 whereas the HP estimator requires covariance stationarity also. To explore the equivalence, algebra shows that the HP estimator βˆ fd can be written in the form
n T
,
+
nT (βˆ l − β) ⇒(T ,n)→∞ N (0, 2(1 + β)),
t =3
Since (1yit −2 )uit −1 is a martingale difference sequence and
σα2 (1−β)2
The asymptotic variance of (5) increases with σα2 /σ 2 , which is natural because the αi are uninformative for the identification of β . As T → ∞
√
n i=1 t =3
n→∞
ρ j ut −j is stationary,
E y2it =
2
2 σ2 −T1 1+β 2 (1 + β) 2 (1 + β)2 σα2 /σ 2 β = N 0, + 2 + . (5) T1 1+β T1 (1 − β) 1 − β
(1)
i=1 t =2
i Since yit = 1−β +
n(βˆ l − β) ⇒ N 0,
2σ 2 T1 1σ+β + 2σ 2
T
2 (1yit −2 )uit −1
n
1 2
i=1
leading terms and are therefore negligible for large T , leading to the asymptotic equivalence of βˆ fd and βˆ l .
t =3
T σ2 σα2 2 + + σ E (1yit −2 )2 1 − β2 (1 − β)2 t =3 2 2 σ σ σ2 α 2 = 2σ 2 + + 2 σ T 2 1 − β2 1+β (1 − β)2 2 2 2 σα σ σ σ2 2 = 2σ 2 + − + 2 σ T 1 1 − β2 1+β 1+β (1 − β)2 2 2 2 σα σ β σ = 2σ 2 + + 2σ 2 T1 . 2 2 1 − β 1 +β (1 − β) T Since E( t =2 yit −2 1uit ) = 0, the numerator of (1) satisfies the CLT n T 1 σ2 yit −2 1uit ⇒ N 0, 2σ 2 T1 √ 1+β n i=1 t =2 σα2 σ 2β 2 + 2σ + 1 − β2 (1 − β)2 = 2σ 2
Difference IV The difference IV estimator βˆ d satisfies √1
√
n
n(βˆ d − β) = 1 n
n T
1yit −2 1uit
i =1 t =3
n T
=:
1yit −2 1yit −1
d NnT
DdnT
.
i =1 t =3
In the stationary case, the denominator satisfies DdnT
=
=
n T 1
n i =1 t =3 T
1yit −2 1yit −1 → a.s. E
T t =3
E (yit −1 yit −2 ) − E y2it −2
t =3
− E (yit −1 yit −3 ) + E (yit −2 yit −3 )
1yit −2 1yit −1
P.C.B. Phillips, C. Han / Economics Letters 127 (2015) 89–92
91
(a) σα2 /σ 2 = 1.
(b) σα2 /σ 2 = 8. Fig. 1. Asymptotic variance ratio Avar (βˆ l )/Avar (βˆ d ).
σα2 βσ 2 σα2 σ2 = T2 2 + − + 1 − β2 1 − β2 (1 − β)2 (1 − β)2 2 2 2 β σ σα − + 1 − β2 (1 − β)2 2 2 σ (1 − β) 1−β 2 = −T2 = −T2 σ 1 − β2 1+β T2 = DlnT (1 − β).
=
2σ 4 1+β
[2 + T3 (3 − β)] =
√
n(βˆ d − β) =
d NnT N d n→∞ DnT
⇒
1yit −2 1uit = (1yiT −2 )uiT − (1yi1 )ui2 −
t =3
= N 0,
E(∆2 yit −2 )2
(∆ yit −2 )uit ,
2(3−β)σ 2 , 1+β
E(1yit −1 1yit −2 ) = −
σ 2 (1−β) , 1+β
2σ 2 , 1+β
3−β 1+β
−T 2
− 2 11−β +β 2
−
1−β 1+β
2
1+β 1−β
T22
,
(9)
nT (βˆ d − β)
⇒
(T ,n)→∞
N
0,
2 (1 + β) (3 − β)
(1 − β)2
.
(10)
and
(∆ yit −2 )uit is a martingale difference, the numerator satisfies d NnT ⇒ N (0, VT ) with VT = 2E(1yit )2 + T3 E(∆2 yit −2 )2 σ 2 4σ 2 2(3 − β)σ 2 = + T3 · σ2 1+β 1+β 2
T2 (1 − β)2
√
t =4
=
0,
2T2
which differs from AH’s result (8.3) and leads to the following sequential limit theory
2
where ∆2 yit −2 = 1yit −2 − 1yit −3 . Because E(1yit )2 =
[T2 (3 − β) − (1 − β)]. (8)
2 (1 + β) (3 − β)
Similar to (4), we have T
1+β
Then
T1
T
2σ 4
Efficiency comparison Comparing (10) with (6), it is clear that the limit variance of βˆ l for large n and T is smaller than that of βˆ d for all β ∈ (−1, 1) since 3−β 2 > 0. It is easy to show, as in Phillips (2014) using the meth(1−β)
ods of Phillips and Moon (1999), that the convergences in (10)with
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P.C.B. Phillips, C. Han / Economics Letters 127 (2015) 89–92
Table 1 Simulated n× variances (n = 400, β = 0.5, 10,000 replications). yit = αi + β yit −1 + uit , αi = σα ai , ui = σ εit , σ = 1, yi,−100 = αi /(1 − β) + ui0 / 1 − β 2 , ai , εit ∼iid N (0, 1). T
HP
AH(L)
AH(D)
(a) σα /σ = 1 2
5 10 20 40 80 160
2
0.7564 0.3308 0.1579 0.0771 0.0378 0.0190
2.1299 0.5943 0.2203 0.0929 0.0415 0.0200
(2.0625) (0.5926) (0.2161) (0.0907) (0.0413) (0.0197)
9.7448 3.7375 1.6798 0.7889 0.3833 0.1924
(9.3333) (3.6562) (1.6481) (0.7853) (0.3836) (0.1896)
11.0850 2.2150 0.5809 0.1792 0.0621 0.0253
(9.9375) (2.1481) (0.5651) (0.1736) (0.0615) (0.0247)
9.7448 3.7375 1.6798 0.7889 0.3833 0.1924
(9.3333) (3.6562) (1.6481) (0.7853) (0.3836) (0.1896)
(b) σα2 /σ 2 = 8 5 10 20 40 80 160
0.7564 0.3308 0.1579 0.0771 0.0378 0.0190
Note: The numbers in parentheses are asymptotic variances.
(6) are both sequential (n → ∞ followed by T → ∞) and joint (n, T → ∞ without restriction on the rates or the path of divergence). For fixed T , from (5) and (9), we evaluate Avar (βˆ l )/Avar (βˆ d ) for various β , T and σα2 /σ 2 . Fig. 1(a) exhibits this ratio for σα2 /σ 2 = 1. The levels estimator is more efficient for all β and T ≥ 3 in this case. Larger σα2 /σ 2 ratios are more favorable to βˆ d but still the levels estimator is more efficient unless T is very small. Numerical evaluations suggest that the levels estimator is better than the difference estimator for all β ∈ (−1, 1) for all T if σα2 /σ 2 ≤ 4. For even larger σα2 /σ 2 , Fig. 1(b) considers σα2 /σ 2 = 8. The difference AH estimator performs better than the levels estimator only for small T and large β . Numerical evaluations show that the levels estimator is more efficient than the difference estimator for all β for T ≥ 8.
3. Simulations Table 1 presents the simulated variances (times n) and the asymptotic variances in (5) and (9) for β = 0.5. For the levels IV estimator, denoted AH(L), and difference IV estimator, denoted AH(D), the simulated variances are close to the asymptotic except for very small T , for which a larger n would be required due to possible correlation between the numerator and the denominator. For all settings except T = 5 and σα2 /σ 2 = 8, AH(L) is more efficient than AH(D), reflecting the asymptotic theory. (See Fig. 1 for the asymptotic variances.) For σα2 /σ 2 = 1, the variance of the HP estimator is uniformly (and considerably) smaller than that of AH(L) for small T ; and for large T , HP and AH(L) perform similarly, just as the large-T asymptotic equivalence suggests. But when σα2 /σ 2 = 8, the discrepancy is larger and the HP estimator is markedly superior to AH(L). From (5), the large n asymptotic variance of AH(L) involves the additional term 2 (1 + β)2 T12 (1 − β)
β σα2 /σ 2 + , 1−β 1+β
which has a substantial impact on variance when σα2 /σ 2 is large. In this event, much larger values of T are required for the variance of AH(L) to be close to that of HP. References Anderson, T.W., Hsiao, C., 1981. Estimation of dynamic models with error components. J. Amer. Statist. Assoc. 76, 598–606. Han, C., Phillips, P.C.B., 2010. GMM estimation for dynamic panels with fixed effects and strong instruments at unity. Econometric Theory 26, 119–151. Han, C., Phillips, P.C.B., 2013. First difference MLE and dynamic panel estimation. J. Econometrics 175, 35–45. Moon, H.R., Phillips, P.C.B., 2004. GMM estimation of autoregressive roots near unity with panel data. Econometrica 72, 467–522. Phillips, P.C.B., 2014. Dynamic panel GMM with roots near unity, Working Paper, Yale University. Phillips, P.C.B., Han, C., 2008. Gaussian inference in AR(1) time series with or without a unit root. Econometric Theory 24, 631–650. Phillips, P.C.B., Moon, H.R., 1999. Linear regression limit theory for nonstationary panel data. Econometrica 67, 1057–1111.
