![On the Estimation and Inference of a Panel Cointegration Model 5
Ω1/2 to be any matrix such that Ω = (Ω1/2)(Ω1/2)′. We use A to de-
note {tr(A′A)}1/2, |A| to denote the determinant of A, ⇒ to denote weak
convergence,
p
→ to denote convergence in probability, [x] to denote the
largest integer ⩽ x, I(0) and I(1) to signify a time-series that is integrated
of order zero and one, respectively, and BM(Ω) to denote Brownian mo-
tion with the covariance matrix Ω. We let M ∞ be a generic positive
number, not depending on T or n.
1.2 The model
Consider the following fixed effect panel regression:
(1.1)
yit = αi + βxit + eit, i = 1, . . . , n, t = 1, . . . , T,
where yit is 1 × 1, β is a 1 × k vector of the slope parameters, αi is the
intercept, and eit is the stationary regression error. We assume that xit is a
k × 1 integrated processes of order one for all i, where
xit = xit−1 + εit.
Under these specifications, (1.1) describes a system of cointegrated re-
gressions, i.e., yit is cointegrated with xit. The initialization of this system
is yi0 = xi0 = Op(1) as T → ∞ for all i. The individual constant
term αi can be extended into general deterministic time trends such as
α0i + α1it + · · · + αpit or other deterministic component. To model the
cross-sectional dependence we assume the error term, eit, follows a factor
model (e.g., Bai and Ng, 2002, 2004):
(1.2)
eit = λ′
iFt + uit,
where Ft is a r ×1 vector of common factors, λi is a r ×1 vector of factor
loadings and uit is the idiosyncratic component of eit, which means
E(eitejt) = λ′
iE(FtF′
t )λj ,
i.e., eit and ejt are correlated due to the common factors Ft.
REMARK 1.1. We could also allow εit to have a factor structure such that
εit = γ ′
i Ft + ηit.
Then we can use Δxit to estimate Ft and γi. Or we can use eit together
with Δxit to estimate Ft, λi and γi. In general, εit can be of the form
εit = γ ′
i Ft + τ′
i Gt + ηit,
where Ft and Gt are zero mean processes, and ηit are usually independent
over i and t.](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-27-320.jpg)
![6 J. Bai and C. Kao
1.3 Assumptions
Our analysis is based on the following assumptions.
ASSUMPTION 1.1. As n → ∞, 1
n
n
i=1 λiλ′
i → Σλ, a r × r positive
definite matrix.
ASSUMPTION 1.2. Let wit = (F′
t , uit, ε′
it)′. For each i, wit = Πi(L)vit
=
∞
j=0 Πij vit−j ,
∞
j=0 jaΠij ∞, |Πi(1)| = 0, for some a 1,
where vit is i.i.d. over t. In addition, Evit = 0, E(vitv′
it ) = Σv 0, and
Evit8 ⩽ M ∞.
ASSUMPTION 1.3. Ft and uit are independent; uit are independent
across i.
Under Assumption 1.2, a multivariate invariance principle for wit holds,
i.e., the partial sum process 1
√
T
[T r]
t=1 wit satisfies:
(1.3)
1
√
T
[T r]
t=1
wit ⇒ B(Ωi) as T → ∞ for all i,
where
Bi =
BF
Bui
Bεi
.
The long-run covariance matrix of {wit} is given by
Ωi =
∞
j=−∞
E(wi0w′
ij )
= Πi(1)ΣvΠi(1)′
= Σi + Γi + Γ ′
i
=
ΩFi ΩFui ΩFεi
ΩuFi Ωui Ωuεi
ΩεFi Ωεui Ωεi
,
where
(1.4)
Γi =
∞
j=1
E(wi0w′
ij ) =
ΓFi ΓFui ΓFεi
ΓuFi Γui Γuεi
ΓεFi Γεui Γεi](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-28-320.jpg)
![On the Estimation and Inference of a Panel Cointegration Model 9
as (n, T → ∞) with n
T → 0 where
δnT =
1
n
n
i=1
λ′
i ΩFεiΩ
1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
−1/2
εi + ΔFεi
+ ΩuεiΩ
1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
−1/2
εi + Δuεi
×
1
n
n
i=1
1
T 2
T
t=1
(xit − xit)(xit − x̄i)′
−1
,
Wi = Wi −
Wi and Ωε = limn→∞
1
n
n
i=1 Ωεi.
REMARK 1.3. It is also possible to construct the bias-corrected OLS by
using the averages of the long run covariances. Note
E[δnT ]
≃
1
n
n
i=1
λ′
i −
1
2
ΩFεi + ΔFεi
−
1
2
Ωuεi + Δuεi
1
6
Ωε
−1
=
1
n
n
i=1
−
1
2
(λ′
iΩFεi + Ωuεi) + λ′
iΔFεi + Δuεi
1
6
Ωε
−1
=
1
n
n
i=1
−
1
2
λ′
iΩFεi +
1
n
n
i=1
Ωuεi +
1
n
n
i=1
λ′
iΔFεi
+
1
n
n
i=1
Δuεi
1
6
Ωε
−1
.
It can be shown by a central limit theorem that
√
n
δnT − E[δnT ]
⇒ N(0, B)
for some B. Therefore,
√
nT (β̂OLS − β) −
√
nE[δnT ]
=
√
nT (β̂OLS − β) −
√
nδnT +
√
n
δnT − E[δnT ]
⇒ N(0, A)
for some A.](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-31-320.jpg)
![On the Estimation and Inference of a Panel Cointegration Model 15
1.7 Hypothesis testing
We now consider a linear hypothesis that involves the elements of the co-
efficient vector β. We show that hypothesis tests constructed using the FM
estimator have asymptotic chi-squared distributions. The null hypothesis
has the form:
(1.12)
H0: Rβ = r,
where r is a m×1 known vector and R is a known m×k matrix describing
the restrictions. A natural test statistic of the Wald test using β̂FM is
W =
1
6
nT 2
(Rβ̂FM − r)′
6
Ω−1
ε lim
n→∞
1
n
n
i=1
(λ̂′
i
ΩF.εiλ̂i
Ωεi
(1.13)
+
Ωu.εi
Ωεi)
Ω−1
ε
−1
(Rβ̂FM − r).
It is clear that W converges in distribution to a chi-squared random
variable with k degrees of freedom, χ2
k , as (n, T → ∞) under the null
hypothesis. Hence, we establish the following theorem:
THEOREM 1.4. If Assumptions 1.1–1.4 hold, then under the null hypoth-
esis (1.12), with (n, T → ∞), W ⇒ χ2
k ,
REMARK 1.6. (1) One common application of Theorem 1.4 is the single-
coefficient test: one of the coefficient is zero; βj = β0,
R = [ 0 0 · · · 1 0 · · · 0 ]
and r = 0. We can construct a t-statistic
(1.14)
tj =
√
nT (β̂jFM − β0)
sj
,
where
s2
j =
6
Ω−1
ε lim
n→∞
1
n
n
i=1
(λ̂′
i
ΩF.εiλ̂i
Ωεi +
Ωu.εi
Ωεi)
Ω−1
ε
jj
,
the jth diagonal element of
6
Ω−1
ε lim
n→∞
1
n
n
i=1
(λ̂′
i
ΩF.εiλ̂i
Ωεi +
Ωu.εi
Ωεi)
Ω−1
ε
.
It follows that
(1.15)
tj ⇒ N(0, 1).](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-37-320.jpg)
![On the Estimation and Inference of a Panel Cointegration Model 17
for i = 1, . . . , n, t = 1, . . . , T , where
(1.17)
Ft
uit
εit
=
F∗
t
u∗
it
ε∗
it
+
0 0 0
0 0.3 −0.4
θ31 θ32 0.6
F∗
t−1
u∗
it−1
ε∗
it−1
with
F∗
t
u∗
it
ε∗
it
i.i.d.
∼ N
0
0
0
,
1 σ12 σ13
σ21 1 σ23
σ31 σ32 1
.
For this experiment, we have a single factor (r = 1) and λi are gen-
erated from i.i.d. N(μλ, 1). We let μλ = 0.1. We generated αi from a
uniform distribution, U[0, 10], and set β = 2. From Theorems 1.1–1.3 we
know that the asymptotic results depend upon variances and covariances
of Ft, uit and εit. Here we set σ12 = 0. The design in (1.17) is a good one
since the endogeneity of the system is controlled by only four parameters,
θ31, θ32, σ31 and σ32. We choose θ31 = 0.8, θ32 = 0.4, σ31 = −0.8 and
θ32 = 0.4.
The estimate of the long-run covariance matrix in (1.11) was obtained
by using the procedure KERNEL in COINT 2.0 with a Bartlett window.
The lag truncation number was set arbitrarily at five. Results with other
kernels, such as Parzen and quadratic spectral kernels, are not reported,
because no essential differences were found for most cases.
Next, we recorded the results from our Monte Carlo experiments that
examined the finite-sample properties of (a) the OLS estimator, β̂OLS in
(1.6), (b) the 2S-FM estimator, β̂2S, in (1.8), (c) the two-step naive FM es-
timator, β̂b
FM, proposed by Kao and Chiang (2000) and Phillips and Moon
(1999), (d) the CUP-FM estimator β̂CUP, in (1.9) and (e) the CUP naive
FM estimator β̂d
FM which is similar to the two-step naive FM except the
iteration goes beyond two steps. The naive FM estimators are obtained
assuming the cross-sectional independence. The maximum number of the
iteration for CUP-FM estimators is set to 20. The results we report are
based on 1,000 replications and are summarized in Tables 1.1–1.4. All the
FM estimators were obtained by using a Bartlett window of lag length five
as in (1.11).
Table 1.1 reports the Monte Carlo means and standard deviations (in
parentheses) of (β̂OLS−β), (β̂2S−β), (β̂b
FM−β), (β̂CUP−β), and (β̂d
FM−β)
for sample sizes T = n = (20, 40, 60). The biases of the OLS estimator,
β̂OLS, decrease at a rate of T . For example, with σλ = 1 and σF = 1, the
bias at T = 20 is −0.045, at T = 40 is −0.024, and at T = 60 is −0.015.
Also, the biases stay the same for different values of σλ and σF .
While we expected the OLS estimator to be biased, we expected FM
estimators to produce better estimates. However, it is noticeable that the](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-39-320.jpg)
![On the Estimation and Inference of a Panel Cointegration Model 23
Acknowledgements
We thank Badi Baltagi, Yu-Pin Hu, Giovanni Urga, Kamhon Kan, Chung-
Ming Kuan, Hashem Pesaran, Lorenzo Trapani and Yongcheol Shin for
helpful comments. We also thank seminar participants at Academia Sinica,
National Taiwan University, Syracuse University, Workshop on Recent
Developments in the Econometrics of Panel Data in London, March 2004
and the European Meeting of the Econometric Society in Madrid, August
2004 for helpful comments and suggestions. Jushan Bai acknowledges fi-
nancial support from the NSF (grant SES-0137084).
Appendix A1
Let
BnT =
n
i=1
T
t=1
(xit − x̄i)(xit − x̄i)′
.
Note
√
nT (β̂OLS − β)
=
√
n
1
n
n
i=1
1
T
T
t=1
eit(xit − x̄i)′
1
n
1
T 2
BnT
−1
=
√
n
1
n
n
i=1
ζ1iT
1
n
n
i=1
ζ2iT
−1
=
√
nξ1nT [ξ2nT ]−1
,
where x̄i = 1
T
T
t=1 xit, ȳi = 1
T
T
t=1 yit, ζ1iT = 1
T
T
t=1 eit(xit − x̄i)′,
ζ2iT = 1
T 2
T
t=1(xit − x̄i)(xit − x̄i)′, ξ1nT = 1
n
n
i=1 ζ1iT , and ξ2nT =
1
n
n
i=1 ζ2iT . Before going into the next theorem, we need to consider
some preliminary results.
Define Ωε = limn→∞
1
n
n
i=1 Ωεi and
θn
=
1
n
n
i=1
λ′
i ΩF.εiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + ΔFεi
+ Ωu.εiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + Δuεi
.
If Assumptions 1.1–1.4 hold, then](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-45-320.jpg)
![24 J. Bai and C. Kao
LEMMA A1.1. (a) As (n, T → ∞),
1
n
1
T 2
BnT
p
→
1
6
Ωε.
(b) As (n, T → ∞) with n
T → 0,
√
n
1
n
1
T
n
i=1
T
t=1
eit(xit − x̄i)′
− θn
⇒ N 0,
1
6
lim
n→∞
1
n
n
i=1
{λ′
iΩF.εiλiΩεi + Ωu.εiΩεi} .
PROOF. (a) and (b) can be shown easily by following Theorem 8 in
Phillips and Moon (1999).
A1.1 Proof of Theorem 1.1
PROOF. Recall that
√
nT (β̂OLS − β) −
√
n
1
n
n
i=1
λ′
i ΩFεiΩ
−1/2
εi
×
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + ΔFεi
+ ΩεuiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + Δεui
1
n
1
T 2
BnT
−1
=
√
n
1
n
n
i=1
ζ1iT − λ′
i ΩFεiΩ
−1/2
εi
×
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + ΔFεi
− ΩuεiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + Δuεi
×
1
n
n
i=1
ζ2iT
−1
=
√
n
1
n
n
i=1
ζ∗
1iT
1
n
n
i=1
ζ2iT
−1
=
√
nξ∗
1nT [ξ2nT ]−1
,](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-46-320.jpg)
![On the Estimation and Inference of a Panel Cointegration Model 25
where
ζ∗
1iT = ζ1iT − λ′
i ΩFεiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + ΔFεi
− ΩuεiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + Δuεi
and
ξ∗
1nT =
1
n
n
i=1
ζ∗
1iT .
First, we note from Lemma A1.1(b) that
√
nξ∗
1nT ⇒ N 0,
1
6
lim
n→∞
1
n
n
i=1
{λ′
iΩF.εiλiΩεi + Ωu.εiΩεi}
as (n, T → ∞) and n
T → 0. Using the Slutsky theorem and (a) from
Lemma A1.1, we obtain
√
nξ∗
1nT [ξ2nT ]−1
⇒ N 0, 6Ω−1
ε lim
n→∞
1
n
n
i=1
(λ′
iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1
ε .
Hence,
√
nT (β̂OLS − β) −
√
nδnT
(A1.1)
⇒ N 0, 6Ω−1
ε lim
n→∞
1
n
n
i=1
(λ′
iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1
ε ,
proving the theorem, where
δnT =
1
n
n
i=1
λ′
i ΩFεiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + ΔFεi
+ ΩuεiΩ
−1/2
εi
1
T
T
t=1
x′
it(xit − x̄i) Ω
1/2
εi + Δuεi
×
1
n
1
T 2
BnT
−1
.
Therefore, we established Theorem 1.1.](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-47-320.jpg)
![26 J. Bai and C. Kao
A1.2 Proof of Theorem 1.2
PROOF. Let
F+
it = Ft − ΩFεiΩ−1
εi εit,
and
u+
it = uit − ΩuεiΩ−1
εi εit.
The FM estimator of β can be rewritten as follows
β̃FM =
n
i=1
T
t=1
y+
it (xit − x̄i)′
− T
λ′
iΔ+
Fεi + Δ+
uεi
B−1
nT
= β +
n
i=1
T
t=1
λ′
iF+
it + u+
it
(xit − x̄i)′
(A1.2)
− T
λ′
iΔ+
Fεi + Δ+
uεi
B−1
nT .
First, we rescale (β̃FM − β) by
√
nT
√
nT (β̃FM − β) =
√
n
1
n
n
i=1
1
T
T
t=1
λ′
iF+
it + u+
it
(xit − x̄i)′
− λ′
iΔ+
Fεi − Δ+
uεi
1
n
1
T 2
BnT
−1
=
√
n
1
n
n
i=1
ζ∗∗
1iT
1
n
n
i=1
ζ2iT
−1
(A1.3)
=
√
nξ∗∗
1nT [ξ2nT ]−1
,
where ζ∗∗
1iT = 1
T
T
t=1[(λ′
iF+
it + û+
it )(xit − x̄i)′ − λ′
iΔ+
Fεi − Δ+
uεi], and
ξ∗∗
1nT = 1
n
n
i=1 ζ∗∗
1iT .
Modifying Theorem 11 in Phillips and Moon (1999) and Kao and Chi-
ang (2000) we can show that as (n, T → ∞) with n
T → 0
√
n
1
n
1
T
n
i=1
T
t=1
λ′
iF+
it (xit − x̄i)′
− λ′
iΔ+
Fεi
⇒ N 0,
1
6
lim
n→∞
1
n
n
i=1
λ′
iΩF.εiλiΩεi](https://image.slidesharecdn.com/70155-250225101621-3712cfcb/85/Panel-Data-Econometrics-Theoretical-Contributions-and-Empirical-Applications-1st-Edition-B-H-Baltagi-48-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
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Title: Index of the Project Gutenberg Works of Pelham Grenville
Wodehouse
Author: P. G. Wodehouse
Editor: David Widger
Release date: December 21, 2018 [eBook #58508]
Most recently updated: July 8, 2019
Language: English
Credits: Produced by David Widger
*** START OF THE PROJECT GUTENBERG EBOOK INDEX OF THE
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Panel Data Econometrics Theoretical Contributions and Empirical Applications 1st Edition B H Baltagi
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- 5. Panel Data Econometrics Theoretical Contributions and Empirical Applications 1st Edition B H Baltagi Digital Instant Download Author(s): B HBaltagi ISBN(s): 9780444521729, 0080462251 Edition: 1 File Details: PDF, 2.39 MB Year: 2006 Language: english
- 7. PANEL DATA ECONOMETRICS Theoretical Contributions and Empirical Applications
- 8. CONTRIBUTIONS TO ECONOMIC ANALYSIS 274 Honorary Editors: D. W. JORGENSON J. TINBERGEN† Editors: B. BALTAGI E. SADKA D. WILDASIN Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo
- 9. PANEL DATA ECONOMETRICS Theoretical Contributions and Empirical Applications Edited by BADI H. BALTAGI Department of Economics and Center for Policy Research Syracuse University, Syracuse, NY 13244-1020 U.S.A. Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo
- 10. ELSEVIER The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2006 Copyright © 2006 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Depart- ment in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; Email: permissions@elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting: Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52172-9 ISBN-10: 0-444-52172-0 For information on all Elsevier publictions visit our website at books.elsevier.com Printed and bound in The Netherlands 06 07 08 09 10 10 9 8 7 6 5 4 3 2 1
- 11. Introduction to the Series This series consists of a number of hitherto unpublished studies, which are introduced by the editors in the belief that they represent fresh contribu- tions to economic science. The term ‘economic analysis’ as used in the title of the series has been adopted because it covers both the activities of the theoretical economist and the research worker. Although the analytical method used by the various contributors are not the same, they are nevertheless conditioned by the common origin of their studies, namely theoretical problems encountered in practical research. Since for this reason, business cycle research and national accounting, research work on behalf of economic policy, and problems of planning are the main sources of the subjects dealt with, they necessarily determine the manner of approach adopted by the authors. Their methods tend to be ‘practical’ in the sense of not being too far remote from application to actual economic conditions. In addition, they are quantitative. It is the hope of the editors that the publication of these studies will help to stimulate the exchange of scientific information and to reinforce international cooperation in the field of economics. The Editors
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- 13. Contents Introduction to the Series v Preface ix List of Contributors xv PART I THEORETICAL CONTRIBUTIONS 1 Chapter 1. On the Estimation and Inference of a Panel Cointegra- tion Model with Cross-Sectional Dependence 3 Jushan Bai and Chihwa Kao Chapter 2. A Full Heteroscedastic One-Way Error Components Model: Pseudo-Maximum Likelihood Estimation and Specification Testing 31 Bernard Lejeune Chapter 3. Finite Sample Properties of FGLS Estimator for Random- Effects Model under Non-Normality 67 Aman Ullah and Xiao Huang Chapter 4. Modelling the Initial Conditions in Dynamic Regression Models of Panel Data with Random Effects 91 I. Kazemi and R. Crouchley Chapter 5. Time Invariant Variables and Panel Data Models: A Generalised Frisch–Waugh Theorem and its Implications 119 Jaya Krishnakumar PART II EMPIRICAL APPLICATIONS 133 Chapter 6. An Intertemporal Model of Rational Criminal Choice 135 Robin C. Sickles and Jenny Williams Chapter 7. Swedish Liquor Consumption: New Evidence on Taste Change 167 Badi H. Baltagi and James M. Griffin Chapter 8. Import Demand Estimation with Country and Prod- uct Effects: Application of Multi-Way Unbalanced Panel Data Models to Lebanese Imports 193 Rachid Boumahdi, Jad Chaaban and Alban Thomas
- 14. viii Contents Chapter 9. Can Random Coefficient Cobb–Douglas Production Functions be Aggregated to Similar Macro Functions? 229 Erik Biørn, Terje Skjerpen and Knut R. Wangen Chapter 10. Conditional Heteroskedasticity and Cross-Sectional De- pendence in Panel Data: An Empirical Study of Inflation Uncertainty in the G7 Countries 259 Rodolfo Cermeño and Kevin B. Grier Chapter 11. The Dynamics of Exports and Productivity at the Plant Level: A Panel Data Error Correction Model (ECM) Ap- proach 279 Mahmut Yasar, Carl H. Nelson and Roderick M. Rejesus Chapter 12. Learning about the Long-Run Determinants of Real Ex- change Rates for Developing Countries: A Panel Data Investigation 307 Imed Drine and Christophe Rault Chapter 13. Employee Turnover: Less is Not Necessarily More? 327 Mark N. Harris, Kam Ki Tang and Yi-Ping Tseng Chapter 14. Dynamic Panel Models with Directors’ and Officers’ Liability Insurance Data 351 George D. Kaltchev Chapter 15. Assessment of the Relationship between Income In- equality and Economic Growth: A Panel Data Analysis of the 32 Federal Entities of Mexico, 1960–2002 361 Araceli Ortega-Díaz
- 15. Preface Panel data econometrics has evolved rapidly over the last decade. Dynamic panel data estimation, non-linear panel data methods and the phenomenal growth in non-stationary panel data econometrics makes this an exciting area of research in econometrics. The 11th international conference on panel data held at Texas A&M University, College Station, Texas, June 2004, witnessed about 150 participants and 100 papers on panel data. This volume includes some of the papers presented at that conference and other solicited papers that made it through the refereeing process. Theoretical econometrics contributions include: Bai and Kao who sug- gest a factor model approach to model cross-section dependence in the panel co-integrated regression setting; Lejeune who proposes new esti- mation methods and some diagnostics tests for a general heteroskedastic error component model with unbalanced panel data; Ullah and Huang who study the finite sample properties of feasible GLS for the random effects model with non-normal errors; Kazemi and Crouchley who suggest a prag- matic approach to the problem of estimating a dynamic panel regression with random effects under various assumptions about the nature of the initial conditions; Krishnakumar who uses a generalized version of the Frisch–Waugh theorem to extend Mundlak’s (1978) results for the error component model. Empirical applications include: Sickles and Williams who estimate a dynamic model of crime using panel data from the 1958 Philadelphia Birth Cohort study; Baltagi and Griffin who find that at least 4 structural breaks in a panel data on liquor consumption for 21 Swedish counties over the period 1956–1999; Boumahdi, Chaaban and Thomas who estimate a flexible AIDS demand model for agricultural imports into Lebanon incorporating a three-way error component model that allows for product, country and time effects as separate unobserved determinants of import demand; Biørn, Skjerpen and Wangen who are concerned with the analysis of heterogeneous log-linear relationships (and specifically Cobb–Douglas production functions) at the firm-level and at the corre- sponding aggregate industry level. They use unbalanced panel data on firms from two Norwegian manufacturing industries over the period 1972– 1993; Cermeño and Grier who apply a model that accounts for conditional heteroskedasticity and cross-sectional dependence to a panel of monthly inflation rates of the G7 over the period 1978.2–2003.9; Yasar, Nelson and Rejesus who use plant level panel data for Turkish manufacturing in- dustries to analyze the relative importance of short-run versus long-run
- 16. x Preface dynamics of the export-productivity relationship; Drine and Rault who focus on developing countries and analyze the long-run relationship be- tween real exchange rate and some macroeconomic variables, via panel unit root and cointegration tests; Harris, Tang and Tseng who quantify the impact of employee turnover on productivity using an Australian busi- ness longitudinal survey over the period 1994/5 to 1997/8; Kaltchev who uses proprietary and confidential panel data on 113 public U.S. compa- nies over the period 1997–2003 to analyze the demand for Directors’ and Officers’ liability insurance; Ortega-Díaz who assesses how income in- equality influences economic growth across 32 Mexican States over the period 1960–2002. Theoretical econometrics contributions Bai and Kao suggest a factor model approach to model cross-section de- pendence in the panel co-integrated regression setting. Factor models are used to study world business cycles as well as common macro shocks like international financial crises or oil price shocks. Factor models offer a significant reduction in the number of sources of cross-sectional depen- dence in panel data and they allow for heterogeneous response to common shocks through heterogeneous factor loadings. Bai and Kao suggest a continuous-updated fully modified estimator for this model and show that it has better finite sample performance than OLS and a two step fully mod- ified estimator. Lejeune proposes new estimation methods for a general heteroskedastic error component model with unbalanced panel data, namely the Gaussian pseudo maximum likelihood of order 2. In addition, Lejeune suggests some diagnostics tests for heteroskedasticity, misspecification testing us- ing m-tests, Hausman type and Information type tests. Lejeune applies these methods to estimate and test a translog production function using an unbalanced panel of 824 French firms observed over the period 1979– 1988. Ullah and Huang study the finite sample properties of feasible GLS for the random effects model with non-normal errors. They study the effects of skewness and excess kurtosis on the bias and mean squared error of the estimator using asymptotic expansions. This is done for large N and fixed T , under the assumption that the first four moments of the error are finite. Kazemi and Crouchley suggest a pragmatic approach to the problem of estimating a dynamic panel regression with random effects under various assumptions about the nature of the initial conditions. They find that the
- 17. Preface xi full maximum likelihood improves the consistency results if the relation- ships between random effects, initial conditions and explanatory variables are correctly specified. They illustrate this by testing a variety of different hypothetical models in empirical contexts. They use information criteria to select the best approximating model. Krishnakumar uses a generalized version of the Frisch–Waugh theo- rem to extend Mundlak’s (1978) results for the error component model with individual effects that are correlated with the explanatory variables. In particular, this extension is concerned with the presence of time invari- ant variables and correlated specific effects. Empirical contributions The paper by Sickles and Williams estimates a dynamic model of crime using panel data from the 1958 Philadelphia Birth Cohort study. Agents are rational and anticipate the future consequence of their actions. The authors investigate the role of social capital through the influence of social norms on the decision to participate in crime. They find that the initial level of social capital stock is important in determining the pattern of criminal involvement in adulthood. The paper by Baltagi and Griffin uses panel data on liquor consump- tion for 21 Swedish counties over the period 1956–1999. It finds that at least 4 structural breaks are necessary to account for the sharp decline in per-capita liquor consumption over this period. The first structural break coincides with the 1980 advertising ban, but subsequent breaks do not appear linked to particular policy initiatives. Baltagi and Griffin inter- pret these results as taste change accounting for increasing concerns with health issues and changing drinking mores. The paper by Boumahdi, Chaaban and Thomas estimate a flexible AIDS demand model for agricultural imports into Lebanon incorporating a three- way error component model that allows for product, country and time effects as separate unobserved determinants of import demand. In their application to trade in agricultural commodities the authors are primarily concerned with the estimation of import demand elasticities. Convention- ally, such estimates are frequently obtained from time series data that ignore the substitution elasticities across commodities, and thus implicitly ignore the cross-sectional dimension of the data. Exhaustive daily trans- actions (both imports and exports) data are obtained from the Lebanese customs administration for the years 1997–2002. Restricting their atten- tion to major agricultural commodities (meat, dairy products, cereals, ani- mals and vegetable fats and sugar), they estimate an import share equation
- 18. xii Preface for European products as a function of own-price and competitors prices. Competition is taking place between European countries, Arab and re- gional countries, North and South America and the rest of the world. The import share equations are estimated by allowing for parameter hetero- geneity across the 5 commodity groups, and tests for the validity of the multi-way error components specification are performed using unbalanced panel data. Estimation results show that this specification is generally sup- ported by the data. The paper by Biørn, Skjerpen and Wangen is concerned with the analysis of heterogeneous log-linear relationships (and specifically Cobb– Douglas production functions) at the firm-level and at the correspond- ing aggregate industry level. While the presence of aggregation bias in log-linear models is widely recognized, considerable empirical analysis continues to be conducted ignoring the problem. This paper derives a de- composition that highlights the source of biases that arise in aggregate work. It defines some aggregate elasticity measures and illustrates these in an empirical exercise based on firm-level data in two Norwegian manu- facturing industries: The pulp and paper industry (2823 observations, 237 firms) and the basic metals industry (2078 observations, 166 firms) ob- served over the period 1972–1993. The paper by Cermeño and Grier specify a model that accounts for conditional heteroskedasticity and cross-sectional dependence within a typical panel data framework. The paper applies this model to a panel of monthly inflation rates of the G7 over the period 1978.2–2003.9 and finds significant and quite persistent patterns of volatility and cross-sectional dependence. The authors use the model to test two hypotheses about the inter-relationship between inflation and inflation uncertainty, finding no support for the hypothesis that higher inflation uncertainty produces higher average inflation rates and strong support for the hypothesis that higher in- flation is less predictable. The paper by Yasar, Nelson and Rejesus uses plant level panel data for Turkish manufacturing industries to analyze the relative importance of short-run versus long-run dynamics of the export-productivity relation- ship. The adopted econometric approach is a panel data error correction model that is estimated by means of system GMM. The data consists of plants with more than 25 employees from two industries, the textile and apparel industry and the motor vehicles and parts industry, observed over the period 1987–1997. They find that “permanent productivity shocks gen- erate larger long-run export level responses, as compared to long-run pro- ductivity responses from permanent export shocks”. This result suggests that industrial policy should be geared toward permanent improvements in plant-productivity in order to have sustainable long-run export and eco- nomic growth.
- 19. Preface xiii The paper by Drine and Rault focuses on developing countries and analyzes the long-run relationship between real exchange rate and some macroeconomic variables, via panel unit root and cointegration tests. The results show that the degrees of development and of openness of the econ- omy strongly influence the real exchange rate. The panels considered are relatively small: Asia (N = 7, T = 21), Africa (N = 21, T = 16) and Latin America (N = 17, T = 23). The paper by Harris, Tang and Tseng consider a balanced panel of medium sized firms drawn from the Australian business longitudinal sur- vey over the period 1994/5 to 1997/8. The paper sets out to quantify the impact of employee turnover on productivity and finds that the optimal turnover rate is 0.22. This is higher than the sample median of 0.14 which raises the question about whether there are institutional rigidities hinder- ing resource allocation in the labor market. The paper by Kaltchev uses proprietary and confidential panel data on 113 public U.S. companies over the period 1997–2003 to analyze the de- mand for Directors’ and Officers’ liability insurance. Applying system GMM methods to a dynamic panel data model on this insurance data, Kaltchev rejects that this theory is habit driven but still finds some role for persistence. He also confirms the hypothesis that smaller companies demand more insurance. Other empirical findings include the following: Returns are significant in determining the amount of insurance and com- panies in financial distress demand higher insurance limits. Indicators of financial health such as leverage and volatility are significant, but not cor- porate governance. The paper by Ortega-Díaz assesses how income inequality influences economic growth across 32 Mexican States over the period 1960–2002. Using dynamic panel data analysis, with both, urban personal income for grouped data and household income from national surveys, Ortega-Díaz finds that inequality and growth are positively related. This relationship is stable across variable definitions and data sets, but varies across regions and trade periods. A negative influence of inequality on growth is found in a period of restrictive trade policies. In contrast, a positive relationship is found in a period of trade openness. I hope the readers enjoy this set of 15 papers on panel data and share my view on the wide spread use of panels in all fields of economics as clear from the applications. I would like to thank the anonymous referees that helped in reviewing these manuscripts. Also, Jennifer Broaddus for her editorial assistance and handling of these manuscripts. Badi H. Baltagi College Station, Texas and Syracuse, New York
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- 21. List of Contributors Numbers in parenthesis indicate the pages where the authors’ contributions can be found. Jushan Bai (3) Department of Economics, New York University, New York, NY 10003, USA and Department of Economics, Tsinghua University, Bei- jing 10084, China. E-mail: jushan.bai@nyu.edu Badi H. Baltagi (167) Department of Economics, and Center for Policy Re- search, Syracuse University, Syracuse, NY 13244-1020, USA. E-mail: bbaltagi@maxwell.syr.edu Erik Biørn (229) Department of Economics, University of Oslo, 0317 Oslo, Norway and Research Department, Statistics Norway, 0033 Oslo, Norway. E-mail: erik.biorn@econ.uio.no Rachid Boumahdi (193) University of Toulouse, GREMAQ and LIHRE, F31000 Toulouse, France. E-mail: rachid.boumahdi@univ-tlse1.fr Rodolfo Cermeño (259) División de Economía, CIDE, México D.F., México. E-mail: rodolfo.cermeno@cide.edu Jad Chaaban (193) University of Toulouse, INRA-ESR, F-31000 Toulouse cedex, France. E-mail: chaaban@toulouse.inra.fr Rob Crouchley (91) Centre for e-Science, Fylde College, Lancaster University, Lancaster LA1 4YF, UK. E-mail: r.crouchley@lancaster.ac.uk Imed Drine (307) Paris I, Masion des Sciences de l’Economie, 75647 Paris cedex 13, France. E-mail: drine@univ-paris1.fr Kevin B. Grier (259) Department of Economics, University of Oklahoma, OK 73019, USA. E-mail: angus@ou.edu James M. Griffin (167) Bush School of Government and Public Service, Texas A&M University, College Station, TX 77843-4220, USA. E-mail: jgriffin@bushschool.tamu.edu Mark N. Harris (327) Department of Econometrics and Business Statistics, Monash University, Melbourne, Vic 3800, Australia. E-mail: mark.harris@buseco.monash.edu.au Xiao Huang (67) Department of Economics University of California, Riverside, CA 92521-0427, USA. E-mail: xiao.huang@email.ucr.edu George D. Kaltchev (351) Department of Economics, Southern Methodist Uni- versity, Dallas, TX 75275-0496, USA. E-mail: gkaltche@mail.smu.edu Chihwa Kao (3) Center for Policy Research and Department of Economics, Syracuse University Syracuse, NY 13244-1020, USA. E-mail: cdkao@maxwell.syr.edu xv
- 22. xvi List of Contributors Iraj Kazemi (91) Centre for Applied Statistics, Lancaster University, Lancaster LA1 4YF, UK. E-mail: i.kazemi@lancaster.ac.uk Jaya Krishnakumar (119) Department of Econometrics, University of Geneva, UNI-MAIL, CH-1211 Geneva 4, Switzerland. E-mail: jaya.krishnakumar@metri.unige.ch Bernard Lejeune (31) HEC-University of Liège, CORE and ERUDITE, 4000 Liège, Belgium. E-mail: b.lejeune@ulg.ac.be Carl H. Nelson (279) Department of Agricultural & Consumer Economics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail: chnelson@uiuc.edu Araceli Ortega-Díaz (361) Tecnológico de Monterrey, 14380 Tlalpan, México. E-mail: araceli.ortega@itesm.mx;aortega@sedesal.gob.mx Chrisophe Rault (307) University of Evry-Val d’Essonne, Department d’économie, 91025 Evry cedex, France. E-mail: chrault@hotmail.com Roderick M. Rejesus (279) Department of Agricultural & Applied Economics, Texas Tech University, Lubbock, TX 79409-2132, USA. E-mail: roderick.rejesus@ttu.edu Robin C. Sickles (135) Department of Economics, Rice University, Houston, TX 77005-1892, USA. E-mail: rsickles@rice.edu Terje Skjerpen (229) Research Department, Statistics Norway, 0033 Oslo, Nor- way. E-mail: terje.skjerpen@ssb.no Kam-Ki Tang (327) School of Economics, University of Queensland, St. Lucia, Qld 4072, Australia. E-mail: kk.tang@uq.edu.au Alban Thomas (193) University of Toulouse, INRA-LERNA, F-31000 Toulouse cedex, France. E-mail: thomas@toulouse.inra.fr Yi-Ping Tseng (327) Melbourne Institute of Applied Economic and Social Re- search, University of Melbourne, Parkville, Vic 3010, Australia. E-mail: y.tseng@unimelb.edu.au Aman Ullah (67) Department of Economics, University of California, River- side, CA 92521-0427, USA. E-mail: aman.ullah@ucr.edu Knut R. Wangen (229) Research Department, Statistics Norway, 0033 Oslo, Norway. E-mail: knut.reidar.wangen@ssb.no Jenny Williams (135) Department of Economics, University of Melbourne, Melbourne, Vic 3010, Australia. E-mail: jenny.williams@unimelb.edu.au Mahmut Yasar (279) Department of Economics, Emory University, Atlanta, GA 30322, USA. E-mail: myasar@emory.edu
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- 25. Panel Data Econometrics B.H. Baltagi (Editor) © 2006 Published by Elsevier B.V. DOI: 10.1016/S0573-8555(06)74001-9 CHAPTER 1 On the Estimation and Inference of a Panel Cointegration Model with Cross-Sectional Dependence Jushan Baia and Chihwa Kaob aDepartment of Economics, New York University, New York, NY 10003, USA and Department of Economics, Tsinghua University, Beijing 10084, China E-mail address: Jushan.Bai@nyu.edu bCenter for Policy Research and Department of Economics, Syracuse University, Syracuse, NY 13244-1020, USA E-mail address: cdkao@maxwell.syr.edu Abstract Most of the existing literature on panel data cointegration assumes cross- sectional independence, an assumption that is difficult to satisfy. This pa- per studies panel cointegration under cross-sectional dependence, which is characterized by a factor structure. We derive the limiting distribution of a fully modified estimator for the panel cointegrating coefficients. We also propose a continuous-updated fully modified (CUP-FM) estimator. Monte Carlo results show that the CUP-FM estimator has better small sample properties than the two-step FM (2S-FM) and OLS estimators. Keywords: panel data, cross-sectional dependence, factor analysis, CUP- FM, 2S-FM JEL classifications: C13, C33 1.1 Introduction A convenient but difficult to justify assumption in panel cointegration analysis is cross-sectional independence. Left untreated, cross-sectional dependence causes bias and inconsistency estimation, as argued by Andrews (2005). In this paper, we use a factor structure to characterize cross-sectional dependence. Factors models are especially suited for this purpose. One major source of cross-section correlation in macroeconomic data is common shocks, e.g., oil price shocks and international financial
- 26. 4 J. Bai and C. Kao crises. Common shocks drive the underlying comovement of economic variables. Factor models provide an effective way to extract the comove- ment and have been used in various studies.1 Cross-sectional correlation exists even in micro level data because of herd behavior (fashions, fads, and imitation cascades) either at firm level or household level. The general state of an economy (recessions or booms) also affects household decision making. Factor models accommodate individual’s different responses to common shocks through heterogeneous factor loadings. Panel data models with correlated cross-sectional units are important due to increasing availability of large panel data sets and increasing inter- connectedness of the economies. Despite the immense interest in testing for panel unit roots and cointegration,2 not much attention has been paid to the issues of cross-sectional dependence. Studies using factor models for nonstationary data include Bai and Ng (2004), Bai (2004), Phillips and Sul (2003), and Moon and Perron (2004). Chang (2002) proposed to use a nonlinear IV estimation to construct a new panel unit root test. Hall et al. (1999) considered a problem of determining the number of common trends. Baltagi et al. (2004) derived several Lagrange Multiplier tests for the panel data regression model with spatial error correlation. Robertson and Symon (2000), Coakley et al. (2002) and Pesaran (2004) proposed to use common factors to capture the cross-sectional dependence in station- ary panel models. All these studies focus on either stationary data or panel unit root studies rather than panel cointegration. This paper makes three contributions. First, it adds to the literature by suggesting a factor model for panel cointegrations. Second, it proposes a continuous-updated fully modified (CUP-FM) estimator. Third, it provides a comparison for the finite sample properties of the OLS, two-step fully modified (2S-FM), CUP-FM estimators. The rest of the paper is organized as follows. Section 1.2 introduces the model. Section 1.3 presents assumptions. Sections 1.4 and 1.5 develop the asymptotic theory for the OLS and fully modified (FM) estimators. Section 1.6 discusses a feasible FM estimator and suggests a CUP-FM estimator. Section 1.7 makes some remarks on hypothesis testing. Sec- tion 1.8 presents Monte Carlo results to illustrate the finite sample proper- ties of the OLS and FM estimators. Section 1.9 summarizes the findings. Appendix A1 contains the proofs of lemmas and theorems. The following notations are used in the paper. We write the integral 1 0 W(s) ds as W when there is no ambiguity over limits. We define 1 For example, Stock and Watson (2002), Gregory and Head (1999), Forni and Reichlin (1998) and Forni et al. (2000) to name a few. 2 See Baltagi and Kao (2000) for a recent survey.
- 27. On the Estimation and Inference of a Panel Cointegration Model 5 Ω1/2 to be any matrix such that Ω = (Ω1/2)(Ω1/2)′. We use A to de- note {tr(A′A)}1/2, |A| to denote the determinant of A, ⇒ to denote weak convergence, p → to denote convergence in probability, [x] to denote the largest integer ⩽ x, I(0) and I(1) to signify a time-series that is integrated of order zero and one, respectively, and BM(Ω) to denote Brownian mo- tion with the covariance matrix Ω. We let M ∞ be a generic positive number, not depending on T or n. 1.2 The model Consider the following fixed effect panel regression: (1.1) yit = αi + βxit + eit, i = 1, . . . , n, t = 1, . . . , T, where yit is 1 × 1, β is a 1 × k vector of the slope parameters, αi is the intercept, and eit is the stationary regression error. We assume that xit is a k × 1 integrated processes of order one for all i, where xit = xit−1 + εit. Under these specifications, (1.1) describes a system of cointegrated re- gressions, i.e., yit is cointegrated with xit. The initialization of this system is yi0 = xi0 = Op(1) as T → ∞ for all i. The individual constant term αi can be extended into general deterministic time trends such as α0i + α1it + · · · + αpit or other deterministic component. To model the cross-sectional dependence we assume the error term, eit, follows a factor model (e.g., Bai and Ng, 2002, 2004): (1.2) eit = λ′ iFt + uit, where Ft is a r ×1 vector of common factors, λi is a r ×1 vector of factor loadings and uit is the idiosyncratic component of eit, which means E(eitejt) = λ′ iE(FtF′ t )λj , i.e., eit and ejt are correlated due to the common factors Ft. REMARK 1.1. We could also allow εit to have a factor structure such that εit = γ ′ i Ft + ηit. Then we can use Δxit to estimate Ft and γi. Or we can use eit together with Δxit to estimate Ft, λi and γi. In general, εit can be of the form εit = γ ′ i Ft + τ′ i Gt + ηit, where Ft and Gt are zero mean processes, and ηit are usually independent over i and t.
- 28. 6 J. Bai and C. Kao 1.3 Assumptions Our analysis is based on the following assumptions. ASSUMPTION 1.1. As n → ∞, 1 n n i=1 λiλ′ i → Σλ, a r × r positive definite matrix. ASSUMPTION 1.2. Let wit = (F′ t , uit, ε′ it)′. For each i, wit = Πi(L)vit = ∞ j=0 Πij vit−j , ∞ j=0 jaΠij ∞, |Πi(1)| = 0, for some a 1, where vit is i.i.d. over t. In addition, Evit = 0, E(vitv′ it ) = Σv 0, and Evit8 ⩽ M ∞. ASSUMPTION 1.3. Ft and uit are independent; uit are independent across i. Under Assumption 1.2, a multivariate invariance principle for wit holds, i.e., the partial sum process 1 √ T [T r] t=1 wit satisfies: (1.3) 1 √ T [T r] t=1 wit ⇒ B(Ωi) as T → ∞ for all i, where Bi = BF Bui Bεi . The long-run covariance matrix of {wit} is given by Ωi = ∞ j=−∞ E(wi0w′ ij ) = Πi(1)ΣvΠi(1)′ = Σi + Γi + Γ ′ i = ΩFi ΩFui ΩFεi ΩuFi Ωui Ωuεi ΩεFi Ωεui Ωεi , where (1.4) Γi = ∞ j=1 E(wi0w′ ij ) = ΓFi ΓFui ΓFεi ΓuFi Γui Γuεi ΓεFi Γεui Γεi
- 29. On the Estimation and Inference of a Panel Cointegration Model 7 and Σi = E(wi0w′ i0) = ΣFi ΣFui ΣFεi ΣuFi Σui Σuεi ΣεFi Σεui Σεi are partitioned conformably with wit. We denote Ω = lim n→∞ 1 n n i=1 Ωi, Γ = lim n→∞ 1 n n i=1 Γi, and Σ = lim n→∞ 1 n n i=1 Σi. ASSUMPTION 1.4. Ωεi is nonsingular, i.e., {xit}, are not cointegrated. Define Ωbi = ΩFi ΩFui ΩuFi Ωui , Ωbεi = ΩFεi Ωuεi and Ωb.εi = Ωbi − ΩbεiΩ−1 εi Ωεbi. Then, Bi can be rewritten as (1.5) Bi = Bbi Bεi = Ω 1/2 b.εi ΩbεiΩ −1/2 εi 0 Ω 1/2 εi Vbi Wi , where Bbi = BF Bui , Vbi = VF Vui , and Vbi Wi = BM(I)
- 30. 8 J. Bai and C. Kao is a standardized Brownian motion. Define the one-sided long-run covari- ance Δi = Σi + Γi = ∞ j=0 E(wi0w′ ij ) with Δi = Δbi Δbεi Δεbi Δεi . REMARK 1.2. (1) Assumption 1.1 is a standard assumption in factor models (e.g., Bai and Ng, 2002, 2004) to ensure the factor structure is identifiable. We only consider nonrandom factor loadings for simplicity. Our results still hold when the λ′ is are random, provided they are indepen- dent of the factors and idiosyncratic errors, and Eλi4 ⩽ M. (2) Assumption 1.2 assumes that the random factors, Ft, and idiosyn- cratic shocks (uit, ε′ it) are stationary linear processes. Note that Ft and εit are allowed to be correlated. In particular, εit may have a factor structure as in Remark 1.1. (3) Assumption of independence made in Assumption 1.3 between Ft and uit can be relaxed following Bai and Ng (2002). Nevertheless, inde- pendence is not a restricted assumption since cross-sectional correlations in the regression errors eit are taken into account by the common factors. 1.4 OLS Let us first study the limiting distribution of the OLS estimator for Equa- tion (1.1). The OLS estimator of β is (1.6) β̂OLS = n i=1 T t=1 yit(xit − x̄i)′ n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 . THEOREM 1.1. Under Assumptions 1.1–1.4, we have √ nT (β̂OLS − β) − √ nδnT ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 (λ′ iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1 ε ,
- 31. On the Estimation and Inference of a Panel Cointegration Model 9 as (n, T → ∞) with n T → 0 where δnT = 1 n n i=1 λ′ i ΩFεiΩ 1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω −1/2 εi + ΔFεi + ΩuεiΩ 1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω −1/2 εi + Δuεi × 1 n n i=1 1 T 2 T t=1 (xit − xit)(xit − x̄i)′ −1 , Wi = Wi − Wi and Ωε = limn→∞ 1 n n i=1 Ωεi. REMARK 1.3. It is also possible to construct the bias-corrected OLS by using the averages of the long run covariances. Note E[δnT ] ≃ 1 n n i=1 λ′ i − 1 2 ΩFεi + ΔFεi − 1 2 Ωuεi + Δuεi 1 6 Ωε −1 = 1 n n i=1 − 1 2 (λ′ iΩFεi + Ωuεi) + λ′ iΔFεi + Δuεi 1 6 Ωε −1 = 1 n n i=1 − 1 2 λ′ iΩFεi + 1 n n i=1 Ωuεi + 1 n n i=1 λ′ iΔFεi + 1 n n i=1 Δuεi 1 6 Ωε −1 . It can be shown by a central limit theorem that √ n δnT − E[δnT ] ⇒ N(0, B) for some B. Therefore, √ nT (β̂OLS − β) − √ nE[δnT ] = √ nT (β̂OLS − β) − √ nδnT + √ n δnT − E[δnT ] ⇒ N(0, A) for some A.
- 32. 10 J. Bai and C. Kao 1.5 FM estimator Next we examine the limiting distribution of the FM estimator, β̂FM. The FM estimator was suggested by Phillips and Hansen (1990) in a different context (nonpanel data). The FM estimator is constructed by making cor- rections for endogeneity and serial correlation to the OLS estimator β̂OLS in (1.6). The endogeneity correction is achieved by modifying the variable yit, in (1.1) with the transformation y+ it = yit − (λ′ iΩFεi + Ωuεi)Ω−1 εi Δxit. The serial correlation correction term has the form Δ+ bεi = Δbεi − ΩbεiΩ−1 εi Δεi = Δ+ Fεi Δ+ uεi . Therefore, the infeasible FM estimator is β̃FM = n i=1 T t=1 y+ it (xit − x̄i)′ − T λ′ iΔ+ Fεi + Δ+ uεi (1.7) × n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 . Now, we state the limiting distribution of β̃FM. THEOREM 1.2. Let Assumptions 1.1–1.4 hold. Then as (n, T → ∞) with n T → 0 √ nT (β̃FM − β) ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 (λ′ iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1 ε . REMARK 1.4. The asymptotic distribution in Theorem 1.2 is reduced to √ nT (β̃FM − β) ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 λ2 i ΩF.ε + Ωu.ε if the long-run covariances are the same across the cross-sectional unit i and r = 1.
- 33. On the Estimation and Inference of a Panel Cointegration Model 11 1.6 Feasible FM In this section we investigate the limiting distribution of the feasible FM. We will show that the limiting distribution of the feasible FM is not af- fected when λi, Ωεi, Ωεbi, Ωεi, and Δεbi are estimated. To estimate λi, we use the method of principal components used in Stock and Watson (2002). Let λ = (λ1, λ2, . . . , λn)′ and F = (F1, F2, . . . , FT )′. The method of principal components of λ and F minimizes V (r) = 1 nT n i=1 T t=1 (êit − λ′ iFt)2 , where êit = yit − α̂i − β̂xit = (yit − ȳi) − β̂(xit − x̄i), with a consistent estimator β̂. Concentrating out λ and using the nor- malization that F′F/T = Ir, the optimization problem is identical to maximizing tr(F′(ZZ′)F), where Z = (ê1, ê2, . . . , ên) is T × n with êi = (êi1, êi2, . . . , êiT )′. The estimated factor matrix, denoted by F, a T × r matrix, is √ T times eigenvectors corresponding to the r largest eigenvalues of the T × T matrix ZZ′, and λ̂′ = ( F′ F)−1 F′ Z = F′Z T is the corresponding matrix of the estimated factor loadings. It is known that the solution to the above minimization problem is not unique, i.e., λi and Ft are not directly identifiable but they are identifiable up to a trans- formation H. For our setup, knowing Hλi is as good as knowing λi. For example in (1.7) using λ′ iΔ+ Fεi will give the same information as using λ′ iH′H′−1Δ+ Fεi since Δ+ Fεi is also identifiable up to a transformation, i.e., λ′ iH′H′−1Δ+ Fεi = λ′ iΔ+ Fεi. Therefore, when assessing the properties of the estimates we only need to consider the differences in the space spanned by, say, between λ̂i and λi. Define the feasible FM, β̂FM, with λ̂i, Ft, Σi, and Ωi in place of λi, Ft, Σi, and Ωi, β̂FM = n i=1 T t=1 ŷ+ it (xit − x̄i)′ − T λ̂′ i Δ+ Fεi + Δ+ uεi × n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 ,
- 34. 12 J. Bai and C. Kao where ŷ+ it = yit − (λ̂′ i ΩFεi + Ωuεi) Ω−1 εi Δxit and Δ+ Fεi and Δ+ uεi are defined similarly. Assume that Ωi = Ω for all i. Let e+ it = eit − (λ′ iΩFε + Ωuε)Ω−1 ε Δxit, Δ+ bεn = 1 n n i=1 Δ+ bεi, and Δ+ bεn = 1 n n i=1 Δ+ bεi. Then √ nT (β̂FM − β̃FM) = 1 √ nT n i=1 T t=1 ê+ it (xit − x̄i)′ − T Δ+ bεn − T t=1 e+ it (xit − x̄i)′ − T Δ+ bεn × 1 nT 2 n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 = 1 √ nT n i=1 T t=1 ê+ it − e+ it (xit − x̄i)′ − T Δ+ bεn − Δ+ bεn × 1 nT 2 n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 . Before we prove Theorem 1.3 we need the following lemmas. LEMMA 1.1. Under Assumptions 1.1–1.4 √ n( Δ+ bεn − Δ+ bεn) = op(1). Lemma 1.1 can be proved similarly by following Phillips and Moon (1999) and Moon and Perron (2004). LEMMA 1.2. Suppose Assumptions 1.1–1.4 hold. There exists an H with rank r such that as (n, T → ∞)
- 35. On the Estimation and Inference of a Panel Cointegration Model 13 (i) 1 n n i=1 λ̂i − Hλi2 = Op 1 δ2 nT . (ii) Let ci (i = 1, 2, . . . , n) be a sequence of random matrices such that ci = Op(1) and 1 n n i=1 ci2 = Op(1) then 1 n n i=1 (λ̂i − Hλi)′ ci = Op 1 δ2 nT , where δnT = min{ √ n, √ T }. Bai and Ng (2002) showed that for a known êit that the average squared deviations between λ̂i and Hλi vanish as n and T both tend to infinity and the rate of convergence is the minimum of n and T . Lemma 1.2 can be proved similarly by following Bai and Ng (2002) that parameter estimation uncertainty for β has no impact on the null limit distribution of λ̂i. LEMMA 1.3. Under Assumptions 1.1–1.4 1 √ nT n i=1 T t=1 ê+ it − e+ it (xit − x̄i)′ = op(1) as (n, T → ∞) and √ n T → 0. Then we have the following theorem: THEOREM 1.3. Under Assumptions 1.1–1.4 and (n, T → ∞) and √ n T → 0 √ nT (β̂FM − β̃FM) = op(1). In the literature, the FM-type estimators usually were computed with a two-step procedure, by assuming an initial consistent estimate of β, say β̂OLS. Then, one constructs estimates of the long-run covariance matrix, Ω(1), and loading, λ̂ (1) i . The 2S-FM, denoted β̂ (1) 2S is obtained using Ω(1) and λ̂(1) i : β̂ (1) 2S = n i=1 T t=1 ŷ +(1) it (xit − x̄i)′ − T λ̂′(1) i Δ +(1) Fεi + Δ +(1) uεi (1.8) × n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 .
- 36. 14 J. Bai and C. Kao In this paper, we propose a CUP-FM estimator. The CUP-FM is con- structed by estimating parameters and long-run covariance matrix and loading recursively. Thus β̂FM, Ω and λ̂i are estimated repeatedly, until convergence is reached. In Section 1.8, we find the CUP-FM has a supe- rior small sample properties as compared with the 2S-FM, i.e., CUP-FM has smaller bias than the common 2S-FM estimator. The CUP-FM is de- fined as β̂CUP = n i=1 T t=1 ŷ+ it (β̂CUP)(xit − x̄i)′ − T λ̂′ i(β̂CUP) Δ+ Fεi(β̂CUP) + Δ+ uεi(β̂CUP) (1.9) × n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ −1 . REMARK 1.5. (1) In this paper, we assume the number of factors, r, is known. Bai and Ng (2002) showed that the number of factors can be found by minimizing the following: IC(k) = log V (k) + k n + T nT log nT n + T . (2) Once the estimates of wit, wit = ( F′ t , ûit, Δx′ it)′, were estimated, we used (1.10) Σ = 1 nT n i=1 T t=1 wit w′ it to estimate Σ, where ûit = êit − λ̂′ i Ft. Ω was estimated by Ω = 1 n N i=1 1 T T t=1 wit w′ it (1.11) + 1 T l τ=1 ̟τl T t=τ+1 ( wit w′ it−τ + wit−τ w′ it) , where ̟τl is a weight function or a kernel. Using Phillips and Moon (1999), Σi and Ωi can be shown to be consistent for Σi and Ωi.
- 37. On the Estimation and Inference of a Panel Cointegration Model 15 1.7 Hypothesis testing We now consider a linear hypothesis that involves the elements of the co- efficient vector β. We show that hypothesis tests constructed using the FM estimator have asymptotic chi-squared distributions. The null hypothesis has the form: (1.12) H0: Rβ = r, where r is a m×1 known vector and R is a known m×k matrix describing the restrictions. A natural test statistic of the Wald test using β̂FM is W = 1 6 nT 2 (Rβ̂FM − r)′ 6 Ω−1 ε lim n→∞ 1 n n i=1 (λ̂′ i ΩF.εiλ̂i Ωεi (1.13) + Ωu.εi Ωεi) Ω−1 ε −1 (Rβ̂FM − r). It is clear that W converges in distribution to a chi-squared random variable with k degrees of freedom, χ2 k , as (n, T → ∞) under the null hypothesis. Hence, we establish the following theorem: THEOREM 1.4. If Assumptions 1.1–1.4 hold, then under the null hypoth- esis (1.12), with (n, T → ∞), W ⇒ χ2 k , REMARK 1.6. (1) One common application of Theorem 1.4 is the single- coefficient test: one of the coefficient is zero; βj = β0, R = [ 0 0 · · · 1 0 · · · 0 ] and r = 0. We can construct a t-statistic (1.14) tj = √ nT (β̂jFM − β0) sj , where s2 j = 6 Ω−1 ε lim n→∞ 1 n n i=1 (λ̂′ i ΩF.εiλ̂i Ωεi + Ωu.εi Ωεi) Ω−1 ε jj , the jth diagonal element of 6 Ω−1 ε lim n→∞ 1 n n i=1 (λ̂′ i ΩF.εiλ̂i Ωεi + Ωu.εi Ωεi) Ω−1 ε . It follows that (1.15) tj ⇒ N(0, 1).
- 38. 16 J. Bai and C. Kao (2) General nonlinear parameter restriction such as H0: h(β) = 0, where h(·), is k∗ × 1 vector of smooth functions such that ∂h ∂β′ has full rank k∗ can be conducted in a similar fashion as in Theorem 1.4. Thus, the Wald test has the following form Wh = nT 2 h(β̂FM)′ V −1 h h(β̂FM), where V −1 h = ∂h(β̂FM) ∂β′ V −1 β ∂h(β̂′ FM) ∂β and (1.16) Vβ = 6 Ω−1 ε lim n→∞ 1 n n i=1 (λ̂′ i ΩF.εiλ̂ Ωεii + Ωu.εi Ωεi) Ω−1 ε . It follows that Wh ⇒ χ2 k∗ as (n, T → ∞). 1.8 Monte Carlo simulations In this section, we conduct Monte Carlo experiments to assess the finite sample properties of OLS and FM estimators. The simulations were per- formed by a Sun SparcServer 1000 and an Ultra Enterprise 3000. GAUSS 3.2.31 and COINT 2.0 were used to perform the simulations. Random numbers for error terms, (F∗ t , u∗ it, ε∗ it) were generated by the GAUSS pro- cedure RNDNS. At each replication, we generated an n(T + 1000) length of random numbers and then split it into n series so that each series had the same mean and variance. The first 1,000 observations were discarded for each series. {F∗ t }, {u∗ it} and {ε∗ it} were constructed with F∗ t = 0, u∗ i0 = 0 and ε∗ i0 = 0. To compare the performance of the OLS and FM estimators we con- ducted Monte Carlo experiments based on a design which is similar to Kao and Chiang (2000) yit = αi + βxit + eit, eit = λ′ iFt + uit, and xit = xit−1 + εit
- 39. On the Estimation and Inference of a Panel Cointegration Model 17 for i = 1, . . . , n, t = 1, . . . , T , where (1.17) Ft uit εit = F∗ t u∗ it ε∗ it + 0 0 0 0 0.3 −0.4 θ31 θ32 0.6 F∗ t−1 u∗ it−1 ε∗ it−1 with F∗ t u∗ it ε∗ it i.i.d. ∼ N 0 0 0 , 1 σ12 σ13 σ21 1 σ23 σ31 σ32 1 . For this experiment, we have a single factor (r = 1) and λi are gen- erated from i.i.d. N(μλ, 1). We let μλ = 0.1. We generated αi from a uniform distribution, U[0, 10], and set β = 2. From Theorems 1.1–1.3 we know that the asymptotic results depend upon variances and covariances of Ft, uit and εit. Here we set σ12 = 0. The design in (1.17) is a good one since the endogeneity of the system is controlled by only four parameters, θ31, θ32, σ31 and σ32. We choose θ31 = 0.8, θ32 = 0.4, σ31 = −0.8 and θ32 = 0.4. The estimate of the long-run covariance matrix in (1.11) was obtained by using the procedure KERNEL in COINT 2.0 with a Bartlett window. The lag truncation number was set arbitrarily at five. Results with other kernels, such as Parzen and quadratic spectral kernels, are not reported, because no essential differences were found for most cases. Next, we recorded the results from our Monte Carlo experiments that examined the finite-sample properties of (a) the OLS estimator, β̂OLS in (1.6), (b) the 2S-FM estimator, β̂2S, in (1.8), (c) the two-step naive FM es- timator, β̂b FM, proposed by Kao and Chiang (2000) and Phillips and Moon (1999), (d) the CUP-FM estimator β̂CUP, in (1.9) and (e) the CUP naive FM estimator β̂d FM which is similar to the two-step naive FM except the iteration goes beyond two steps. The naive FM estimators are obtained assuming the cross-sectional independence. The maximum number of the iteration for CUP-FM estimators is set to 20. The results we report are based on 1,000 replications and are summarized in Tables 1.1–1.4. All the FM estimators were obtained by using a Bartlett window of lag length five as in (1.11). Table 1.1 reports the Monte Carlo means and standard deviations (in parentheses) of (β̂OLS−β), (β̂2S−β), (β̂b FM−β), (β̂CUP−β), and (β̂d FM−β) for sample sizes T = n = (20, 40, 60). The biases of the OLS estimator, β̂OLS, decrease at a rate of T . For example, with σλ = 1 and σF = 1, the bias at T = 20 is −0.045, at T = 40 is −0.024, and at T = 60 is −0.015. Also, the biases stay the same for different values of σλ and σF . While we expected the OLS estimator to be biased, we expected FM estimators to produce better estimates. However, it is noticeable that the
- 40. 18 J. Bai and C. Kao Table 1.1. Means biases and standard deviation of OLS and FM estimators σλ = 1 σλ = √ 10 σλ = √ 0.5 OLS FMa FMb FMc FMd OLS FMa FMb FMc FMd OLS FMa FMb FMc FMd σF = 1 T = 20 −0.045 −0.025 −0.029 −0.001 −0.006 −0.046 −0.025 −0.029 −0.001 −0.006 −0.045 −0.025 −0.029 −0.001 −0.006 (0.029) (0.028) (0.029) (0.034) (0.030) (0.059) (0.054) (0.059) (0.076) (0.060) (0.026) (0.026) (0.026) (0.030) (0.028) T = 40 −0.024 −0.008 −0.011 −0.002 −0.005 −0.024 −0.009 −0.012 −0.003 −0.005 −0.024 −0.008 −0.011 −0.002 −0.005 (0.010) (0.010) (0.010) (0.010) (0.010) (0.020) (0.019) (0.019) (0.021) (0.018) (0.009) (0.009) (0.009) (0.009) (0.009) T = 60 −0.015 −0.004 −0.005 −0.001 −0.003 −0.015 −0.003 −0.005 −0.001 −0.002 −0.015 −0.004 −0.005 −0.001 −0.003 (0.006) (0.005) (0.005) (0.005) (0.005) (0.011) (0.010) (0.010) (0.011) (0.010) (0.005) (0.005) (0.005) (0.005) (0.004) σF = √ 10 T = 20 −0.054 −0.022 −0.036 0.011 −0.005 −0.057 −0.024 −0.038 0.013 −0.003 −0.054 −0.022 −0.036 0.011 −0.005 (0.061) (0.054) (0.061) (0.078) (0.062) (0.176) (0.156) (0.177) (0.228) (0.177) (0.046) (0.042) (0.047) (0.059) (0.047) T = 40 −0.028 −0.007 −0.015 0.001 −0.007 −0.030 −0.009 −0.017 −0.001 −0.009 −0.028 −0.007 −0.014 0.001 −0.007 (0.021) (0.019) (0.019) (0.021) (0.019) (0.059) (0.054) (0.057) (0.061) (0.053) (0.016) (0.015) (0.015) (0.016) (0.015) T = 60 −0.018 −0.002 −0.007 0.001 −0.004 −0.017 −0.001 −0.006 0.002 −0.003 −0.018 −0.002 −0.007 0.001 −0.004 (0.011) (0.011) (0.011) (0.011) (0.010) (0.032) (0.029) (0.030) (0.031) (0.029) (0.009) (0.008) (0.008) (0.009) (0.008) σF = √ 0.5 T = 20 −0.044 −0.025 −0.028 −0.003 −0.006 −0.045 −0.026 −0.028 −0.002 −0.006 −0.044 −0.026 −0.028 −0.003 −0.006 (0.026) (0.026) (0.026) (0.030) (0.028) (0.045) (0.041) (0.045) (0.056) (0.046) (0.024) (0.025) (0.025) (0.028) (0.026) T = 40 −0.023 −0.009 −0.010 −0.003 −0.004 −0.023 −0.009 −0.011 −0.003 −0.005 −0.023 −0.009 −0.010 −0.003 −0.004 (0.009) (0.009) (0.009) (0.009) (0.009) (0.016) (0.015) (0.015) (0.016) (0.014) (0.009) (0.009) (0.009) (0.009) (0.008) T = 60 −0.015 −0.004 −0.005 −0.001 −0.003 −0.015 −0.004 −0.005 −0.001 −0.002 −0.015 −0.004 −0.005 −0.001 −0.003 (0.005) (0.005) (0.005) (0.005) (0.005) (0.009) (0.008) (0.008) (0.008) (0.008) (0.005) (0.005) (0.005) (0.005) (0.004) Note: (a) FMa is the 2S-FM, FMb is the naive 2S-FM, FMc is the CUP-FM and FMd is the naive CUP-FM. (b) μλ = 0.1, σ31 = −0.8, σ21 = −0.4, θ31 = 0.8, and θ21 = 0.4.
- 41. On the Estimation and Inference of a Panel Cointegration Model 19 Table 1.2. Means biases and standard deviation of OLS and FM estimators for different n and T (n, T ) OLS FMa FMb FMc FMd (20, 20) −0.045 −0.019 −0.022 −0.001 −0.006 (0.029) (0.028) (0.029) (0.034) (0.030) (20, 40) −0.024 −0.006 −0.009 −0.001 −0.004 (0.014) (0.014) (0.013) (0.014) (0.013) (20, 60) −0.017 −0.004 −0.006 −0.001 −0.003 (0.010) (0.009) (0.009) (0.009) (0.009) (20, 120) −0.008 −0.001 −0.002 −0.000 −0.001 (0.005) (0.004) (0.005) (0.004) (0.004) (40, 20) −0.044 −0.018 −0.021 −0.002 −0.006 (0.021) (0.019) (0.019) (0.023) (0.021) (40, 40) −0.024 −0.007 −0.009 −0.002 −0.004 (0.010) (0.010) (0.010) (0.010) (0.010) (40, 60) −0.015 −0.003 −0.005 −0.001 −0.002 (0.007) (0.007) (0.007) (0.007) (0.007) (40, 120) −0.008 −0.001 −0.002 −0.001 −0.001 (0.003) (0.003) (0.003) (0.003) (0.003) (60, 20) −0.044 −0.018 −0.022 −0.002 −0.007 (0.017) (0.016) (0.016) (0.019) (0.017) (60, 40) −0.022 −0.006 −0.008 −0.002 −0.004 (0.009) (0.008) (0.008) (0.008) (0.008) (60, 60) −0.015 −0.003 −0.005 −0.001 −0.003 (0.006) (0.005) (0.005) (0.005) (0.005) (60, 120) −0.008 −0.001 −0.002 −0.001 −0.001 (0.003) (0.002) (0.002) (0.002) (0.002) (120, 20) −0.044 −0.018 −0.022 −0.002 −0.007 (0.013) (0.011) (0.012) (0.013) (0.012) (120, 40) −0.022 −0.006 −0.008 −0.002 −0.004 (0.006) (0.006) (0.006) (0.006) (0.006) (120, 60) −0.015 −0.003 −0.005 −0.001 −0.003 (0.004) (0.004) (0.004) (0.004) (0.004) (120, 120) −0.008 −0.001 −0.002 −0.001 −0.002 (0.002) (0.002) (0.002) (0.002) (0.002) Note: μλ = 0.1, σ31 = −0.8, σ21 = −0.4, θ31 = 0.8, and θ21 = 0.4. 2S-FM estimator still has a downward bias for all values of σλ and σF , though the biases are smaller. In general, the 2S-FM estimator presents the same degree of difficulty with bias as does the OLS estimator. This is probably due to the failure of the nonparametric correction procedure. In contrast, the results in Table 1.1 show that the CUP-FM, is distinctly superior to the OLS and 2S-FM estimators for all cases in terms of the mean biases. Clearly, the CUP-FM outperforms both the OLS and 2S-FM estimators.
- 42. 20 J. Bai and C. Kao Table 1.3. Means biases and standard deviation of t-statistics σλ = 1 σλ = √ 10 σλ = √ 0.5 OLS FMa FMb FMc FMd OLS FMa FMb FMc FMd OLS FMa FMb FMc FMd σF = 1 T = 20 −1.994 −1.155 −1.518 −0.056 −0.285 −0.929 −0.546 −0.813 −0.006 −0.122 −2.248 −1.299 −1.656 −0.071 −0.321 (1.205) (1.267) (1.484) (1.283) (1.341) (1.149) (1.059) (1.495) (1.205) (1.254) (1.219) (1.325) (1.490) (1.314) (1.366) T = 40 −2.915 −0.941 −1.363 −0.227 −0.559 −1.355 −0.465 −0.766 −0.128 −0.326 −3.288 −1.056 −1.474 −0.250 −0.602 (1.202) (1.101) (1.248) (1.054) (1.141) (1.127) (0.913) (1.207) (0.912) (1.049) (1.221) (1.151) (1.253) (1.096) (1.159) T = 60 −3.465 −0.709 −1.158 −0.195 −0.574 −1.552 −0.308 −0.568 −0.074 −0.261 −3.926 −0.814 −1.280 −0.229 −0.643 (1.227) (1.041) (1.177) (0.996) (1.100) (1.146) (0.868) (1.113) (0.851) (1.016) (1.244) (1.091) (1.189) (1.042) (1.118) σF = √ 10 T = 20 −1.078 −0.484 −0.984 0.180 −0.096 −0.373 −0.154 −0.350 0.085 −0.006 −1.427 −0.639 1.257 0.229 −0.138 (1.147) (1.063) (1.501) (1.220) (1.271) (1.119) (0.987) (1.508) (1.194) (1.223) (1.163) (1.117) (1.498) (1.244) (1.301) T = 40 −1.575 −0.355 −0.963 0.042 −0.407 −0.561 −0.152 −0.397 −0.014 −0.190 −2.082 −0.453 −1.211 0.073 −0.506 (1.131) (0.917) (1.214) (0.926) (1.063) (1.097) (0.844) (1.179) (0.871) (1.008) (1.154) (0.967) (1.232) (0.967) (1.096) T = 60 −1.809 −0.155 −0.776 0.111 −0.390 −0.588 −0.041 −0.247 0.049 −0.111 −2.424 −0.212 −1.019 0.143 −0.523 (1.158) (0.879) (1.131) (0.867) (1.035) (1.108) (0.812) (1.078) (0.811) (0.983) (1.192) (0.929) (1.162) (0.909) (1.069) σF = √ 0.5 T = 20 −2.196 −1.319 −1.606 −0.137 −0.327 −1.203 −0.734 −1.008 −0.054 −0.176 −2.367 −1.421 −1.692 −0.157 −0.351 (1.219) (1.325) (1.488) (1.307) (1.362) (1.164) (1.112) (1.488) (1.217) (1.273) (1.231) (1.363) (1.492) (1.324) (1.379) T = 40 −3.214 −1.093 −1.415 −0.311 −0.576 −1.752 −0.619 −0.922 −0.188 −0.385 −3.462 −1.176 −1.481 −0.333 −0.599 (1.226) (1.057) (1.155) (1.104) (1.169) (1.148) (0.962) (1.222) (0.944) (1.087) (1.236) (1.185) (1.255) (1.121) (1.168) T = 60 −3.839 −0.868 −1.217 −0.296 −0.602 −2.037 −0.446 −0.712 −0.139 −0.331 −4.149 −0.949 −1.295 −0.329 −0.646 (1.239) (1.088) (1.183) (1.037) (1.112) (1.169) (0.908) (1.131) (0.881) (1.038) (1.249) (1.123) (1.190) (1.069) (1.122) Note: (a) FMa is the 2S-FM, FMb is the naive 2S-FM, FMc is the CUP-FM and FMd is the naive CUP-FM. (b) μλ = 0.1, σ31 = −0.8, σ21 = −0.4, θ31 = 0.8, and θ21 = 0.4.
- 43. On the Estimation and Inference of a Panel Cointegration Model 21 Table 1.4. Means biases and standard deviation of t-statistics for different n and T (n, T ) OLS FMa FMb FMc FMd (20, 20) −1.994 −0.738 −1.032 −0.056 −0.286 (1.205) (1.098) (1.291) (1.283) (1.341) (20, 40) −2.051 −0.465 −0.725 −0.105 −0.332 (1.179) (0.999) (1.126) (1.046) (1.114) (20, 60) −2.129 −0.404 −0.684 −0.162 −0.421 (1.221) (0.963) (1.278) (0.983) (1.111) (20, 120) −2.001 −0.213 −0.456 −0.095 −0.327 (1.222) (0.923) (1.083) (0.931) (1.072) (40, 20) −2.759 −1.017 −1.404 −0.103 −0.402 (1.237) (1.116) (1.291) (1.235) (1.307) (40, 40) −2.915 −0.699 −1.075 −0.227 −0.559 (1.202) (1.004) (1.145) (1.054) (1.141) (40, 60) −2.859 −0.486 −0.835 −0.173 −0.493 (1.278) (0.998) (1.171) (1.014) (1.154) (40, 120) −2.829 −0.336 −0.642 −0.181 −0.472 (1.209) (0.892) (1.047) (0.899) (1.037) (60, 20) −3.403 −1.252 −1.740 −0.152 −0.534 (1.215) (1.145) (1.279) (1.289) (1.328) (60, 40) −3.496 −0.807 −1.238 −0.255 −0.635 (1.247) (1.016) (1.165) (1.053) (1.155) (60, 60) −3.465 −0.573 −0.987 −0.195 −0.574 (1.227) (0.974) (1.111) (0.996) (1.100) (60, 120) −3.515 −0.435 −0.819 −0.243 −0.609 (1.197) (0.908) (1.031) (0.913) (1.020) (120, 20) −4.829 −1.758 −2.450 −0.221 −0.760 (1.345) (1.162) (1.327) (1.223) (1.308) (120, 40) −4.862 −1.080 −1.679 −0.307 −0.831 (1.254) (1.022) (1.159) (1.059) (1.143) (120, 60) −4.901 −0.852 −1.419 −0.329 −0.846 (1.239) (0.964) (1.097) (0.978) (1.077) (120, 120) −5.016 −0.622 −1.203 −0.352 −0.908 (1.248) (0.922) (1.059) (0.927) (1.048) Note: μλ = 0.1, σ31 = −0.8, σ21 = −0.4, θ31 = 0.8, and θ21 = 0.4. It is important to know the effects of the variations in panel dimen- sions on the results, since the actual panel data have a wide variety of cross-section and time-series dimensions. Table 1.2 considers 16 different combinations for n and T , each ranging from 20 to 120 with σ31 = −0.8, σ21 = −0.4, θ31 = 0.8, and θ21 = 0.4. First, we notice that the cross- section dimension has no significant effect on the biases of all estimators. From this it seems that in practice the T dimension must exceed the n
- 44. 22 J. Bai and C. Kao dimension, especially for the OLS and 2S-FM estimators, in order to get a good approximation of the limiting distributions of the estimators. For example, for OLS estimator in Table 1.2, the reported bias, −0.008, is sub- stantially less for (T = 120, n = 40) than it is for either (T = 40, n = 40) (the bias is −0.024), or (T = 40, n = 120) (the bias is −0.022). The re- sults in Table 1.2 again confirm the superiority of the CUP-FM. Monte Carlo means and standard deviations of the t-statistic, tβ=β0 , are given in Table 1.3. Here, the OLS t-statistic is the conventional t-statistic as printed by standard statistical packages. With all values of σλ and σF with the exception σλ = √ 10, the CUP-FM t-statistic is well approxi- mated by a standard N(0, 1) suggested from the asymptotic results. The CUP-FM t-statistic is much closer to the standard normal density than the OLS t-statistic and the 2S-FM t-statistic. The 2S-FM t-statistic is not well approximated by a standard N(0, 1). Table 1.4 shows that both the OLS t-statistic and the FM t-statistics become more negatively biased as the dimension of cross-section n in- creases. The heavily negative biases of the 2S-FM t-statistic in Tables 1.3–1.4 again indicate the poor performance of the 2S-FM estimator. For the CUP-FM, the biases decrease rapidly and the standard errors converge to 1.0 as T increases. It is known that when the length of time series is short the estimate Ω in (1.11) may be sensitive to the length of the bandwidth. In Tables 1.2 and 1.4, we first investigate the sensitivity of the FM estimators with re- spect to the choice of length of the bandwidth. We extend the experiments by changing the lag length from 5 to other values for a Barlett window. Overall, the results (not reported here) show that changing the lag length from 5 to other values does not lead to substantial changes in biases for the FM estimators and their t-statistics. 1.9 Conclusion A factor approach to panel models with cross-sectional dependence is use- ful when both the time series and cross-sectional dimensions are large. This approach also provides significant reduction in the number of vari- ables that may cause the cross-sectional dependence in panel data. In this paper, we study the estimation and inference of a panel cointe- gration model with cross-sectional dependence. The paper contributes to the growing literature on panel data with cross-sectional dependence by (i) discussing limiting distributions for the OLS and FM estimators, (ii) suggesting a CUP-FM estimator and (iii) investigating the finite sam- ple proprieties of the OLS, CUP-FM and 2S-FM estimators. It is found that the 2S-FM and OLS estimators have a nonnegligible bias in finite samples, and that the CUP-FM estimator improves over the other two estimators.
- 45. On the Estimation and Inference of a Panel Cointegration Model 23 Acknowledgements We thank Badi Baltagi, Yu-Pin Hu, Giovanni Urga, Kamhon Kan, Chung- Ming Kuan, Hashem Pesaran, Lorenzo Trapani and Yongcheol Shin for helpful comments. We also thank seminar participants at Academia Sinica, National Taiwan University, Syracuse University, Workshop on Recent Developments in the Econometrics of Panel Data in London, March 2004 and the European Meeting of the Econometric Society in Madrid, August 2004 for helpful comments and suggestions. Jushan Bai acknowledges fi- nancial support from the NSF (grant SES-0137084). Appendix A1 Let BnT = n i=1 T t=1 (xit − x̄i)(xit − x̄i)′ . Note √ nT (β̂OLS − β) = √ n 1 n n i=1 1 T T t=1 eit(xit − x̄i)′ 1 n 1 T 2 BnT −1 = √ n 1 n n i=1 ζ1iT 1 n n i=1 ζ2iT −1 = √ nξ1nT [ξ2nT ]−1 , where x̄i = 1 T T t=1 xit, ȳi = 1 T T t=1 yit, ζ1iT = 1 T T t=1 eit(xit − x̄i)′, ζ2iT = 1 T 2 T t=1(xit − x̄i)(xit − x̄i)′, ξ1nT = 1 n n i=1 ζ1iT , and ξ2nT = 1 n n i=1 ζ2iT . Before going into the next theorem, we need to consider some preliminary results. Define Ωε = limn→∞ 1 n n i=1 Ωεi and θn = 1 n n i=1 λ′ i ΩF.εiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + ΔFεi + Ωu.εiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + Δuεi . If Assumptions 1.1–1.4 hold, then
- 46. 24 J. Bai and C. Kao LEMMA A1.1. (a) As (n, T → ∞), 1 n 1 T 2 BnT p → 1 6 Ωε. (b) As (n, T → ∞) with n T → 0, √ n 1 n 1 T n i=1 T t=1 eit(xit − x̄i)′ − θn ⇒ N 0, 1 6 lim n→∞ 1 n n i=1 {λ′ iΩF.εiλiΩεi + Ωu.εiΩεi} . PROOF. (a) and (b) can be shown easily by following Theorem 8 in Phillips and Moon (1999). A1.1 Proof of Theorem 1.1 PROOF. Recall that √ nT (β̂OLS − β) − √ n 1 n n i=1 λ′ i ΩFεiΩ −1/2 εi × 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + ΔFεi + ΩεuiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + Δεui 1 n 1 T 2 BnT −1 = √ n 1 n n i=1 ζ1iT − λ′ i ΩFεiΩ −1/2 εi × 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + ΔFεi − ΩuεiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + Δuεi × 1 n n i=1 ζ2iT −1 = √ n 1 n n i=1 ζ∗ 1iT 1 n n i=1 ζ2iT −1 = √ nξ∗ 1nT [ξ2nT ]−1 ,
- 47. On the Estimation and Inference of a Panel Cointegration Model 25 where ζ∗ 1iT = ζ1iT − λ′ i ΩFεiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + ΔFεi − ΩuεiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + Δuεi and ξ∗ 1nT = 1 n n i=1 ζ∗ 1iT . First, we note from Lemma A1.1(b) that √ nξ∗ 1nT ⇒ N 0, 1 6 lim n→∞ 1 n n i=1 {λ′ iΩF.εiλiΩεi + Ωu.εiΩεi} as (n, T → ∞) and n T → 0. Using the Slutsky theorem and (a) from Lemma A1.1, we obtain √ nξ∗ 1nT [ξ2nT ]−1 ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 (λ′ iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1 ε . Hence, √ nT (β̂OLS − β) − √ nδnT (A1.1) ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 (λ′ iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1 ε , proving the theorem, where δnT = 1 n n i=1 λ′ i ΩFεiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + ΔFεi + ΩuεiΩ −1/2 εi 1 T T t=1 x′ it(xit − x̄i) Ω 1/2 εi + Δuεi × 1 n 1 T 2 BnT −1 . Therefore, we established Theorem 1.1.
- 48. 26 J. Bai and C. Kao A1.2 Proof of Theorem 1.2 PROOF. Let F+ it = Ft − ΩFεiΩ−1 εi εit, and u+ it = uit − ΩuεiΩ−1 εi εit. The FM estimator of β can be rewritten as follows β̃FM = n i=1 T t=1 y+ it (xit − x̄i)′ − T λ′ iΔ+ Fεi + Δ+ uεi B−1 nT = β + n i=1 T t=1 λ′ iF+ it + u+ it (xit − x̄i)′ (A1.2) − T λ′ iΔ+ Fεi + Δ+ uεi B−1 nT . First, we rescale (β̃FM − β) by √ nT √ nT (β̃FM − β) = √ n 1 n n i=1 1 T T t=1 λ′ iF+ it + u+ it (xit − x̄i)′ − λ′ iΔ+ Fεi − Δ+ uεi 1 n 1 T 2 BnT −1 = √ n 1 n n i=1 ζ∗∗ 1iT 1 n n i=1 ζ2iT −1 (A1.3) = √ nξ∗∗ 1nT [ξ2nT ]−1 , where ζ∗∗ 1iT = 1 T T t=1[(λ′ iF+ it + û+ it )(xit − x̄i)′ − λ′ iΔ+ Fεi − Δ+ uεi], and ξ∗∗ 1nT = 1 n n i=1 ζ∗∗ 1iT . Modifying Theorem 11 in Phillips and Moon (1999) and Kao and Chi- ang (2000) we can show that as (n, T → ∞) with n T → 0 √ n 1 n 1 T n i=1 T t=1 λ′ iF+ it (xit − x̄i)′ − λ′ iΔ+ Fεi ⇒ N 0, 1 6 lim n→∞ 1 n n i=1 λ′ iΩF.εiλiΩεi
- 49. On the Estimation and Inference of a Panel Cointegration Model 27 and √ n 1 n 1 T n i=1 T t=1 û+ it (xit − x̄i)′ − Δ+ uεi ⇒ N 0, 1 6 lim n→∞ 1 n n i=1 Ωu.εiΩεi and combing this with Assumption 1.4 that Ft and uit are independent and Lemma A1.1(a) yields √ nT (β̃FM − β) ⇒ N 0, 6Ω−1 ε lim n→∞ 1 n n i=1 (λ′ iΩF.εiλiΩεi + Ωu.εiΩεi) Ω−1 ε as required. A1.3 Proof of Lemma 1.3 PROOF. We note that λi is estimating Hλi, and ΩFε is estimating H−1′ ΩFε. Thus λ̂′ i ΩFε is estimating λ′ iΩFε, which is the object of in- terest. For the purpose of notational simplicity, we shall assume H being a r × r identify matrix in our proof below. From ê+ it = eit − (λ̂′ i ΩFε + Ωuε) Ω−1 ε Δxit and e+ it = eit − (λ′ iΩFε + Ωuε)Ω−1 ε Δxit, ê+ it − e+ it = − (λ̂′ i ΩFε + Ωuε) Ω−1 ε − (λ′ iΩFε + Ωuε)Ω−1 ε Δxit = − λ̂′ i ΩFε Ω−1 ε − λ′ iΩFεΩ−1 ε + Ωuε Ω−1 ε − ΩuεΩ−1 ε Δxit . Then, 1 √ n 1 T n i=1 T t=1 Ωuε Ω−1 ε − ΩuεΩ−1 ε Δxit(xit − x̄i)′ = Ωuε Ω−1 ε − ΩuεΩ−1 ε 1 √ n 1 T n i=1 T t=1 Δxit(xit − x̄i)′ = op(1)Op(1) = op(1)
- 50. 28 J. Bai and C. Kao because Ωuε Ω−1 ε − ΩuεΩ−1 ε = op(1) and 1 √ n 1 T n i=1 T t=1 Δxit(xit − x̄i)′ = Op(1). Thus 1 √ nT n i=1 T t=1 ê+ it − e+ it (xit − x̄i)′ = 1 √ n n i=1 1 T T t=1 (λ′ iΩFε + Ωuε)Ω−1 ε − (λ̂′ i ΩFε + Ωuε) Ω−1 ε Δxit (xit − x̄i)′ = 1 √ n 1 T n i=1 T t=1 λ′ iΩFεΩ−1 ε − λ̂′ i ΩFε Ω−1 ε Δxit(xit − x̄i)′ + 1 √ n 1 T n i=1 T t=1 ΩuεΩ−1 ε − Ωuε Ω−1 ε Δxit(xit − x̄i)′ = 1 √ n 1 T n i=1 T t=1 λ′ iΩFεΩ−1 ε − λ̂′ i ΩFε Ω−1 ε Δxit(xit − x̄i)′ + op(1). The remainder of the proof needs to show that 1 √ n 1 T n i=1 T t=1 λ′ iΩFεΩ−1 ε − λ̂′ i ΩFε Ω−1 ε Δxit(xit − x̄i)′ = op(1). We write A for ΩFεΩ−1 ε and A for ΩFε Ω−1 ε respectively and then 1 √ n 1 T n i=1 T t=1 λ′ iΩFεΩ−1 ε − λ̂′ i ΩFε Ω−1 ε Δxit(xit − x̄i)′ = 1 √ n 1 T n i=1 T t=1 (λ′ iA − λ̂′ i A)Δxit(xit − x̄i)′ = 1 √ n 1 T n i=1 T t=1 λ′ i(A − A) + (λ′ i − λ̂′ i) A Δxit(xit − x̄i)′
- 51. Another Random Scribd Document with Unrelated Content
- 55. The Project Gutenberg eBook of Index of the Project Gutenberg Works of Pelham Grenville Wodehouse
- 56. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Index of the Project Gutenberg Works of Pelham Grenville Wodehouse Author: P. G. Wodehouse Editor: David Widger Release date: December 21, 2018 [eBook #58508] Most recently updated: July 8, 2019 Language: English Credits: Produced by David Widger *** START OF THE PROJECT GUTENBERG EBOOK INDEX OF THE PROJECT GUTENBERG WORKS OF PELHAM GRENVILLE WODEHOUSE ***
- 57. INDEX OF THE PROJECT GUTENBERG WORKS OF PELHAM GRENVILLE WODEHOUSE Compiled by David Widger
- 59. CONTENTS Click on the ## before each title to view a linked table of contents for that volume. Click on the title itself to open the original online file. ## PICCADILLY JIM ## PSMITH, JOURNALIST ## INDISCRETIONS OF ARCHIE ## LOVE AMONG THE CHICKENS ## THE INTRUSION OF JIMMY ## PSMITH IN THE CITY ## THE MAN UPSTAIRS ## THE HEAD OF KAY'S ## THE COMING OF BILL
- 60. ## THE WHITE FEATHER ## THE PRINCE AND BETTY ## TALES OF ST. AUSTIN'S ## THE POTHUNTERS ## A PREFECT'S UNCLE ## CLICKING OF CUTHBERT ## HOW CLARENCE SAVED ENGLAND ## NOT GEORGE WASHINGTON ## MIKE ## THE ADVENTURES OF SALLY ## THE MAN WITH TWO LEFT FEET ## MY MAN JEEVES ## THE POLITENESS OF PRINCES ## A WODEHOUSE MISCELLANY ## A MAN OF MEANS ## MIKE AND PSMITH
- 61. ## LOVE AMONG THE CHICKENS ## JILL THE RECKLESS ## THE GIRL ON THE BOAT EBOOKS WITHOUT LINKED CONTENTS: SOMETHING NEW A DAMSEL IN DISTRESS THE LITTLE NUGGET UNEASY MONEY THREE MEN AND A MAID THE LITTLE WARRIOR THE GOLD BAT WILLIAM TELL TOLD AGAIN THE GEM COLLECTOR RIGHT HO, JEEVES
- 62. TABLES OF CONTENTS OF VOLUMES PICCADILLY JIM
- 63. By Pelham Grenville Wodehouse CONTENTS CHAPTER I A RED-HAIRED GIRL CHAPTER II THE EXILED FAN CHAPTER III FAMILY JARS CHAPTER IV JIMMY'S DISTURBING NEWS CHAPTER V THE MORNING AFTER CHAPTER VI JIMMY ABANDONS PICCADILLY CHAPTER VII ON THE BOAT-DECK CHAPTER VIII PAINFUL SCENE IN A CAFE CHAPTER IX MRS. PETT IS SHOCKED CHAPTER X INSTRUCTION IN DEPORTMENT CHAPTER XI JIMMY DECIDES TO BE HIMSELF CHAPTER XII JIMMY CATCHES THE BOSS'S EYE CHAPTER XIII SLIGHT COMPLICATIONS CHAPTER XIV LORD WISBEACH CHAPTER XV A LITTLE BUSINESS CHAT CHAPTER XVI MRS. PETT TAKES PRECAUTIONS CHAPTER XVII MISS TRIMBLE, DETECTIVE CHAPTER XVIII THE VOICE PROM THE PAST CHAPTER XIX BETWEEN FATHER AND SON CHAPTER XX CELESTINE IMPARTS INFORMATION CHAPTER XXI CHICAGO ED.
- 64. CHAPTER XXII IN THE LIBRARY CHAPTER XXIII STIRRING TIMES FOR THE PETTS CHAPTER XXIV SENSATIONAL TURNING OF A WORM CHAPTER XXV NEARLY EVERYBODY HAPPY CHAPTER XXVI EVERYBODY HAPPY PSMITH, JOURNALIST
- 65. By Pelham Grenville Wodehouse CONTENTS PREFACE CHAPTER I COSY MOMENTS CHAPTER II BILLY WINDSOR CHAPTER III AT THE GARDENIA CHAPTER IV BAT JARVIS CHAPTER V PLANNING IMPROVEMENTS CHAPTER VI THE TENEMENTS CHAPTER VII VISITORS AT THE OFFICE CHAPTER VIII THE HONEYED WORD CHAPTER IX FULL STEAM AHEAD CHAPTER X GOING SOME CHAPTER XI THE MAN AT THE ASTOR CHAPTER XII A RED TAXIMETER CHAPTER XIII REVIEWING THE SITUATION CHAPTER XIV THE HIGHFIELD CHAPTER XV AN ADDITION TO THE STAFF CHAPTER XVI THE FIRST BATTLE CHAPTER XVII GUERILLA WARFARE CHAPTER XVIII AN EPISODE BY THE WAY CHAPTER XIX IN PLEASANT STREET CHAPTER XX CORNERED
- 66. CHAPTER XXI THE BATTLE OF PLEASANT STREET CHAPTER XXII CONCERNING MR. WARING CHAPTER XXIII REDUCTIONS IN THE STAFF CHAPTER XXIV A GATHERING OF CAT-SPECIALISTS CHAPTER XXV TRAPPED CHAPTER XXVI A FRIEND IN NEED CHAPTER XXVII PSMITH CONCLUDES HIS RIDE CHAPTER XXVIII STANDING ROOM ONLY CHAPTER XXIX THE KNOCK-OUT FOR MR. WARING CONCLUSION INDISCRETIONS OF ARCHIE
- 67. By P. G. Wodehouse CONTENTS CHAPTER I. DISTRESSING SCENE CHAPTER II. A SHOCK FOR MR BREWSTER CHAPTER III. MR BREWSTER DELIVERS SENTENCE CHAPTER IV. WORK WANTED CHAPTER V. STRANGE EXPERIENCES OF AN ARTIST'S MODEL CHAPTER VI. THE BOMB CHAPTER VII. MR ROSCOE SHERRIFF HAS AN IDEA CHAPTER VIII. A DISTURBED NIGHT FOR DEAR OLD SQUIFFY CHAPTER IX. A LETTER FROM PARKER CHAPTER X. DOING FATHER A BIT OF GOOD CHAPTER XI. SALVATORE CHOOSES THE WRONG MOMENT CHAPTER XII. BRIGHT EYES—AND A FLY CHAPTER XIII. RALLYING ROUND PERCY CHAPTER XIV. THE SAD CASE OF LOONEY BIDDLE CHAPTER XV. SUMMER STORMS CHAPTER XVI. ARCHIE ACCEPTS A SITUATION CHAPTER XVII. BROTHER BILL'S ROMANCE CHAPTER XVIII. THE SAUSAGE CHAPPIE CHAPTER XIX. REGGIE COMES TO LIFE CHAPTER XX. THE-SAUSAGE-CHAPPIE-CLICKS
- 68. CHAPTER XXI. THE GROWING BOY CHAPTER XXII. WASHY STEPS INTO THE HALL OF FAME CHAPTER XXIII. MOTHER'S KNEE CHAPTER XXIV. THE MELTING OF MR CONNOLLY CHAPTER XXV. THE WIGMORE VENUS CHAPTER XXVI. A TALE OF A GRANDFATHER LOVE AMONG THE CHICKENS
- 69. By P. G. Wodehouse CONTENTS I A LETTER WITH A POSTSCRIPT II MR. AND MRS. S. F. UKRIDGE III WATERLOO STATION, SOME FELLOW-TRAVELLERS, AND A GIRL WITH BROWN HAIR IV THE ARRIVAL V BUCKLING TO VI MR. GARNET'S NARRATIVE—HAS TO DO WITH A REUNION VII THE ENTENTE CORDIALE IS SEALED VIII A LITTLE DINNER AT UKRIDGE'S IX DIES IRAE X I ENLIST THE SERVICES OF A MINION XI THE BRAVE PRESERVER XII SOME EMOTIONS AND YELLOW LUPIN XIII TEA AND TENNIS XIV A COUNCIL OF WAR XV THE ARRIVAL OF NEMESIS XVI A CHANCE MEETING XVII OF A SENTIMENTAL NATURE XVIII UKRIDGE GIVES ME ADVICE XIX ASKING PAPA
- 70. XX SCIENTIFIC GOLF XXI THE CALM BEFORE THE STORM XXII THE STORM BREAKS XXIII AFTER THE STORM THE INTRUSION OF JIMMY
- 71. By P.G. Wodehouse CONTENTS CHAPTER I JIMMY MAKES A BET CHAPTER II PYRAMUS AND THISBE CHAPTER III MR. McEACHERN CHAPTER IV MOLLY CHAPTER V A THIEF IN THE NIGHT CHAPTER VI AN EXHIBITION PERFORMANCE CHAPTER VII GETTING ACQUAINTED CHAPTER VIII AT DREEVER CHAPTER IX FRIENDS, NEW AND OLD CHAPTER X JIMMY ADOPTS A LAME DOG CHAPTER XI AT THE TURN OF THE ROAD CHAPTER XII MAKING A START CHAPTER XIII SPIKE'S VIEWS CHAPTER XIV CHECK AND A COUNTER MOVE CHAPTER XV MR. MCEACHERN INTERVENES CHAPTER XVI A MARRIAGE ARRANGED CHAPTER XVII JIMMY REMEMBERS SOMETHING CHAPTER XVIII THE LOCHINVAR METHOD CHAPTER XIX ON THE LAKE CHAPTER XX A LESSON IN PICQUET CHAPTER XXI LOATHSOME GIFTS
- 72. CHAPTER XXII TWO OF A TRADE DISAGREE CHAPTER XXIII FAMILY JARS CHAPTER XXIV THE TREASURE SEEKER CHAPTER XXV EXPLANATIONS CHAPTER XXVI STIRRING TIMES FOR SIR THOMAS CHAPTER XXVII A DECLARATION OF INDEPENDENCE CHAPTER XXVIII SPENNIE'S HOUR OF CLEAR VISION CHAPTER XXIX THE LAST ROUND CHAPTER XXX CONCLUSION PSMITH IN THE CITY
- 73. By P. G. Wodehouse CONTENTS 1. Mr Bickersdyke Walks behind the Bowler's Arm 2. Mike Hears Bad News 3. The New Era Begins 4. First Steps in a Business Career 5. The Other Man 6. Psmith Explains 7. Going into Winter Quarters 8. The Friendly Native 9. The Haunting of Mr Bickersdyke 10. Mr Bickersdyke Addresses His Constituents 11. Misunderstood 12. In a Nutshell 13. Mike is Moved On 14. Mr Waller Appears in a New Light 15. Stirring Times on the Common 16. Further Developments 17. Sunday Supper 18. Psmith Makes a Discovery 19. The Illness of Edward 20. Concerning a Cheque 21. Psmith Makes Inquiries
- 74. 22. And Take Steps 23. Mr Bickersdyke Makes a Concession 24. The Spirit of Unrest 25. At the Telephone 26. Breaking The News 27. At Lord's 28. Psmith Arranges his Future 29. And Mike's 30. The Last Sad Farewells THE MAN UPSTAIRS AND OTHER STORIES
- 75. By P. G. Wodehouse CONTENTS THE MAN UPSTAIRS SOMETHING TO WORRY ABOUT DEEP WATERS WHEN DOCTORS DISAGREE BY ADVICE OF COUNSEL ROUGH-HEW THEM HOW WE WILL THE MAN WHO DISLIKED CATS RUTH IN EXILE ARCHIBALD'S BENEFIT THE MAN, THE MAID, AND THE MIASMA THE GOOD ANGEL POTS O'MONEY OUT OF SCHOOL THREE FROM DUNSTERVILLE THE TUPPENNY MILLIONAIRE AHEAD OF SCHEDULE SIR AGRAVAINE THE GOAL-KEEPER AND THE PLUTOCRAT IN ALCALA
- 76. THE HEAD OF KAY'S
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