Multi-dimensional Panel Data Modelling in the Presence of Cross Sectional Error Dependence

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Multi-dimensional Panel Data Modelling in the Presence of Cross Sectional Error Dependence

George Kapetanios King’s College, London

Camilla Mastromarco University of Salento Laura Serlenga

University of Bari

Yongcheol Shin University of York February 12, 2017

Abstract

Given the growing availability of the big dataset which contain information on multiple dimensions and following the recent research trend on the multidimensional modelling, we develop the 3D panel data models with three-way error components that allow for strong cross-section dependence (CSD) thorough unobserved heterogeneous global factors, and propose the consistent estimation procedure. We also discuss the extent of CSD in 3D models and provide a diagnostic test for cross-section dependence. We provide the extensions to unbalanced panels and 4D models.

The validity of the proposed approach is con…rmed by the Monte Carlo simulation results. We also demonstrate the empirical usefulness through the application to the 3D panel gravity model of the intra-EU trade ‡ows.

Address for correspondence: Y. Shin, A/EC/119 Department of Economics and Re- lated Studies, University of York, Heslington, York, YO10 5DD, United Kingdom. Email:

yongcheol.shin@york.ac.uk. Tel: +44 (0)1904 323757. We are grateful for the insightful comments of the editor, Laszlo Matyas. This work has beneted greatly from the stimulating discussion of delegates at the PANDA Conference at University of York, July 2016 as well as the many detailed comments raised by seminar participants at Universities of Sogang and York. The usual disclaimer applies.

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1 Introduction

Given the growing availability of the big dataset which contain information on multiple dimensions, the recent literature on the panel data has focused more on extending the two-way error components models to the multidimensional set- ting. Balazsi, Matyas and Wansbeek (2015, BMW hereafter) introduce the appropriate within estimators for the most frequently used three-dimensional (3D)

…xed e¤ects panel data models, and Balazsi, Baltagi, Matyas and Pus (2016, BBMP hereafter) consider the random e¤ects approach and propose a sequence of optimal GLS estimators. This multi-dimensional approach is expected to become an essential tool for the analysis of complex interconnectedness of the big dataset, and it can be not only applied to the number of bilateral (origin- destination) ‡ows such as trade, FDI, capital or migration ‡ows (e.g. Feenstra, 2004; Bertoli and Fernandez-Huertas Moraga, 2013; Gunnella et al., 2015), but also to a variety of matched dataset which may link the employer-the employee and the pupils-teachers (e.g. Abowd et al., 1999; Kramarz et al., 2008).

However, there has been no study attempting to address an important issue of explicitly controlling cross-sectional error dependence in 3D or higher- dimensional panel data, even though the cross-section dependence (CSD) seems pervasive in 2D panels because it seems rare that the cross-section covariance of the errors is zero (e.g. Pesaran, 2015). Recently, there has been much progress in modelling CSD in 2D panels by two main approaches, the factor-based approach (e.g. Pesaran, 2006; Bai, 2009) and the spatial econometrics techniques (e.g. Behrens et al., 2012; Mastromarco et al., 2016b). Chudik et al. (2011) show that the factor-based models exhibit the strong CSD whilst the spatial- based models can deal with weak CSD only. See also Bailey, Kapetanios and Pesaran (2016) for more general discussions.

Chapter 3 by Le Gallo and Pirotte (2016) reviews the current state-of-art in the analysis of multi-dimensional nested spatial panels, highlighting a range of issues related to the speci…cation, estimation, testing procedures and predic- tions. Chapter 10 by Baltagi, Egger and Erhardt (2016) provides a survey of empirical issues in the analysis of gravity-model estimation of international trade

‡ows, proceeds with the modelling of the multidimensional stochastic structure, focusing on …xed-e¤ects estimation, and describes how the spatial autocorrela- tion and spillovers can be introduced into such models. Chapter 12 by Baltagi and Bresson (2016) surveys hedonic housing models and discrete choice models using multi-dimensional panels, also focussing on the spatial econometrics approach.

Following this research trend, we develop the 3D panel data models with strong CSD. In particular, we generalise the multi-dimensional error components speci…cation by modelling residual CSD via unobserved heterogeneous global factors. The multidimensional country-time …xed (CTFE) and random e¤ects (CTRE) estimators proposed by BMW and BBMP fail to remove heterogenous global factors, suggesting that they are biased in the presence of the nonzero correlation between the regressors and unobserved global factors. In this regard, we develop the two-step consistent estimation procedure. First,

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we follow Pesaran (2006) and augment the 3D model with the cross-section averages of dependent variable and regressors over double cross-section units, which are shown to provide valid proxies for unobserved heterogenous global factors. Next, we apply the 3D-within transformation to the augmented speci-

…cation and obtain consistent estimators, called the 3D-PCCE estimator. Our approach is the …rst attempt to accommodate strong CSD in multi-dimensional panels, and expected to make timely contribution to the growing literature.

We discuss the extent of CSD within the 3D panel data models under the three di¤erent error components speci…cations respectively with the CTFE, with the two-way heterogeneous factor, and with both components. We also distinguish between three types of CSD under the hierarchical multi-factor error components speci…cation recently advanced by Kapetanios and Shin (2017). First, the global factor tends to display strong CSD as it in‡uences the (ij) pairwise interactions for i = 1; :::; N1 and j = 1; :::; N2 (of N1N2 dimension). Next, the local factors show semi-strong or semi-weak CSD, as they in‡uence origin and destination countries separately (each of N1 or N2 dimension). Finally, idiosyncratic errors are characterised with the weak or no CSD.

We then develop a diagnostic test for the null hypothesis of (pairwise) residual cross-section independence or weak dependence in the 3D panels, which is a modi…ed counterpart of an existing CD test in the 2D panels proposed by Pe- saran (2015). (CSD exponent) Furthermore, we provide a couple of extensions into unbalanced panels and 4D or higher dimensional models.

We have conducted the Monte Carlo studies to investigate the small sample properties of the 3D-PCCE estimators relative to the CTFE estimator. We …nd strong evidence that the 3D-PCCE estimators perform well when the 3D panel data is subject to the strong CSD through heterogeneous global factors. On the contrary, the CTFE estimator tends to display severe biases and size distortions.

We apply our proposed 3D PCCE estimation techniques, together with the two-way …xed e¤ects and the CTFE estimators, to the dataset over the period 1960-2008 (49 years) for 91 country-pairs amongst 14 EU countries. Based on the CD test results, estimates of CSD exponent, and the predicted signs and statistical signi…cance of the coe¢ cients, we come to a conclusion that the 3D PCCE estimation results are mostly satisfactory and reliable. In particular, when we explicitly control for strong CSD in the 3D panels, we …nd that the trade e¤ect of currency union is rather modest. This evidence provides strong support for the thesis that the trade increase within the Euro area may re‡ect a continuation of a long-run historical trend linked to the broader set of EU’s economic integration policies.

This Chapter proceeds in 7 Sections. Section 16.2 introduces 3D models with three-way error components that allow for strong cross-section dependence, and develops the consistent estimation procedure. Section 16.3 discusses the nature of CSD in 3D models and provide a diagnostic test for cross-section dependence.

Section 16.4 presents the extension to 4D models. Section 16.5 discusses the Monte Carlo simulation results. The empirical results for the gravity model of EU exports ‡ows are presented in Section 16.6, while Section 16.7 concludes.

Throughout the chapter we adopt the following standard notations. IN is

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an N N identity matrix, JN the N N identity matrix of ones, and N the N 1 vector of ones, respectively. MA projects the N N matrix A into its null-space, i.e., M_A = I_N A(A⁰A) ¹A⁰. Finally, y_:jt = N₁¹PN1

i=1y_ijt, y_i:t= N₂¹PN2

j=1y_ijt and y_ij:= T ¹PT

t=1y_ijt denote the average of y over the index i, j and t, respectively, with the de…nition extending to other quantities such as y::t, y:j:, yi:: and y:::.

2 3D Models with Cross-sectional Error Depen- dence

Following Balazsi, Matyas and Wansbeek (2015, BMW hereafter) and Balazsi, Baltagi, Matyas and Pus (2016, BBMP hereafter), we consider the following three-dimensional country-time …xed e¤ects panel data model:

yijt= ⁰xijt+ ⁰sit+ ⁰djt+ ⁰qt+'⁰zij+uijt; i = 1; :::; N1; j = 1; :::; N2; t = 1; :::; T;

(1) with the error components:

u_ijt = _ij+ v_it+ _jt+ "_ijt (2) where y_ijt is the dependent variable observed across three indices (e.g. the import of country j from country i at period t), x_ijt, s_it, d_jt, q_t, z_ij are the k_x 1, k_s 1, k_d 1, k_q 1, k_z 1 vectors of covariates covering all possible measurements observed across three indices, and , , , , '; are the associ- ated vectors of the parameters. The multiple error components in (2) contain bilateral pair-…xed e¤ects _ij as well as origin and destination country-time

…xed e¤ects (CTFE), vitand _jt, respectively.¹

To remove all unobserved …xed e¤ects, _ij, vit and _jt, BMW derive the following 3D within transformation:²

~

yijt= yijt yij: y:jt yi:t+ y::t+ y:j:+ yi:: y::: (3) Applying the 3D within transformation to (1), we can estimate consistently only from the following regression:

~

y_ijt= ⁰x~_ijt+ ~"_ijt; i = 1; :::; N₁; j = 1; :::; N₂; t = 1; :::; T; (4) where ~x_ijt = x_ijt x_ij: x_:jt x_i:t+ x_::t+ x_:j:+ x_i:: x_::: and similarly for

~"_ijt. We write (4) compactly as

Y~ij= ~Xij + ~Eij (5)

1Notice that the error component speci…cation (2) is proposed by Baltagi et al. (2003).

2Baltagi et al. (2015) also derive the same projection by applying Davis’ (2002) Lemma twice (see Corollary 1).

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where

Y~_ij

T 1

= 2 64

~ y_ij1

...

~ y_ijT

3 75 ; ~X_ij

T kx

= 2 64

~ x⁰_ij1

...

~ x⁰_ijT

3 75 ; ~E_ij

T 1

= 2 64

~

"_ij1 ...

~

"_ijT 3 75 :

The 3D-within estimator of is obtained by

^W =

N1

P

i=1 N2

P

j=1

X~⁰_ijX~_ij

! 1 N1

P

i=1 N2

P

j=1

X~⁰_ijY~_ij

!

: (6)

Then, it follows that, as (N1; N2; T ) ! 1 (see also BBMP), pN₁N₂T ^

W

a N 0

@0; ²_" lim

(N1;N2;T )!1

1 N₁N₂T

N1

P

i=1 N2

P

j=1

X~⁰_ijX~_ij

! 11 A :

By construction, the within transformation in (3) wipes out all other covariates, x_it, x_jt, x_t, and x_ij in (1). But, we may be interested in uncovering the e¤ects of those covariates (e.g. the impacts of measured trade costs in the structural gravity model). In order to recover those coe¢ cients, it would be worthwhile to develop an extension of the Hausman-Taylor (1981) estimation, which has been popular in the two-way panel data models even in the presence of cross sectionally correlated errors (e.g. Serlenga and Shin, 2007). Chapter 3 by Balazsi, Bun, Chan and Harris (2016) develops an extended Hausman-Taylor estimator for multi-dimensional panel data models.

BMW also show that the CTFE error components in (2) nests the number of special cases by applying suitable restrictions to (2).³ Notice, however, that the model (1) with (2) does not address an important issue of cross-sectional error dependence. In the presence of such cross-section dependence (CSD), the 3D-within estimator would be likely to be biased. In this regard, we consider a couple of alternative 3-way error components speci…cations that can accommodate CSD, and develop the appropriate estimation techniques.

Given that v_it and _jt are supposed to measure the (local) origin and destination country-time …xed e¤ects, it is natural to add the global factor _t to (2):

uijt = _ij+ vit+ _jt+ t+ "ijt

But, the 3D-within transformation, (3) removes _t together with _ij, v_it and

jt, because _tis shown to be proportional to

N1

P

i=1

v_itor

N2

P

j=1 jt(see also footnote 6 below).

To introduce strong CSD explicitly in the 3D model, (1), we …rst consider the following error components speci…cation:

uijt = _ij+ ij t+ "ijt: (7)

3Baltagi et al. (2003), Baldwin and Taglioni (2006), and Baier and Bergstrand (2007) consider several forms of …xed e¤ects such as uijt= i+ _j+ t+ "ijt, uijt= _ij+ t+ "ijt, uijt= _jt+ "ijt, uijt= vit+ "ijt, and uijt= vit+ _jt+ "ijt:

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This is similar to the two-way heterogeneous factor model considered by Serlenga and Shin (2007). We follow Pesaran (2006) and apply the cross-section averages of (1) and (7) over i and j to obtain:

y::t = 1 N₁

N1

X

i=1

1 N₂

N2

X

j=1

0xijt+ ⁰sit+ ⁰djt+ ⁰qt+ '⁰zij+ _ij+ ij t+ "ijt

= ⁰x::t+ ⁰s:t+ ⁰d:t+ ⁰qt+ '⁰z:: + _::+ :: t+ "::t (8) where s_:t= N₁¹PN1

i=1s_it, d_:t= N₂¹PN2

j=1d_jt, z_::= (N₁N₂) ¹PN1

i=1

PN2

j=1z_ij,

::= (N1N2) ¹PN1

i=1

PN2

j=1 ij and ::= (N1N2) ¹PN1

i=1

PN2

j=1 ij. Hence,

t= 1

::

y_::t ⁰x::t+ ⁰s:t+ ⁰d:t+ ⁰qt+ '⁰z::+ _::+ "_::t

Using these results we can augment the model (1) with the cross-section averages as follows:

yijt= ⁰xijt+ ⁰sit+ ⁰djt+ ⁰_ijft+ ij+ _ij+ "_ijt; (9) where

0ij= _0ij; ⁰_1ij; ⁰_2ij; ⁰_3ij; ⁰_4ij = ^ij

::

; ^ij

0 ::

; ^ij ⁰

::

; ^ij

0 ::

; 1 ^ij

::

0

ft= y::t; x⁰_::t; s⁰_:t; d⁰_:t; q⁰_t ⁰ (10)

ij = '⁰zij ij ::

'⁰z::; _ij = _ij ^ij ^::

::

; "_ijt= "_ijt ^ij

::

"_::t: We write (9) compactly as

Yij = Xij + Si + Dj + F _ij+ ij T + _{ij T}+ E_ij (11)

= Wij + H _ij+ E_ij; i = 1; :::; N1; j = 1; :::; N2

where

Y_ij

T 1

= 2 64

yij1

... yijT

3 75 ; Xij

T kx

= 2 64

x⁰_ij1 ... x⁰_ijT

3 75 ; Si

T ks

= 2 64

s⁰_i1 ... s⁰_iT

3 75 ;

D_j

T kd

= 2 64

d⁰_j1 ... d⁰_jT

3 75 ; F

T kf

= 2 64

f₁⁰ ... f_T⁰

3 75 ; Eij

T 1

= 2 64

"_ij1 ...

"_ijT 3 75 ;

Wij = Xij; S_i; D_j , = ⁰ ⁰ ⁰ ⁰, _ij = ⁰_ij; ij+ _ij ⁰ and H = [F; T]. Then, we derive the consistent estimator of (called 3D-PCCE) by⁴

^_{P CCE}=

N1

P

i=1 N2

P

j=1

W⁰_ijMHWij

! 1 N1

P

i=1 N2

P

j=1

W⁰_ijMHYij

!

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4 and ' cannot be identi…ed due to the factor approximations and the within transformation.

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where MH = IT H(H⁰H) ¹H⁰. Following Pesaran (2006), it is straightforward to show that as (N1; N2; T ) ! 1, the PCCE estimator, (12) follows the asymptotic normal distribution (see also Kapetanios and Shin, 2017):

pN₁N₂T ^_{P CCE} ^a N (0; ) ;

where the (robust) consistent estimator of is given by

^ = 1

N1N2

S ¹R S ¹;

R = 1

N1N2 1

N1

P

i=1 N2

P

j=1

W⁰_ijMHWij

T ^_ij ^_{M G} ^_ij ^_{M G} ⁰ W_ij⁰ MHWij

T ;

S = 1

N₁N₂

N1

P

i=1 N2

P

j=1

W⁰_ijMHWij

T ; ^_{M G}= 1

N₁N₂

N1

P

i=1 N2

P

j=1

^_ij;

where ^ij is the (ij) pairwise OLS estimator obtained from the individual regression of Yij on (Wij; H) in (11) for i = 1; :::; N1 and j = 1; :::; N2.

Next, we consider the 3D model (1) with the following general error components by combining CTFEs and heterogeneous global factors:

uijt = _ij+ vit+ _jt+ ij t+ "ijt: (13) It is straightforward to show that the 3D-within transformation (3) fails to remove heterogeneous factors _ij _t, because it is easily seen that

~

u_ijt = ~_ij~

t+ ~"_ijt where ~t = t with = T ¹PT

t=1 tand ~ij = ij :j i:+ :: with

:j = N₁¹PN1

i=1 ij and i:= N₂¹PN2

j=1 ij.⁵ It is clear in the presence of the nonzero correlation between xijt and t that the 3D-within estimator of is biased.

We develop the two-step consistent estimation procedure. First, taking the cross-section averages of (1) and (13) over i and j, we have:

y::t= ⁰x::t+ ⁰s:t+ ⁰d:t+ ⁰qt+ '⁰z::+ _::+ v:t+ _:t+ :: t+ "::t (14) where v:t= N₁¹PN1

i=1vit, _:t= N₂¹PN2

j=1 jtand see (8) for other de…nitions.

Hence, we augment the model (1) with the cross-section averages as:

y_ijt= ⁰x_ijt+ ⁰s_it+ ⁰d_jt+ ⁰_ijf_t+ _ij+ _ij+ v_ijt+ _ijt+ "_ijt; (15) where v_ijt= vit ^ijv:t

:: ; _ijt = _jt ^ij ^:t

:: ; and see (11) for other de…nitions.

We rewrite (15) as

y_ijt= ⁰x_ijt+ ⁰s_it+ ⁰d_jt+ ⁰_ijf_t+ _ij+ _ij+ v_it+ _jt+ "_ijt; (16)

5Unless ~ij= 0,~uijt6= ~"^ijt. This holds only if factor loadings, ijare homogeneous for all (i; j)pairs.

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where "_ijt= "ijt ^ij_::"::t ^ijv:t ::

ij :t

:: . Notice that as N1; N2! 1, "ijt!^p

"ijt since v:t !^p 0, _:t !^p 0 and "::t !^p 0. Next, we apply the 3D-within transformation (3) to (16), and obtain:⁶

~

yijt = ⁰~xijt+ ~⁰_ij~ft+ ~"_ijt; (17) where ~_ij = _ij _:j _j:+ _::, ~ft= ft f with f = T ¹PT

t=1ft, and ftis de…ned in (10). Rewriting (17) compactly as

Y~ij= ~Xij + ~F ~_ij+ ~E_ij; i = 1; :::; N1; j = 1; :::; N2 (18) where

Y~ij T 1

= 2 64

~ yij1

...

~ yijT

3 75 ; ~Xij

T kx

= 2 64

~ x⁰_ij1

...

~ x⁰_ijT

3 75 ; F~

T kf

= 2 64

~f₁⁰ ...

~f_T⁰ 3 75 ; ~E_ij

T 1

= 2 64

~"_ij1 ...

~

"_ijT 3 75 :

Then, the 3D-PCCE estimator of is obtained by

^P CCE =

N1

P

i=1 N2

P

j=1

X~⁰_ijMF~X~_ij

! 1 N1

P

i=1 N2

P

j=1

X~⁰_ijMF~Y~_ij

!

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where M_F_~ = I_T F ~~ F⁰F~

1F~⁰ is the T T idempotent matrix. Following Pesaran (2006) and Kapetanios and Shin (2017), it is also straightforward to show that as (N1; N2; T ) ! 1, the PCCE estimator, (19) follows the asymptotic normal distribution:

pN1N2T ^

P CCE

a N (0; ) ;

where the (robust) consistent estimator of is given by

^ = 1

N²S ¹R S ¹;

R = 1

N₁N₂ 1

N1

P

i=1 N2

P

j=1

X~⁰_ijMF~X~ij

T

!

^ij ^

M G ^

ij ^

M G

0 X~⁰_ijMF~X~ij

T

!

;

S = 1

N1N2 N1

P

i=1 N2

P

j=1

X~⁰_ijMF~X~ij

T

!

; ^_{M G}= 1 N1N2

N1

P

i=1 N2

P

j=1

^ij;

6It is clear that ; ; , and ' cannot be identi…ed due to the 3D-within transformation and the factor approximation. De…ne ijt= ⁰_ijft, then it is straightforward to show that

~_ijt= ijt ij:+ :jt+ i:t + ::t+ i::+ :j: :::= _ij _:j _j:+ _:: ⁰ ft f :

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where ^_ij is the (ij) pairwise OLS estimator obtained from the individual regression of ~Yij on X~ij; ~F in (18) for i = 1; :::; N1 and j = 1; :::; N2.

We can extend the proposed approach to the 3D panels with heterogeneous slope parameters:

yijt= ⁰_ijxijt+ ⁰_jsit+ ⁰_idjt+ ⁰_ijqt+'⁰zij+uijt; i = 1; :::; N1; j = 1; :::; N2; t = 1; :::; T:

(20) In this case we can develop the mean group estimators for (2), (7) and (13) in a straightforward manner (e.g. Pesaran, 2006; Kapetanios and Shin, 2017):

^W;M G= 1

N1N2 N1

P

i=1 N2

P

j=1

X~⁰_ijX~_ij ¹ X~⁰_ijY_ij

^_{M GCCE}= 1

N₁N₂

N1

P

i=1 N2

P

j=1

W⁰_ijMHWij

1 W⁰_ijMHYij

^M GCCE= 1

N₁N₂

N1

P

i=1 N2

P

j=1

X~⁰_ijMF~X~_ij ¹ X~⁰_ijMF~Y~_ij

3 Cross-section Dependence (CD) Test

We discuss the extent of cross-section dependence in the 3D panel data models.

Following Pesaran (2015) and Bailey et al. (2016, hereafter BKP), we can show that the extent of CSD is captured by non-zero covariance between u_ijt and u_i0j⁰t for i 6= i⁰ and j 6= j, denoted as ijt;u. Here, the extent of CSD involves both N1and N2, and thus relates to the rate at which_N¹

1N2

N1

P

i=1 N2

P

j=1

ijt;udeclines with the product, N1N2.

First, we consider the 3D model (1) with country-time e¤ects (2). Following BBMP, we make the following random e¤ects assumptions:

ij iid 0; ² ; vit iid 0; ²_v ; _jt iid 0; ² ; "ijt iid 0; ²_"

and _ij, v_it, _jt and "_ijt are pairwise uncorrelated. Rewrite (2) sequentially as uij

T 1

= _{ij T} + vi+ _j+ "ij; i = 1; :::; N1; j = 1; :::; N2; ui

N2T 1

= _i T + N2 vi+ + "i; i = 1; :::; N1;

N1Nu2T 1= _T + V + _N₁ + " (21) where

uij T 1

= 2 64

uij1

... u_ijT

3 75 ; vⁱ

T 1

= 2 64

vi1

... v_iT

3 75 ; j

T 1

= 2 64

j1...

jT

3 75 ; "^ij

T 1

= 2 64

"ij1

...

"_ijT 3 75 ;

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u_i

N2T 1

= 2 64

u_i1 ... uiN2

3 75 ; i

N2 1

= 2 64

..i1

.

iN2

3 75 ;

N2T 1

= 2 64

..1

.

N2

3 75 ; "_i

N2T 1

= 2 64

"_i1 ...

"iN2

3 75 ;

N1Nu2T 1= 2 64

u₁ ... u_N₁

3 75 ;

N1N2 1

= 2 64

..1

.

N1

3 75 ; V

N1N2T 1= 2 64

N2 v₁ ...

N2 v_N₁ 3 75 ; "

N1N2T 1= 2 64

"₁ ...

"_N₁ 3 75

(22) Then, it is easily seen that (see also BBMP):

Cov (u)

N1N2T N1N2T

= IN1N2

2JT +IN1 JN2

2

vIT +JN1

2IN2T + ²_"IN1N2T

(23) Notice that the CTRE model imposes very limited structure of CSD, because for i 6= i⁰ and j 6= j⁰, we have:

E[u_ijtu_ij0t] = ²_v; E[u_ijtu_ij0t] = ² and E[u_ijtu_i0j⁰t] = 0: (24) This suggests that the covariance between uijt and ui⁰jt is common ²_v for all i = 1; :::; N1 while the covariance between uijt and uij⁰t is common ² for all j = 1; :::; N2. Further, it imposes zero covariance between uijt and uij⁰t⁰.

Next, we consider the 3D model with the two-way heterogeneous factor spec- i…cation, (7). In this case it is straightforward to derive:

N1Nu2T 1= _T + _T+ " (25)

where

N1N2 1= 2 64

1

...

N1

3 75 ; ⁱ

N2 1

= 2 64

i1

...

iN2

3 75 ; T

T 1

= 2 64

1

...

T

3 75 ;

and see (22) for other de…nitions. The covariance matrix for u in (25) is:

Cov (u)

N1N2T N1N2T

= IN1N2

2JT + ( ⁰) ²IT + ²_"IN1N2T (26) Thus, the speci…cation (25) can capture CSD by non-zero covariances between uijt and ui⁰j⁰tfor i 6= i and j 6= j by

E[u_ijtu_ij0t] = _ij _ij0 2; E[u_ijtu_i0jt] = _ij _i0j 2; E[u_ijtu_i0j⁰t] = _ij _i0j⁰ 2: (27) Next, we consider the 3D model with more general error components, (13).

Combining the above results, it is straightforward to derive:

N1Nu2T 1= T+ V + N1 + T + ": (28) Thus, the covariance matrix for u in (28) is given by

Cov (u)

N1N2T N1N2T

= I_N₁_N₂ ²J_T + I_N₁ J_N₂ ²_vI_T (29) +JN1

2IN2T + ( ⁰) ²IT + ²_"IN1N2T

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This model can capture CSD by non-zero covariances between uijtand ui⁰j⁰tfor i 6= i⁰ and j 6= j⁰, given by

E[u_ijtu_ij0t] = _ij _ij0 2+ ²_v; E[u_ijtu_i0jt] = _ij _i0j 2+ ²; E[u_ijtu_i0j⁰t] = _ij _i0j⁰ 2: (30) Comparing (24), (27) and (30), we …nd that the CTFE speci…cation in (2) can only accommodate non-zero covariances locally, but it also imposes the same covariance for all i = 1; :::; N₁ and j = 1; :::; N₂, respectively. Such restrictions are too strong to hold in practice.⁷ On the contrary our proposed error components speci…cation (13) can accommodate non-zero covariances both locally and globally.

Notice that vit and _jt are related to the local-time factors. In order to examine whether they exhibit weak or strong CSD, we consider the following heterogeneous local factors speci…cations:

vit= vi t and _jt= _{j t}

where _tand _tare the origin and the destination-speci…c local common factors, respectively. Then, (13) can be replaced by

u_ijt= _ij+ v_{i t}+ _{j t} + "_ijt: (31) This speci…cation implies that the exporter, i, reacts heterogeneously to the common import market conditions, t and the importer, j, reacts heterogeneously to the common export market conditions, _t. Recently, Kapetanios and Shin (2017) propose a more general hierarchical multi-factor error components speci…cation:

uijt= _ij+ vi it+ _{j jt}+ ij t+ "ijt: (32) Within this model, we can distinguish between three types of CSD: (i) the strong global factor, _twhich in‡uences the (ij) pairwise interactions (of N₁N₂ dimension); (ii) the semi-strong local factors, _itand _jt, which in‡uence origin or destination countries separately (each of N₁ or N₂ dimension); and (iii) the weak CSD idiosyncratic errors, "_ijt. We expect that this kind of generalisation would be most natural within the 3D panel data models. Following BBMP, we assume:

ij iid 0; ² ; it iid 0; ² ; _jt iid 0; ² ; t iid 0; ² ; "ijt iid 0; ²_"

and _ij, it, _jt, t and "ijt are mutually independent.⁸ It is clear that the model (32) can capture CSD by non-zero covariances between uijt and ui⁰j⁰tfor i 6= i⁰ and j 6= j, given by

E[u_ijtu_ij0t] = v_i² ²+ _ij _ij0 2; E[u_ijtu_i0jt] = ²_j ² + _ij _i0j 2 7In the two-way error components with individual e¤ects and time e¤ects, the cross-section correlation is the same for all cross-section pairs. Serlenga and Shin (2007) show that such speci…cation would produce very misleading results in the presence of heterogeneous strong CSD in the 2D panel data model.

8This assumption is still more general than the random e¤ects assumptions made in BBMP.

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E[uijtui⁰j⁰t] = ij i⁰j⁰ 2: (33) The covariance structure in (33) is clearly more general than (30).

We now develop a diagnostic test for the null hypothesis of residual cross- section independence and the estimation of the exponent of cross-sectional dependence in the triple-index panel data models. Those are evaluated using the residuals obtained respectively from (5), (11) and (18), which we denote as e_ij = (e_ij1; :::; e_ijT)⁰. In particular, we have e_ij = ~Y_ij X~_ij^

W for the model (5), eij = MHYij MHWij^_{P CCE} for the model (11), and …nally eij= MF~Y~ij MF~X~ij^

P CCE for the model (18).

The proposed cross-section dependence (CD) test is a modi…ed counterpart of an existing CD test proposed by Pesaran (2015). For convenience, we rep- resent eij as the (ij) pair using the single index n = 1; :::; N1N2, and compute the pair-wise residual correlations between n and n⁰ cross-section units by

^_nn0(= _n0n) = e⁰_nen⁰

p(e⁰_nen) (e⁰_n0en⁰); n; n⁰ = 1; :::; N1N2 and n 6= n⁰:

Then, we construct the CD statistic by

CD =

s 2

N1N2(N1N2 1)

N1XN2 1 n=1

NX1N2

n⁰=n+1

pT ^_nn0 (34)

Pesaran derives that the CD test has the limiting N (0; 1) distribution under the null hypothesis of cross-sectional error independence, namely H0: ^_nn0 = 0 for all n; n⁰ = 1; :::; N1N2 and n 6= n⁰. Following BKP, Pesaran further shows that the CD statistic, (34) can also be applicable to testing the null hypothesis of weak cross-sectional error dependence.⁹ As an extension one can also construct hierarchical CD tests based on 2D sub-dataset out of the 3D dataset.

Following BKP, we introduce the exponent of cross-sectional dependence based on the double cross sectional averages de…ned as u::t= (N1N2) ¹PN1

i=1

PN2

j=1uijt. If uijt’s are cross sectionally correlated across (i; j) pairs, V ar (u::t) declines at a rate that is a function of , where is de…ned as

N1Nlim2!1(N1N2) max( u)

and _u is the N₁N₂ N₁N₂ covariance matrix of u_t= (u_11t; :::; u_N₁_N₂_t)⁰ with

max( _u) denoting the largest egeinlaue. Clearly, V ar (u_::t) cannot decline at rate faster than (N1N2) ¹ as well as it cannot decline at rate slower than (N₁N₂) ¹with 0 1. Given that

V ar (u::t) (N1N2) ¹ max( u) ;

9Consider the e¤ects of the jth factor on the ith error, and suppose that the factor loadings take nonzero values for Mj out of the N cross-section units. The degree of cross-sectional dependence due to the jth factor can be measured by j = ln(Mj)= ln(N ), and the overall degree of CSD by = maxj j, and 2 [0; 1], with 1 indicating the highest CSD. Then, the implicit null is given by H₀^w: < ¹₂. See BKP for more details.

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we …nd that de…ned by (N1N2) ¹ max( u) = O N ¹ will provide an upper bound for V ar (u::t). The extent of CSD depends on the nature of factor loadings in the following factor-based errors:

u_ijt= _ij _t+ "_ijt:

If the average of the heterogenous loading parameters, _ij, denoted , is bounded away from zero, the cross-sectional dependence will be strong, in which case, (N₁N₂) ¹ _max( _u) and V ar (u_::t) are both O (1), which yields = 1.

Furthermore, for 1=2 < 1, BKP propose the following bias-adjusted estimator to consistently estimate :

= 1 +1 2

ln ^²_u_::t ln(N1N2)

ln ^² 2 ln(N1N2)

^ cN1N2

2 N1N2ln (N1N2) ^²_u::_t (35) where ^²_u_::t = T ¹ ^T_t=1u²_::t, ^cN1N2 = d²

N1N2 = (N1N2) ¹PN1

i=1

PN2

j=1^²_ij and

^²_ij = T ¹ ^T_t=1^"²_ijt is the ijth diagonal element of the estimated covariance matrix, ^" with ^"ijt = uijt ^_iju::t and ^ij is the OLS estimator from the regression of uijton u::t. BKP also discuss that a suitable estimation of ² can be derived, noticing that is the mean of the population regression coe¢ cient of uijt on ~u::t= u::t=^u::t for units uijt that have at least one non-zero loading, and those units are selected using Holm’s (1979) multiple testing approach.

In the empirical section we apply the above 3D extension of the BKP estimation and testing techniques directly to the residuals e_ijt obtained respectively from (5), (11) and (18). We also evaluate the con…dence band for the estimated CSD exponent by employing the test statistic de…ned in (B47) in BKP’s Supplementary Appendix VI.

4 Extensions

We provide two extensions of the proposed estimation techniques into unbalanced panels and four dimensional (4D) models. Such extensions would be challenging as they involve several layers of factor speci…cations.

4.1 Unbalanced Panels

In practice, we may be faced with the unbalanced panel data. Notice, however, that an issue of unbalanced panels or missing data has been almost neglected even in literature on the 2D panels with unobserved factors or interactive ef- fects. Kapetanios and Pesaran (2005) brie‡y deal with it in their Monte Carlo studies. Bai et al. (2015) investigate the unbalanced 2D panel data model with interactive e¤ects, and propose the functional principal components analysis and the EM algorithm. Via simulation studies they …nd that the EM-type estimators are consistent for both smooth and stochastic factors, though no asymptotic analysis is provided. For the error components model (2), the 3D within

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transformation fails to fully eliminate the …xed e¤ects. BMW thus extend the Wansbeek and Kapteyn (1989) approach and derive the complex within transformation, which is computationally quite demanding as it involves an inversion of N T N T matrices. Thus, we expect that such extension of our proposed 3D PCCE estimation into unbalanced panels would be more challenging.

We now introduce a vector of selection indicators for each pair (i; j), s_ij = (s_ij;1; :::; s_ij;T)⁰, where s_ij;t = 1 if time period t for pair (i; j) can be used in estimation. We only use information on units where a full set of data are observed. Therefore, sij;t= 1 if and only if (xijt; yijt) is fully observed; otherwise, sij;t = 0. Following Wooldridge (2010), we assume that selection is ignorable conditional on xijt; sit; djt; qt; zij; _ij; t :

E yitjx^ijt; sit; djt; qt; zij; _ij; t; si = E yitjx^ijt; sit; djt; qt; zij; _ij; t : Let n = PN1

i=1

PN2

j=1

PT

t=1s_ij;t be the total number of observations. Also de-

…ne nt=PN1

i=1

PN2

j=1sij;t and nij =PT

t=1sij;t as the number of cross-section pairs observed for time period t and the number of time periods observed for pair (i; j). Similarly, de…ne ni = PN2

j=1

PT

t=1sij;t, nj = PN1

i=1

PT t=1sij;t, n_it=PN2

j=1s_ij;t and n_jt =PN1

i=1s_ij;t, respectively. To simplify further analysis we maintain the assumption: minini; minjnj; mintnt; min_(ij)nij ! 1 or

mintnt; min(ij)nij ! 1.¹⁰

Consider the 3D model (1) with the error components speci…cation (7). We multiply the model by the selection indicator to get:

y_ijt^s = ⁰x^s_ijt+ ⁰s^s_it+ ⁰d^s_jt+ ⁰q^s_t+ '⁰z^s_ij+ ^s_ij+ ^s_ij t+ "^s_ijt; (36) where yijt = sij;tyijt, x^s_ijt = sij;txijt, s^s_it= sij;tsit, d^s_jt = sij;tdjt, q^s_t = sij;tqt, z^s_ij = sij;tzij, ^s_ij = sij;t ij, ^s_ij = sij;t ij, and "^s_ijt = sij;t"ijt. Applying the cross-section averages of (36) over i and j, we obtain:

y^s_::t= ⁰x^s_::t+ ⁰s^s_:t+ ⁰d^s_:t+ ⁰qt+ '⁰z^s_::t+ ^s_::t+ ^s_{::t t}+ "^s_::t (37) where y_::t^s = _n¹

t

PN1

i=1

PN2

j=1sij;tyijt =PN1

i=1wity_i:t^s is expressed as a weighted average with wit = nit=nt and y^s_i:t = n_it¹PN2

j=1sij;tyijt.¹¹ Similarly for x^s_::t, z^s_::t, ^s_::t, ^s_::t and "^s_::t. Further, s^s_:t = PN1

i=1w_its_it, d^s_:t = PN2

j=1w_jtd_jt with w_jt= n_jt=n_t, and q^s_t= q_t. As n_t! 1,

z^s_::t= z + op(1) ; ^s_::t= + op(1) ; ^s_::t= + op(1) and "^s_::t= "::t+ op(1) (38) where z = (N₁N₂) ¹PN1

i=1

PN2

j=1z_ij !pE (z_ij), = (N₁N₂) ¹PN1

i=1

PN2

j=1 ij !p

0; = (N1N2) ¹PN1

i=1

PN2

j=1 ij !^pE ( ij) 6= 0 and "^::t= (N1N2) ¹PN1

i=1

PN2

j=1"ijt!^p 0. Using (38), we rewrite (37) as

y_::t^s = ⁰x^s_::t+ ⁰s^s_:t+ ⁰d^s_:t+ ⁰qt+ '⁰z+ + t+ "::t+ op(1)

1 0For factor approximation by cross-section averages or the principal components to be valid, we still require min(ij)nij! 1, see Pesaran (2006) and Bai (2009).

1 1y^s_::tcan be expressed as a (column sum) weighted averagePN₂

j=1wjty^s_:jtwith wjt= njt=nt

and y^s_:jt= n_jt¹PN₁

i=1sij;tyijt.