Handbook of Empirical Economics and Finance _14 doc

13.7 Models with Both Individual- and Time-Specific Additive Effects When time-specific effects also appear in v itas in Equation 13.2, the estimatorsignoring the presence oft like those

Conditional on h, the MLE of

step, we substitute estimated ˆh for h in Equation 13.43 However, the feasible

GLS is not as efficient as GLS (for detail, see Hsiao, Pesaran, and Tahmiscoglu2002)

13.7 Models with Both Individual- and Time-Specific

Additive Effects

When time-specific effects also appear in v itas in Equation 13.2, the estimatorsignoring the presence oft like those discussed in Sections 13.13 to 13.6 are

no longer consistent when T is finite For notational ease and without loss of

generality, we illustrate the fundamental issues of dynamic model with bothindividual- and time-specific additive effects model by restricting

= 0inEquation 13.1, thus the model becomes

v it = i+ t+ it , i = 1, , N, t = 1, , T, y i0observable. (13.46)The panel data estimators discussed in Sections 13.5 and 13.6 assume nopresence oft (i.e.,t = 0∀t) When  t are indeed present, those estimators

are not consistent if T is finite when N → ∞ For instance, the consistency

of GMM (Equation 13.33) is based on the assumption that N1 N

i=1 y i,t− j v it

converges to the population moments (Equation 13.32) However, if t arealso present as in Equation 13.46, this condition is likely to be violated To seethis, taking first difference of Equation 13.45 yields

y it = y i,t−1 + v it

= y i,t−1 +  t +  it , (13.47)

i = 1, , N,

t = 2, , T.

converges to ¯y t− j  t , which in general is not equal to zero, in particular, if y it

has mean different from zero,5where ¯y t = 1

conditional on y i0or to complete the system (Equation 13.45) by deriving the

marginal distribution of y i0 By continuous substitutions, we have

assuming the process started at period−m.

Under Equation 13.4, E y i0 = Ev i0 = 0, Var (y i0)= 2

 I T

+ 2

0, c, d) as N and T increase

There-fore, the MLE (or quasi-MLE or GLS) of Equation 13.51 is consistent andasymptotically normally distributed

5For instance, if yitis also a function of exogenous variables as Equation 13.1.

When i andt are fixed constants, we note that first differencing onlyeliminates i from the specification The time-specific effects,  t, remain

at Equation 13.47 To further eliminate t, we note that the cross-sectionalmeany t = 1

N

N

i=1 y itis equal to

y t = y t−1 +  t +  t , (13.54)where  t = 1

it = (y it −y t) and∗

it = ( it − t) The system (Equation 13.55)

no longer involvesiandt

, t = 2, 3, , T Following Arellano and Bond

(1991), we can propose a generalized method of moments (GMM) estimator,6

−1

1

−1

1

Remark 13.7 The GLS with 

present is basically of the same form asthe GLS without the time-specific effects (i.e.,

= 0) (Hsiao, Pesaran, andTahmiscioglu 2002), (Equation 13.25) However, there is an important dif-ference between the two The estimator (Equation 13.63) usesy∗

i,t−1 as theregressor for the equationy∗

it(Equation 13.62), not usesy i,t−1as the sor for the equationy it(Equation 13.47) If there are indeed common shocksthat affect all the cross-sectional units, then the estimator Equation 13.25

regres-is inconsregres-istent while Equation 13.63 regres-is consregres-istent (for detail, see Hsiao andTahmiscioglu 2008) Note also that even though when there are no time-specific effects, Equation 13.63 remains consistent, although it will not be

as efficient as Equation 13.25

Remark 13.8 The estimator (Equation 13.63) and the estimator Equation 13.58remain consistent and asymptotically normally distributed when the effectsare random because the transformation (Equation 13.54) effectively removesthe individual- and time-specific effects from the specification However, ifthe effects are indeed random,then the MLE or GLS of Equation 13.51 is moreefficient

Remark 13.9 The GLS (Equation 13.63) assumes known If  is unknown,one may substitute it by a consistent estimator ˆ, then apply the feasibleGLS However, there is an important difference between the GLS and thefeasible GLS in a dynamic setting The feasible GLS is not asymptotically

equivalent to the GLS when T is finite However, if both N and T → ∞ andlim (N T)= c > 0, then the FGLS will be asymptotically equivalent to the GLS.

(Hsiao and Tahmiscioglu 2008)

Remark 13.10 The MLE or GLS of Equation 13.63 can also be derived bytreating  t as fixed parameters in the system (Equation 13.47) Throughcontinuous substitution, we have

y i1= ∗

where∗

1 =m j=0j 1− j and˜ i1 =m

normally distributed across i with mean 0

and covariance matrix

2

A, and˜

 = Var (˜i1)

 2

The log-likelihood function ofy

takes the form

and solving for

Remark 13.11 When approaches to 1 and 2

 is large relative to2

, theGMM estimator of the form (Equation 13.68) suffers from the weak instru-mental variables issues and performs poorly (e.g., Binder, Hsiao, and Pesaran2005) On the other hand, the performance of the likelihood or GLS estimator(Equation 13.63) is not affected by these problems

Remark 13.12 Hahn and Moon (2006) propose a bias corrected estimator as

the BC rapidly improves as T increase In terms of the closeness of actual

size to the nominal size, again FGLS dominates and rapidly approaches the

nominal size when N or T increases The GMM also has actual sizes close

to nominal sizes except for the cases when is close to unity (here  = 0.8).

The BC has significant size distortion, presumably because of the correction

of bias being based on ˆ∗

cvand the use of asymptotic covariance matrix which

is significantly downward biased in finite sample

Remark 13.13 Hsiao and Tahmiscioglu (2008) also compared the FGLS andGMM with and without the correction of time-specific effects in the presence

of both and time-specific effects or in the presence of specific effects only It is interesting to note that when both individual- andtime-specific effects are present, the biases and root mean squares errors arelarge for estimators assuming no time-specific effects On the other hand, even

individual-in the case of no time-specific effects individual-in the true data generatindividual-ing process, there

is hardly any efficiency loss for the FGLS or GMM that makes the correction

of presumed presence of time-specific effects Therefore, if an investigator isnot sure if the assumption of cross-sectional independence is valid or not, itmight be advisable to use estimators that take account both individual- andtime-specific effects

13.8 Estimation of Multiplicative Models

In this section we consider the estimation of Equation 13.1, where v it is sumed to be of the form

estimator is consistent and asymptotically normally distributed either N or

T or both tend to infinity.

Wheni andtare treated as fixed constants, the MLE are inconsistent if

T is finite for the same basic reason as the additive model (Equation 13.2).

Ahn, Lee, and Schmidt (2001), Bai (2007), Kiefer (1980), etc., have proposed anonlinear GMM and iterative LS estimators for the static model with multi-plicative effects Their nonlinear GMM approach can be similarly generalized

to obtain a consistent estimator of (e.g., Hsiao 2008)

E[W i(

i − i,−1)]= 0. (13.76)The nonlinear GMM estimators of and  amount to applying nonlinear

three-stage least squares to the system

y

i = [I T−1+ ]yi,−1 − yi,−2+ i − i,−1 , i = 1, , N, (13.77)

using W i as instruments, where y

normally distributed as N→ ∞ From the t, we can solve fortthrough thenormalization rule1 = 1 orT

The estimator (Equation 13.78) is consistent if T → ∞

The implementation of nonlinear GMM is quite complicated, Pesaran (2006,2007) notes that

¯y t =  ¯y t−1+ ¯t+ ¯t , (13.79)where

When N → ∞, ¯ t t = ¯−1( ¯y t −  ¯y t−1)into Equation 13.45 yields,

y it = y i,t−1+ 1i ¯y t+ 2i ¯y t−1+ it (13.80)Therefore, Pesaran (2006, 2007) suggests estimating the cross-sectional mean

augment regression (Equation 13.80) and shows that as both N and T → ∞,the least squares estimator of Equation 13.80 yields consistent and asymptot-ically normally distributed ˆ

13.9 Test of Additive versus Multiplicative Model

Multiplicative model implies departure from additivity in their effects onoutcomes It is shown by Bai (2007) that the additive model is embeddedinto the model of multiple common factors with heterogeneous response byletting

When N−→ ∞, one may solve 

tfrom Equation 13.79 that yieldsˆ

t = ( ¯

¯)−¯(υ t −  ¯y t−1 ), (13.82)where ( ¯

¯)− denotes the generalized inverse of ( ¯¯) Substituting tion 13.82 into Equation 13.45 again yields Equation 13.80 Therefore, thePesaran cross-sectional mean augmented regression of Equation 13.80 is con-sistent whether the unobserved heterogeneity is additive or multiplicative,but Equation 13.80 is inefficient if the unobserved heterogeneities are additivecompared to Equation 13.58 or Equation 13.63 However, if the underlyingmodel is multiplicative, Equation 13.80 is consistent, but not Equation 13.58 orEquation 13.63 Therefore, a Hausman type specification test can be proposed

Equa-to test the null:

H0: Equation 13.2 holds

versus

H1: Equation 13.2 does not hold

by considering the test statistic

ˆ

A− ˆm

√Var ( ˆm)− Var(ˆA) ∼ N(0, 1), (13.83)

where ˆAdenotes the efficient estimator of Equation 13.1 under the additiveassumption (Equation 13.2) and ˆmis the estimator (Equation 13.1) under themultiplicative assumption (Equation 13.73).

13.10 Concluding Remarks

In this chapter we review three fundamental issues of modeling dynamicpanel data in the presence of unobserved heterogeneity across individuals andover time—the fixed effects of modeling unobserved individual- and time-specific heterogeneity versus random effects; additive versus multiplicativeeffects and the likelihood versus methods of moments approach

We have not discussed issues of modeling multivariate dynamic panel els (e.g., Binder, Hsiao, and Pesaran (2005), panel unit root tests (e.g., Breitungand Pesaran 2008; Moon and Perron 2004; Phillips and Sul 2003); parameterheterogeneity (e.g., Hsiao and Pesaran 2008), etc However, in principle, thoseissues can also be put in these perspectives

mod-The advantage of the fixed effects specification is that there is no need tospecify the relations between the unobserved effects and observed condi-tional (or explanatory) variables The disadvantages are that (1) unless bothcross-sectional dimension and time dimension of panels are large, the fixed ef-fects specification introduces incidental parameters issues on the individual-specific effects,i, if the time dimension is fixed and on the time-specific ef-fects, t if the cross-sectional dimension is small; (2) the impact oftime-invariant but individual-specific variables such as gender or socio-demographic background variables with the presence of additive individual-specific effects and the impact of time-specific but individual invariant such

as price and some macro-variables with the presence of additive time-specificeffects are unidentified; and (3) the fixed effects inference only makes use ofwithin-group variation The between group information is ignored

The advantages of random-effects specification are (1) there are no tal parameter issues; (2) the impacts of observed individual-specific but time-invariant and individual-invariant but time-varying variables can be iden-tified; (3) both the within-group and between group information are usedfor inference Since the between group variation in general is much largerthan the within group variation, the RE specification can lead to much moreefficient use of sample information The disadvantage is that the relationshipbetween the unobserved effects and observed conditional variables need to

inciden-be specified In short, the advantages of random effects specification are thedisadvantage of fixed effects specification and the advantages of fixed effectsspecification are the disadvantages of random effects specification

Statistical inference procedures for additive effects models are simpler thanthe multiplicative effects models However, if the data generating process callsfor a multiplicative effects specification, statistical inference procedures based

on additive effects specification will be misleading On the other hand, if the

effects are additive, statistical procedures based on multiplicative effects willalso be misleading In this chapter, we have proposed a testing procedure foradditive versus multiplicative effects.

Inference procedures based on the likelihood and moments approaches arereviewed The likelihood approach uses a fixed number of moment condi-tions The moment conditions used in the moments approach increase at theorder of square of time series dimension of the panel In finite sample the mo-ments approach is likely to generate larger bias than the likelihood approach

as shown in the Monte Carlo by Binder, Hsiao and Pesaran (2005), Hsiao andTahmiscioglu (2008), Hsiao, Pesaran, and Tahmiscioglu (2002), Ziliak (1997),etc Moreover, if the observed outcomes in the time dimension is persistent(when the coefficient of lagged variables,, is close to one) or if the variance

of individual-specific effects is large relative to overall variance, the momentsapproach either breaks down or suffers from the weak instrumental variablesissue, but the performance of the likelihood approach is not affected

13.11 Acknowledgment

I would like to thank a referee for helpful comments

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A Unified Estimation Approach for Spatial Dynamic Panel Data Models: Stability,

Spatial Co-integration, and Explosive Roots

Lung-fei Lee and Jihai Yu

CONTENTS

14.1 Introduction 397

14.2 The Model 398

14.2.1 The DGP 398

14.2.2 Data Transformation 401

14.2.3 The Log-Likelihood Function 403

14.3 Asymptotic Properties of QMLE 404

14.3.1 Consistency 405

14.3.2 Asymptotic Distribution 407

14.3.3 Bias Correction 409

14.3.4 Testing 410

14.4 Monte Carlo Results 411

14.5 Conclusion 417

Appendices 418

References 432

14.1 Introduction

In recent decades, there is growing literature on the estimation of dynamic panel data models (see Phillips and Moon 1999; Hahn and Kuersteiner 2002; Alvarez and Arellano 2003; Hahn and Newey 2004, etc.) For the panel data with spatial interactions, Kapoor, Kelejian, and Prucha (2007) extend the asymptotic analysis of the method of moments estimators to a spatial panel

model with error components, where T is finite Baltagi, Song, Jung, and

Koh (2007) consider the testing of spatial and serial dependence in an ex-tended model, where serial correlation on each spatial unit over time and spatial dependence across spatial units are allowed in the disturbances Su and Yang (2007) study the dynamic panel data with spatial error and random

397

effects These panel models specify the spatial correlation by including tially correlated disturbances but do not incorporate a spatial autoregres-

spa-sive term in the regression equation With large n and moderate or large T,

Korniotis (2005) studies a time-space recursive model where only an ual time lag and a spatial time lag are present but not a contemporaneousspatial lag A general model could be the spatial dynamic panel data (SDPD)where a contemporaneous spatial lag is also included Yu, de Jong, and Lee(2007, 2008) and Yu and Lee (2010) study, respectively, the spatial cointegra-tion, stable, and unit root SDPD models, where the individual time lag, spatialtime lag and contemporaneous spatial lag are all included

individ-When the SDPD model has time dummy effects, we might need to form the data to reduce the possible bias caused by the estimation of time

trans-effects (see Lee and Yu, 2010a), especially, when n is proportional to T, or n

is small relative to T Yu, de Jong, and Lee (2007) have a different bias

cor-rection procedures from that of the stable case in Yu, de Jong, and Lee (2008)

In this chapter, we propose a data transformation approach based on a tial difference operator, which can eliminate the time dummy effects as well

spa-as possible unstable and/or explosive components After the data mation, we can estimate the model by the method of maximum likelihood(ML) or quasi-maximum likelihood (QML) similar to Yu, de Jong, and Lee(2008), where there are neither time dummy effects, nor unstable and explo-sive components We derive the asymptotics for the ML estimator (MLE) andQML estimator (QMLE) We propose a bias correction procedure that can beapplied to different types of DGPs

transfor-This chapter is organized as follows In Section 14.2, the model is presented

We show that the stochastic process can be decomposed into stable, unstable

or explosive, and time components A spatial difference operator motivated

by the spatial co-integration can provide a unified data transformation toeliminate the time component and the possible unstable or explosive compo-nents We explain our method of estimation, which is a concentrated QML.Section 14.3 establishes the consistency and asymptotic distribution of theQMLE of the unified transformation approach A bias correction procedure

is also proposed A Monte Carlo study is conducted in Section 14.4 to tigate finite sample performance of the estimators under different DGPs, andalso the power of hypothesis testing of spatial co-integration using this uni-fied approach Section 14.5 concludes the chapter Some useful lemmas andproofs are collected in the appendices