Research Article Estimation of Nonlinear Dynamic Panel...

Research ArticleEstimation of Nonlinear Dynamic Panel Data Models withIndividual Effects

Yi Hu1 Dongmei Guo2 Ying Deng3 and Shouyang Wang4

1 School of Management University of Chinese Academy of Sciences Beijing 100190 China2 School of Economics Central University of Finance and Economics Beijing 100081 China3 School of International Trade and Economics University of International Business and Economics Beijing 100029 China4Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Correspondence should be addressed to Dongmei Guo guodongmeicufe163com

Received 6 December 2013 Accepted 22 January 2014 Published 25 February 2014

Academic Editor Chuangxia Huang

Copyright copy 2014 Yi Hu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper suggests a generalized method of moments (GMM) based estimation for dynamic panel data models with individualspecific fixed effects and threshold effects simultaneously We extend Hansenrsquos (Hansen 1999) original setup to models includingendogenous regressors specifically lagged dependent variables To address the problem of endogeneity of these nonlinear dynamicpanel data models we prove that the orthogonality conditions proposed by Arellano and Bond (1991) are valid The threshold andslope parameters are estimated byGMM and asymptotic distribution of the slope parameters is derived Finite sample performanceof the estimation is investigated throughMonte Carlo simulations It shows that the threshold and slope parameter can be estimatedaccurately and also the finite sample distribution of slope parameters is well approximated by the asymptotic distribution

1 Introduction

Since many economic relationships are dynamic and non-linear nonlineardynamic panel data models could obtainmore information from data sources than traditional models[1 2] For example many researchers suggest that economicgrowth is a nonlinear process [3ndash5] and anumber of empiricalanalyses of economic growth entail dynamic econometricmodels [6ndash9] with lagged dependent variable among theregressors However few researchers consider the dynamicand nonlinear relationships simultaneously and the purposeof this paper is to combine these two factors in one model

Many results exist in the theoretical literature concerningthe estimation and inference for dynamic panel data modelsSince the lagged dependent variables and the disturbanceterm are correlated due to the unobserved effects standardleast square methods could not obtain consistent estimatorswhen the model is dynamic To overcome this problemAnderson and Hsiao [6] suggested that we difference themodel first to get rid of the unobserved effects and then useinstrumental variable (IV) estimation for the transformed

model Nevertheless this IV estimation method leads toconsistent but not necessarily efficient estimates of theparameters because it does not use all the available momentconditions Arellano and Bond [10] proposed a generalizedmethod of moments (GMM) procedure that is more efficientthan the Anderson and Hsiao [6] estimator This literatureis generalized and extended by Arellano and Bover [11] andBlundell and Bond [12] which are called forward orthogonaldeviation and system GMM respectively For the latestdevelopment of dynamic panel data models see Baltagi [13]and Han and Phillips [14] for more details

Several models could be chosen to describe the nonlinearrelationship such as mixture models switching modelssmooth transition threshold models and threshold modelsIn this paper threshold model is used because of wide appli-cations in empirical researches This model splits the sampleinto classes based on an observed variablemdashwhether or not itexceeds some thresholds In most situations the complexityof the problem increases because the exact threshold isunknown and needed to be estimated The estimation andinference are fairly well developed for linear models with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 672610 7 pageshttpdxdoiorg1011552014672610

2 Mathematical Problems in Engineering

exogenous regressors [15ndash17] in which only the nondynamiccase is considered

The dynamic panel threshold models have been usedin empirical literature Cheng et al [18] examined the evi-dence on the conditional convergence growth theory whichextended dynamic panel data growth model to control boththreshold effects and cross-section dependence Chong et al[19] studied the relationship between the depletion rate offoreign reserves and currency crises using threshold autore-gressive model Ho [20] applied a dynamic panel thresholdmodel to examine whether the low-income countries catchupwith the rich ones Kremer et al [21] considered a dynamicpanel threshold model to study inflation thresholds for long-term economic growth As Hansenrsquos model required that allregressors are exogenous the method of Hansen [17] usedin these papers to estimate the dynamic models may notbe suitable due to the lagged dependent variables So farthe theory of dynamic panel threshold model has not beenavailable as we know except for Dang et al [22] Howeverthe validity of the instrumental matrices is not proved Thispaper proposes an estimation method for dynamic panelthreshold model and our analysis mainly relies on Hansen[17] Arellano and Bond [10] and Caner and Hansen [16]First we prove that the orthogonality conditions consideredin ordinary dynamic panel data models are also valid innonlinear dynamic models Second we develop a GMMestimator of the threshold and slope parameters based on theabove moment conditions

The remainder of the paper proceeds as follows Section 2introduces the model and notations Section 3 discusses theestimation for the threshold and slope coefficients Section 4reports a Monte Carlo simulation and Section 5 concludes

2 Model

Consider a simple AR(1)model without exogenous variablesbut with individual and threshold effects as shown in thefollowing structural equation

119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905

for 119894 = 1 119873 119905 = 1 119879(1)

where 119894 denotes cross-sections and 119905 denotes time 119910119894119905denotes

the observable dependent variable 119902119894119905denotes the exogenous

threshold variable 120574 denotes the threshold parameter whichis assumed to be unknown and needs to be estimated 120572

119895

denotes a parameter that satisfies |120572119895| lt 1 (119895 = 1 2) 120583

119894

is the unobserved individual effect 120576119894119905is the idiosyncratic

error which is assumed to be independent and identicallydistributed (iid) withmean zero and variance 1205902 conditionalon 120583119894 119910119894119905minus1

1199101198940 119868(sdot) is indicator function

One can also write (1) in the form

119910119894119905= 120583119894+ 1205721119910119894119905minus1

+ 120576119894119905 119902119894119905le 120574

119910119894119905= 120583119894+ 1205722119910119894119905minus1

+ 120576119894119905 119902119894119905gt 120574

(2)

For simplicity we assume that 1199101198940is observed and let 119909

119894119905equiv

119910119894119905minus1

Alternatively (1) can also be written compactly as

119910119894119905= 120583119894+ 119909119894119905(120574)1015840120572 + 120576119894119905 (3)

where 119909119894119905(120574) = (119909

119894119905119868(119902119894119905le 120574) 119909

119894119905119868(119902119894119905gt 120574))1015840 120572 = (120572

1 1205722)1015840

3 Estimation

In this section we first consider a simple model withoutexogenous covariates and derive GMM based estimator forthe threshold parameter 120574 and slope parameters 120572 Then weextend the simple model to cases with strictly exogenouscovariates

31 Estimation of Threshold and Slope Parameters In tradi-tional dynamic panel data model two methods are com-monly used to remove individual effect 120583

119894 One is first-

difference approach suggested by Arellano and Bond [10]the other one is forward orthogonal deviation proposed byArellano and Bover [11] We will utilize the first-differenceapproach in the following derivation due to the fact that itis more convenient for computation

First we take first-difference for model (3) to get rid ofthe time invariant individual effects

Δ119910119894119905= Δ119909119894119905(120574)1015840120572 + Δ120576

119894119905 (4)

where Δ denotes difference operator If 1205721

= 1205722 that is

there is no threshold effect then additional instruments canbe obtained in dynamic panel data models if one utilizes theorthogonality conditions that exist between lagged values of119910119894119905and the disturbance 120576

119894119905according to Arellano and Bond

[10] Here we prove that these orthogonality conditions arealso valid in model (4) when 120572

1= 1205722

For any given t we have either 119909119894119905(120574) = (119909

119894119905 0)1015840 or 119909119894119905(120574) =

(0 119909119894119905)1015840 Consider the former one without loss of generality

Similarly there must be two cases in the period 119905 minus 1

Case 1 119909119894119905minus1

(120574) = (119909119894119905minus1

0)1015840

Case 2 119909119894119905minus1

(120574) = (0 119909119894119905minus1

)1015840

(5)

Then first difference yields

Δ119909119894119905(120574) =

(Δ119909119894119905 0)1015840 for case 1

(119909119894119905 minus119909119894119905minus1

)1015840 for case 2

(6)

Correspondingly

Δ119909119894119905(120574)1015840120572 =

1205721Δ119909119894119905= 1205721119909119894119905minus 1205721119909119894119905minus1

for case 1

1205721119909119894119905 minus1205722119909119894119905minus1

for case 2(7)

For any given 119905 119909119894119905minus1

is a valid instrument (A valid instru-ment means that it should have two basic conditions firsterogeneity ie it should be independent of (or at leastuncorrelated with) the disturbance term in the equation ofinterest second relevance ie it should be correlated withthe included endogenous explanatory variables for which it

Mathematical Problems in Engineering 3

is supposed to serve as an instrument) for both case 1 andcase 2 since it is correlated with Δ119909

119894119905(120574)1015840120572 that is 120572

1119909119894119905

minus

1205721119909119894119905minus1

or 1205721119909119894119905 minus1205722119909119894119905minus1

but not correlated with Δ120576119894119905 that

is 120576119894119905minus 120576119894119905minus1

as long as 120576119894119905rsquos are serially uncorrelated Given

the autoregressive nature of the model and the assumptionthat there is no serial correlation in 120576

119894119905 it can be easily shown

that 1199091198941 119909

119894119905minus2are also valid instruments Therefore the

orthogonality conditions are given by

119864 (119909119894119905minus119904

Δ120576119894119905) = 0 for 119904 = 1 119905 minus 1 119905 = 2 119879 (8)

Define

119885119894= (

1199091198941

0 sdot sdot sdot 0

0 1199091198941 1199091198942

sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1199091198941 1199091198942 119909119894119879minus1

) (9)

then for each 119894 the 119879(119879 minus 1)2moment conditions describedabove can be written as

119864 (1198851015840

119894Δ120576119894) = 0 (10)

Note that Δ120576119894119905

= 120576119894119905

minus 120576119894119905minus1

is MA(1) Define Δ120576119894

=

(Δ1205761198942 sdot sdot sdot Δ120576

119894119879)1015840 then

119864 (Δ120576119894Δ1205761015840

119894) = 1205902119866 (11)

where

119866 = (

2 minus1 0 sdot sdot sdot 0 0

minus1 2 minus1 sdot sdot sdot 0 0

d

0 0 0 sdot sdot sdot 2 minus1

0 0 0 sdot sdot sdot minus1 2

) (12)

Let GMM(120574) be the GMMestimator withmoment conditionsgiven by (10) The GMM estimator can be written as

GMM (120574)

=

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119883119894(120574)]

minus1

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119884119894]

(13)

where Δ119883119894(120574) = [Δ119909

1198942(120574) Δ119909

119894119879(120574)]1015840 Δ119884119894

=

(Δ1199101198942 Δ119910

119894119879)1015840

Stacking over individuals (13) can be written compactlyas

GMM (120574) = [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119883 (120574)]

minus1

times [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119884]

(14)

where Ω = 119868119873

otimes 119866 otimes is Kronecker product and Δ119883(120574) =

[Δ1198831(120574)1015840 Δ119883

119873(120574)1015840]1015840 Δ119884 = [Δ1198841015840

1 Δ1198841015840

119873]1015840 and 119885 =

(11988510158401 1198851015840

119873)1015840

In fact this estimator is infeasible in empirical studiessince it depends on an unknown parameter 120574 Therefore ournext step is to estimate 120574 from the regression residuals

119890 (120574) = Δ119884 minus Δ119883 (120574) GMM (120574) (15)

We apply the estimator suggested by Chan [23] and Hansen[15 17] then 120574 can be estimated by

120574 = argmin120574

119878 (120574) (16)

where 119878(120574) = 119890(120574)1015840119890(120574) is the sum of squared errors

Once 120574 is obtained we substitute the true parameter 120574

with its estimate 120574 yielding the feasible GMM estimator ofslope coefficient estimate

= GMM (120574) (17)

According to Hansen [24] under the case of known120574 GMM estimator GMM(120574) is efficient and asymptoticallynormal

radic119873(GMM (120574) minus 120572) 997904rArr 119873(0 119881) as 119873 997888rarr infin (18)

where

119881 = 119873[Δ119883(120574)1015840119885(1205902119866)minus1

1198851015840Δ119883 (120574)]

minus1

(19)

Hansen [17] and Caner and Hansen [16] show that thedependence on the threshold estimate is not of first-orderasymptotic importance so inference on could proceed as ifthe estimated threshold parameter 120574 was the true parameter120574 Then

radic119873( minus 120572) 997904rArr 119873(0 119881) as 119873 997888rarr infin (20)

The estimated covariance matrix becomes (One can alsoconsiderWindmeijerrsquos bias-corrected estimator (Windmeijer[25]) for the robust VCE of two-step GMM estimators)

= 119873[Δ119883(120574)1015840119885119866minus11198851015840Δ119883 (120574)]

minus1

(21)

where 119866 = sum119873

119894=11198851015840119894(1198901(120574)1198901(120574)1015840)119885119894and 1198901(120574) = Δ119884 minus Δ119883(120574)

Similarly one can prove that converges in probability to119881asin Caner and Hansen [16]

32 Estimation of the Model with Exogenous Variables Nowwe extend the results in the previous subsection to cases withstrictly exogenous variables Consider additional regressors119898119894119905in model (1)

119910119894119905= 120583119894+ (1205721119910119894119905minus1

+ 1198981015840

1198941199051205731) 119868 (119902119894119905le 120574)

+ (1205722119910119894119905minus1

+ 1198981015840

1198941199051205732) 119868 (119902119894119905gt 120574) + 120576

119894119905

(22)


Table 1 Quantiles of distribution 120579 = 01

Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 010 209 412 091 205 317 139 200 234119873 = 200 089 200 340 116 200 284 160 200 233119873 = 300 minus022 188 326 154 200 258 172 200 236

1205721= 05

119873 = 100 025 044 061 037 046 055 038 046 052119873 = 200 033 045 057 040 048 054 043 048 053119873 = 300 019 046 056 043 049 054 045 049 053

1205722= 06

119873 = 100 037 055 084 047 056 065 050 057 063119873 = 200 046 057 070 050 057 064 053 058 063119873 = 300 045 056 066 053 059 064 054 058 063


Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 184 202 217 190 200 202 198 199 200119873 = 200 193 200 206 194 200 203 198 200 202119873 = 300 193 201 204 196 200 204 200 200 202

1205721= 05

119873 = 100 026 042 057 037 046 054 040 047 053119873 = 200 032 045 058 040 047 054 043 048 053119873 = 300 037 047 058 042 048 054 044 049 053

1205722= 08

119873 = 100 054 071 086 065 074 083 070 076 082119873 = 200 062 074 085 070 077 083 073 078 083119873 = 300 064 076 087 070 077 084 073 079 084

for 119894 = 1 119873 119905 = 1 119879 Since 119898119894119905are strictly

exogenous they are valid instruments for the first differencedform of (22) Therefore (1198981015840

1198942 1198981015840

119894119879) should be added to

each diagonal element of 119885119894in (9) Hence the matrix of

instruments is

119885119894

= (

1199091198941 11989810158401198942

0 sdot sdot sdot 0

0 1199091198941 1199091198942 11989810158401198943

sdot sdot sdot 0


d


1198981015840119894119879

)

(23)

then the estimators of 120574 and slope coefficients (1205721015840 1205731015840)1015840 can beobtained accordingly as in (16) and (17)

4 Monte Carlo Experiments

In this section Monte Carlo experiments are implemented toexamine the finite sample performance of our estimator Forthis purpose we consider the following design

41 Simulation Design The data generating process (DGP) isgiven by

119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905 (24)

for 119894 = 1 119873 and 119905 = 1 119879 where 120583119894sim iid119873(0 1)

119902119894119905

sim iid119873(2 1) 120576119894119905

sim iid119873(0 1) and 120583119894 119902119894119905 120576119894119905are

mutually independent random variables Let 1199101198940

= 0 120574 = 21205721= 05 and 120579 = 120572

2minus 1205721= 01 03 and 119873 varies among

100 200 300 and119879 varies among 10 15 20 All results arebased on 1000 replications

The computation of the threshold 120574 involves the mini-mization problem in (16) which can be reduced to searchingfor the values of 120574 that minimizes the sum of squared errorsamong all distinct values of 119902

119894119905in the sample Obviously

there are at most119873119879 distinct values of 119902119894119905 and the minimum

value of 119873119879 considered in the simulation is 1000 Thus


02 03 04 05 06 070

2

4

6

8

10

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 10

(a)

03 04 05 060

2

4

6

8

10

12

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 15

(b)

0

2

4

6

8

10

12

14

16

Density

03 04 05 06x

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 20

(c)

03 04 05 06 07 080

2

4

6

8

10

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 10

(d)

04 05 06 070

2

4

6

8

10

12

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 15

(e)

04 05 06 07x

0

2

4

6

8

10

12

14

16

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 20

(f)

Figure 1 Density distribution of slope parameters (small threshold)

the searching could take a fair amount of time when thenumber of possible values is large To reduce the computationload we employ themethod proposed byHansen [17] Specif-ically instead of searching over all values of 119902

119894119905 we limit it to

some specific quantiles 001 00125 0015 099 whichcontain only 393 different valuesHowever this approachmaynot be as appealing as searching over all possible values of 120574when the number of distinct value of 119902

119894119905is small

42 Simulation Results Tables 1 and 2 represent the 5 50and 95 quantiles of the simulation distribution of 120574

1 and

2for 119879 varying among 10 15 and 20 and 119873 varying among

100 200 and 300Table 1 reports the results of 120579 = 01 corresponding

to the case when threshold is small The estimates of thethreshold 120574 perform fairly well for all cases considered sincethe medians of 120574 are around the true value 120574 = 2 As 119879

increases the distribution of 120574 is becoming more and moreconcentrated around the true value For example when119873 =

100 and119879 = 10 the length of the quantile range between 005

and 095 is 402 while when 119879 = 20 the length decreasesto 095 The distribution of the slope coefficient estimator

1

exhibits a little downward bias as it has been shown in someof the existing Monte Carlo studies for dynamic panel datamodels For 119873 = 100 and 119879 = 10 the median bias of

1

is 006 but this bias is reduced as N andor 119879 increases forexample this bias is only 001 for 119873 = 300 and 119879 = 20Similarly the length of the quantile range between 005 and095 for

1is getting smaller as 119879 increases which means that

the performance improves The quantiles of the distributionof 2also performs well in all cases although it is relatively

weak in cases with small 119879 and small NTable 2 presents the results for the case when threshold is

big that is 120579 = 03 Compared to the small threshold case theperformance of the distribution of 120574 is improvedThemedianbias of 120574 is zero for almost all cases and the length of thequantile range between 005 and 095 is getting smaller as thethreshold effect increases Meanwhile Table 2 reports similarresults as Table 1 for the parameters of

1and

2 In Table 2

they also perform fairly well in the big threshold case


0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)

Figure 2 Density distribution of slope parameters (big threshold)

Figure 1 displays kernel estimates of the distribution ofthe slope parameters

1and

2based on 1000 replications

with 119873 = 100 200 300 119879 = 10 15 20 and small threshold(120579 = 01) The estimates are slightly biased downwards when119879 is small or 119873 is small This bias is common in dynamicpanel data model as mentioned earlier One could also usesome bias-corrected methods to improve the finite sampleproperties of the estimators which is beyond the scope ofthis paper The estimates are gradually centered around thetrue values as119873 andor 119879 increases which is consistent withthe above analyses and confirms the validity of our proposedestimation procedure again

Figure 2 shows the distribution of the same parameters asFigure 1 and based on the same number of replications andsample size but with bigger threshold (120579 = 03) In this casethe same conclusion can be found as in Figure 1 In particularthe performance of the estimators in this case is better thanthat in the smaller threshold for all cases

5 Conclusion

This paper extends the estimation of threshold models innondynamic panels to dynamic panels and presents practicalestimation methods for these econometric models withindividual-specific effects and threshold effectsThe foremostfeature of these models is that they allow the econometri-cian to consider the dynamic and threshold relationshipsin economic system simultaneously As mentioned in theintroductionmany applicationsmay have such relationshipsUsing the first-difference to eliminate the individual-specificeffects we prove that the orthogonality conditions proposedby Arellano and Bond [10] for nonthreshold models are alsovalid in our models Then we estimate the threshold andslope parameters by GMM Monte Carlo simulations revealthat our method has very good finite sample performance

There are several possible extensions to this work Theasymptotic properties of the threshold parameter would be


an interesting topic Also testing for one or multiple thresh-olds is also worth studying which is saved for future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the editor and three anonymous refereesfor many constructive and helpful comments This work waspartially supported by the National Natural Science Foun-dation of China (Grant nos 71301160 and 71301173) ChinaPostdoctoral Science Foundation funded project (Grant nos2012M520419 2012M520420 and 2013T60186) Beijing Plan-ning Office of Philosophy and Social Science (13JGB018) andProgram for Innovation Research in Central University ofFinance and Economics

References

[1] C X Huang C L Peng X H Chen and F HWen ldquoDynamicsanalysis of a class of delayed economic modelrdquo Abstract andApplied Analysis vol 2013 Article ID 962738 12 pages 2013

[2] F Wen and Z Dai ldquoModified Yabe-Takano nonlinear conjugategradient methodrdquo Pacific Journal of Optimization vol 8 no 2pp 347ndash360 2012

[3] O Galor and D N Weil ldquoPopulation technology and growthfrom malthusian stagnation to the demographic transition andbeyondrdquoTheAmerican Economic Review vol 90 no 4 pp 806ndash828 2000

[4] AMas-colell andA Razin ldquoAmodel of intersectioralmigrationand growthrdquo Oxford Economic Papers vol 25 no 1 pp 72ndash791973

[5] P F Peretto ldquoIndustrial development technological change andlong-run growthrdquo Journal of Development Economics vol 59no 2 pp 389ndash417 1999

[6] T W Anderson and C Hsiao ldquoEstimation of dynamic modelswith error componentsrdquo Journal of the American StatisticalAssociation vol 76 no 375 pp 598ndash606 1981

[7] A Ciarreta and A Zarraga ldquoEconomic growth-electricityconsumption causality in 12 European countries a dynamicpanel data approachrdquo Energy Policy vol 38 no 7 pp 3790ndash3796 2010

[8] T S Eicher and T Schreiber ldquoStructural policies and growthtime series evidence from a natural experimentrdquo Journal ofDevelopment Economics vol 91 no 1 pp 169ndash179 2010

[9] B-NHuangM J Hwang andCW Yang ldquoCausal relationshipbetween energy consumption and GDP growth revisited adynamic panel data approachrdquo Ecological Economics vol 67 no1 pp 41ndash54 2008

[10] M Arellano and S Bond ldquoSome tests of specification for paneldata Monte Carlo evidence and an application to employmentequationsrdquoTheReview of Economic Studies vol 58 pp 277ndash2971991

[11] M Arellano and O Bover ldquoAnother look at the instrumentalvariable estimation of error-components modelsrdquo Journal ofEconometrics vol 68 no 1 pp 29ndash51 1995

[12] R Blundell and S Bond ldquoInitial conditions andmoment restric-tions in dynamic panel data modelsrdquo Journal of Econometricsvol 87 no 1 pp 115ndash143 1998

[13] B H Baltagi Econometric Analysis of Panel Data JohnWiley ampSons Chichester UK 2008

[14] C Han and P C B Phillips ldquoGMM estimation for dynamicpanels with fixed effects and strong instruments at unityrdquoEconometric Theory vol 26 no 1 pp 119ndash151 2010

[15] B E Hansen ldquoSample splitting and threshold estimationrdquoEconometrica vol 68 no 3 pp 575ndash603 2000

[16] M Caner and B E Hansen ldquoInstrumental variable estimationof a threshold modelrdquo Econometric Theory vol 20 no 5 pp813ndash843 2004

[17] B E Hansen ldquoThreshold effects in non-dynamic panels esti-mation testing and inferencerdquo Journal of Econometrics vol 93no 2 pp 345ndash368 1999

[18] J Cheng C Lin and C Wang ldquoEstimation of growth conver-gence using common correlated effects approachesrdquo WorkingPaper 2009

[19] T T L Chong Q He and M J Hinich ldquoThe nonlineardynamics of foreign reserves and currency crisesrdquo Studies inNonlinear Dynamics amp Econometrics vol 12 no 4 article 22008

[20] T-W Ho ldquoIncome thresholds and growth convergence a paneldata approachrdquo Manchester School vol 74 no 2 pp 170ndash1892006

[21] S Kremer A Bick and D Nautz ldquoInflation and growth newevidence from a dynamic panel threshold analysisrdquo EmpiricalEconomics vol 44 pp 861ndash878 2013

[22] V A Dang M Kim and Y Shin ldquoAsymmetric capital structureadjustments new evidence from dynamic panel thresholdmodelsrdquo Journal of Empirical Finance vol 19 no 4 pp 465ndash482 2012

[23] K S Chan ldquoConsistency and limiting distribution of the leastsquares estimator of a threshold autoregressive modelrdquo TheAnnals of Statistics vol 21 no 1 pp 520ndash533 1993

[24] L P Hansen ldquoLarge sample properties of generalized methodof moments estimatorsrdquo Econometrica vol 50 no 4 pp 1029ndash1054 1982

[25] F Windmeijer ldquoA finite sample correction for the variance oflinear efficient two-stepGMMestimatorsrdquo Journal of Economet-rics vol 126 no 1 pp 25ndash51 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


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Journal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


exogenous regressors [15ndash17] in which only the nondynamiccase is considered

The dynamic panel threshold models have been usedin empirical literature Cheng et al [18] examined the evi-dence on the conditional convergence growth theory whichextended dynamic panel data growth model to control boththreshold effects and cross-section dependence Chong et al[19] studied the relationship between the depletion rate offoreign reserves and currency crises using threshold autore-gressive model Ho [20] applied a dynamic panel thresholdmodel to examine whether the low-income countries catchupwith the rich ones Kremer et al [21] considered a dynamicpanel threshold model to study inflation thresholds for long-term economic growth As Hansenrsquos model required that allregressors are exogenous the method of Hansen [17] usedin these papers to estimate the dynamic models may notbe suitable due to the lagged dependent variables So farthe theory of dynamic panel threshold model has not beenavailable as we know except for Dang et al [22] Howeverthe validity of the instrumental matrices is not proved Thispaper proposes an estimation method for dynamic panelthreshold model and our analysis mainly relies on Hansen[17] Arellano and Bond [10] and Caner and Hansen [16]First we prove that the orthogonality conditions consideredin ordinary dynamic panel data models are also valid innonlinear dynamic models Second we develop a GMMestimator of the threshold and slope parameters based on theabove moment conditions

The remainder of the paper proceeds as follows Section 2introduces the model and notations Section 3 discusses theestimation for the threshold and slope coefficients Section 4reports a Monte Carlo simulation and Section 5 concludes

2 Model

Consider a simple AR(1)model without exogenous variablesbut with individual and threshold effects as shown in thefollowing structural equation

119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905

for 119894 = 1 119873 119905 = 1 119879(1)

where 119894 denotes cross-sections and 119905 denotes time 119910119894119905denotes

the observable dependent variable 119902119894119905denotes the exogenous

threshold variable 120574 denotes the threshold parameter whichis assumed to be unknown and needs to be estimated 120572

119895

denotes a parameter that satisfies |120572119895| lt 1 (119895 = 1 2) 120583

119894

is the unobserved individual effect 120576119894119905is the idiosyncratic

error which is assumed to be independent and identicallydistributed (iid) withmean zero and variance 1205902 conditionalon 120583119894 119910119894119905minus1

1199101198940 119868(sdot) is indicator function

One can also write (1) in the form

119910119894119905= 120583119894+ 1205721119910119894119905minus1

+ 120576119894119905 119902119894119905le 120574

119910119894119905= 120583119894+ 1205722119910119894119905minus1

+ 120576119894119905 119902119894119905gt 120574

(2)

For simplicity we assume that 1199101198940is observed and let 119909

119894119905equiv

119910119894119905minus1

Alternatively (1) can also be written compactly as

119910119894119905= 120583119894+ 119909119894119905(120574)1015840120572 + 120576119894119905 (3)

where 119909119894119905(120574) = (119909

119894119905119868(119902119894119905le 120574) 119909

119894119905119868(119902119894119905gt 120574))1015840 120572 = (120572

1 1205722)1015840

3 Estimation

In this section we first consider a simple model withoutexogenous covariates and derive GMM based estimator forthe threshold parameter 120574 and slope parameters 120572 Then weextend the simple model to cases with strictly exogenouscovariates

31 Estimation of Threshold and Slope Parameters In tradi-tional dynamic panel data model two methods are com-monly used to remove individual effect 120583

119894 One is first-

difference approach suggested by Arellano and Bond [10]the other one is forward orthogonal deviation proposed byArellano and Bover [11] We will utilize the first-differenceapproach in the following derivation due to the fact that itis more convenient for computation

First we take first-difference for model (3) to get rid ofthe time invariant individual effects

Δ119910119894119905= Δ119909119894119905(120574)1015840120572 + Δ120576

119894119905 (4)

where Δ denotes difference operator If 1205721

= 1205722 that is

there is no threshold effect then additional instruments canbe obtained in dynamic panel data models if one utilizes theorthogonality conditions that exist between lagged values of119910119894119905and the disturbance 120576

119894119905according to Arellano and Bond

[10] Here we prove that these orthogonality conditions arealso valid in model (4) when 120572

1= 1205722

For any given t we have either 119909119894119905(120574) = (119909

119894119905 0)1015840 or 119909119894119905(120574) =

(0 119909119894119905)1015840 Consider the former one without loss of generality

Similarly there must be two cases in the period 119905 minus 1

Case 1 119909119894119905minus1

(120574) = (119909119894119905minus1

0)1015840

Case 2 119909119894119905minus1

(120574) = (0 119909119894119905minus1

)1015840

(5)

Then first difference yields

Δ119909119894119905(120574) =

(Δ119909119894119905 0)1015840 for case 1

(119909119894119905 minus119909119894119905minus1

)1015840 for case 2

(6)

Correspondingly

Δ119909119894119905(120574)1015840120572 =

1205721Δ119909119894119905= 1205721119909119894119905minus 1205721119909119894119905minus1

for case 1

1205721119909119894119905 minus1205722119909119894119905minus1

for case 2(7)

For any given 119905 119909119894119905minus1

is a valid instrument (A valid instru-ment means that it should have two basic conditions firsterogeneity ie it should be independent of (or at leastuncorrelated with) the disturbance term in the equation ofinterest second relevance ie it should be correlated withthe included endogenous explanatory variables for which it



119894119905(120574)1015840120572 that is 120572

1119909119894119905

minus

1205721119909119894119905minus1

or 1205721119909119894119905 minus1205722119909119894119905minus1


is 120576119894119905minus 120576119894119905minus1




that 1199091198941 119909



119864 (119909119894119905minus119904

Δ120576119894119905) = 0 for 119904 = 1 119905 minus 1 119905 = 2 119879 (8)

Define

119885119894= (

1199091198941

0 sdot sdot sdot 0

0 1199091198941 1199091198942

sdot sdot sdot 0


d


) (9)


119864 (1198851015840

119894Δ120576119894) = 0 (10)

Note that Δ120576119894119905

= 120576119894119905

minus 120576119894119905minus1

is MA(1) Define Δ120576119894

=

(Δ1205761198942 sdot sdot sdot Δ120576

119894119879)1015840 then

119864 (Δ120576119894Δ1205761015840

119894) = 1205902119866 (11)

where

119866 = (



d



) (12)


GMM (120574)

=

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119883119894(120574)]

minus1

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119884119894]

(13)

where Δ119883119894(120574) = [Δ119909

1198942(120574) Δ119909

119894119879(120574)]1015840 Δ119884119894

=

(Δ1199101198942 Δ119910

119894119879)1015840


GMM (120574) = [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119883 (120574)]

minus1

times [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119884]

(14)

where Ω = 119868119873


[Δ1198831(120574)1015840 Δ119883

119873(120574)1015840]1015840 Δ119884 = [Δ1198841015840

1 Δ1198841015840

119873]1015840 and 119885 =

(11988510158401 1198851015840

119873)1015840


119890 (120574) = Δ119884 minus Δ119883 (120574) GMM (120574) (15)


120574 = argmin120574

119878 (120574) (16)




= GMM (120574) (17)



where

119881 = 119873[Δ119883(120574)1015840119885(1205902119866)minus1

1198851015840Δ119883 (120574)]

minus1

(19)




= 119873[Δ119883(120574)1015840119885119866minus11198851015840Δ119883 (120574)]

minus1

(21)

where 119866 = sum119873

119894=11198851015840119894(1198901(120574)1198901(120574)1015840)119885119894and 1198901(120574) = Δ119884 minus Δ119883(120574)



119910119894119905= 120583119894+ (1205721119910119894119905minus1

+ 1198981015840

1198941199051205731) 119868 (119902119894119905le 120574)

+ (1205722119910119894119905minus1

+ 1198981015840

1198941199051205732) 119868 (119902119894119905gt 120574) + 120576

119894119905

(22)



Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 010 209 412 091 205 317 139 200 234119873 = 200 089 200 340 116 200 284 160 200 233119873 = 300 minus022 188 326 154 200 258 172 200 236

1205721= 05

119873 = 100 025 044 061 037 046 055 038 046 052119873 = 200 033 045 057 040 048 054 043 048 053119873 = 300 019 046 056 043 049 054 045 049 053

1205722= 06

119873 = 100 037 055 084 047 056 065 050 057 063119873 = 200 046 057 070 050 057 064 053 058 063119873 = 300 045 056 066 053 059 064 054 058 063


Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 184 202 217 190 200 202 198 199 200119873 = 200 193 200 206 194 200 203 198 200 202119873 = 300 193 201 204 196 200 204 200 200 202

1205721= 05

119873 = 100 026 042 057 037 046 054 040 047 053119873 = 200 032 045 058 040 047 054 043 048 053119873 = 300 037 047 058 042 048 054 044 049 053

1205722= 08

119873 = 100 054 071 086 065 074 083 070 076 082119873 = 200 062 074 085 070 077 083 073 078 083119873 = 300 064 076 087 070 077 084 073 079 084



1198942 1198981015840



instruments is

119885119894

= (

1199091198941 11989810158401198942

0 sdot sdot sdot 0

0 1199091198941 1199091198942 11989810158401198943

sdot sdot sdot 0


d


1198981015840119894119879

)

(23)





119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905 (24)


119902119894119905

sim iid119873(2 1) 120576119894119905

sim iid119873(0 1) and 120583119894 119902119894119905 120576119894119905are


= 0 120574 = 21205721= 05 and 120579 = 120572








02 03 04 05 06 070

2

4

6

8

10

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 10

(a)

03 04 05 060

2

4

6

8

10

12

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 15

(b)

0

2

4

6

8

10

12

14

16

Density

03 04 05 06x

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 20

(c)

03 04 05 06 07 080

2

4

6

8

10

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 10

(d)

04 05 06 070

2

4

6

8

10

12

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 15

(e)

04 05 06 07x

0

2

4

6

8

10

12

14

16

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 20

(f)





119894119905is small


1 and







1


1






1and

2 In Table 2



0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)



1and




5 Conclusion







Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119905(120574)1015840120572 that is 120572

1119909119894119905

minus

1205721119909119894119905minus1

or 1205721119909119894119905 minus1205722119909119894119905minus1


is 120576119894119905minus 120576119894119905minus1




that 1199091198941 119909



119864 (119909119894119905minus119904

Δ120576119894119905) = 0 for 119904 = 1 119905 minus 1 119905 = 2 119879 (8)

Define

119885119894= (

1199091198941

0 sdot sdot sdot 0

0 1199091198941 1199091198942

sdot sdot sdot 0


d


) (9)


119864 (1198851015840

119894Δ120576119894) = 0 (10)

Note that Δ120576119894119905

= 120576119894119905

minus 120576119894119905minus1

is MA(1) Define Δ120576119894

=

(Δ1205761198942 sdot sdot sdot Δ120576

119894119879)1015840 then

119864 (Δ120576119894Δ1205761015840

119894) = 1205902119866 (11)

where

119866 = (



d



) (12)


GMM (120574)

=

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119883119894(120574)]

minus1

[

119873

sum119894=1

Δ119883119894(120574)1015840119885119894][

119873

sum119894=1

1198851015840

119894119866119885119894]

minus1

[

119873

sum119894=1

1198851015840

119894Δ119884119894]

(13)

where Δ119883119894(120574) = [Δ119909

1198942(120574) Δ119909

119894119879(120574)]1015840 Δ119884119894

=

(Δ1199101198942 Δ119910

119894119879)1015840


GMM (120574) = [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119883 (120574)]

minus1

times [Δ119883(120574)1015840119885(1198851015840Ω119885)minus1

1198851015840Δ119884]

(14)

where Ω = 119868119873


[Δ1198831(120574)1015840 Δ119883

119873(120574)1015840]1015840 Δ119884 = [Δ1198841015840

1 Δ1198841015840

119873]1015840 and 119885 =

(11988510158401 1198851015840

119873)1015840


119890 (120574) = Δ119884 minus Δ119883 (120574) GMM (120574) (15)


120574 = argmin120574

119878 (120574) (16)




= GMM (120574) (17)



where

119881 = 119873[Δ119883(120574)1015840119885(1205902119866)minus1

1198851015840Δ119883 (120574)]

minus1

(19)




= 119873[Δ119883(120574)1015840119885119866minus11198851015840Δ119883 (120574)]

minus1

(21)

where 119866 = sum119873

119894=11198851015840119894(1198901(120574)1198901(120574)1015840)119885119894and 1198901(120574) = Δ119884 minus Δ119883(120574)



119910119894119905= 120583119894+ (1205721119910119894119905minus1

+ 1198981015840

1198941199051205731) 119868 (119902119894119905le 120574)

+ (1205722119910119894119905minus1

+ 1198981015840

1198941199051205732) 119868 (119902119894119905gt 120574) + 120576

119894119905

(22)



Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 010 209 412 091 205 317 139 200 234119873 = 200 089 200 340 116 200 284 160 200 233119873 = 300 minus022 188 326 154 200 258 172 200 236

1205721= 05

119873 = 100 025 044 061 037 046 055 038 046 052119873 = 200 033 045 057 040 048 054 043 048 053119873 = 300 019 046 056 043 049 054 045 049 053

1205722= 06

119873 = 100 037 055 084 047 056 065 050 057 063119873 = 200 046 057 070 050 057 064 053 058 063119873 = 300 045 056 066 053 059 064 054 058 063


Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 184 202 217 190 200 202 198 199 200119873 = 200 193 200 206 194 200 203 198 200 202119873 = 300 193 201 204 196 200 204 200 200 202

1205721= 05

119873 = 100 026 042 057 037 046 054 040 047 053119873 = 200 032 045 058 040 047 054 043 048 053119873 = 300 037 047 058 042 048 054 044 049 053

1205722= 08

119873 = 100 054 071 086 065 074 083 070 076 082119873 = 200 062 074 085 070 077 083 073 078 083119873 = 300 064 076 087 070 077 084 073 079 084



1198942 1198981015840



instruments is

119885119894

= (

1199091198941 11989810158401198942

0 sdot sdot sdot 0

0 1199091198941 1199091198942 11989810158401198943

sdot sdot sdot 0


d


1198981015840119894119879

)

(23)





119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905 (24)


119902119894119905

sim iid119873(2 1) 120576119894119905

sim iid119873(0 1) and 120583119894 119902119894119905 120576119894119905are


= 0 120574 = 21205721= 05 and 120579 = 120572








02 03 04 05 06 070

2

4

6

8

10

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 10

(a)

03 04 05 060

2

4

6

8

10

12

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 15

(b)

0

2

4

6

8

10

12

14

16

Density

03 04 05 06x

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 20

(c)

03 04 05 06 07 080

2

4

6

8

10

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 10

(d)

04 05 06 070

2

4

6

8

10

12

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 15

(e)

04 05 06 07x

0

2

4

6

8

10

12

14

16

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 20

(f)





119894119905is small


1 and







1


1






1and

2 In Table 2



0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)



1and




5 Conclusion







Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 010 209 412 091 205 317 139 200 234119873 = 200 089 200 340 116 200 284 160 200 233119873 = 300 minus022 188 326 154 200 258 172 200 236

1205721= 05

119873 = 100 025 044 061 037 046 055 038 046 052119873 = 200 033 045 057 040 048 054 043 048 053119873 = 300 019 046 056 043 049 054 045 049 053

1205722= 06

119873 = 100 037 055 084 047 056 065 050 057 063119873 = 200 046 057 070 050 057 064 053 058 063119873 = 300 045 056 066 053 059 064 054 058 063


Quantiles 119879 = 10 119879 = 15 119879 = 20

005 05 095 005 05 095 005 05 095120574 = 2

119873 = 100 184 202 217 190 200 202 198 199 200119873 = 200 193 200 206 194 200 203 198 200 202119873 = 300 193 201 204 196 200 204 200 200 202

1205721= 05

119873 = 100 026 042 057 037 046 054 040 047 053119873 = 200 032 045 058 040 047 054 043 048 053119873 = 300 037 047 058 042 048 054 044 049 053

1205722= 08

119873 = 100 054 071 086 065 074 083 070 076 082119873 = 200 062 074 085 070 077 083 073 078 083119873 = 300 064 076 087 070 077 084 073 079 084



1198942 1198981015840



instruments is

119885119894

= (

1199091198941 11989810158401198942

0 sdot sdot sdot 0

0 1199091198941 1199091198942 11989810158401198943

sdot sdot sdot 0


d


1198981015840119894119879

)

(23)





119910119894119905= 120583119894+ 1205721119910119894119905minus1

119868 (119902119894119905le 120574) + 120572

2119910119894119905minus1

119868 (119902119894119905gt 120574) + 120576

119894119905 (24)


119902119894119905

sim iid119873(2 1) 120576119894119905

sim iid119873(0 1) and 120583119894 119902119894119905 120576119894119905are


= 0 120574 = 21205721= 05 and 120579 = 120572








02 03 04 05 06 070

2

4

6

8

10

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 10

(a)

03 04 05 060

2

4

6

8

10

12

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 15

(b)

0

2

4

6

8

10

12

14

16

Density

03 04 05 06x

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 20

(c)

03 04 05 06 07 080

2

4

6

8

10

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 10

(d)

04 05 06 070

2

4

6

8

10

12

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 15

(e)

04 05 06 07x

0

2

4

6

8

10

12

14

16

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 20

(f)





119894119905is small


1 and







1


1






1and

2 In Table 2



0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)



1and




5 Conclusion







Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 070

2

4

6

8

10

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 10

(a)

03 04 05 060

2

4

6

8

10

12

x

Density

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 15

(b)

0

2

4

6

8

10

12

14

16

Density

03 04 05 06x

x = 05

N = 100

N = 200

N = 300

1205721 = 05 T = 20

(c)

03 04 05 06 07 080

2

4

6

8

10

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 10

(d)

04 05 06 070

2

4

6

8

10

12

x

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 15

(e)

04 05 06 07x

0

2

4

6

8

10

12

14

16

Density

x = 06

N = 100

N = 200

N = 300

1205722 = 06 T = 20

(f)





119894119905is small


1 and







1


1






1and

2 In Table 2



0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)



1and




5 Conclusion







Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10Density

x

1205721 = 05 T = 10

02 03 04 05 06 07

x = 05

N = 100

N = 200

N = 300

(a)

0

2

4

6

8

10

12

x

Density

1205721 = 05 T = 15

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(b)

0

2

4

6

8

10

12

14

16

Density

1205721 = 05 T = 20

x

03 04 05 06

x = 05

N = 100

N = 200

N = 300

(c)

x

0

2

4

6

8

10

Density

1205722 = 08 T = 10

x = 08

N = 100

N = 200

N = 300

05 06 07 08 09 1

(d)

0

2

4

6

8

10

12

Density

1205722 = 08 T = 15

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(e)

0

2

4

6

8

10

12

14

16

Density

1205722 = 08 T = 20

x

x = 08

N = 100

N = 200

N = 300

06 07 08 09

(f)



1and




5 Conclusion







Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Estimation of Nonlinear Dynamic Panel...

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