ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING

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1 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 ESTIMATION OF DYNAMIC PANEL DATA MODELS WHEN REGRESSION COEFFICIENTS AND INDIVIDUAL EFFECTS ARE TIME-VARYING Chung-ki Min Professor Hankuk Universi of Foreign Sudies Seoul Korea ABSTRACT This sud suggess an esimaion mehod for dnamic panel daa models when heir coefficiens and individual effecs are ime-varing. The quasi-differencing ransformaion is emploed o eliminae he ime-varing individual effecs. Empirical resuls from simulaed daa herein suppor he performance of he quasi-differencing approach for esimaion and es of dnamic panel daa models paricularl for small-sized samples. Kewords: dnamic panel daa model; ime-varing individual effecs; quasi-differencing; orhogonali condiions. INTRODUCTION The objecive of his sud is o sugges an esimaion mehod for dnamic panel daa models when heir regression coefficiens and individual effecs are ime-varing. As he regression coefficiens are likel o var over ime in reali paricularl during a long period of ime his sud is expeced o provide applied researchers wih useful implicaions. Afer appling he quasi-differencing ) ransformaion o eliminae he ime-varing individual effecs we esimae he ime-varing parameers b emploing he generalized mehod of momen GMM) Arellano and Bond 99; Arellano and Bover 995; and Blundell and Bond 998). We also appl a Wald es o examine wheher he regression coefficiens varied over ime. Empirical resuls from simulaed daa herein suppor he performance of he approach for esimaion and es of dnamic panel daa models paricularl for small-sized samples. 2. ESTIMATION METHODS Coprigh IJSSER 207 All righ reserved Page 244

2 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 The model considered in his sud is from a wo-variable vecor auoregressive regression of lag order one VAR). For cross-secional uni i =... M) and ime period =... T) his model allows for ime-specific and individual effecs. i i x f u ) i i where he error erm u i is uncorrelaed beween unis and beween ime periods and also saisfies he orhogonali condiions E u ] E x u ] 0 is i is i s ). The ime-specific effecs ) are common o all cross-secional unis. This model allows individual effecs o var over ime as he ime-invarian individual effecs f i are muliplied b a ime-varing coefficien Holz-Eakin e al. 988). Ahn e al. 200) lis various applicaions of panel daa models wih such ime-varing individual effecs. This sud emplos he ransformaion used in Chamberlain 983) and Holz-Eakin e al. 988). Afer mulipling Eq.) for ime period - b r / he resul is subraced from he equaion for ime period. or i i x d v 2) x 2 w v ' i i i where r r r w x x ]' and d ]'. i d r and v i ui r u The orhogonali condiions in Eq.) impl ha he error erm v i saisfies E isvi] E xisvi] 0 for s because of he presence of u in v i. Thus he insrumenal variables which can be used o esimae he parameers of Eq.2) are included in he following vecor. z i 2 i x 2 xi ]' Because of he ime-varing coefficiens he orhogonali condiions are defined separael for each Holz-Eakin e al. 988). Coprigh IJSSER 207 All righ reserved Page 245

3 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 M M z i v i i M 0 3) are consan over ime hen Eq.2) becomes he firs- If he regression coefficiens and differenced FD) specificaion wih r. or i ' i i h u i x i u 4) i where denoes he difference beween ime period and - h i x D DT ]' ]' and D DT ) are ime dummies wih heir corresponding coefficiens T T ). Since he regression coefficiens are consan over ime he insrumenal variables for he FD specificaion ) saisf he orhogonali condiions defined for he enire period no period b period. Z FD MT M i T z FD i u i M 0 5) 3. EMPIRICAL RESULTS USING SIMULATED DATA To evaluae he esimaion performance of he and he FD approach daa are generaed using he following VAR) specificaion. x i i x x 2 f u w i i i 6) Afer daa are generaed for = -9 o 2 he firs 20 observaions = -9 o 0) are discarded o minimize an effecs of he saring s. The s assigned for he ime-varing regression coefficiens are repored in Table 2. Values for he ime-specific effecs s) and for he individual effecs f i ) are independenl drawn from uniform disribuions ~ Uniform ) and ~ Uniform 2 2) respecivel. f i Coprigh IJSSER 207 All righ reserved Page 246

4 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 ) Case : ime-varing regression coefficiens ) We appl he and FD approaches wih assuming consan coefficiens in his secion. This is o compare how well he wo approaches can accoun for he ime variaion in. Table shows ha as he ime variaion in ) increases wih a larger ) he FD esimaes are severel biased. When he averages of he rue s ) are ) he FD esimaes are ˆ ˆ) 0.05 = ) for ) for 0.03 and ) for. Even when he insrumenal variables are used for esimaion he FD esimaes are sill biased; ) for for ) for 0.03 and ) In conras he esimaes are no biased when he variaion in ) is small wih 0.0 ˆ ˆ ) = ). For larger s of he esimaes are much less biased; ˆ ˆ ) = ) for and ) for This is because he can pariall accoun for he ime variaion in ) hrough he ime-varing individual effecs. Table : and FD specificaions esimaed wih assuming consan regression coefficiens when he varied over ime 0 consan) ransformaion FD ransformaion esimaed b Z esimaed b esimaed b esimae se esimae se esimae se Z FD Z Coprigh IJSSER 207 All righ reserved Page 247

5 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 Noe: The ime-varing regression coefficiens were generaed b where and were randoml drawn from a normal disribuion. 7 0 and N0 The rue s of he ime-varing regression coefficiens are repored in Table 2. 2 ) ) Esimaion and es for ime-varing regression coefficiens The ime-varing regression coefficiens ) are esimaed b he approach wihou assuming consan coefficiens. Table 2 shows ha for all of he cases he approach correcl esimaed he ime-varing coefficiens; he rue coefficiens are included in he 95% confidence inervals. Table 2: Esimaes of he ime-varing regression coefficiens b he approach rue 0 consan) esimae rue 0.0 esimae rue 0.03 esimae rue 0.05 esimae Wald es of a null hpohesis ha he regression coefficiens ) are consan over ime. p See he noe in Table. We also applied a Wald es o examine wheher he regression coefficiens significanl varied over ime. When he coefficiens were consan 0) or varing a lile 0.0) he Wald es failed o rejec he null hpohesis of consan coefficiens; he p- is and respecivel. As he ime variaion in ) increased wih a larger he null hpohesis was more significanl rejeced; he p- is for and for Coprigh IJSSER 207 All righ reserved Page 248

6 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 These es resuls are consisen wih he esimaion resuls in Table. When he assumpion of consan coefficiens was no rejeced for 0 and 0. 0 he regression coefficiens were correcl esimaed b he approach wih assuming consan coefficiens; ˆ ˆ) ) and ) as shown in he second column of Table. However for and wih which he assumpion of consan coefficiens was rejeced b he Wald es he esimaes were biased; ˆ ˆ) = ) and ). I is because he coefficiens were esimaed assuming consan when he varied significanl. These resuls in Tables and 2 provide useful implicaions for applied researchers as he regression coefficiens are likel o var over ime in reali paricularl during a long period of ime. If an econom experiences big changes such as global financial crisis and bad weaher he relaions beween variables of a model migh no be consan. Based on he above resuls his sud suggess ha we firs es abou he consanc of regression coefficiens using he esimaes of ime-varing coefficiens as shown in Table 2. If he ime variaion in ) is no significan he approach ogeher wih assuming consan regression coefficiens is expeced o produce correc esimaes and small sandard errors as shown in Table. 4. CONCLUSIONS In man panel daa ses here are a large number of cross-secional unis wih a shor period of ime. Therefore how o conrol for individual effecs becomes a main issue. In his sud he quasi-differencing ) approach was applied o eliminae he ime-varing individual effecs. The firs-differencing FD) approach which is valid onl when individual effecs and regression coefficiens are consan over ime was also applied. Overall he approach produced more reliable esimaes han he FD approach in all of he cases considered in his sud. As he regression coefficiens in reali are likel o var over ime his sud emploed he approach o esimae ime-varing regression coefficiens and o es wheher he are consan over ime. The resuls indicae ha if he Wald es fails o rejec a null hpohesis of consan coefficiens we ma assume consan regression coefficiens and use he approach bu no he FD). Therefore we conclude ha in all of he cases considered in his sud he approach dominaes over he FD for he esimaion and es of panel daa models. ACKNOWLEDGEMENT This work was suppored b he Hankuk Universi of Foreign Sudies Research Fund. = Coprigh IJSSER 207 All righ reserved Page 249

7 Inernaional Journal of Social Science and Economic Research Volume:02 Issue:0 REFERENCES Ahn S.C. Lee Y.H. and Schmid P. 200). GMM esimaion of linear panel daa models wih ime-varing individual effecs. Journal of Economerics Arellano M. and Bond S. 99). Some ess of specificaion for panel daa: Mone Carlo evidence and an applicaion o emplomen equaions. Review of Economics Sudies Arellano M. and Bover O. 995). Anoher look a he insrumenal variables esimaion of error componens models. Journal of Economerics 68) Blundell R. and Bond S. 998). Iniial condiions and Momen Resricions in dnamic panel daa models. Journal of Economerics Chamberlain G. 983). Panel daa. Ch. 22 in The Handbook of Economerics Volume II Ed. b Z. Griliches and M. Inrilligaor Amserdam: Norh-Holland Publishing Compan. Holz-Eakin D. Newe W. and Rosen H.S. 988). Esimaing vecor auoregressions wih panel daa. Economerica 56 6) Coprigh IJSSER 207 All righ reserved Page 250