1 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551, USA b Hmilton College, 198 College Hill Rod, Clinton, NY 1333, USA Received 11 December 1998; ccepted 1 My 1999 Abstrct Using Monte Crlo pproch, we find tht the bis of LSDV for dynmic pnel dt models cn be sizeble, even when T 5 0. A corrected LSDV estimtor is the best choice overll, but prcticl considertions my limit its pplicbility. GMM is second best solution nd, for long pnels, the computtionlly simpler Anderson Hsio estimtor performs well Elsevier Science S.A. All rights reserved. Keywords: Pnel dt; Simultion; Dynmic model JEL clssifiction: C3; O11; E00 1. Introduction The revitliztion of interest in long-run growth nd the vilbility of mcroeconomic dt for lrge pnels of countries hs generted interest mong mcroeconomists in estimting dynmic pnel models. However, microeconomists hve generlly been more vid users of pnel dt, nd, thus, existing pnel techniques hve been devised nd tested with the typicl dimensions of microeconomic dtsets in mind. These dtsets usully hve time dimension fr smller nd n individul (country) dimension fr greter thn the typicl mcroeconomic pnel. This difference is importnt in choosing n estimtion technique for two resons. First, it is well known tht the LSDV (lest squres dummy vrible) model with lgged dependent vrible genertes bised estimtes when the time dimension of the pnel (T ) is smll. Thus, for mny q This pper represents the views of the uthors nd should not be interpreted s reflecting those of the Bord of Governors of the Federl Reserve System or other members of its stff. *Corresponding uthor. Tel.: ; fx: E-mil ddress: (A.L. Owen) / 99/ $ see front mtter 1999 Elsevier Science S.A. All rights reserved. PII: S (99)
2 10 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) 9 15 mcroeconomists, the question, How big should T be before the bis cn be ignored?, is criticl one. A second reson tht mcro pnels my require different estimtion techniques thn those used on micro pnels is tht recent work investigting the ppropriteness of competing estimtors hs generted conflicting results, showing tht the chrcteristics of the dt influence the performnce of 1 n estimtor. We evlute severl different techniques for estimting dynmic models with pnels chrcteristic of mny mcroeconomic pnel dtsets; our gol is to provide guide to choosing pproprite techniques for pnels of vrious dimensions. Our work most closely follows Kiviet s (1995); however, we focus our ttention on dt with the qulities normlly encountered by mcroeconomists while he focuses on the short (smll T ), wide (lrge N) pnels typicl of micro dt. We hve three min conclusions. First, mcroeconomists should not dismiss the LSDV bis s insignificnt. Even with time dimension s lrge s 30, we find tht the bis my be equl to s much s 0% of the true vlue of the coefficient of interest. However, using n RMSE criterion, the LSDV performs just s well or better thn mny lterntives when T530. With smller time dimension, LSDV does not dominte the lterntives. Second, for pnels of ll sizes, corrected LSDV estimtor generlly hs the lowest RMSE. However, implementtion of the corrected LSDV for n unblnced pnel hs not been derived nd therefore lterntives my be needed. When the corrected LSDV is not prcticl, GMM procedure usully produces lower RMSEs reltive to the Anderson Hsio estimtor. Finlly, we find tht restricted GMM estimtor tht uses subset of the vilble lgged vlues s instruments increses computtionl efficiency without significntly detrcting from its effectiveness.. The problem nd proposed solutions We consider the dynmic fixed effects model yi,t5 gyi,t1 1 x9i,tb 1hi1 e i,t; ug u, 1 (1) i i,t i,t e where h is fixed-effect, x is (K 1) 3 1 vector of exogenous regressors nd e N(0, s )is rndom disturbnce. We ssume e s. 0, E(e i,t, e j,s) 5 0 E(x, e ) 5 0 i,t j,s i ± j or t ± s ; i, j, t, s () 1 Arellno nd Bond (1991) find tht GMM procedures re more efficient thn the Anderson Hsio estimtor. However, Kiviet (1995), using slightly different experimentl design, finds tht the Anderson Hsio estimtor compres fvorbly to GMM nd concludes tht no estimtor is pproprite in ll circumstnces. We confine ourselves to models nd techniques most likely to be of prcticl use in mcro pnels. We thus limit our study to sttionry dt, GMM with first-moment instruments, nd T between 5 nd 30. Other Monte Crlo studies hve considered models beyond these prmeters: Ko (1997) nd Pedroni (1997) explore unit roots; Blundell nd Bond (1998) investigte restricting initil conditions; Zilik (1997) reviews vriety of methods, but only for T #11. Ahn nd Schmidt (1995) nd Kene nd Runkle (199) consider exploiting dditionl moment restrictions. Since the writing of this pper, Bun nd Kiviet (1998) hve done further work in which they consider moderte T nd N.
3 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) The fixed effects model we hve chosen is common choice for mcroeconomists, nd is generlly more pproprite thn rndom effects model for two resons. First, if the individul effect represents omitted vribles, it is likely tht these country-specific chrcteristics re correlted with the other regressors. Second, it is lso likely tht typicl mcro pnel will contin most countries of interest nd, thus, will not likely be rndom smple from much lrger universe of countries (e.g. n OECD pnel contins most OECD countries). The model in Eq. (1), however, includes s one of the regressors lgged dependent vrible. In this cse, the usul pproch to estimting fixed-effects model, LSDV, genertes bised estimte of the coefficients. Nickell (1981) derives n expression for the bis of g when there re no exogenous regressors, showing tht the bis pproches zero s T pproches infinity. Thus, LSDV only performs well when the time dimension of the pnel is lrge. Severl estimtors hve been proposed to estimte Eq. (1) when T is moderte. With typicl mcro dtset in mind, we implement Monte Crlo study to consider four estimtors: n instrumentl vribles estimtor proposed by Anderson nd Hsio (1981), two GMM estimtors 3 discussed in Arellno nd Bond (1991), nd corrected LSDV estimtor derived in Kiviet, Henceforth, we cll the Anderson Hsio estimtor, AH, Arellno nd Bond s one-step estimtor 4 GMM1 nd their two-step estimtor GMM, nd Kiviet s corrected LSDV estimtor, LSDVC. 3. Methodology Our dt genertion process closely follows Kiviet (1995). The model for y ws generted with it is given in Eq. (1); x x 5 rx 1 j j N(0, s ). (3) i,t i,t1 i,t i,t j j it it Thus, in ddition to b, r nd s lso determine the correltion between y nd x. Kiviet defines signl to noise rtio, s s 1 it s s 5 vr(vit e i,t), v i,t; yi,t]] h i (4) 1 g nd shows tht it cn be clculted from other prmeters of the model s follows F 1 (g 1 r) g s j ]]] ]] e G s 5 b s 1 1 [gr 1] (gr) 1 s. (5) 1 1 gr 1 g The higher the signl-to-noise rtio, the more useful xit is in explining y it. Kiviet (1995) finds tht vrying the signl-to-noise rtio significntly lters the reltive bis of the estimtors. We lso choose b 5 1 g so tht chnge in g ffects only the short-run dynmic reltionship 3 We consider here only the AH estimtor tht uses the lgged level s n instrument becuse Arellno (1989) shows tht using the lgged difference is inefficient. 4 Anderson nd Hsio (1981), Arellno nd Bond (1991), nd Kiviet (1995) offer more thorough discussion of ech of these estimtors. GAUSS progrms re vilble from the uthors.
4 1 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) 9 15 e s between x nd y nd not the stedy-stte reltionship. Thus, given choices for g, s, s, nd r, ll of the other prmeters of the model re determined. Our prmeter choices cn be summrized s follows: s e is normlized to 1, r is set t the intermedite vlue of 0.5, s s lterntes between vlue of nd 8, nd g lterntes between 0. nd 0.8. For ech combintion of prmeters we vry the size of our pnel. N, the cross-sectionl dimension, tkes on vlues of 0 or 100, nd T, the time dimension, is ssigned vlues of 5, 10, 0 nd 30. In totl, we hve 3 different prmeter combintions. We generte the dt by choosing x, y 5 0 nd then discrding the first 50 observtions before i,0 i,0 selecting our smple. We performed 1000 replictions with fixed seeds for the rndom number genertor so tht our results cn be replicted. 4. Results We first exmine the bis of the OLS nd LSDV estimtors for vrious pnel sizes. Tble 1 5 summrizes the results from this initil experiment for subset of prmeter vlues. These results confirm severl well-documented conclusions bout these estimtors: (1) in both cses, the bis of g is more severe thn tht for b, () OLS provides bised estimtes even for lrge T, nd (3) the bis of the LSDV estimtor increses with g nd decreses with T. In ddition, these results show tht the bis of the LSDV estimte is not unsubstntil, even t T50. When T530, the verge bis becomes smller, lthough the LSDV does not become more efficient. Bsed on the results in Tble 1, one could expect n LSDV estimte with bis from 3% to 0% of the true vlue of the coefficient even when T530. Errors of this mgnitude, however, would still result in n estimte with the correct sign. Since LSDV is often inpproprite, we explore the properties of other estimtors. Before mking n overll comprison, we first nrrow our selection by compring vrious GMM procedures. Arellno Tble 1 OLS nd LSDV bis estimtes T g g bis b b bis OLS (S.E.) LSDV (S.E.) OLS (S.E.) LSDV (S.E.) (0.039) (0.040) (0.044) (0.045) (0.06) (0.058) (0.055) 0.07 (0.070) (0.03) (0.03) (0.031) (0.06) (0.017) 0.3 (0.03) (0.037) 0.00 (0.045) (0.08) 0.07 (0.015) (0.03) (0.017) (0.01) (0.019) (0.06) (0.08) (0.06) (0.01) (0.019) (0.014) (0.011) (0.014) (0.00) (0.0) 1000 drws; N 5 100; se 5 1; sj 5 ; r Results for full set of prmeter vlues for this experiment nd ll others described in this pper re qulittively similr to those reported nd re vilble from the uthors.
5 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) nd Bond (1991) discuss two vrints of GMM procedure tht use ll lgged vlues s instruments. When T gets lrge, however, computtionl requirements increse substntilly nd GMM estimtion using ll vilble instruments my not be prcticl to implement. Results from Monte Crlo simultions (vilble from the uthors) indicte tht (1) the one-step GMM estimtor outperforms the two-step estimtion, nd () using restricted GMM procedure (the number of vlues of the lgged dependent vrible nd the exogenous regressors used s instruments is reduced 6 to three, five, nd seven instruments) does not mterilly reduce the performnce of this technique. Finlly, we mke n overll comprison between OLS, LSDV, AH, GMM nd LSDVC. Bsed on the GMM comprisons reported bove, we focus on two restricted GMM estimtors GMM13 nd GMM17 which re GMM1 with three nd seven instruments, respectively. Tble shows the verge bis, stndrd devitions nd RMSEs of our estimtes of g (the bis of the estimtes of b re reltively smll nd cnnot be used to distinguish between estimtors). The results in Tble show tht ll the estimtors (with the exception of OLS) generlly perform better with lrger N nd T. Thus, for sufficiently lrge N nd T, the differences in efficiency, bis nd RMSEs of the different techniques become quite smll. Even so, the results in Tble do highlight one technique tht consistently outperforms the others LSDVC. Unfortuntely, while LSDVC my produce superior results, it is not lwys prcticl to implement. In prticulr, method of implementing LSDVC for n unblnced pnel hs not yet been developed. This is prticulrly importnt considertion for mcro dt since the likelihood of hving n unblnced pnel my increse s the time dimension gets lrge. Our results indicte tht if LSDVC cnnot be implemented tht (1) when T530, LSDV performs just s well or better thn the vible lterntives, () when T #10, GMM is best nd (3) when T50, GMM or AH my be chosen. While GMM still produces the lowest RMSEs even when T50, the difference in performnce is not tht gret nd computtionl issues my become more importnt. Becuse the efficiency of the AH estimtor increses substntilly s T gets lrger, the computtionlly simpler AH method my be justified when T is lrge enough. 5. Conclusion The recommendtions from our Monte Crlo nlysis re summrized below. Summry of recommendtions T #10 T50 T530 Blnced pnel LSDVC LSDVC LSDVC Unblnced pnel GMM1 GMM1 or AH LSDV 6 For exmple, when we use seven instruments, we use seven lgged vlues of the dependent vrible (if vilble). For the exogenous regressors, x, we use the seven closest vlues three previous vlues, the contemporneous vlue nd three future vlues (when vilble). We lso checked if there were gins to iterting GMM or to using more thn seven instruments for T50 (where 18 instruments would be vilble). There ws miniml gin to either of these procedures.
6 14 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) 9 15 Tble Bis estimtes for g using vrious estimtors T N g OLS LSDV LSDVC A-HIV GMM13 GMM17 (S.E.) (S.E.) (S.E.) (S.E.) (S.E.) (S.E.) [RMSE] [RMSE] [RMSE] [RMSE] [RMSE] [RMSE] (0.09) (0.089) (0.110) (0.194) (0.111) (0.105) [0.6] [0.173] [0.109] [0.194] [0.1] [0.15] (0.06) (0.130) (0.161) (1.86) (0.56) (0.15) [0.071] [0.54] [0.49] [1.86] [0.439] [0.458] (0.039) (0.040) (0.046) (0.077) (0.05) (0.050) [0.8] [0.15] [0.046] [0.077] [0.053] [0.05] (0.06) (0.058) (0.080) (0.0) (0.150) (0.141) [0.055] [0.508] [0.154] [0.0] [0.190] [0.08] (0.07) (0.049) (0.05) (0.098) (0.058) (0.053) [0.] [0.078] [0.05] [0.098] [0.069] [0.071] (0.04) (0.07) (0.081) (0.18) (0.19) (0.097) [0.056] [0.48] [0.095] [0.19] [0.49] [0.55] (0.03) (0.03) (0.04) (0.043) (0.09) (0.07) [0.7] [0.063] [0.04] [0.043] [0.030] [0.09] (0.017) (0.03) (0.041) (0.088) (0.059) (0.05) [0.05] [0.34] [0.05] [0.088] [0.081] [0.097] (0.061) (0.033) (0.034) (0.063) (0.038) (0.035) [0.] [0.043] [0.034] [0.063] [0.050] [0.050] (0.030) (0.040) (0.046) (0.118) (0.073) (0.057) [0.051] [0.115] [0.047] [0.118] [0.150] [0.154] (0.08) (0.015) (0.015) (0.07) (0.018) (0.017) [0.7] [0.031] [0.015] [0.07] [0.019] [0.018] (0.01) (0.019) (0.03) (0.050) (0.033) (0.08) [0.051] [0.106] [0.03] [0.050] [0.044] [0.049] (0.057) (0.06) (0.07) (0.049) (0.030) (0.08) [0.] [0.03] [0.07] [0.049] [0.043] [0.04] (0.05) (0.030) (0.034) (0.088) (0.05) (0.04) [0.050] [0.074] [0.034] [0.088] [0.10] [0.1] (0.06) (0.01) (0.01) (0.01) (0.015) (0.013) [0.7] [0.01] [0.01] [0.01] [0.016] [0.015] (0.011) (0.014) (0.015) (0.037) (0.05) (0.00) [0.050] [0.067] [0.015] [0.037] [0.035] [0.036] 1000 drws; se 5 1; sj 5 ; r 5 0.5; m 5 1.
7 R.A. Judson, A.L. Owen / Economics Letters 65 (1999) References Ahn, S.C., Schmidt, P., Efficient estimtion of models for dynmic pnel dt. Journl of Econometrics 68, 5 7. Anderson, T.W., Hsio, C., Estimtion of dynmic models with error components. Journl of the Americn Sttisticl Assocition 76, Arellno, M., A note on the Anderson Hsio estimtor for pnel dt. Economic Letters 31, Arellno, M., Bond, S., Some tests of specifiction for pnel dt: Monte Crlo evidence nd n ppliction to employment equtions. Review of Economic Studies 58, Blundell, R., Bond, S., Initil conditions nd moment restrictions in dynmic pnel dt models. Journl of Econometrics 87, Bun, M.J.G., Kiviet, J.F., On the Smll Smple Accurcy of Vrious Inference Procedures in Dynmic Pnel Dt Models, University of Amsterdm, Mimeo. Ko, C., Spurious regression nd residul-bsed tests for cointegrtion in pnel dt. Journl of Econometrics, forthcoming. Kene, M.P., Runkle, D.E., 199. On the estimtion of pnel-dt models with seril correltion when instruments re not strictly exogenous. Journl of Business nd Economic Sttistics 10, 1 9. Kiviet, J.F., On bis, inconsistency, nd efficiency of vrious estimtors in dynmic pnel dt models. Journl of Econometrics 68, Nickell, S., Bises in dynmic models with fixed effects. Econometric 49, Pedroni, P., Pnel Cointegrtion: Asymptotics nd Finite Smple Properties of Pooled Time Series Tests With n Appliction To the PPP Hypothesis, Indin University, Working Ppers in Economics, No Zilik, J.P., Efficient estimtion with pnel dt when instruments re predetermined: n empiricl comprison of moment-condition estimtors. Journl of Business nd Economic Sttistics 15,