A Noe on Missing Daa Effecs on he Hausman (978) Simulaneiy Tes: Some Mone Carlo Resuls. Dikaios Tserkezos and Konsaninos P. Tsagarakis Deparmen of Economics, Universiy of Cree, Universiy Campus, 7400, Rehymno, Greece Absrac This shor paper demonsraes he effecs of using missing daa on he power of he well-known Hausman (978) es for simulaneiy in srucural economeric models. This es is a reliable es and is widely used for esing simulaneiy in linear and nonlinear srucural models. Using Mone Carlo echniques, we find ha he exisence of missing daa could affec seriously he power of he es. As heir number is geing larger, he probabiliy of rejecing simulaneiy wih Hausman es is increasing significanly especially in small samples. A Full Informaion Maximum Likelihood Missing Daa correcion echnique is used o overcome he problem and hen we find ou ha ha he es is more effecive when we rerieve hese daa and include hem in he sample. JEL Classificaion Numbers: C0,C,C5. Keywords: Hausman (978) simulaneiy es, srucural economeric models, FIML, missing daa, simulaion *Address for correspondence: Dikaios Tserkezos Deparmen of Economics, Universiy of Cree, Universiy Campus, 7400 Rehymno, Greece Tel. +30 830 7745 Fax. +30 830 77406 E-mail: serkez@econ.soc.uoc.gr
I. Inroducion I is a common pracice for developmens concerning mos economic variables o be analyzed conaining some missing daa. Very ofen, however, using only he available daa and ignoring he problem of missing daa, frequenly leads o erroneous resuls wih respec o he ime dependence in various economic variables. Similar effecs should also be expeced in he case of he applicaion of various ess based on he available daa each ime. Given he above, i is he aim of his shor paper o sudy he effecs of missing daa in esing simulaneiy in srucural economeric models. More specifically, we repor he resuls of a Mone Carlo experimen, which examines he effecs of missing daa on he power of he Hausman (978) es for simulaneiy. On he basis of hese Mone Carlo resuls i appears ha as he number of missing daa increases, he probabiliy of acceping he null hypohesis of simulaneiy using he Hausman es is decreased, especially in small samples of daa. As he number of he available daa increases, he problem is geing less problemaic and afer a cerain level of available daa i disappears. To overcome his problem we use a Full Informaion Maximum Likelihood missing daa approach (Sargan and Dreakis, 974) o esimae he missing daa and hen use hem for esing for simulaneiy. According o our Mone Carlo resul, he probabiliy of acceping he null hypohesis of simulaneiy increases significanly when we use simulaneously he available daa and he esimaed (rerieved) daa for esing for simulaneiy, especially in a small sample. The power of he es is more effecive as compared o applying he Hausman es in he case where he missing daa would have been ignored. Finally he effecs of missing daa on he power of he Hausman 978 simulaneiy es disappear as he number of he oal observaions increases, independenly of wheher we ignore he missing daa or we use he suggesed missing observaions echnique o rerieve hem.
The paper is organised as follows. Secion II presens he Hausman 978 srucural sysems simulaneiy es and he Full Informaion Maximum Likelihood approach o esimae he missing daa. Secion III presens he simulaion resuls. Secion IV sums up and provides some concluding remarks. II.The Hausman 978 Simulaneiy Tes. In order o apply he Hausman 978 simulaneiy es we follow a simple wo-sep procedure beween wo endogenous variables (y, y ) of a srucural sysem of equaions. In he firs sep we regress he firs endogenous variable wih he exogenous and predeermined variables of he sysem and esimae he residuals, say v. In he second sep we include he esimaed residuals in he second srucural equaion wih dependen variable being he y and we perform a -es on he coefficien of he esimaed residuals v. If he esimaed coefficien is saisically significan, we accep he null hypohesis of simulaneiy. The esimaed missing daa were obained using a Full Informaion Maximum approach based on he Sargan-Dreakis approach (Dreakis, 973), as follows: T min = g V g () mis sin g Daa Parameers where g is he (mx) vecor of he residuals of he srucural model wih m endogenous variables and σ σ. σ m σ σ σ m Var( g = ) = δ Ω s () σ m σ m σ m 3
Ω = V and δ s is he Kronecker dela. The missing observaions y i will be obained solving he normal equaions: ( T g V = y i g ) = 0 (3) III. The Mone Carlo Experimen Our simulaion experimen is based on he following srucural dynamic model: y β y + β x + ε = (4) y γ y + γ x + ε = (5) * * x τx + ( ( τ) ) w = (6) * * x τ x, + ( ( τ ) ) w = (7) ε σ σ. Cov = = ε σ σ.76.3 (8) w NID(.5) w NID(.5) (9) τ =.90 =. 5 τ (0) For differen numbers of available daa beween 60 and 700 we generae he dependen variables hrough he relaions of he sysem aking ino accoun he relaions (6)-(0). Five housand If we are concerned wih he endogeneiy of more han one variable, he analysis become somewha more complicaed, bu a similar es can applied (Pindyck and Rubinfeld p. 54). 4
replicaions of he wo dependen variables y and y are generaed. For each applicaion we assumed differen numbers of missing daa for he variable y. The number of missing daa is a percenage ranging from 6% o 60% on he number of he available daa. The missing daa for he dependen variable y were esimaed using he FIML approach which, in he case of he sysem described by (4)-(0) can be obained using he relaions: P P P 3 4 y = x + y + x P P P () wih: = ω+ γ ω γ ω P () ( β ω γ γ ) P = ω (3) 3 ( β ω γ ω + ω + β β ) P = ω (4) 4 ( γ γ ω γ ) P = ω (5) A wo-sep ieraive procedure is pursued o obain he missing daa: In he firs sep we apply a FIML procedure o obain esimaes of he parameers of he sysem using only he se of complee observaions. In he second sep using he esimaes of he parameers and he variance covariance marix of he esimaed residuals, we obained he missing daa using he relaion () wih ()-(5). Then he missing daa were included in he full se of daa o apply he Hausman 978 simulaneiy es. The resuls are summarized in Table and. In Table we presen he effecs missing daa have on he power of he Hausman 978 simulaneiy es for differen numbers of available and missing daa. In Table we presen he power of he Hausman 978 simulaneiy es having aken ino accoun he suggesed mehod of For a singe equaion missing daa approach applied o simulaneous sysems see: Dagenais (975). 5
recovering missing daa and applying he es o he full se. [TABLE ABOUT HERE] [TABLE ABOUT HERE] The following conclusions regarding he effecs of ime aggregaion on he power of he Hausman 978 simulaneiy es can be drawn from an analysis of he daa in Table and. On he basis of our Mone Carlo resuls in Table, i emerges ha as he number of missing daa increase, he probabiliy of acceping simulaneiy using he Hausman 978 es is decreasing especially in small samples of daa. As he number of he available daa is increasing, he siuaion is geing less problemaic and afer a level of available (N=700) daa i seems o disappear. To overcome his problem we use a Full Informaion Maximum Likelihood approach o esimae he missing daa and hen use hem, ogeher wih he iniial available daa, for esing r simulaneiy. According o Mone Carlo resuls, he probabiliy of acceping he rue hypohesis increases significanly when we use simulaneously he available daa and he esimaed missing daa for esing for simulaneiy, especially in small samples. The power of he es is more effecive compared o he case of leaving ou he missing daa compleely. The effecs of missing daa on he power of he Hausman 978 simulaneiy es disappear as he number of he oal observaions becomes large independenly of wheher we ignore he missing daa or we use hem (hrough he suggesed missing observaions echnique o rerieve hem). IV. Conclusions The resuls of his noe show he implicaions of missing daa in applied ime series work. Using 6
Mone Carlo echniques, we found ha he exisence of missing daa could someimes lead o erroneous conclusions abou he exisence of simulaneiy in he variables of a srucural economeric sysem. Using Hausman 978 srucural sysems simulaneiy es, we show ha, as he number of missing daa increases, he power of he es declines, leading o wrong conclusions abou he exisence of causal ineracion beween he endogenous variables of he srucural sysem. On he basis of our Mone Carlo resuls, i emerges ha as he number of missing daa increases, he probabiliy of rejecing simulaneiy using he Hausman 978 es is increasing especially in small samples of daa. As he number of he available daa increases, he siuaion is geing less problemaic and as daa increase beyond a cerain number (N= 700), i disappears. To overcome his problem we used a Full Informaion Maximum Likelihood approach o esimae he missing daa and hen used hem for esing for simulaneiy. According o our Mone Carlo resuls, he probabiliy of acceping he rue hypohesis is increasing significanly, especially in small samples and he es is more effecive compared wih he case where he missing daa are ignored. Las, he conclusions of his paper are in line wih he more general findings on he negaive effecs of missing daa and specially he case of ime aggregaion (Tserkezos e al., 998) on he effeciveness of he saisical crieria for conrolling he inerdependence beween economic magniudes. References 7
Anderson, T. W. (957) Maximum likelihood esimaes for a mulivariae normal disribuion when some observaions are missing, Journal of he American Saisical Associaion, 5(78), pp. 00-03. Dreakis, E. (973) Missing daa in economeric esimaion, Review of Economic Sudies, 40(4): 537-55. Dagenais, M. (975) Incomplee Observaions and simulaneous equaions models, Journal of Economerics, 4(3), pp. 3-4. Hausman, J.A. (978) Specificaion Tess in Economerics, Economerica 46(6), pp. 5-7. Gilber, C.L. (977) Regression using mixed annual and quarerly daa, Journal of Economerics, 5(), pp. -39. Pindyck R. S. & Rubinfeld D. L. (997) Economeric Models and Economic Forecass, Irwin/McGraw Hill Massachuses. Oguchi, N. and Fukuchi, T. (990). On emporal aggregaion of linear dynamic models, Inernaional Economic Review, 3(), pp. 87-93. Sargan, J.D. and Dreakis, E.S. (974) Missing daa in an auoregressive model, Inernaional Economic Review, 5(): 39-58. Tserkezos, D., Georgusos, D. & Koureas, G. (998) Temporal Aggregaion in Srucure VAR Models, Applied Sochasic Models and Daa Analysis, 4(), pp.9-34. Tserkezos, D. (99) Simulaneous Use of Annual and Quarerly Daa in Economeric Models, Working Paper,. Deparmen of Economics, Universiy of Piraeus. 8
Table. Probabiliy of ype I error (differen numbers of missing daa and available daa). Percenage of Number of available daa missing daa 60 0 00 300 400 600 600 700 6% 34.7 93.6 96.4 97.6 97.9 98. 98.4 98.9 % 0.8 9.8 96. 97.4 97.8 97.9 98.3 98.9 8% 0.7 9.3 95.5 96.7 97.6 97.7 98. 98.8 4% 0.5 88.0 94.8 96.4 97.3 97.5 98. 98.8 30% 0.4 79. 93.7 95.9 96.9 97. 98. 98.7 36% 0.3 45.8 9.5 95.0 96.4 96.7 97.8 98.6 4% 0.7 0. 89. 94. 95.9 96.3 97.4 98.3 48% 0. 0. 78.5 9.6 95. 95.9 96.9 98.0 54% 0.3 9.8 8.0 89. 93.7 95.0 96.4 97.6 60% 0.5 0. 8.7 68. 90.4 93.3 95. 96.6 Source: Daa enries are probabiliies of rejecing he wrong hypohesis, i.e. he exisence of measuremen errors in he independen variable of model ()-(4) The Hausman 978 es was replicaed 5000 imes for he specificaion (4)-(0) The size of he es is a=0.05. Daa enries are given by n/5000 where n is he number of imes he null hypohesis is acceped. 9
Table. Probabiliy of ype I error (differen numbers of missing daa and oal daa). Percenage of Number of available daa missing daa 60 0 00 300 400 600 600 700 6% 36.7 9.5 95.4 96.8 97.0 97.0 97.3 97.8 % 30.6 9. 94.6 96.0 96.9 96.5 96.8 97.0 8% 3.5 9.7 94.5 95.6 96.0 96.4 96.4 96.9 4% 3.8 89. 94.6 95.6 96.3 96.0 96.0 96.7 30% 3.9 8.6 94. 95.7 96.3 96. 97. 97.5 36% 33.0 70.7 93.3 95.7 96.7 96.9 97.4 98. 4% 33. 63.3 9.4 95. 96.8 96.8 97.7 98. 48% 3.9 64.3 86.3 94.4 96.4 96.7 97.4 98.3 54% 3.4 65.4 75.0 9.5 95.6 96. 97. 98.0 60% 3. 65.6 74.5 84. 93.6 95. 96.5 97.6 Source: Daa enries are probabiliies of rejecing he wrong hypohesis, i.e. he exisence of measuremen errors in he independen variable of he model ()-(4).The Hausman 978 es was replicaed 5000 imes for he specificaion (4)-(0). The size of he es is a=0.05. Daa enries are given by n/5000 where n is he number of imes he null hypohesis is acceped. 0