Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070 Kyiv, Ukraine Email: coupe@eerc.kiev.ua Tel: 38-044 492-8012 * I hank he referees and Ehan Kaz for helpful commens and he Economics Research and Oureach Cener (EROC), Kyiv for financial suppor. The STATA do-file is available on he Poliical Analysis Web sie. 1
Absrac In a recen paper published in his journal, Kaz (2001) compares he bias in condiional and uncondiional fixed effecs logi esimaion using Mone Carlo Simulaion. This noe shows ha while Kaz s (2001) specificaion has wrong fixed effecs (in he sense ha he fixed effecs are he same for all individuals), his conclusions sill hold if I correc his specificaion (so ha he fixed effecs do differ over individuals). This noe also illusraes he danger, when using logi, of including dummies when no fixed effecs are presen. 2
1. Inroducion In a recen paper published in his journal, Kaz (2001) sudies he esimaion of he binary choice logi model when panel daa are available, using Mone Carlo Simulaion. He specifies he relaionship beween he binary variable y and X as follows (1) P(y i, exp( γ + αi + xi, β) = 1γ, αi, β ) = 1+ exp( γ + α + x β) i i, Where γ is a consan, α i an individual fixed effec and β he coefficien of he explicaive variable. Daa are available for many individuals i and ime periods, so I have a panel daa se-up. Kaz hen aims o compare wo ways of conrolling for fixed effecs, using uncondiional maximum likelihood bu including individual specific dummies and esimaing he model by condiional maximum likelihood. Condiional maximum likelihood is known o give consisen esimaes of β (Chamberlain, 1980) bu does no provide esimaes of he individual fixed effecs α i, which are needed if one wans o compue saisics like marginal effecs. The uncondiional-wih-dummies esimaor provides esimaes of he individual fixed effecs α i bu has been proven o lead o inconsisency in β due o he incidenal parameer problem for =2, Abrevaya (1997) has shown ha he uncondiional - wih - dummies esimaor of β equals wo imes he condiional maximum likelihood esimaor of β. Mone Carlo simulaions lead Kaz o conclude ha: The implicaion for applied researchers is ha he condiional esimaor is always safe when T<20 and ha he uncondiional esimaor is safe when 16 T<20. When 8<T<16, he bias in he uncondiional esimaor is small and may be accepable o he researcher bu when T is close o wo, he bias is subsanial. Even in his case, however, he bias akes a predicable direcion and magniude. 3
Unforunaely, Kaz s Mone Carlo simulaions are no wha hey preend o be. On p. 381, Kaz wries: I conrolled p by adjusing he consan erm γ, and I se α i o 1 for all i 1. This is a crucial misake. Indeed, if all α i s ake he same value, here is basically no fixed effec anymore. Tha is, he subscrip i can be removed from α i and he new consan erm γ =γ+α brings us o a simple logi model wihou fixed effecs. (2) P(y i, exp( γ' + xi, β) = 1,γ', β) = 1+ exp( γ' + x β) i, Hence, in Kaz s paper he rue model has no fixed effecs in conras o wha Kaz claims 2,3. In Table 1, I esimae he se-up of Kaz s paper using condiional logi and uncondiional logi wih dummies, and hen add a hird esimaion mehod, uncondiional logi wihou dummies (ha is, simple logi) 4. Noe ha he real underlying model is hree imes a simple logi, bu only he hird esimaion mehod realizes his. [Table 1 Here] Resuls for he condiional logi and he uncondiional logi wih dummies are similar o Kaz s resuls. Imporan o noe, however, is column 3, a simple logi gives resuls similar o he condiional esimaor even for small and hence, is much beer han he uncondiional wih dummies esimaor for small. Hence, in Kaz s se-up i is never useful o use he uncondiional wih dummies specificaion. Noe ha Kaz s sudy is an ineresing illusraion of he fac ha in he logi case, including irrelevan variables as regressors can cause biased esimaes, even if hese irrelevan variables are no correlaed wih he rue explicaive variables indeed, including individual specific 4
dummies, which by consrucion are uncorrelaed wih he (random) X, in he case where here are NO fixed effecs will cause a bias in he esimaes of he coefficien of X. I menioned before ha for =2, Abrevaya (1997) has shown ha he uncondiional - wih - dummies esimaor of β equals wo imes he condiional maximum likelihood esimaor of β. In our case, I found exacly he same resul, he coefficien of he dummy-variables specificaion is double he coefficien of he condiional specificaion hence, he bias ha I cause by including he dummies has nohing o do wih he presence of fixed effecs bu is solely due o he mehod of esimaion. To correc Kaz s sudy I generaed α i from N(0,1) insead of seing all α i s o one. To ge fixed effecs, I furher need a correlaion beween X and he fixed effecs. I propose he 5 following specificaion ha models X as a funcion of α i. 2 (3) x i = ( ai + ε i ),ε i : N(0,1) 2 In his way, I have a correlaion beween x i and α i ha is independen of ime T, while x i is sandard normally disribued 6,7. [Table 2 Here] Our conclusion from Table 2 is essenially he same as Kaz s conclusions: from =8, he uncondiional logi wih dummies esimaors gives resuls ha on average are quie close o he rue β. Bu noe ha in conras o Table 1, here, he use of simple logi (column 3) is clearly inferior as one would expec since he simple logi compleely ignores he fixed effecs. 5
2. Conclusions This noe shows ha while Kaz s (2001) specificaion has no meaningful fixed effecs, his conclusions sill hold if I modify his specificaion such as o incorporae fixed effecs hence, when one has a panel daa se wih a relaively large number of ime periods, using dummies o ake ino accoun he fixed effecs is no unreasonable. However, his noe also poins ou he danger, when using logi, of including dummies when no fixed effecs are presen. 6
3. References Abrevaya, Jason. 1997. The Equivalence of Two Esimaors of he Fixed-Effecs Logi Model. Economics Leers 55:41-43. Chamberlain, Gary. 1980. Analysis of Covariance wih Qualiaive Daa. Review of Economic Sudies 47:225-238. Greene, William. 2004. The Behaviour of he Maximum Likelihood Esimaor of Limied Dependen Variable Models in he Presence of Fixed Effecs. Economerics Journal 7: 98-119 Kaz, Ehan. 2001. Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion. Poliical Analysis 9:379-384. 7
4. Endnoes 1 Ialics added. 2 Of course, one could claim ha here are fixed effecs, bu hey jus happen o be he same. However, in such case he fixed effecs are meaningless. 3 This misake is also presen in he compuer program wrien by Kaz and posed on he Poliical Analysis websie (hp://polmeh.wusl.edu/pa/vol9no4.hml), hough in a slighly differen form. In he compuer program, he fixed effec of he firs individual is dropped, hence he formula is P(y i, exp( γ + αi + xi, β) = 1γ, αi, β ) = 1+ exp( γ + α + x β) i i, wih α i =1 for all i>1 and zero for α 1. Hence in his case, I jus make a difference beween he fixed effec of he firs individual (γ) and he fixed effecs of all oher individuals (γ+ α i ). 4 The parameers are he same as for column 1 in Table 2 of Kaz (2001): β=0.5, γ=-1 5 I could generae α i from N(0,1) and combine his wih a randomly generaed X. This leads o wha economiss would call a random effec since by consrucion here is no correlaion beween he individual specific effecs and he explicaive variable X. The esimaion procedure in his case would be o use he random effecs esimaor, raher han including dummies o capure he fixed effecs. 6 Insead, one could follow Greene (2004) by generaing individual specific effecs ha are correlaed wih X as follows: 8
α i = T xi + a, a N(0,1) i i However, his specificaion implies ha he correlaion beween he fixed effecs and he explicaive variable will decrease as T grows since he influence of an individual x i on he average decreases wih. Hence, using his specificaion one canno disinguish beween he effec of increasing and decreasing he correlaion beween he fixed effecs and he explicaive variable. 7 Noe ha his implies ha I use a sandard normal disribuion for X raher han a uniform disribuion (which was used by Kaz). The disribuion of he X generally, however, has no influence on he resuls (see e.g. Greene (2004)). 9
5. Tables Table 1: Kaz s specificaion under differen esimaion mehods N=100 Condiional Logi Uncondiional Logi wih dummies Uncondiional Logi wihou dummies =2 0.472 (0.032) 0.944 (0.064) 0.488 (0.022) =4 0.520 (0.017) 0.696 (0.023) 0.514 (0.015) =8 0.506 (0.012) 0.579 (0.014) 0.504 (0.011) =12 0.505 (0.095) 0.551 (0.010) 0.498 (0.009) =16 0.495 (0.082) 0.529 (0.008) 0.500 (0.008) Based on 500 Mone Carlo simulaions. Parameers: p=0.56, β=0.5, γ =-1. Mone Carlo sandard errors in brackes. 10
Table 2: Fixed Effec Specificaion under differen esimaion mehods N=100 Condiional Logi Uncondiional Logi wih dummies Uncondiional Logi wihou dummies T=2 0. 535 (0.020) 1.070 (0.041) 1.122 (0.009) T=4 0.502 (0.009) 0.675 (0.013) 1.145 (0.006) T=8 0.508 (0.005) 0.583 (0.006) 1.030 (0.004) =12 0.501 (0.005) 0.549 (0.005) 1.183 (0.003) =16 0.496 (0.004) 0.529 (0.004) 1.165 (0.003) Based on 500 Mone Carlo simulaions. Parameers: β=0.5, γ =-1. Mone Carlo sandard errors in brackes. 11