This note points out to applied researchers what adjustments are needed to the coefficient estimates in a random effects probit model in order to make valid comparisons in terms of coefficient estimates and marginal effects across different specifications. These adjustments are necessary because of the normalisation that is used by standard software in order to facilitate easy estimation of the random effects probit model

Arulampalam, Wiji

English

Warwick Research Archives Portal Repository

A NOTE ON ESTIMATED COEFFICIENTS IN RANDOM EFFECTS PROBIT MODELS*Wiji Arulampalam**Department of Economics,University of Warwick,Coventry, CV4 7AL,U.K.email: wiji.arulampalam@warwick.ac.uktel: (+44) (01203) 523471fax: (+44) (01203) 523032August, 1996.Revised October, 1998.Abstract:  This note points out to applied researchers what adjustments are needed to thecoefficient estimates in a random effects probit model in order to make valid comparisons interms of coefficient estimates and marginal effects across different specifications. Theseadjustments are necessary because of the normalisation that is used by standard software inorder to facilitate easy estimation of the random effects probit model.Keywords: Random effects probit.JEL Classification: C13, C23.* I should like to thank Alison Booth, Stephen Jenkins, Gordon Kemp and Chris Orme forhelpful comments.**  nee Narendranathan.1I.  INTRODUCTIONRandom effects (RE) probit models have recently become very popular due to the availabilityof data such as the British Household Panel Survey (Arulampalam and Booth (1998), Booth etal. (1996)).  This model imposes the restriction that the correlation between successive errorterms for the same individual is a constant and thus known in the literature as the‘equicorrelation’ model.  But a static model can be estimated by pooling the data and ignoringthis particular correlation structure to obtain c nsistent parameter estimates (Robinson (1982),Maddala (1987)). Hence any discussion between the pooled probit model estimates and a REprobit model estimates is not very illuminating under the maintained assumption that the REprobit specification is correct. Estimated coefficients between these models produced bypopular software such as Limdep (Greene (1998)), Stata (1997) etc. look different because ofdifferent normalisations that are used by these programs to facilitate easy estimation. This notepoints out to applied researchers what adjustments are needed in order to be able to makevalid comparisons in terms of coefficient estimates and marginal effects across differentspecifications.Econometric model is presented and the issues discussed in the next section.  Section IIIprovides the necessary calculations for the marginal effects and is followed by an illustrationusing an artificially generated data.  Final section concludes.II.  ECONOMETRIC MODEL AND ISSUESConsider the following model,yit* = xit’b + vit                     i=1,2,...,n  and  t=1,...,T (1)2vit = ai + uit (2)andyit = 1   if yit* > 0   and  = 0   else,where, y* denotes the unobservable variable, y is the observed outcome, x is observable timevarying and time invariant vector of strictly exogenous characteristics which influence y*, b isthe vector of coefficients associated with the x, ai denotes the individual specific unobservableeffect and the uit is a random error.  In the case of probit random effects (RE) it is alsoassumed that uit ~ IN(0,s2u).  In order to marginalise the likelihood we assume that,conditional on the xit, ai s are IN(0, s2a) and are independent of the uit s and the xit s.  Thisimplies that the correlation between two successive error terms for the same individual is aconstant given by,r = corr(vit, vit-1) =ss saa22 2+ u. (3)As shown in Heckman (1981), the parameters of this model are easily estimated by noting thatthe distribution of yit* conditional on ai are independent normal.  We have,Prob   Probu i itituiu( , ) ( )y it = = >- -=1 |'asasxxzit itbF( ) (4)wherez xitit i u = - +( )' /b a s (5)3and F is the distribution function of the standard normal variate.  We then marginalise theappropriate likelihood function with respect to a, which is given by{ [ ] [ ] } (  ) (  ) ( )t 1Tiy yit it11 11-ÕòÕ +-+-=-¥¥ -F Fx xit*it* * *' 'b b* *rrarra f a ad (6)where b *  = b /su  and  a* = a / sa .  Software programs such as Limdep and Stata returnestimates of  b * and r.  But note the fact that the coefficient estimates here are normalised onsu.In the above model, one could obtain consistent parameter estimates of the bcoefficients by ignoring the correlation structure (Robinson (198 ), Maddala (1987)).1  Thus, asimple pooled probit will provide consistent parameter estimates.  But, because of the binarynature of the data, a simple probit model estimation does not allow one to estimate the scaleparameter.  One only obtains parameter estimates of  b /sv.  Hence it is clear that pooledprobit parameter estimates will be equal to the RE probit model parameter estimates onlywhen s2a=0.  When s2a¹0 the consistent pooled probit estimates will not look similar to theRE probit estimates because of the normalisation.  A simple calculation will show that toconvert RE probit coefficients from b /su to b /sv, RE probit coefficient estimates need to bemultiplied by the factor 1- $r . This will enable one to make valid comparisons not onlybetween univariate and RE probit estimated effects, but also across estimates obtained fromdifferent specifications of the vector xit.                                         1 But the standard errors will be wrongly calculated.  The covariance matrix of the estimated parametersproduced by a standard software will be biased.  As discussed in Guilkey and Murphy (1993), a consistentestimate of the covariance matrix is obtained as  H-1GH-1  where, H is the hessian and the G the outerproduct of the score matrix.   For example, one can obtain a consistent estimate of the covariance matrix inLimdep using the ‘choice based sampling’ option with weights set equal to one for each observation.4III  MARGINAL EFFECTSBut how do we interpret the various estimated effects in these models with unobservableindividual specific components?  In standard cross-sectional univariate probits, it is customaryto provide expected changes in the outcome probability when particular characteristics arechanged one at a time,  known as marginal effects.Consider the mean effect of changing one particular continuous covariate xj, j=1,...k,by a small amount, on the outcome probability.  In the pooled probit, under the normalisationsv=1, this effect will be given by( )[ ] ( )[ ] ( )[ ]¶¶¶¶¶¶f b  y =1|   y    ( )  ProbxEx'x'it itjititjititjitit jx xx= = =F bb      j=1,..,k  (7)As this varies with the values of x we can either evaluate this at the means of the regressors orevaluate this separately for each individual in the sample and then average over the sample.But remember the implicit normalisation of sv=1 in the above evaluation.The associated covariance matrix may be calculated using the so-called ‘Delta-Method’ (see Greene (1997)) which uses a first order Taylor-series expansion to calculate thecovariance matrix in the case of non-linear functions of random variables.  This is provided inGreene (1997) for the probit model,Asy. Var [f( ) x' $ $b b ] = Asy. Var[$g ] = ¶¶¶¶$$$$'gbgbéëêêùûúúéëêêùûúúV (8)where V=Asy.var[$b ] and the marginal effects $g  are evaluated at the means of the variablesand hence f( )x' $b = $f  is a scalar.  The matrix of derivatives is5¶¶$$gbéëêêùûúú= $f I - $f  ( $ 'bx ).(9)where, I is a k dimensional Identity matrix.But in the case of a RE probit model, as shown above, taking the normalisation intoconsideration implies that in the calculation of the marginal effects, the coefficient estimatesneed to be multiplied by 1- $r  prior to using (7) giving( )[ ] ( ) ( ) ( )¶ ¶ b r r b  y =1|  = ( ) 1-    1-     Probx' 'it itjitit j it j*xx xf fb b *=            (10)Software will return estimates of  b* and r, and their covariance matrix W.  But$ $ $b b *= -1 r   which implies thatCovar( $b ) =A W A’               (11)where   A = 1 0 0 10 1 0 10 00 0 1 112- - -- - -- - -éëêêêêêùûúúúúú$ . . $ / $$ . . $ / $. . . .. . $ $ / $r b rr b rr b r(2 )(2 )(2 )k,             (12)k by (k+1) matrix which contains in its j’th row the derivative of $b j  with respect to all the $ *b j sas well as $r . To calculate the covariance matrix for the correct marginal effects given by (10),we can still use equation (8) but we need to replace the covariance matrix V by A W A’ (see(11)).6IV  ILLUSTRATIONThe above points are iluustrated using artificially generated data.  The data are generated fromthe model,yit = 1 if  (1.0 - 0.5 * xit + ai + uit) > 0    = 0 else, I=1,...,500,   and    t=1,2,and  xit , ai  and  uit  are all drawn as iidN(0,1).  This implies that  r given in (3) is 0.5 and thusthe theoretical correction factor 1- r  is  0.7071.  Results from the pooled probit and the REprobit model estimation using the above data, along with the marginal effects calculated withand without the corrections, are given in Table 1.  From these results, it is easy to see thatonce the correction is made to the RE probit coefficient estimates, they do not look verydifferent to those from a simple pooled probit model.  This is also true with regard to themarginal effects.  It is also seen that when the corrections are not made to the coefficients andmarginal effects the results can be very misleading.Table 1 - Coefficient Estimates and Marginal Effects (std. errors)Pooled Probit Random Effects ProbitUnadjusted AdjustedCoefficient MarginalEffectCoefficient MarginalEffectCoefficient MarginalEffectIntercept 0.705 (0.045)0.216 (0.007) 1.092 (0.112)0.230 (0.006)0.706 (0.052) 0.216 (0.008)x -0.402 (0.048)-0.123 (0.014)-0.651 (0.083)-0.137 (0.017)-0.421 (0.045)-0.129 (0.013)r 0.581 (0.065)LogLikelihood -523.6514 -497.4611Notes:  The marginal effects are calculated at the means (1, -0.058).7V  CONCLUSIONNot all commercially available software provide calculations of marginal effects in veryroutinely estimated models such as the random effects probit model.  These are of practicalimportance to applied researchers.  Comparison between different random effects probit modelcoefficient estimates (marginal effects), and between these and the pooled probit coefficientestimates (marginal effects) can be very misleading for the very simple reason of thenormalisation that is implemented in software to facilitate easy estimation.  This note discussesthis point and provides the necessary adjustments to enable applied researchers to make validcomparisons.8ReferencesArulampalam, W. and Booth, A. L. (1998) - ‘Training and labour market flexibility:  Is there atrade-off?’, forthcoming, British Journal of Industrial Relations, Dec.Booth, A.L., Garcia-Serrano, C. and Jenkins, S. P. (1996) - ‘New men and new women:  Isthere convergence in patterns of labour market transition?’, Institute of LabourResearch, Discussion Paper 96/01, University of Essex.Greene, W. H. (1997) - Econometric Analysis, Prentice-Hall International, Inc.Greene, W. H. (1998) - LIMDEP, Version 7.0: User’s Manual, Bellport, NY: EconometricSoftware.Guilkey, D. K. and Murphy, J. L. (1993) - ‘Estimation and testing in the random effects probitmodel’, Journal of Econometrics, 59, 301-317.Heckman, J. J. (1981) - ‘Statistical models for discrete panel data’, in Structural Analysis ofDiscrete Data with Econometric Applications, ed. C. F. Manski and D. McFadden, 114-178, Cambridge: MIT Press.Maddala, G. S. (1987) - ‘Limited dependent variable models using panel data’, Journal ofHuman Resources, XX11,3, 307-337.Robinson, P. M. (1982) - ‘On the asymptotic properties of estimators of models containinglimited dependent variables’, Econometrica, 50(1), 27-42.STATA (1997) - Statistical Software, Release 5, Stata Corporation.

Econometric Analysis,

A note on estimated coefficients in random effects probit models

Abstract

Similar works

Full text

Available Versions

Warwick Research Archives Portal Repository