Corresponding author Kaddour Hadri Department of Economics City University London, EC1 0HB, UK

Transcription

1 Misspeci cationinstochastic Frontier Models. Backpropagation Neural Network Versus Translog Model: A Monte Carlo Comparison 1 by C. Guermat University of Exeter, Exeter EX4 4RJ, UK C.Guermat@exeter.ac.uk and K. Hadri City University, Northampton Square, London EC1V 0HB, UK K.Hadri@city.ac.uk First version: April 1999 Corresponding author Kaddour Hadri Department of Economics City University London, EC1 0HB, UK K.Hadri@City.ac.uk JEL classi cation: C15; C21; C24; D24; Q12. Keywords: Stochastic Frontier Production; Backpropagation Neural Network; Technical E ciency; Monte Carlo; Maximum likelihood Estimation. 1 We gratefully acknowledge the hospitality of the School of Business and Economics at Exeter University and my thanks gotothe Head oftheschool, Mr. Martin Timbrell.

2 Summary: Little attention has been given to the e ects of misspeci cation on the estimation of stochastic frontier models andto the possibility of using backpropagation neural network as a exible functional form to approximate the production and cost functions. This paper has two aims. First, it uses Monte Carlo experimentation to investigate the e ect of misspeci cation on the nite sample properties ofthe maximum likelihood (ML) estimators ofthe half-normal stochastic frontier production functions. Second, it compares the performance of backpropagation neural network with that of translog. It is foundthat whenthe neural network is a serious alternative to the translog formulation, but that both options produce poor e ciency rankings when the data is generated by a Leon Tief or CES functions. Hence, when estimating e ciency, one should be aware of the possibilities of misspeci cation and alternative models should be considered. 2

3 1. Introduction Stochastic frontier production for cross-section data was independently suggested by Aigner, Lovell andschmidt (1977), Battese and Corra (1977) and Meeusen and van den Broeck (1977). Although there has been considerable research to extend and apply the initial model, stochastic frontier modeling has been centeredaround the Cobb-Douglas and the Translog formulations. A recent survey of this research is providedby Green (1993). The rst step in modeling a production function is to assume a structure of technology. This has usually meant specifying a production function with few parameters. However, recent advances in computer power and statistical techniques have made it easier to hanlde more complicated, but more exbile, functional forms. To the best of our knowledge, there has been no attempt to apply neural network methods as an alternative to the translog model. There is also a lack of research on the e ect of misspeci cation on the nite sample properties of e ciency estimation. Caudill (1995) conducted an extensive Monte Carlo study on the nite sample properties of stochastic frontier production functions assuming only a constant in the production function. The e ects of heteroscedasticity in stochastic frontier models were investigated by Caudill and Ford (1993) and Guermat and Hadri (1999) using Monte Carlo experiments. The e ects of misspeci cation have not been investigated yet and we intend to ll this gap. At the same time we attempt to see whether backpropagation neural network can be a useful addition to the modeling of e ciency using stochastic productionfrontiers. The remainder of this paper is organized as follows. Section 2 outlines the models and notation while in section 3 we describe the Monte Carlo experiment. In section 4 we present and discuss the results. The nal section presents the main ndings and remarks. 2. Models and Notation The basic model used in the literature to describe a frontier production function can be writtenas follows: y i =g(x i ; )+w i v i ; (1) wherey i denotes the logarithm of the production for the ith sample farm (i = 1;:::;N) ;g(:) is a function such as the Cobb-Douglas or the Translog function; X i is a (1 k) vector of the logarithm of the inputs associated with theith sample 3

4 farm (the rst element would be one when an intercept term is included); is a (k 1) vector of unknown parameters to be estimated;w i is a two-sided error term with E[w i ]= 0, E[w i w j ] =0for all i andj, i 6=j; var(w i ) =¾ 2 w ; v i is a non-negative one-sided error term withe[v i ]>0,E[v i v j ]=0for alli andj,i6=j; and var(v i )=¾ 2 v : Furthermore, it is assumed thatw andvare uncorrelated. The one-sided disturbance v re ects the fact that each rm s production must lie on or below its frontier. Such a term represents factors under the rm s control. The two-sidederror term represents factors outside the rm s control. If we assume thatv i is half-normal andw i is normal, then the density function of their sum, derived by Weinstein (1964), takes the form: f(² i )=(2=¾)f (² i =¾)(1 F ( ² i =¾)); 1<² i <+1; (2) where² i = w i +v i ;¾ 2 = ¾ 2 w+¾ 2 v; =¾ v =¾ w andf (:) andf (:) are, respectively, the standard normal density and distribution functions. The advantage of stochastic frontier estimation is that it permits the estimation of rm-speci c ine ciency. The most widely used measure of rm-speci c ine ciency suggested by Jondrow, Lovell, Materov and Schmidt (1982) based on the conditional expected value ofv i given² i is given by where¾ =¾ v ¾ w =¾: E[v i j² i ]=¾ [ ² i =¾+f (² i =¾)=F ( ² i =¾)]; (3) The log-likelihood function is logl( ; ;µ)= NX i=1 log(f i (² i )): (6) In this paper we consider the following functional forms forg(:): 1. Cobb-Douglass: 2. Translog: kx g(x)= 0 ky i=1 g(x)= 0+ j=1 jlnx j x i i XX i j ij lnx i lnx j

5 ( ij = ii ): 3. Backpropagation Neural Network: We use the three layer (neurons) backpropagation function: g(x)= 3X j=1 j=(1+exp( kx i=1 ij x i )) (in the simulation we take lnx instead of x for comparison purpose). 4. CES: g(x)= 0( kx i=1 ix p i) 1=p 5. Generalised Leontief: g(x)= X i X j ij (x i x j ) 1=2 ( ij = ii ): 3. The Monte Carlo Design In order to investigate the performance of the neural network speci cation against the translog and in order to examine the e ects of misspeci cation on the nite sample properties of the maximum likelihood (ML) estimators of the half-normal stochastic frontier production functions we use a Monte Carlo experiment. We use two inputs (k=2) to simulate frontier data using a simple Cobb-Douglass function, the CES function (with p=0.5), and the Generalised Leontief function. We also use the translog function to compare the performance of the neural network model with the simple Cobb-Douglass model. The general form of the models is as follows: lnq i =g(l i ;K i ; )+W i V i ; where W and V are a normal error variable and a half normal error respectively. We generate data using the following procedure: 5

6 fl i ;K j g are pairs(i;j),i=1,:::; p N, andj=1;:::; p N recursively. The two error terms were generated asw»n(0;¾ 2 w ) andv» jn(0;¾2 v )j. The variances were modeled as follows: ¾ v = 0 ¾ w = 0 The parameters were set as follows: 1. For the Cobb-Douglass and Translog models: 0=1; 1= 2=0:5; 0 =0:3; 0 =0:01: We also used 0 =3; 0 =0:1 for the Cobb-Douglass model. 2. For the CES andthe Generalised Leontief models: 0=2; 1= 2=1; 0 =0:3; 0 =0:01: We also considered di erent sample sizes: 100, 200, 300 and 400 observations. To nd out if the number of replication of 1000 (see Hadri and Garry (1998) on the importance of the number of replications) was su cient, we carried out some experiments with replications. We found that the results were not signi cantly di erent. For each replication, and for both models, we compute the mean estimated e ciency, its standard deviation, the rank correlation coe cient between the estimated e ciencies and the true e ciencies, and the maximum and minimum estimated e ciencies. At the endofthe replications, these statistics are averaged and comparedto the true mean, maximum andminimum e ciencies. 4. The Results The main results of the Monte Carlo experiment are presented intable 1-5. The mean e ciency, the standard deviations, the mean rank correlation coe cients between the true rank and the estimated one, and the mean maximum and minimum are shown. In Table 1 the data was generated by a Cobb-Douglass function with two di erent sets of variances. This shows the case of high mean e ciency as in Table 6

7 1 as well as the case of low mean e ciency as in Table 2. Both models performed well. The mean estimated e ciency is very close to the true one and the standard deviation is very small for all N. The maxima and minima are also very close to the true ones. The mean rank correlations are all over 93%. However, the tables reveal that the translog model is slightly more accurate than the neural network model. This is expected since the translog model is nested with the Cobb-Douglass model. Table 3 shows the result for the CES case. The estimators are slightly less accurate compared to the previous case, but the meane ciency, standard deviation, maximum and minimum e ciencies are close enough to the true ones. We nd no signi cant di erence between the translog and the neural network models in terms of these statistics. However, the ranking of e ciencies deteriorates signi cantly, varying between 68.7% and 26.8%. We note that the neural network produces a higher ranking than the translog. Table 4 displays the results for the generalised Leontief case. The accuracy is similar to the CES case, except the ranking which has worsened. In this case the highest mean rank correlation is only 42.1%, with the translog model being marginally better for N=100 and 200 while the neural network has higher ranking for N=300 and 400. In general there is little di erence between the two models when the true data are generated by a generalised Leontief model. Finally, Table 5 shows the results when the data were gerated by the translog function. The comparison is now made between the neural network model and the Cobb-Douglass model. The superiority of the neural network is clear in every aspect. 5. Conclusion The results of the present Monte Carlo experiment show that misspeci cation can have a serious e ect on the ranking of rms. However, if the researcher is only interested in the mean e ciency, the simulation shows that misspeci cation does not a ect the mean, maximum or mimimum signi cantly. The estimation remains acceptable for both the translog and the neural network models. On the other hand, if the aim is to rank the rms in terms of e ciency, we note that when using the wrong speci cation, the ranking could be misleading. The rank correlation between the true and estimated e ciencies can drop to less than 30% when we use the wrong model. The neural network speci cation is a good alternative to the most commonly used translog function. In the Cobb-Douglass 7

8 case the translog is only marginally superior. In the generalised Leontief case, the two models are similar. But in the CES case the neural network produces higher mean rank correlations. Finally, when we compared the neural network with the Cobb-Douglass models on data generated by a translog function, the neural network model was well ahead. This paper shows that there is a need for considering various speci cations when modeling production and cost functions. The general practice of relying solely on a Cobb-Douglass or Translog functions may lead to serious misrepresentation of the ranking of rms, although the mean e ciency remains accurate. We have also shown that the neural network speci cation should also be considered among other speci cations. 8

9 References Aigner, D.J., Lovell, C.A.K., and Schmidt, P. (1977), Formulation and Estimation ofstochastic Frontier ProductionFunctionModels, Journal of Econometrics, 6, Battese, G.E., and Corra, G.S. (1977), Estimation of a Production Frontier Model: With Application to the Pastoral Zone of Eastern Australia, Australian Journal of Agricultural Economics, 21, Caudill, S.B. and Ford, J.M. (1993) Biases in frontier estimation due to heteroscedasticity, Economics Letters, 41, Caudill, S.B, Ford, J.M., and Gropper, D.M. (1995), Frontier Estimation and Firm- Speci c Ine ciency Measures in the Presence of Heteroscedasticity, Journal of Business & Economic Statistics, 13, Coelli, T.J. (1995) Estimation and hypothesis tests for a stochastic frontier function: a Monte Carlo analysis. J. Prod. Anal., 6, Green, W.H. (1993), The econometric Approach to E ciency Analysis, in The measurement of Productive E ciency, eds. H.O. Fried, C.A.K. Lovell, and S.S. Schmidt, New York: Oxford University Press, pp Hadri, K., and J. Whittaker. (1995) e ciency, environmental contaminants and farm size: testing for links using stochastic production frontiers discussion paper in economics 95/05, Exeter University. Hadri, K. (1997). A frontier Approachto Disequilibrium Models, Applied Economic Letters, No. 4, pp Hadri, K., C. Guermat, and J. Whittaker. (1999) Doubly heteroscedastic stochastic production frontiers with application to English cereals farms Discussion paper in economics, Exeter University. Hadri, K. and G.D.A. Phillips. (1999) The accuracy of the higher order bias approximation for the 2SLS estimator, Economics Letters, 62, Hadri, K. (1999) Estimation of a doubly heteroscedastic stochastic frontier cost function, forthcoming in the Journal of Business & Economic Statistic. Jondraw, J., Lovell, C.A.K., Materov, I., andschmidt, P. (1982), On the Estimation of Technical Ine ciency in Stochastic Production Function Model, Journal of Econometrics, 19, Meeusen, W., and van den Broeck, J. (1977), E ciency Estimation from Cobb- Douglas Production Function with Composed Error, International Economic Review, 18, Prais, S.J. and H.S. Houthaker. (1955), The analysis of family budgets, Cambridge University Press, Cambridge. 9

10 Weistein, M.A. (1964), The Sum of Values From a Normal anda Truncated Normal Distribution, Technometrics, 6, (with some additional material, ). White, H. (1982), Maximum Likelihood Estimation of Misspeci ed Models, Econometrica, 50,

11 Table 1. Simulation Results (DGP=Cobb Douglass) (σv=0.3, σw =0.01). Translog Neural Net. True N=100 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=200 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=300 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=400 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min Table 2. Simulation Results (DGP=Cobb Douglass) (σv=3, σw =0.1). Translog Neural Net. True N=100 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=200 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=300 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=400 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min

12 Table 3. Simulation Results (DGP=CES) (σv=0.3, σw =0.01). Translog Neural Net. True N=100 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=200 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=300 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=400 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min Table 4. Simulation Results (DGP=Leontief) (σv=0.3, σw =0.01). Translog Neural Net. True N=100 Eff. Mean Eff. SD Mean Rank Eff. Max Eff. Min N=200 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=300 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=400 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min

13 Table 5. Simulation Results (DGP=Translog) (σv=0.3, σw =0.01). Cobb Douglass Neural Net. True N=100 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=200 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N= 300 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min N=400 Eff. mean Eff. SD Mean Rank Eff. Max Eff. Min