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PPt of econometrics

1. Doctoral Seminar-I Department of Social Sciences Dr. Yashwant Singh Parmar University of Horticulture and Forestry, Nauni-Solan, Himachal Pradesh India-173230 AgEcon-691 (1+0) Speaker Amit Guleria Adm. No . F-14-01-D 1

2. Production Function Analysis for Resource use and Technical efficiency: A Case of Translog and Stochastic Frontier Production Function Seminar Title 2

3. Key Points Production function Translog production function Stochastic frontier Analysis Current research Comparison Between CD, Translog and SFA General finding 3

4. 4 Production Function It is a relationship between output and a set of input quantities. We use this when we have a single output In case of multiple outputs: people often use revenue (adjusted for price differences) as an output measure It is possible to use multi-output distance functions to study production technology. The functional relationship is usually written in the form:

5. 5 Production Function Specification A number of different functional forms are used in the literature to model production functions : Linear Cobb-Douglas (linear logs of outputs and inputs) Quadratic Translog function Translog function is very commonly used – it is a generalization of the Cobb-Douglas function It is a flexible functional form providing a second order approximation Cobb-Douglas and Translog functions are linear in parameters and can be estimated using least squares methods.

6. Translog production function The limitation of the CES production function, namely the constant elasticity of substitution and restriction on its general applicability, are partially responsible for the development of more flexible forms of the production function. The transcendental logarithmic , also called the translog production function, is one such form. 6

7. The function was first proposed by L. Christensen, D. Jorgenson and L. Lace in 1972. This form of the production function is becoming quite popular, mathematical simplicity to the applications of the Shephard’s Duality Theory and translog cost functions. 7

8. y=f(x 1 , x 2, x 3........ x n ) where, y is the output, a is the efficiency parameter, x i and x j are the i th and j th input quantities ( i,j = 1,2,.....n), a i and b ij are the unknown parameter. Just as in the case of other exponential type of production function is also often written in its logarithmic form. 8

9. Thus, there are two estimation problems: The number of inputs is increased, the number of parameter to be estimated increased rapidly. The additional terms are squares and cross products of inputs variables, thus making multicollinerity a difficult problem. Removal of squared and cross products terms t-ratios are non-significant or below certain critical value or modifying the functional form by excluding the square terms may be one way to get rid of the problem. Such approaches may, however, destroy flexibilities in the relationships. 9

10. Marginal product MP i = b ij >0, x j →0 MP i can be positive for a range of value of x j Ep i = j=1,2,3,......n Thus, it is obvious from eq. the elasticity of production w.r.t . the i th input is not a constant but varies with the level of input j (j=1,2,3,...n) Elasticity of production 10

11. Return to scale For a homogenous production, the scale economies may be less than equal to the sum of the production elasticities. R. Frish and C.E. Ferguson have establishment that even for homogenous function, the return to scale are equal to the sum of elasticity of production with respect to the different inputs. Thus, for a translog function which is non-homogenous the return of scale (function coefficient) are not invariant with input levels. The function coefficient for eq. given by: 11

12. 12 Cobb-Douglas Functional form for SFA Linear in logs Advantages: easy to estimate and interpret requires estimation of few parameters: K +3 Disadvantages: simplistic - assumes all firms have same production elasticities and that substitution elasticities equal 1

13. 13 Translog Functional form for SFA Quadratic in logs Advantages: flexible functional form - less restrictions on production elasticities and substitution elasticities Disadvantages: more difficult to interpret requires estimation of many parameters: K +3+ K ( K +1)/ 2

14. 14 Functional forms for Production Frontier Cobb-Douglas: lnq i =  +  1 ln x 1i +  2 ln x 2i + v i - u i Translog : lnq i =  +  1 ln x 1i +  2 ln x 2i + 0.5 11 ( ln x 1i ) 2 + 0.5 22 ( ln x 2i ) 2 +  12 ln x 1i ln x 2i + v i - u i

15. 15 Interpretation of estimated parameters Cobb-Douglas: Production elasticity for j- th input is: E j =  j Scale elasticity is:  = E 1 +E 2 Translog : Production elasticity for i-th firm and j- th input is: E ji =  j +  j1 ln x 1i +  j2 ln x 2i Scale elasticity for i-th firm is:  i = E 1i +E 2i Note: If we use transformed data where inputs are measured relative to their means, then Translog elasticities at means would simply be  i .

16. 16 Test for Cobb-Douglas versus Translog Using sample data file which comes with the FRONTIER program H :  11 = 22 = 12 =0, H 1 : H false Compute -2[LLF o -LLF 1 ] which is distributed as Chi-square (r) under H o . For example, if: LLF 1 =-14.43, LLF =-17.03 LR=-2[-17.03-(-14.43)]=5.20 Since  3 2 5% table value = 7.81 => do not reject H

17. Production frontier Two major problems It is not the average function, so interest has centred on determining in examining the shape and location of the frontier. Interest in examining the nature of the relationship between the production frontier and fitted average function. 17

18. Average function permits ranking of observations by technical efficiency, but not provide any qualitative information. An estimated frontier carries the promise of shedding light on the actual magnitude of technical efficiency. 18

19. Types of Production Frontiers Deterministic production frontier Probabilistic production frontier Stochastic production frontier 19

20. Deterministic Production Frontier Computed by mathematical programming techniques But it involves various statistical problems, although some disturbance must implicitly be assumed, no assumption are made about its properties; as result, the parameters are not estimated in the statistical sense. And it is extremely sensitive to the outliers 20

21. Probabilistic Production Frontier Deterministic production frontier is computed by mathematical programming techniques, after which supporting data points are discarded and a new deterministic frontier is computed. The process continues until the computed frontier stabilizes. The probabilistic frontier approach thus solves the outlier problem by discarding outliers from the sample, and no attempt is made to reconcile their placement above the final frontier with the theoretical concept of a frontier as an upper bound on output. Since probabilistic production frontier is computed from a subset of the original sample, it remains vulnerable, it is computed rather than estimated, hypothesis testing is impossible. 21 Timmer , 1971

22. Stochastic Production Frontier Ameliorate the problem associated with both deterministic and probabilistic production frontier It permits firms to be technically inefficient relative to their own frontier rather than to some sample norms. Inter-firm variation of the frontier presumably captures the effects of exogenous shocks, favourable and unfavourable, beyond the control of the firms. Errors of observation and measurement on output constitute another source of the variation in the frontier. 22

23. 23 Stochastic Frontier Analysis It is a parametric technique that uses standard production function methodology. The approach explicitly recognizes that production function represents technically maximum feasible output level for a given level of output. The Stochastic Frontier Analysis (SFA) technique may be used in modeling functional relationships where you have theoretical bounds: Estimation of cost functions and the study of cost efficiency Estimation of revenue functions and revenue efficiency This technique is also used in the estimation of multi-output and multi-input distance functions Potential for applications in other disciplines

24. The statistical specification is that the disturbance term is made up of two parts : A symmetric (normal) component capturing randomness outside the control of the firm (v) And a one sided (non-positive) component randomness under the control of the firm (i.e. Inefficiency, u). 24

25. Technical Inefficiency Production process can be technical inefficient, in the sense that it fails to produce maximum output from a given input; technical inefficiency results in an equi -proportionate overutilization of the all inputs; Allocatively inefficient in the sense that the marginal revenue product of an input might not be equal to marginal cost of that input; allocative inefficiency results in utilization of inputs in the wrong proportions , given input prices. Since estimation of production frontiers is carried out with observations on output and inputs only, such an exercise cannot provide evidence bearing on the matter of allocative efficiency and hence cannot be used to draw inferences about total or economic efficiency. 25

26. 26 Deterministic Frontier models In this model all the errors are assumed to be due to technical inefficiency – no account is taken of noise. Consider the following simple specification: x i ’s are in logs and include a constant u i is a non-negative random variable. Therefore ln y i  x i  . Given the frontier nature of the model we can measure technical efficiency using:

27. 27 Deterministic Frontier models We have Frontier is deterministic since it is given by exp(x i ) which is non-random. Estimation of Parameters: Linear programming approach

28. 28 Deterministic Frontier models Aigner and Chu suggested a Quadratic prrogramming approach Afriat (1972) suggested the use of a Gamma distribution for u i and the use of maximum likelihood estimation. Corrected ordinary least squares [COLS] If we apply OLS, intercept estimate is biased downwards, all other parameters are unbiased.

29. 29 Deterministic Frontier models So COLS suggests that the OLS estimator from OLS be corrected. If we do not wish to make use of any probability distribution for y i then where is the OLS residual for i-th firm. If we assume that u i is distributed as Gamma then It is a bit more complicated if u i follows half-normal distribution.

30. 30 Production functions/frontiers OLS x × × × × × × × × q × Deterministic SFA ×

31. 31 Production functions/frontiers OLS: q i =  +  1 x i + v i Deterministic : q i =  +  1 x i - u i SFA: q i =  +  1 x i + v i - u i where v i = “noise” error term - symmetric u i = “inefficiency error term” - non-negative

32. 32 SFA......Some history…. Much of the work on stochastic frontiers began in 70’s. Major contributions from Aigner , Schmidt, Lovell, Battese and Coelli and Kumbhakar . Ordinary least squares (OLS) regression production functions: fit a function through the centre of the data assumes all firms are efficient Deterministic production frontiers: fit a frontier function over the data assumes there is no data noise SFA production frontiers are a “mix” of these two .

33. 33 Stochastic Frontier: Model Specification We start with the general production function as before and add a new term that represents technical inefficiency. This means that actual output is less than what is postulated by the production function specified before. We achieve this by subtracting u from the production function Then we have In the Cobb-Douglas production function with one input we can write the stochastic frontier function for the i-th firm as :

34. 34 Stochastic frontiers

35. 35 Stochastic Frontier: Model Specification…cont…. deterministic component noise inefficiency We stipulate that u i is a non-negative random variable By construction the inefficiency term is always between 0 and 1. This means that if a firm is inefficient, then it produces less than what is expected from the inputs used by the firm at the given technology. We can define technical efficiency as the ratio of “observed” or “realized output” to the stochastic frontier output In general, we write the stochastic frontier model with several inputs and a general functional form (which is linear in parameters) as

36. 36 Stochastic Specifications The SF model is specified as: The following are the assumptions made on the distributions of v and u . Standard assumptions of zero mean, homoskedasticity and independence is assumed for elements of v i . We assume that u i ’s are identically and independently distributed non-negative random variables. Further we assume that v i and u i are independently distributed. The distributional assumptions are crucial to the estimation of the parameters. Standard distributions used are: H alf-normal (truncated at zero) Exponential Gamma distribution

37. 37 Estimation of SF Models Parameters to be estimated in a standard SF model are: Likelihood methods are used in estimating the unknown parameters. Coelli (1995)’s Montecarlo study shows that in large samples MLE is better than COLS. Usually variance parameters are re- parametrized in the following forms . Testing for the presence of technical inefficiency depends upon the parametrization used. and Aigner , Lovell and Schmidt and Battese and Corra

38. 38 Estimation of SF Models In the case of translog model, it is a good idea to transform the data – divide each observation by its mean Then the coefficients of ln X i can be interpreted as elasticities . Most standard packages such as SHAZAM and LIMDEP . FRONTIER by Coelli is a specialised program for purposes of estimating SF models. Available for free downloads from CEPA website: www.uq.edu.au/economics/cepa

39. 39 FRONTIER output the final mle estimates are : coefficient standard-error t-ratio beta 0 0.27436347E+00 0.39600416E-01 0.69282978E+01 beta 1 0.15110945E-01 0.67544802E-02 0.22371736E+01 beta 2 0.53138167E+00 0.79213877E-01 0.67081892E+01 beta 3 0.23089543E+00 0.74764329E-01 0.30883101E+01 beta 4 0.20327381E+00 0.44785423E-01 0.45388387E+01 beta 5 -0.47586195E+00 0.20221150E+00 -0.23532883E+01 beta 6 0.60884085E+00 0.16599693E+00 0.36677839E+01 beta 7 0.61740289E-01 0.13839069E+00 0.44613038E+00 beta 8 -0.56447322E+00 0.26523510E+00 -0.21281996E+01 beta 9 -0.13705357E+00 0.14081595E+00 -0.97328160E+00 beta10 -0.72189747E-02 0.92425705E-01 -0.78105703E-01 sigma-squared 0.22170997E+00 0.24943636E-01 0.88884383E+01 gamma 0.88355629E+00 0.36275231E-01 0.24357013E+02 mu is restricted to be zero eta is restricted to be zero log likelihood function = -0.74409920E+02

40. 40 FRONTIER output technical efficiency estimates : firm eff.-est. 1 0.77532384 2 0.72892751 3 0.77332991 341 0.76900626 342 0.92610064 343 0.81931012 344 0.89042718 mean efficiency = 0.72941885 Mean efficiency can be interpreted as the “industry efficiency”.

41. 41 Tests of hypotheses e.g., Is there significant technical inefficiency? H : =0 versus H 1 : >0 Test options: t-test t-ratio = (parameter estimate) / (standard error) Likelihood ratio (LR) test [note that the above hypothesis is one-sided - therefore must use Kodde and Palm critical values (not chi-square) for LR test LR test “safer”

42. 42 Likelihood ratio (LR) tests Steps : 1) Estimate unrestricted model (LLF 1 ) 2) Estimate restricted model (LLF ) ( eg . set =0) 3) Calculate LR=-2(LLF -LLF 1 ) 4) Reject H if LR> R 2 table value, where R = number of restrictions (Note: Kodde and Palm tables must be used if test is one-sided)

43. 43 Example - estimate translog production function using sample data file which comes with the FRONTIER program - 344 firms t-ratio for  = 24.36, and N(0,1) critical value at 5% = 1.645 => reject H Or the LR statistic = 28.874, and Kodde and Palm critical value at 5% = 2.71 => reject H The LR statistic has mixed Chi-square distribution

44. 44 Scale efficiency For a Translog Production Function (Ray, 1998) An output-orientated scale efficiency measure is: SE i = exp[(1-  i ) 2 /2] where  i is the scale elasticity of the i-th firm and If the frontier is concave in inputs then <0. Then SE is in the range 0 to 1.

45. 45 Stochastic Frontier Models: Some Comments We note the following points with respect to SFA models It is important to check the regularity conditions associated with the estimated functions – local and global properties This may require the use of Bayesian approach to impose inequality restrictions required to impose convexity and concavity conditions. We need to estimate distance functions directly in the case of multi-output and multi-input production functions. It is possible to estimate scale efficiency in the case of translog and Cobb-Douglas specifications

46. Resource use and Technical Efficiency of Apple by Translog and Stochastic frontier Production Function in Himachal Pradesh 46 Current research

47. 47 Multistage random sampling design was used for the selection of sampling units (households engaged in apple cultivation). Apple cultivation is done mostly inShimla , Kinnaur , Kullu , Mandi, Sirmour and Lahaul Spiti districts of Himachal Pradesh. In the first stage, out of these districts four were selected randomly. In the second stage, three blocks from each district were selected In the third stage, five villages were selected from each block In the fourth stage, only 3 household were selected from each village Production areas in the each district were identified through list of apple cultivars given by directorate of horticulture and agriculture, each district and a sample of 180 farmers having different sizes of orchards were selected from the study area for the collection of primary data. Material and methods

48. 48 Total Yield in MT (Kg/1000) Y Area under Apple cultivation in Acres X1 Average Cost of Agri-inputs in Rs. per acre for FYM X2 Average Cost of Agri-inputs in Rs. per acre for Fertilizers X3 Average Cost of Agri-inputs in Rs. per acre for pesticides X4 Average Cost of Agri-inputs in Rs. per acre for labour X5 Variables taken for the present study:

49. Translog Production Function 49 Source Value Standard error t Pr > |t| Lower bound (95%) Upper bound (95%) Intercept 63.53 27.72 2.29 2.29 7.60 119.47 X1 3.27 6.35 0.51 0.51 -9.54 16.07 X2 - 6.78** 3.43 -1.98 0.05 -13.69 0.14 X3 -0.51 7.21 -0.07 0.94 -15.06 14.04 X4 - 22.08** 8.63 -2.56 0.01 -39.50 -4.65 X5 -4.20 7.30 -0.57 0.57 -18.92 10.53 X1*X2 -0.51 0.55 -0.92 0.07 -2.13 0.10 X1*X3 -0.53 1.29 -0.41 0.42 -3.66 1.55 X1*X4 1.47 1.52 0.96 0.06 -0.14 6.01 X1*X5 - 0.77* 1.41 -0.55 0.05 -4.38 1.30 X2*X3 -0.79 0.87 -0.92 0.07 -3.33 0.16 X2*X4 0.86 0.89 0.96 0.06 -0.08 3.52 X2*X5 0.05 0.73 0.07 0.89 -1.37 1.57 X3*X4 1.08* 1.51 0.05 0.04 -0.88 5.22 X3*X5 -0.12 0.86 -0.14 0.79 -1.96 1.50 X4*X5 0.75 1.77 0.43 0.40 -2.05 5.07 X1*X1 0.04 0.79 0.06 0.91 -1.52 1.69 X2*X2 0.39* 0.40 0.96 0.05 -0.04 1.59 X3*X3 0.00 0.00         X4*X4 0.00 0.00         X5*X5 0.00 0.00         R 2 76.4 Adjusted R 2 66.9

50. Equation of the model 50 Log y=63.53+ 3.27logX1-6.78logX2-0.51logX3-22.08X4-4.20X5-0.5(0.51)Log X1 Log X2-0.5(0.53)Log X1Log X3+0.5(1.47)Log X1 Log X4-0.5(0.77)LogX1 Log X5-0.5(0.79)LogX2 Log X3+0.86 Log X2 Log X4 + 0.5(0.05)Log X2 Log X5 +0.5 (1.08) Log X3 Log X4- 0.5(0.12)Log X3 Log X5 + 0.5(0.75) Log X4 Log X5 +0.5(0.04)(LogX1) 2 + 0.5(0.39)(Log X2) 2

51. Stochastic frontier production function 51 Parameters Value Log likelihood function 13.666* N 60 Sigma Square (v) 0.00000 Sigma (v) 0.00013 Sigma Square (u) 2.7807 Sigma (u) 1.66754 Sigma 1.66754 Gamma 1.00 Chi square 24.463

52. 52 Parameters Coefficient Standard Error Z Constant 0.8905 0.74721 1.20 X1 0.98405*** 0.07307 13.47 X2 -0.27336 *** 0.07606 -3.59 X3 -0.03151 0.17694 -0.18 X4 -0.16876 0.23202 -0.73 X5 0.44506*** 0.08329 5.34 Lambda 12515.6 0.1288 0.01 Mean technical efficiency 81.82% ***,**,*=significance at 1%, 5%, 10% level Logy= 0.8905+ 0.98405LogX1 -0.27336LogX2 -0.03151LogX3 -0.16879LogX4 +0.44506LogX5+v-u Technical efficiency by SFA

53. 53 Technical efficiency (%) TE Age of the farmer (years) Z1 Education of the farmer (No. of years in school) Z2 Farm Size (acres) Z3 Family size (No.) Z4 Experience in apple cultivation (years) Z5 Variables taken to see the effect on technical efficiency for the present study:

54. 54 Parameters Value Standard error t value Intercept 4.82 8.273 0.583 Age 0.03 0.100 0.275 Education 3.12* 0.656 4.753 Farm Size -0.16 0.406 -0.394 Family size 4.91* 0.776 6.331 Experience 0.52* 0.172 3.036 R 2 89.2% Adjusted R 2 88.2 % TE=4.82+0.03 Z1+3.12 Z2 -0.16 Z3 + 4.91 Z4 + 0.52 Z5 +e Variable responsible for technical efficiency

55. Comparison between CD, Translog , SFA 55 Source CD (OLS) Translog (OLS) SFA (MLE) Intercept 2.605 63.53 0.8905 X1 0.719 3.27 0.98405*** X2 -0.561 -6.78 -0.27336*** X3 -0.054 -0.51 -0.03151 X4 0.436 -22.08 -0.16876 X5 -0.265 -4.20 0.44506*** X1*X2 -0.51 X1*X3 -0.53 X1*X4 1.47 X1*X5 -0.77 X2*X3 -0.79 X2*X4 0.86 X2*X5 0.05 X3*X4 1.08 X3*X5 -0.12 X4*X5 0.75 X1*X1 0.04 X2*X2 0.39 X3*X3 0.00 X4*X4 0.00 X5*X5 0.00 R square 55.9 76.4 Adjusted R square 51.8 66.9 Log Likelihood 13.666

56. 56 General findings Three interaction terms were found significant in translog production function. R 2 value in translog was found 76.4 per cent which was more than Cobb Douglas production function. Mean technical efficiency was found 81.82 per cent by Stochastic frontier analysis. Through Linear regression, it was found that TE was positively related with age of farmer, education, family size and experience in farming and negatively with the farm size. R 2 value in linear regression was found 89.2 per cent which indicates that 89.2 per cent of variation in the TE was due to the variable under study.

57. References Aigner D.J., C.A.K. Lovell and P.J. Schmidt, 1977, Formulation and estimation of stochastic frontier production function models Journal of econometrics 6(1): 21-37. Timmer , C.P., 1971. Using a probabilistic frontier production frontier to measure technical efficiency, Journal of Political Economy 79(4): 776-794. Christensen, L., D. Jorgerson and L. Lau, 1972. Conjugate duality and transcendental logarithmic production fountiers , University of Wisconsin, Review of Economics and Statistics , 54(1): 28-45. Sankhayan P.L. 1998. Introduction to the Economics of Agricultural Production. New Delhi: Prentice Hall of India. pp. 41-83. 57

58. 58 Thanks…