Subhash C. Sharma Department of Economics Southern Illinois University Carbondale And Anil K. Bera

1. Estimation of Random Components and Prediction in One- and Two-Way Error Component Regression Models Subhash C. Sharma Department of Economics Southern Illinois University Carbondale And Anil K. Bera Department of Economics University of Illinois at Urbana-Champaign July 2007

2. Outline • Problem • One-Way Error Component Model • Two-Way Error Component Model • Empirical Illustration • Summary

3. Problem • Prediction in Panel Data Models. Consider the one-way error component model: are randomly distributed across m cross-sectional units One problem:

4. Idea borrowed from Bera and Sharma (1999), “Estimating Production Uncertainty in Stochastic Frontier Function Models,” Journal of Productivity Analysis, 12, pp. 187-210. • Model: : represents technical inefficiency Estimation of ?

5. We can recover , which is a “sufficient statistic” for • Use “Rao-Blackwellization” and obtain Then,

6. It is easy to derive the conditional density Using that we can obtain conditional moments of any order. gives a point estimate of (inefficiency) Jondrow, Lovell, Materov and Schmidt (1982), JE. can be viewed as the production uncertainty due to technical inefficiency.

7. Using expression for the conditional mean and variance, we can construct confidence intervals for firm specific inefficiency. Can also use higher-order conditional moments (of given ) to obtain conditional skewness and kurtosis measures.

8. For the panel data model Can construct confidence interval using

9. Introduction We have the panel regression model (for m cross sectional units over n time periods) or or i = 1,2…m, t = 1,2…n (1) yit is an observation on the dependent variable, is a unit specific term, is the i-th observation on (k-1) non-stochastic explanatory variables at time t, is a (k-1)x1 vector of unknown parameters

10. When is fixed it yields the fixed effect model. In other situations it might be more appropriate to consider where α is the mean intercept and is randomly distributed across cross-sectional units. Thus, model (1) can be written as (2) Model (2) is known as the one-way random effect or the one-way error component model.

11. Instead of just cross sectional effect one should also capture the time effect, i.e., where α is the overall effect, is an individual cross sectional effect, and is the time effect, So, for this case the model becomes i = 1,2…m , t = 1,2…n (3) • When and are fixed, (3) is called a two-way fixed effect model. • However, when and are random, (3) is called a two-way error component model, or two-way random effect model.

12. One-Way Error Component Model Consider the random effect model given by equation (2) (2) where for all t and j; if if and are assumed to be distributed as normal. Let or (4)

13. The joint density of δi and ui = (ui1 ui2…uin)´ is given by (5) From (5), the joint density of δi and εi can be easily obtained as (6) where

14. with From (6), the marginal density of εi is (7)

15. Finally, the conditional density function of δi given εi is f(δi | εi) = (8) Thus, E(δi | εi) = and Var (δi | εi) =

16. Thus, we propose that the random coefficient, δi, be estimated by an estimate of E(δi | εi), i.e. (9) where, and are the corresponding GLS estimates of respectively.

17. Confidence Interval: We can estimate Var (δi|εi) as Using these estimate we can easily construct the (1-α) 100% confidence interval for δi, as follows. (10) We can also test the hypothesis Ho: δi = 0, using the approximate t-statistic given by

18. Prediction In the one way random effect model, we propose the predicted value of yi by the expected value of yi conditional on the composed error term, εi, i.e,. E(yi|εi), where E(yi | εi) = Xi β + E(δi | εi) in + E(ui | εi), E(yi | εi) = Xi β + (11) One can easily obtain that f(ui | εi) (12) where, = E(ui | εi) (13) and (14)

19. Thus, following (11) the prediction for the i-th unit is (15) where are the consistent estimates of For this model, the best linear unbiased prediction of the i-th unit, is also given by Lee and Griffiths (1979), Taub (1979) and Baltagi (2001, p. 22), which is (16) where

20. Two-Way Error Component Model , (3) where δi’s are iid, as normal with mean zero and variance, λt’s are iid normal with mean zero and variance and uit are iid normal with mean zero and variance Moreover, δi, λt and uit are assumed to be independent of each other. Let (19) or, (20)

21. 3.1 Estimation of Cross Sectional Effect, δi We propose an estimate of δi by E(δi|εi) using the conditional density f(δi|εi). The joint density of δi, λ and ui is given by (21) From (21), one can easily obtain the joint density of f(δi, λ, εi) by substituting i.e., (22)

22. where (23) and (24) From (22), one can obtain (25)

23. Further, from (25) (26) (27) (28) and (29)

24. Finally, from (25) and (26), we get f(δi | εi) = (30) Thus, we propose that δi be estimated by E(δi | εi) = (31) And Var (δi | εi) = (32) By using one can also obtain the (1-α) 100% confidence interval for δi.

25. 3.2 Estimation of the time effect, λt. We propose an estimate of λt by E(λt | εt), which is the mean of f(λt | εt). To obtain f(λt | εt) we rearrange the observations in (19), as (33) where εt = (ε1t ε2t ….εmt)́, δ = (δ1 δ2 ….δm)́, ut = (u1t u2t….umt)́, and im = (111…11)´. The joint density of δ, λt and ut is given by f(δ, λt, ut) = (34)

26. From (34), one can obtain f(δ, λt, εt) = (35) and f(λt,εt) = (36) where (37) and (38)

27. Further, from (36) we get (39) where (40) and (41) with

28. From (36) and (39), f(λt | εt) = (42) Thus, we propose that λt be estimated by E(λt | εt), i.e. E(λt | εt) = (43) and Var (λt | εt) = (44) By using we can obtain the (1 – α) 100% confidence interval for λt and test the hypothesis H0: λt = 0.

29. 3.3 Prediction In model we propose prediction of yi by an estimate of E(yi | εi). Thus (45) where, are estimates of δi and λt and = (ui | εi) is an estimate of ui.

30. Empirical Illustration We consider an example, first used by Baltagi and Griffin (1983) and later by Baltagi (2001), the demand for gasoline in a panel of 18 OECD countries covering the period 1960-1978. The gasoline demand equation considered by Baltagi and Griffin (1983) and Baltagi (2001, p. 21) is (66) Gas/Car is motor gasoline consumption per auto, Y/N is real per capita income, PMG/PGDP is real motor gasoline price and (Car/N) denotes the stock of cars per capita.

32. : denotes first stage estimates. : are ANOVA type estimates based on GLS residuals. : are ANOVA type estimates adjusted for degrees of freedom, based on GLS residuals.

35. Table 4

39. 5. Summary We consider the one-way error component model, i.e. and propose an estimate of δi by the estimate of E(δi | εi), where, is like a composite error term. We also provide expression for Var (δi | εi). Using E(δi | εi) and Var (δi | εi) one can obtain the confidence interval for the random component. Next, an expression for the prediction of the i-th cross- sectional unit is provided, i.e., where = E(ui | εi) , is also included besides E(δi | εi).