90 [, ] p Panel nested error structure) : Lagrange-multiple LM) Honda [3] LM ; King Wu, Baltagi, Chang Li [4] Moulton Randolph ANOVA) F p Panel,, p Z

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1 00 Chinese Journal of Applied Probability and Statistics Vol6 No Feb 00 Panel, 3,, 0034;,, 38000) 3,, 000) p Panel,, p Panel : Panel,, p,, : O,,, nuisance parameter), Tsui Weerahandi [] Weerahandi [] p generalized p value) generalized confidence interval) F, p, [ 0] Weerahandi [3] p Zhou Mathew [4],, Chi Weerahandi [5], Weerahandi Berger [6] p, Lin Lee [7] Ye Wang [8], Tsui Weerahandi [] p Weerahandi [, 9], Krishnamoorthy Mathew [0] 0679) ) 007 3, 009 9

2 90 [, ] p Panel nested error structure) : Lagrange-multiple LM) Honda [3] LM ; King Wu, Baltagi, Chang Li [4] Moulton Randolph ANOVA) F p Panel,, p Z η, δ), η, δ, δ H 0 : η η 0 H : η > η 0, ) η 0 z Z T Z, z, η, δ) Z, Z z, η, δ) T ) T Z, z, η 0, δ), ) T Z, z, η, δ) T z, z, η, δ), 3) z δ, T Z, z, η, δ) η stochastically increasing) stochastically decreasing), T Z, z, η, δ) T Z, z, η, δ) η, ) p p = PT Z, z, η, δ) T z, z, η, δ) η = η 0 ) T Z, z, η, δ) η, ) p p = PT Z, z, η, δ) T z, z, η, δ) η = η 0 ) α, p α, )), p, T Z, z, η, δ) Z, Z z, η, δ) T

3 : Panel 9 ) T Z, z, η, δ), ) T Z, z, η, δ) T z, z, η, δ), T Z, z, η, δ) T Z, z, η, δ), T z, z, η, δ) = η, T α) T Z, z, η, δ) α, T α) η Tusi Weerahandi [], Weerahandi [] 3 Panel y it = β 0 + x it β + + x itk β k + µ i + λ t + ε it, i =,, N; t =,, T 3) y it i t ; x itj i j t ; β = β,, β k ) ; µ i i, N, µ i ; λ t t, ; ε it µ i, λ t ε it, µ i N0; σµ), λ t N0, σλ ), ε it N0, σε) 3) y = NT β 0 + Xβ + u 3) y = y,, y T, y,, y T,, y N,, y NT ) ; X = x,, x T, x,, x T,, x N,, x NT ) ; x it = x it,, x itk ); u = I N T )µ + N I T )λ + ε; µ = µ,, µ N ) ; λ = λ,, λ T ) ; ε = ε,, ε T,, ε N,, ε NT ) Σ = Cov u) = σµi N J T ) + σλ J N I T ) + σεi NT, J T = t t, J N = N N, I NT = I N I T J T = J T /T, J N = J N /N, E N = I N J N, E T = I T J T, Cov u) = σ εe N E T ) + T σ µ + σ ε)e N J T ) + Nσ λ + σ ε)j N E T ) + T σ µ + Nσ λ + σ ε)j N J T ) σ Q + σ Q + σ 3Q 3 + σ 4Q 4 33)

4 9, σ = σ ε, σ = T σ µ + σ ε, σ 3 = Nσ λ + σ ε, σ = T σ µ + Nσ λ + σ ε, Q = E N E T, Q = E N J T, Q 3 = J N E T, Q 4 = J N J T 3 ) Q, Q, Q 3, Q 4, ) Q, Q, Q 3, Q 4, 3) Q, Q, Q 3, Q 4 N )T ), N, T, [5] Q, Q, Q 3, E T T = 0, E N N = 0, y = Q y = Q Xβ + Q u Q Xβ + u, y = Q y = Q Xβ + Q u Q Xβ + u, y 3 = Q 3 y = Q 3 Xβ + Q 3 u Q 3 Xβ + u 3 34) Eu i = Q i Eu = 0, Cov u i ) = σ i Q i, i =,, 3 X Q i X), 34) β β i = X Q i X) X Q i y, i =,, 3 β i β, Cov β i ) = σ i X Q i X) Σ i σ i ), i =,, 3 û i = Q i y Q i X β i, i =,, 3 σ i n i = rkq i ) k, 3 σ i = S i = û iûi n i = y Q i Q i XX Q i X) X Q i )y n i 35) V i = n is i σ i β, β, β 3, S, S, S 3 χ n i, i =,, 3 β i, S i y, y [5] 4 Panel 3) p, 3

5 : Panel 93 Panel 3) c 0 s i S i, i =,, H 0 : σ µ c 0 H : σ µ > c 0, 4) T = n S σ n s T σ µ + σ s S T : T t = 0 ; S, S, V = n S σ χ n, V = n S σ n s = V T σµ + n s V 4) χ n, T ; T σ µ 4), T T p p = PT t σµ = c 0 ) = P V = E V [ F χ n n s T c 0 + n s V E V V, F χ n n β, σ, σ ) aβ, aσ, aσ ) n s ) T c 0 + n s V )] 43) β, S, S ) a β, as, as ), a > 0) 44) T 44), 4) 44) H 0 : σ µ s T = n S σ = θ µ θ 0 = c 0 s n s s T θ µ + σ εs H : θ µ > θ 0 45) = V n s s T θ µ + n V T 45) T 45) p p = P T t θ µ = θ 0 ) β, σ, θ µ ) aβ, aσ, aθ µ ) β, S, S ) a β, as, as ), a > 0) 46)

6 94 46) 45) T, 45) T p 46) c 0 0, 4), 4) H 0 : σµ = 0 H : σµ > 0 4) p F p F, p F σ µ T = T σ s S σ s ) S = n s n s ) T V V T : T σ µ,, V χ n, V χ n, T T T σ µ α) T α) T α) T α), T α) T α α θ µ T T = σ s T s S σ s ) S c 0 4) T = n 3S 3 σ 3 T 3 = σ εs S H 0 : σ λ c 0 H : σ λ > c 0, 47) H 0 : σ ε c 0 H : σ ε > c 0, 48) n 3 s 3 Nσ λ + σ s S σ ε = n s V σ ε n 3 s 3 = V 3 Nσλ + n σ V, s i S i, i =,, 3 T, T 3 47) 48) p p = PT t σ λ = c 0) = E V [ F χ n3 p 3 = PT 3 t 3 σε n s ) = c 0 ) = F χ n c 0 n 3 s 3 Nc 0 + n s V p σ λ, σ ε, T = σ 3 s 3 N S 3 σ s ) S, T 3 = σ s S )],

7 : Panel 95,,, H 0 : lσµ + mσλ + nσ ε c 0 H : lσµ + mσλ + nσ ε > c 0 49) l, m, n R, c 0 49), l =, m =, n = c 0 = 0, σ µ σ λ ); l = m =, n =, c 0 = 0, σ µ + σ λ σ ε);, l = m = 0, n = l = n = 0, m = m = n = 0, l =, 49) 49) p, T 4 = = l T l T σ s S n s V + m N + m N σ 3 s 3 S 3 n 3 s 3 V 3 + : T 4 t 4 = 0 ; V i = n is i σ i + n l T m N n l T m N ) σ s S χ ni, i =,, 3 ) lσ µ + mσ λ + nσ ε) ) n s ) lσµ + mσλ V + nσ ε) 40) T 4 ; T 4 lσ µ + mσ λ + nσ ε T 4, T 4 p p 4 = PT 4 t 4 lσµ + mσλ + nσ ε = c 0 ) = P c 0 l n s m n 3 s 3 T V N V 3 = P V l T n s c 0 m n 3 s 3 N V 3 l = E V,V 3 F χ n T n s c 0 m N lσ µ + mσ λ + nσ ε, T 4 = l T σ s S + m N σ 3 s 3 S 3 n l T m N n l T m N n 3 s 3 V 3 ) n s ) V + n l T m N ) n s V ) ) n l T m N ) n s V ) )) ) σ s 4) S : T4, T 4 t 4 = lσ µ + mσλ + nσ ε T4, T 4 lσ µ + mσλ + nσ ε T4 α) T 4 α), T 4 α) lσ µ + mσλ + nσ ε α)

8 96 β, σ, σ, σ 3 ) aβ, aσ, aσ, aσ 3 ) β, S, S, S 3 ) a β, as, as, as 3 ), a > 0) 4) 40) T 4, H 0 : lσ µ + mσ λ + nσ ε s = θ θ 0 = c 0 s l σ T 4 = θ s T s + m σ3 s 3 S N s + n l S 3 T m N H : θ > θ 0 43) 43), T 4, p p 4 = P T 4 t 4 θ = θ 0 ) σ, σ, σ 3, θ) aσ, aσ, aσ 3, θ) ) σ S, S, S 3 ) as, as, as 3 ), a > 0) 44) T 4 43) 44), 43) T 4 p 44) p lσ µ + mσ λ + nσ ε, 44) θ T 4 = l T σ s s S + m N σ 3 s 3 s S 3 + n l T m N S ) ) σ 45) T 4, θ T 4 θ T 4 α) T 4 α) T 4 α) α α), T 4 α) θ α), T 4 α) θ α) 5 49), l = m = n = 0 Panel 3), y it = x it β + + x itk β k + µ i + λ t + ε it, i =,, N; t =,, T, λ t N0, ), ε it N0, 0); µ i N0, σ µ), N = 5, T = 0, x it, x it,, x itk ) k 5, 40) p 4), S

9 : Panel 97 T 4 p α = 005) σ µ k β c , ) ,, ) ,,, ) ,,,, ) T4 α = 005) σ µ k β , ) ,, ) ,,, ) ,,,, ) p, α) p [] Tusi, KW and Weerahandi, S, Generalized p value in significiance testing of hytheses in the presence of nuisance parameters, J Amer Statist Assoc, 84989), [] Weerahandi, S, Generalized confidence intervals, J Amer Statist Assoc, 88993), [3] Weerahandi, S, Testing variance components in mixed models with generalized p values, J Amer Statist Assoc, 8699), 5 53

10 98 [4] Zhou, LP and Mathew, T, Some tests for variance components using generalized p values models, Technometrics, 36994), [5] Chi, EM and Weerahandi, S, Comparing treatments under growth curve models: exact tests using generalized p values, J Statist Plan Inference, 7998), [6] Weerahandi, S and Berger, VW, Exact inference for growth curves with intraclass correlation structure, Biometrics, 55999), 9-94 [7] Lin, SH and Lee, JC, Exact tests in simple growth curve models and one-way ANOVA with equicorrelation structure, J Multivariate Anal, 84003), [8] Ye, RD and Wang, SG, Generalized p values and generalized confidences intervals for variance components in genreal random effects model with balanced data, Journal of Systems Science and Complexity, 0007), [9] Weerahandi, S, ANOVA under unequal error variances, Bimetrics, 5995), [0] Krishnamoorthy, K and Mathew, T, Inference on the means of lognormal distributions using generalized p-values and generalized confidence intervals, J Statist Plan Inference, 5003), 03 [],,, P Fiducial, A ), 6007), [] Tang, SJ and Tusi, KW, Distributional properties for the generalized p value for Behren-Fisher problem, Statistics & Probability Letters, 77007), 8 [3] Honda, Y, Testing the error compontents model with non-nonmal distributions, Review of Econometric Studies, 55985), [4] Baltagi, BH, Chang, YJ and Li, Q, Monte Carlo results on several new and existing tests for the error compontent model, Journal of Econometrics, 5499), 95 0 [5],,,,,, 004 Exact Tests of Variance Components in Panel Data Model Cheng Jing, Wang Songgui 3 Yue Rongxian Mathematics and Science College, Shanghai Normal University, Shanghai, 0034 ) Department of Mathematics, Chaohu college, Chaohu, ) 3 College of Applied Sciences, Beijing University of Technology, Beijing, 000 ) In this paper, some new exact tests and confidence intervals of variance components in Panel data model with three random effects are constructed base on the concepts of generalized p value and generalized confidence interval Invariance of these tests and confidence intervals under scale transformation is also discussed in the paperthe powers of these tests and coverage probabilities are obtained by numerical simulations It is shown that generalized p value method is feasible and effective to resolve the hypothesis testing problems with nuisance parameters Keywords: Panel data model, random effect, generalized p value, generalized confidence interval, variance component AMS Subject Classification: 6J0

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