Seemingly Unrelated Regressions in Panel Models. Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker

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1 Seemingly Unrelated Regressions in Panel Models Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker

2 Arnold Zellner An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias Journal of the American Statistical Association, 1962 Different regression equations that seem to be unrelated and indivdually satisfy the classical OLS assumption, but are interdependent in the error term. OLS unbiased and consistent but more efficient estimates by using FGLS to account for interdependence.

3 y 1 = X 1 β 1 + u 1 y 2 = X 2 β 2 + u 2. y i = X i β i + u ị. y m = X m β m + u m y i is an n 1 vector of observations on variable i. X i is an n k i matrix of observations on explanatory variables β i is a k i 1 vector of coefficients u i is an n 1 vector of disturbances

4 y 1 X β 1 u 1 y 2. = 0 X β u 2. y m X m β m u m y = X β + u y is an nm 1 vector of observations on variable i. X is an n i k i matrix of observations on explanatory variables β is a i k i 1 vector of coefficients u is an nm 1 vector of disturbances

5 E(u 1 u 1 ) E(u 1 u 2 )... E(u 1 u m ) E(u 2 u 1 ) E(u 2 u 2 )... E(u 2 u m ) E[uu ] = E(u m u 1 ) E(u m u 2 )... E(u m u m ) VCV 1 E(u 1 u 2 )... E(u 1 u m ) E(u 2 u 1 ) VCV 2... E(u 2 u m ) E[uu ] = E(u m u 1 ) E(u m u 2 )... VCV m

6 σ 11 I N σ 12 I N... σ 1m I N σ 21 I N σ 22 I N... σ 2m I N E[uu ] = σ m1 I N σ m2 I N... σ mm I N E[u i u j ] = Σ I n for a m m symmetric matrix with positive entries Σ.

7 James G. Beierlein, James W. Dunn, James C. McConnon, Jr. The Demand for Electricity and Natural Gas in the Northeastern United States The Review of Economics and Statistics, 1981 Simultaneous estimation of the 1. residential gas 2. residential electric 3. commercial gas 4. commercial electric 5. industrial gas 6. industrial electric demand in logarithmized quantities in the northern US. Both cross-sectional and time dimension for each equation. Estimation as a Two-Way Random Effects SUR model.

8 The individuals are nine states, the time is yearly from Explanatory variables are fuel prices, per capita income and disposable income, value of retail sales and value added by manufacturing. No qualitative differences between OLS, two-way random effects and two-way random effects SUR, but the latter gives much lower standard errors.

9 Notation and assumptions set of M equations y i = X j β j + u j (j = 1,..., M) y j is NT 1, X j is NT k j, β j is k j 1 additive error components u j = Z µ µ j + Z λ λ j + ν j two-way random effects with Z µ = I N e t, Z λ = e N I T µ j, λ j and ν j are random vectors with expectation 0 and a variance-covariance matrix given by µ j E λ j ( ) σ µjl I N 0 0 µ l λ l ν l = 0 σ λjl I T 0 ν j 0 0 σ νjl I NT

10 Combining all M equations Combining all M equations yields y = X β + u with E(uu ) = Ω = [Ω jl ] and X = I M Xj

11 rewriting Ω jl Ω jl = E(u j u l ) = σ2 µ jl A + σ 2 λ jl B + σ 2 ν jl I NT with A = I N e t e t and B = e N e N I T this can be rewritten as Ω jl = σ3jl 2 J NT NT + σ2 1jl ( A T J NT NT ) + σ2 2jl ( B N J NT NT ) + σ2 ν jl Q with Q = I NT A T B N + J NT NT and J NT = e NT e NT, σ 2 3jl = σ 2 ν jl + Nσ 2 λ jl + T σ 2 µ jl σ 2 1jl = σ 2 ν jl + T σ 2 µ jl σ 2 2jl = σ 2 ν jl + Nσ 2 λ jl

12 Eigenvalues of Ω it can be shown that (Nerlove(1971)) σ 2 1jl, σ2 2jl, σ2 3jl and σ 2 ν jl form the eigenvalues of Ω jl. plugging them in yields the following expression for Ω Ω = Ω 3 J NT NT + Ω 1 ( A T J NT NT ) + Ω 2 ( B N J NT NT ) + Ω νq where Ω 3 = [σ3jl 2 ], Ω 2 = [σ2jl 2 ], Ω 1 = [σ1jl 2 ] and Ω ν = [σν 2 jl ]

13 in search of Ω 1 the expression for Ω is of the structure Ω = k Ω i D i where Ω i are nonsingular matrizes and D i are symmetric, sum up to I n and are mutual orthogonal (i.e. D i D j = 0 for i j and idempotent. it can be shown that for a matrix Ω of this structure the unique inverse of the following form can be obtained: Ω 1 = r i=1 Ω 1 i D i closer inspection of the structure of Ω as derived above shows that indeed Ω is of the form: Ω = k Ω i D i with i=1 D 1 = ( A T J NT NT ),D 2 = ( B N J NT NT ), D 3 = J NT NT i=1 and D 4 = Q

14 the GLS-Estimator for β therefore Ω 1 is given by a weighted sum of four matrizes Ω 1 = Ω 1 3 J NT NT +Ω 1 1 ( A T J NT NT )+Ω 1 2 ( B N J NT NT )+Ω 1 ν Q note that X (Ω 1 3 (J NT /NT ))X = 0 as J NT X j = 0 as usual the GLS estimator is then given by: ˆβ GLS = (X Ω 1 X ) 1 )X Ω 1 Y note also that for σµjl 2 = σ λjl 2 = 0, Ω 1 = Ω 2 = Ω ν which results in Ω 1 = Ω 1 ν I NT standard SUR estimator without a two-way random effects error specification. but unlike the standard SUR estimator, even when all matrizes Xj contain the same information, i.e. X 1 = X 2 =...X M = ˆX performing GLS on each system seperately is not equivalent to GLS perform on the whole system.

15 References Robert B. Avery Error Components and Seemingly Unrelated Regressions Econometrica, 1977 Badi Baltagi On Seemingly Unrelated Regressions with Error Componentscomponentss Econometrica, 1980 Marc Nerlove A Note on Error Components Models Econometrica, 1971

16 Ingmar R. Prucha On the Asymptotic Efficiency of Feasible Aitken Estimators for Seemingly Unrelated Regression Models with Error Components Econometrica, 1984 What estimators of the variance-covariance-matrix are good asymptotically?

17 Probabilistic limits are with respect to both N and T. Theorem: Let ˆβ FGLS be a fesible GLS-estimator satisfying (i) plim ˆΩ ν = Ω ν, (ii) plim ˆΩ µ = Ω µ and (iii) plim ˆΩ λ = Ω λ, where Ω µ and Ω λ are any finite, positive definite matrices. Then plim NT ( ˆβ GLS ˆβ FGLS ) = 0.

18 A sufficient condition for plim NT ( ˆβFGLS ˆβ GLS ) = 0 is that (i) plim X (ˆΩ 1 Ω 1 )X /NT = 0 and (ii) plim X (ˆΩ 1 Ω 1 )u NT = 0.

19 Using Baltagis formula for the inverse of Ω, this can be rewritten as (i) plim(ˆσ jl 1 σjl 1 ) X j (A/T J/NT )X l (ˆσ jl ν σ jl ν ) X j QX l NT NT = 0 for j, l = 1,..., M. (ii) plim l T 3/4 (ˆσ jl 1 σjl 1 ) X j (A/T J/NT )u l NT 5/2 σ jl 2 ) X j (B/N J/NT )u l N 5/2 T + (ˆσ jl 2 σjl 2 ) X j (B/N J/NT )X l + N 3/4 (ˆσ jl 2 NT + + (ˆσ jl ν σ jl ν ) X j Qu l NT = 0 for j = 1,..., M.

20 By assumption and converge, so and converge too. X j X l /NT X j QX l /NT X j (A/T J/NT )X l /NT X j (B/T J/NT )X l /NT Write our first condition as (i) plim(ˆσ jl 1 σjl 1 )[ ] + (ˆσjl j, l = 1,..., M. 2 σjl 2 )[ ] + (ˆσjl ν σν jl )[ ] = 0 for

21 Now plim X j (A/T J/NT )u l = plim X j (B/N J/NT )u l NT 5/2 N 5/2 T = 0. X j Qu l/ NT = X j Qv l/ NT can be shown to converge in distribution to a random variable by a CLT. Write our second condition as (ii) plim l T 3/4 (ˆσ jl 1 σjl 1 )0 + N3/4 (ˆσ jl for j = 1,..., M. 2 σjl 2 )0 + (ˆσjl ν σν jl )[ ] = 0

22 We only have to show that (i) plim(ˆσ jl 1 σjl 1 )[ ] + (ˆσjl 2 σjl 2 j, l = 1,..., M. (ii) plim l T 3/4 (ˆσ jl 1 σjl 1 )0 + N3/4 (ˆσ jl for j = 1,..., M. )[ ] + (ˆσjl ν σν jl )[ ] = 0 for 2 σjl 2 )0 + (ˆσjl ν σν jl )[ ] = 0

23 That s all Folks!