The Two-way Error Component Regression Model

Transcription

1 The Two-way Error Component Regression Model Badi H. Baltagi October 26, 2015 Vorisek Jan

2 Highlights Introduction Fixed effects Random effects Maximum likelihood estimation Prediction Example(s)

3 Two-way Error Component y it = α + X itβ + u it (1) i = 1,..., N, individual, cross-section, t = 1,..., T, time-series K explanatory variables u it = µ i + λ t + ν it (2) µ i, individual effect N i=1 µ i = 0 λ t time effect not included in the regression: strike year oil embargo laws restricting smoking T t=1 λ t = 0 Wallace and Hussain (1969), Nerlove (1971) and Amemiya (1971)

4 Vector form u = (u 11,..., u 1T,..., u NT ) u = Z µ µ + Z λ λ + ν (3) Z µ = I N ι T matrix of individual dummies Z λ = ι N I T matrix of time dummies, dim NT T ι N vector of ones, dim N IT identity matrix, dim T ν = (ν 11,..., ν 1T,..., ν NT ) Z λ Z λ = J N I T JN matrix of ones, dim N Z λ (Z λ Z λ) 1 Z λ = J N I T projection matrix JN = J N /N ( J N I T )y predicted values: ȳ.t = N i=1 y it/n

5 Fixed effects µ i and λ t fixed parameters ν it IID(0, σ 2 ν) X it independent of ν it for all i and t regressing the original problem (1) yields several troubles enormous loss in df due to {(N 1) + (T 1)} dummies blurring multicollinearity among regressors inverting large (N + T + K 1) matrix to obtain the fixed effect estimates of β serves Within transformation given by (Wallace, Hussain: 1969): Q = E N E T = (I N J N ) (I T J T ) = (4) = I N I T I N J T J N I T + J N J T

6 Within estimator ỹ = Qy has elements ỹ it = (y it ȳ i. ȳ.t + ȳ.. ) regression of ỹ on X = QX yields the Within estimator β = (X QX ) 1 X Qy (5) α = ȳ.. β x.. µ i = (ȳ i. + ȳ.. ) β( x i. + x.. ) λ t = (ȳ.t + ȳ.. ) β( x.t + x.. ) Within estimator can not estimate the effect of time- and individualinvariant variables since Q transformation wipes out these variables if estimating (5) with the standard regression package to get the proper variance-covariance matrix one needs to divide the estimated one by (NT N T + 1 K) and multiply by (NT K)

7 Testing for fixed effects joint significance of the dummy variables H 0 : µ i = 0 and λ t = 0 RRSS from pooled OLS: yit = α + X it β + ν it URSS from Within regression F 1 = (RRSS URSS)/(N + T 2) URSS/(NT N T + 1 K) F (N+T 2),(NT N T +1 K) existence of individual effects: H 2 : µ i = 0 RRSS from regression with time dummies only y it ȳ.t = (x it x.t )β + (u it ū.t ) F 2 F (N 1),(NT N T +1 K)

8 Random effects µ i IID(0, σ 2 µ) λ t IID(0, σ 2 λ ) ν it IID(0, σ 2 ν) mutualy independent as well as with X it inference for random sample from large population Ω = E(uu ) = Z µ E(µµ )Z µ + Z λ E(λλ )Z λ + σ2 νi NT = σ 2 µ(i N J T ) + σ 2 λ (J N I T ) + σ 2 ν(i N I T ) (6) σµ 2 + σλ 2 + σ2 ν i = j, s = t σµ cov(u it, u js ) = 2 i = j, s t σ λ 2 i j, s = t 0 i j, s t (7) Ω = σ 2 µ T ((E N + J N ) J T )+σ 2 λ N( J N (E T + J T ))+σ 2 ν ((E N+ J N ) (E T + J T ))

9 Matrix spectral decomposition Ω = 4 λ i Q i (8) i=1 i λ i Q i # λ i 1 σν 2 E N E T NT N T T σµ 2 + σν 2 E N J T N 1 3 Nσλ 2 + σ2 ν J N E T T 1 4 T σµ 2 + Nσλ 2 + σ2 ν J N J T 1 λ i are the characteristic roots of Ω Q i are the matrices of eigenprojectors, symetric and idempotent, pairwise orthogonal and 4 i=1 Q i = I NT Ω r = 4 λ r i Q i (9) i=1

10 Transformation σ ν Ω 1/2 = 4 i=i element of y = σ ν Ω 1/2 y is given by θ 1 = 1 (σ ν /λ 1/2 2 ) θ 2 = 1 (σ ν /λ 1/2 3 ) (σ ν /λ 1/2 i )Q i (10) y it = y it θ 1 ȳ i. θ 2 ȳ.t + θ 3 ȳ.. (11) θ 3 = θ 1 + θ 2 + (σ ν /λ 1/2 4 ) 1 GLS estimator of (1) can be obtained as OLS of y on Z = σ ν Ω 1/2 Z ( Fuller, Battese: 1974)

11 Best quadratic unbiased estimators from Q i u (0, λ i Q i ) follows that ˆλ i = u Q i u/tr(q i ) is the BQU estimator for i = 1, 2, 3 estimators are minimum variance unbiased under normality of the disturbances (Graybill: 1961) estimates of variance by OLS residuals (Wallace, Hussain: 1696) are unbiased and consistent, but inefficient Within residuals has the same asymptotic distribution as the true disturbances (Amemiya: 1971): NT (ˆσ 2 ν σ 2 ν) N(ˆσ 2 µ σ 2 µ) T (ˆσ 2 λ σ 2 λ ) N 0, 2σν σµ σλ 4 (12) both OLS and Within residuals introduces bias into estimates. the correction of df depend upon traces given by authors

12 Three LS regressions to estimate the variance components from corresponding MSE of the regressions (Swamy, Arora: 1972) ˆλ i = [y Q i y y Q i X (X Q i X ) 1 X Q i y]/df i (13) Within regression for σ 2 ν = λ 1, df 1 = NT N T + 1 K Between individuals regression of (ȳ i. ȳ.. ) on ( X i. X.. ) for σ 2 µ = (λ 2 σ 2 ν)/t, df 2 = N 1 K Between time-periods regression of (ȳ.t ȳ.. ) on ( X.t X.. ) for σ 2 λ = (λ 3 σ 2 ν)/t, df 3 = T 1 K

13 Stacked regressions Q 1 y Q 2 y Q 3 y = Q 1 X Q 2 X Q 3 X β + Q 1 u Q 2 u Q 3 u (14) since Q i ι NT = 0 and the transformed error has mean 0 and covariance matrix diag[λ i Q i ] for i = 1, 2, 3 ˆβ GLS = W 1 β W + W 2 ˆβ B + W 3 ˆβ C (15) matrix weighted average of Within, Between individuals and Between time-periods estimator if σ 2 µ = σ 2 λ = 0, ˆβ GLS reduces to ˆβ OLS as T and N, λ 2 and λ 3, ˆβ GLS tends to β W

14 Maximum likelihood estimation normality needed for errors log L 1/2 log Ω 1/2(y Zγ) Ω 1 (y Zγ) (16) log L γ log L σ 2 ν log L σµ 2 log L σλ 2 = Z Ω 1 y (Z Ω 1 Z)γ = 0 (17) = 1/2trΩ 1 + 1/2u Ω 2 u = 0 = 1/2trΩ 1 (I N J T ) + 1/2u Ω 2 (I N J T )u = 0 = 1/2trΩ 1 (J N I T ) + 1/2u Ω 2 (J N I T )u = 0 iterative solution needed for normal equations (Amemiya: 1971). estimates of variance consistent with asymptotic distribution (12)

15 Concentrated likelihood from Ω 1 = (σ 2 ν) NT (φ 2 2 )N 1 (φ 2 3 )T 1 φ 2 4 LL(α, β, σ 2 ν, φ 2 2, φ 2 3) (NT /2) log σ 2 ν + (N/2) log φ (18) +(T /2) log φ 2 3 (1/2) log[φ φ 2 3 φ 2 2φ 2 3] 1/(2σ 2 ν)u Σ 1 u where φ i = σ 2 ν/λ i and Σ = Ω/σ 2 ν define d = y X β, u = d ι NT α follows ˆα = ι NT d/nt and ˆσ2 ν = u Σ 1 u/nt u = d ι NT ˆα = (I NT J NT )d (I NT J NT )Σ 1 (I NT J NT ) = Q 1 + φ 2 2Q 2 + φ 2 3Q 3 LL C (β, φ 2 2, φ 2 3) (NT /2) log[d (Q 1 + φ 2 2Q 2 + φ 2 3Q 3 )d] + (19) +(N/2) log φ (T /2) log φ 2 3 (1/2) log[φ φ 2 3 φ 2 2φ 2 3]

16 Maximizing iteration maximizing LL C over β, given φ 2 2 and φ2 3 ˆβ = [X (Q 1 + φ 2 2Q 2 + φ 2 3Q 3 )X ] 1 X (Q 1 + φ 2 2Q 2 + φ 2 3Q 3 )y (20) maximizing LL C over φ 2 2, given β and φ2 3 log L σ 2 λ = NT 2 can be expressed as d Q 2 d d [Q 1 + φ 2 2 Q 2 + φ 2 3 Q 3]d +N 2 1 φ (1 φ 2 3 ) [φ φ2 3 φ2 2 φ2 3 ] (21) aφ bφ c = 0 (22) a = [NT N + 1](1 φ 2 3 )(d Q 2 d) b = (1 φ 2 3 )(N 1)d [Q 1 + φ 2 3 Q 3]d φ 2 3 (T 1)N(d Q 2 d) c = N φ 2 3 d [Q 1 + φ 2 3 Q 3]d

17 Iteration properties for fixed 0 < φ 2 3 < 1 if φ 2 2 = 0, (20) ˆβ BW = [X (Q 1 + φ 2 3 Q 3)X ] 1 X (Q 1 + φ 2 3 Q 3)y ˆβW = [X Q 1 X ] 1 X Q 1 y Within ˆβ C = [X Q 3 X ] 1 X Q 3 y Between time if φ 2 2, (20) ˆβ B = [X Q 2 X ] 1 X Q 2 y Between individual ˆφ 2 2 = [ b ] b 2 4ac /(2a) (23) is the unique positive root since a < 0 and c > 0 (Baltagi, Li: 1992) and since φ 2 3 is fixed using ˆQ 1 = Q 1 + φ 2 3 Q 3: ˆβ = [X ( ˆQ 1 + φ 2 2Q 2 )X ] 1 X ( ˆQ 1 + φ 2 2Q 2 )y (24) if started from ˆβ BW, sequence φ 2 2 (it) is strictly increasing if started from ˆβ B, sequence φ 2 2 (it) is strictly decreasing

18 Best linear unbiased predictor u i,t +S = µ i + λ T +S + ν i,t +S (25) w = E(u i,t +S u) = σ 2 µ(l i ι T ) where l i is to ith column of I N E(u i,t +S, u jt ) = σ 2 µ i = j (26) w Ω 1 = σ 2 µ(l i ι T ) [ 4 i=1 ] 1 Q i λ i (27) = σ2 µ [ (l λ i ι T ) ι NT /N] + σ2 µ (ι NT /N) (28) 2 λ 4 (l i ι T )Q 1 = 0 (l i ι T )Q 2 = (l i ι T ) ι NT /N (l i ι T )Q 3 = 0 (l i ι T )Q 4 = ι NT /N

19 typical element of w Ω 1 û GLS where û GLS = y Z ˆδ GLS is T σµ 2 T σµ T σµ 2 + σν 2 ( û i.,gls û 2..,GLS ) + T σµ 2 + Nσλ 2 + û..,gls (29) σ2 ν or T σ 2 µ T σ 2 µ + σ 2 ν û i.,gls + T σ 2 µ [ 1 λ 4 1 λ 2 ] û..,gls if Z contains a constant, than ι NT Ω 1 û GLS = 0 and using ι NT Ω 1 = ι NT /λ 4 one gets û..,gls = 0 and the BLUP corrects the GLS prediction by a fraction of the mean of GLS residuals of ith individual ( ) T σ 2 µ ŷ i,t +S = Z i,t +S ˆδGLS + û T σµ 2 + σν 2 i.,gls (30)

20 Software LIMDEP, RATS, SAS, TSP and GAUSS ( Blanchard: 1996) SAS, Stata for larga data sets OX, GAUSS for programming estimation and tests LIMDEP, TSP, EViews or Stata for simple panel data R packages: plm for LS procedures, contains example datasets splm for MLE procedure (for one way error only)

21 Two-way error estimation methods OLS Within estimator Swamy and Arora (1972) Wallace and Hussain (1969) Amemiya (1971) negative variance estimates indicates zero or small value

22 Public capital productivity Cobb-Douglas production function ln Y = α + β 1 ln K 1 + β 2 ln K 2 + β 3 ln l + β 4 U + u (31) Y, gross state product, gsp K 1, public capital, pcap K 1, private capital, pc L, labour input as payrolls, emp U, unemployment rate, unemp annual observations for 48 states over (Munnell: 1990) dataset Produc.prn on the Wiley web site

23 log.pcap. log.pc. log.emp. unemp idios id time ols wit bin btp swa wah ame Table: Estimated coefficients from Produc data log.pcap. log.pc. log.emp. unemp ols wit bin btp swa wah ame Table: Estimated std. errors from Produc data

24 Twoways e f f e c t s P o o l i n g Model C a l l : plm ( f o r m u l a = l o g ( gsp ) l o g ( pcap ) + l o g ( pc ) + l o g (emp) + unemp, data = Produc, e f f e c t = twoways, model = p o o l i n g, i n d e x = c ( s t a t e, y e a r ) ) Balanced Panel : n=48, T=17, N=816 R e s i d u a l s : Min. 1 s t Qu. Median 3 rd Qu. Max C o e f f i c i e n t s : E s t i m a t e Std. E r r o r t v a l u e Pr ( > t ) ( I n t e r c e p t ) < 2. 2 e 16 l o g ( pcap ) < 2. 2 e 16 l o g ( pc ) < 2. 2 e 16 l o g ( emp) < 2. 2 e 16 unemp e 06 S i g n i f. c o d e s : T o t a l Sum o f S q u a r e s : R e s i d u a l Sum o f S q u a r e s : R Squared : Adj. R Squared : F s t a t i s t i c : on 4 and 811 DF, p v a l u e : < e 16

25 Twoways e f f e c t s Within Model C a l l : plm ( f o r m u l a = l o g ( gsp ) l o g ( pcap ) + l o g ( pc ) + l o g (emp) + unemp, data = Produc, e f f e c t = twoways, model = w i t h i n, i n d e x = c ( s t a t e, y e a r ) ) Balanced Panel : n=48, T=17, N=816 R e s i d u a l s : Min. 1 s t Qu. Median 3 rd Qu. Max C o e f f i c i e n t s : E s t i m a t e Std. E r r o r t v a l u e Pr ( > t ) l o g ( pcap ) l o g ( pc ) e 09 l o g ( emp) < 2. 2 e 16 unemp S i g n i f. c o d e s : T o t a l Sum o f S q u a r e s : R e s i d u a l Sum o f S q u a r e s : R Squared : Adj. R Squared : F s t a t i s t i c : on 4 and 748 DF, p v a l u e : < e 16

26 Twoways e f f e c t s Random E f f e c t Model (Swamy Arora s t r a n s f o r m a t i o n ) C a l l : plm ( f o r m u l a = l o g ( gsp ) l o g ( pcap ) + l o g ( pc ) + l o g (emp) + unemp, data = Produc, e f f e c t = twoways, model = random, random. method = swar, i n d e x = c ( s t a t e, y e a r ) ) Balanced Panel : n=48, T=17, N=816 E f f e c t s : v a r s t d. dev s h a r e i d i o s y n c r a t i c e e i n d i v i d u a l e e time e e t h e t a : ( i d ) ( time ) ( t o t a l ) R e s i d u a l s : Min. 1 s t Qu. Median 3 rd Qu. Max C o e f f i c i e n t s : E s t i m a t e Std. E r r o r t v a l u e Pr ( > t ) ( I n t e r c e p t ) < 2. 2 e 16 l o g ( pcap ) l o g ( pc ) < 2. 2 e 16 l o g ( emp) < 2. 2 e 16 unemp e 06

27 S i g n i f. c o d e s : T o t a l Sum o f S q u a r e s : R e s i d u a l Sum o f S q u a r e s : R Squared : Adj. R Squared : F s t a t i s t i c : on 4 and 811 DF, p v a l u e : < e 16

28 Amemiya, T., 1971, The estimation of the variances in a variance-components model, International Economic Review 12, 113. Baltagi, B.H. and Q. Li, 1992, A monotonic property for iterative GLS in the two-way random effects model, Journal of Econometrics 53, 4551 Blanchard, P., 1996, Software review, Chapter 33 in L. Mtys and P. Sevestre, eds., The Econometrics of Panel Data: A Handbook of the Theory With Applications (Kluwer Academic Publishers, Dordrecht), Fuller, W.A. and G.E. Battese, 1974, Estimation of linear models with cross-error structure, Journal of Econometrics 2, Graybill, F.A., 1961, An Introduction to Linear Statistical Models (McGraw-Hill, New York).

29 Munnell, A., 1990, Why has productivity growth declined? Productivity and public investment, New England Economic Review, 322. Nerlove, M., 1971, A note on error components models, Econometrica 39, Swamy, P.A.V.B. and S.S. Arora, 1972, The exact finite sample properties of the estimators of coefficients in the error components regression models, Econometrica 40, Wallace, T.D. and A. Hussain, 1969, The use of error components models in combining cross-section and time-series data, Econometrica 37, 5572.