Econometrics of Panel Data
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1 Econometrcs of Panel Data Jakub Mućk Meetng # 8 Jakub Mućk Econometrcs of Panel Data Meetng # 8 1 / 17
2 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random coeffcent model 4 Mean group estmaton Jakub Mućk Econometrcs of Panel Data Heterogenety n the slope coeffcents Meetng # 8 2 / 17
3 Heterogenety n the slope coeffcents I In the standard lnear panel data model we control for unobserved heterogenety: y = β X + u, (1) Where u s the sum of ndvdual-specfc component (n the RE model) and the dosyncratc component. In the FE model, the ndvdual-specfc ntercepts are ntroduced whle the u contans only the dosyncratc shock. At the same tme, we have assumed that all slope coeffcents (vector β) are the same for all unt and all perods. In the above formulaton, we don t allow for any nteracton between ndvdual effects and explanatory varable. Consder the followng formulaton: y t = β tx t + u t, (2) where all slope coeffcents captured by β t are now tme-varyng and ndvdualspecfc. Although the above general formulaton seems to be more realstc t lacks any explanatory power and s not useful for predcton. Jakub Mućk Econometrcs of Panel Data Heterogenety n the slope coeffcents Meetng # 8 3 / 17
4 Heterogenety n the slope coeffcents II The above model s not estmable snce the number of parameters exceeds the number of observatons. More applcable formulatons: y = β X + u, (3) y t = β tx t + u t (4) Whch knd of heterogenety n the slopes should ntroduce? In general, we pay more attenton to ndvdual effects but t depends on T and N, the research queston. To account for the ndvduals dfferences n the slope coeffcents we wll ntroduce: Seemngly Unrelated Regresson (SUR), Swamy s random coeffcent model, Mean group estmaton. Jakub Mućk Econometrcs of Panel Data Heterogenety n the slope coeffcents Meetng # 8 4 / 17
5 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random coeffcent model 4 Mean group estmaton Jakub Mućk Econometrcs of Panel Data Seemngly Unrelated Regresson (SUR) Meetng # 8 5 / 17
6 Seemngly Unrelated Regresson (SUR) I Seemngly Unrelated Regresson (SUR) s estmaton method that s desgned to estmate a system of lnear equaton (wth potentally dfferent set of explanatory varables) and whch accounts for the cross-equaton correlaton of the error term. Consder the followng set of equatons: y = X β + ε for {1,..., m} (5) where the ndex denotes the -th equaton n the consdered system. In the matrx form: y 1 y 2. y m = X X X m β 1 β 2. β m + ε 1 ε 2. ε m. (6) In -th equaton, K parameters are estmated. It yelds the total number of coeffcent K = m K. In addton, the K > T. =1 Strctly exogenety s assumed,.e., E(ε X 1,..., X m) = 0. Jakub Mućk Econometrcs of Panel Data Seemngly Unrelated Regresson (SUR) Meetng # 8 6 / 17
7 Seemngly Unrelated Regresson (SUR) II In the SUR framework, t s possble to assume that the covarance matrx of the error term s not dagonal: Ω = E ( εε X 1,..., X m ) = σ11i 2 σ12i 2... σ1mi 2 σ21i 2 σ22i 2... σ2mi σm1i 2 σm2i 2... σmmi 2. (7) Gven the above structure of the varance-covarance matrx of the error term, the system of equatons can be estmated wth FGLS (feasble generalzed least squares). Conventonally, the two-step estmaton ncludes the followng steps 1 Runnng the OLS regresson for the consdered system of equatons to get consstent and unbased estmates of the varance-covarance matrx of the error term (ˆΩ). 2 Based on the estmates of the ˆΩ, standard GLS estmator can be appled: ˆβ SUR = ( X ˆΩ 1 X ) 1 X ˆΩ 1 y. (8) Note that f Ω s dagonal then β SUR wll be close to the OLS estmator. Jakub Mućk Econometrcs of Panel Data Seemngly Unrelated Regresson (SUR) Meetng # 8 7 / 17
8 The SUR estmaton and panel data In the context of long and narrow panel data, the SUR can be appled to account for a potental heterogenety n the slopes. Consder the case of long (relatvely large T) and narrow (not so large N) panel. Then, the standard lnear model can be expressed as a set of equatons: y 1 = β 1X 1 + ε 1, y 2 = β 2X 2 + ε 2,. =. y N = β NX N + ε N, where β N s the ndvdual-specfc vector of the structural parameters. The SUR method accounts for cross-equaton correlaton. In the above case, ths correlaton s equvalent to cross-sectonal dependence. It s possble to test heterogenety of slopes. The standard Wald test can be used to verfy the hypothess about: homogenety of all slopes,.e., H0 : β 1 =... = β N, where β stands for vector of parameters for -th unt. homogenety of some slopes,.e., H0 : β 1,j =... = β N,j, where β,j stands for j-th parameter for -th unt. Jakub Mućk Econometrcs of Panel Data Seemngly Unrelated Regresson (SUR) Meetng # 8 8 / 17
9 Testng cross-equaton (cross-sectonal) correlaton of the error term The SUR method provdes more effcent estmates snce t accounts for crossequaton dependence. Cross-equaton dependence can be tested wth the LM statstc (Breusch and Pagan, 1980): N 1 N LM = T ˆρ 2,j, (9) =1 j=+1 where ρ,j s cross-sectonal correlaton coeffcent: T t=1 ˆρ,j = ˆε tˆε jt ( T ) 1 ( t=1 ˆε 2 ) 1. (10) T t t=1 ˆε 2 jt The LM statstc s vald for fxed N as T and s asymptomatcally dstrbuted as χ 2 wth N(N 1)/2 degrees of freedom. Jakub Mućk Econometrcs of Panel Data Seemngly Unrelated Regresson (SUR) Meetng # 8 9 / 17
10 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random coeffcent model 4 Mean group estmaton Jakub Mućk Econometrcs of Panel Data Swamy s random coeffcent model Meetng # 8 10 / 17
11 Swamy s random coeffcent model I Swamy (1970) proposes the random coeffcent model. Consder the followng model: y = X β + ε (11) where the ndvdual-specfc slope β s the sum of common (β) and untspecfc (α ) components: β = β + α, (12) where 1 E(α ) = 0, 2 E(α α ) = Σ. Queston: how to estmate β and Σ? The dependent varable can be expressed: y = X β + ε = X β + X α + ε = X β + ν, where ν = X α + ε and E(ν ) = 0. Jakub Mućk Econometrcs of Panel Data Swamy s random coeffcent model Meetng # 8 11 / 17
12 Swamy s random coeffcent model II The varance-covarance of the error term ν for -th unt s the followng: E (ν ν ) = E ((X α + ε )(X α + ε ) ) = E (ε ε ) + X E (α α ) X. If the dosyncratc error term s sphercal then: E (ν ν ) = σ 2 I + X ΣX = Π. The Π varance-covarance matrx for the error term wll be block-dagonal. Fnally, the GLS estmator can be appled: ˆβ RC = ( X Π 1 X ) 1 X Π 1 y = W ˆβOLS. (13) where ˆβ OLS s the unt-specfc OLS estmates and W : [ ] 1 W = (Σ + V ) 1 (Σ + V ) 1 (14) where V s the panel-specfc varance-covarance of ˆβ OLS,.e., ˆV = σ 2 (X X ) 1. Jakub Mućk Econometrcs of Panel Data Swamy s random coeffcent model Meetng # 8 12 / 17
13 Swamy s random coeffcent model III The varance of β can be calculated as: Var(β) = (Σ + V ) 1. (15) Fnally, the remander element of the varance-covarance components whch captures the varaton of the slope coeffcents,.e., Σ, can be estmated based on the varaton n the panel-specfc β OLS estmates: ( ) ˆΣ = 1 ˆβ OLS OLS ( ˆβ ) N N 1 β OLS ( β OLS ) 1 ˆV N where β OLS s the average from the OLS estmates. Swamy (1970) postulates to omt the last component because t s neglgble n large samples and t can be not postve defnte. Jakub Mućk Econometrcs of Panel Data Swamy s random coeffcent model Meetng # 8 13 / 17
14 Testng homogenety n slopes To test whether the random coeffcent model s statstcally motvated one mght compare the panel-specfc estmates wth ther weghted (by V 1 ) average. Test statstc: where The null hypothess: χ = N =1 β = ( ˆβOLS ( N =1 β ˆV 1 ) ˆV 1 ) 1 N =1 ( ˆβOLS ˆV 1 ˆβ OLS. H 0 β 1 = β 2 =... = β N. ) β, (16) The test statstc χ s asymptotcally χ 2 dstrbuted wth k (m 1) degrees of freedom. Jakub Mućk Econometrcs of Panel Data Swamy s random coeffcent model Meetng # 8 14 / 17
15 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random coeffcent model 4 Mean group estmaton Jakub Mućk Econometrcs of Panel Data Mean group estmaton Meetng # 8 15 / 17
16 Mean group estmaton I The Mean Group estmator (MG) was proposed by Pesaran and Smth (1995) to deal wth dynamc random coeffcent model. The MG estmator s defned as the average of the unt-specfc OLS estmators ˆβ : OLS ˆβ MG = 1 N ˆβ OLS, (17) N where =1 ˆβ OLS = (X X ) 1 X y. (18) It s assumed that all explanatory varables are strctly exogenous. The MG estmaton s possble when both T and N are suffcently large. The MG estmaton can be appled rrespectvely of the nature of heterogenety n the slope coeffcent. It can be appled f the dfferences n slopes are random (as n the Swamy estmator), dversty n the slopes can be captured by the fxed effects. Jakub Mućk Econometrcs of Panel Data Mean group estmaton Meetng # 8 16 / 17
17 Mean group estmaton II The varance of the MG estmator: Var( ˆβ MG ) = 1 N(N 1) N =1 ( ˆβOLS ˆβ MG) ( ˆβOLS ˆβ MG). (19) The MG estmator wll be very close to the Swamy s estmator f T tends to nfnty and there s some heterogenety n the slopes: ( lm ˆβMG ˆβ RC) = 0 T Jakub Mućk Econometrcs of Panel Data Mean group estmaton Meetng # 8 17 / 17
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