Chapter 18 Seemingly Unrelated Regression Equations Models

Transcription

1 Chapter 8 Seemngly Unrelated Regresson Equatons Models A basc nature of multple regresson model s that t descrbes the behavour of a partcular study varable based on a set of explanatory varables When the objectve s to explan the whole system, there may be more than one multple regresson equatons For example, n a set of ndvdual lnear multple regresson equatons, each equaton may explan some economc phenomenon One approach to handle such a set of equatons s to consder the set up of smultaneous equatons model s whch one or more of the explanatory varables n one or more equatons are tself the dependent (endogenous) varable assocated wth another equaton n the full system On the other hand, suppose that none of the varables s the system are smultaneously both explanatory and dependent n nature here may stll be nteractons between the ndvdual equatons f the random error components assocated wth at least some of the dfferent equatons are correlated wth each other hs means that the equatons may be lnked statstcally, even though not structurally through the jontness of the dstrbuton of the error terms and through the non-dagonal covarance matrx Such a behavour s reflected n the seemngly unrelated regresson equatons (SURE) model n whch the ndvdual equatons are n fact related to one another, even though superfcally they may not seem to be he basc phlosophy of the SURE model s as follows he jontness of the equatons s explaned by the structure of the SURE model and the covarance matrx of the assocated dsturbances Such jontness ntroduces addtonal nformaton whch s over and above the nformaton avalable when the ndvdual equatons are consdered separately So t s desred to consder all the separate relatonshps collectvely to draw the statstcal nferences about the model parameters Example: Suppose a country has 20 states and the objectve s to study the consumpton pattern of the country here s one consumpton equaton for each state So all together there are 20 equatons whch descrbe 20 consumpton functons It may also not necessary that the same varables are present n all the models Dfferent equatons may contan dfferent varables It may be noted that the consumpton pattern of the neghbourng states may have the characterstcs n common Apparently, the equatons may look dstnct ndvdually but there may be some knd of relatonshp that may be exstng among the equatons Such equatons can be used to examne the jontness of the dstrbuton of dsturbances It seems reasonable to Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur

2 assume that the error terms assocated wth the equatons may be contemporaneously correlated he equatons are apparently or seemngly unrelated regressons rather than ndependent relatonshps Model: We consder here a model comprsng of M multple regresson equatons of the form k y = x β + ε, t =,2,, ; =,2,, M; j =,2,, k t tj j t j= y t s the t th observaton on the th dependent varable whch s to be explaned by the th regresson equaton, x tj s the t th observaton on j th explanatory varable appearng n the th equaton, β j s the coeffcent assocated wth x tj at each observaton and ε t s the t th value of the random error component assocated wth th equaton of the model hese M equatons can be compactly expressed as y = β + ε, =,2,, M y s vector wth elements t y ; s ( K ) observatons on an explanatory varable n the th equaton; ε s a matrx whose columns represent the β s a vector of dsturbances hese M equatons can be further expressed as y 0 0 β ε y β2 ε2 = + ym 0 0 M βm εm or y = β + ε the orders of y s ( M ), s ( M k *), β s ( k *, ) ε s k vector wth elements j M and k* = k β ; and Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 2

3 reat each of the M equatons as the classcal regresson model and make conventonal assumptons for =,2,, M as s fxed rank = k lm = Q Q E u = 0 E( uu ) the sample s nonsngular wth fxed and fnte elements = I s the varance of dsturbances n th equaton for each observaton n Consderng the nteractons between the M equatons of the model, we assume lm j = Qj E uu ;,,2,, j = ji j= M Q j s non-sngular matrx wth fxed and fnte elements and j s the covarance between the dsturbances of th and th j equatons for each observaton n the sample Compactly, we can wrte E E ( ε ) = 0 ( εε ) I I I I I I I I I 2 M M = =Σ I = M M 2 MM denotes the Kronecker product operator, ψ s ( M M ) ψ matrx and ( j ) s ( M M ) Σ= postve defnte symmetrc matrx he defnteness of Σ avods the possblty of lnear dependences among the contemporaneous dsturbances n the M equatons of the model he structure E ( uu ) =Σ I mples that varance of ε t s constant for all t contemporaneous covarance between ε and ε s constant for all t ntertemporal covarance between t and * t ε ε ( t t* ) t j tj are zero for all and j Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 3

4 By usng the termnologes contemporaneous and ntertemporal covarance, we are mplctly assumng that the data are avalable n tme seres form but ths s not restrctve he results can be used for crosssecton data also he constancy of the contemporaneous covarances across sample ponts s a natural generalzaton of homoskedastc dsturbances n a sngle equaton model It s clear that the M equatons may appear to be not related n the sense that there s no smultanety between the varables n the system and each equaton has ts own explanatory varables to explan the study varable he equatons are related stochastcally through the dsturbances whch are serally correlated across the equatons of the model hat s why ths system s referred to as SURE model he SURE model s a partcular case of smultaneous equatons model nvolvng M structural equatons wth M jontly dependent varable and k( k for all ) dstnct exogenous varables and n whch nether current nor logged endogenous varables appear as explanatory varables n any of the structural equatons he SURE model dffers from the multvarate regresson model only n the sense that t takes account of pror nformaton concernng the absence of certan explanatory varables from certan equatons of the model Such exclusons are hghly realstc n many economc stuatons OLS and GLS estmaton: he SURE model s y = β + ε, E ε = 0, V ε =Σ I = ψ Assume that ψ s known he OLS estmator of β s Further b0 = y ( 0 ) = β ( 0) = ( 0 β)( 0 β) = ( ) ψ ( ) E b V b E b b he generalzed least squares (GLS) estmator of β Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 4

5 ( ) y ( ) ( ) ˆ β = ψ ψ E V = Σ I Σ I y ( ˆ β) = β ( β) = E( β β)( β β) ˆ ˆ ˆ = ( ψ ) ( ) = I Σ Defne ( ψ ) G = ψ then G = 0 and we fnd that ( ˆ 0 β) V b V = GψG Snce ψ s postve defnte, so Gψ G s atleast postve semdefnte and so GLSE s, n general, more effcent than OLSE for estmatng β In fact, usng the result that GLSE best lnear unbased estmator of β, so we can conclude that ˆβ s the best lnear unbased estmator n ths case also Feasble generalzed least squares estmaton: When Σ s unknown, then GLSE of β cannot be used hen Σ can be estmated and replaced by ( M M) matrx S Wth such replacement, we obtan a feasble generalzed least squares (FGLS) estmator of β as ( ) ( ) ˆ βf = S I S I y Assume that S ( s j ) = s nonsngular matrx and s j s some estmator of j Estmaton of Σ here are two possble ways to estmate s j Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 5

6 Use of unrestrcted resduals Let K be the total number of dstnct explanatory varables out of k, k2,, k m varables n the full model y = β + ε, E ε = 0, V ε =Σ I and let Z be a K observaton matrx of these varables Regress each of the M study varables on the column of Z and obtan ( ) resdual vectors ˆ ε = y Z Z Z Z y =,2,, M = H y Z H = I Z Z Z Z Z hen obtan s ˆˆ j = εε j = yh y Z j and construct the matrx S ( s j ) = accordngly Snce s a submatrx of Z, so we can wrte and thus Hence = ZJ J s a K k selecton matrx hen ( ) H = Z Z Z Z = ZJ = 0 Z ( β ε ) ( β ε ) yh y = + H + Z j Z j j j = ε H ε Z j Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 6

7 E s E H = jtr H K = j E sj = j K ( j ) = ( ε Zε j ) ( Z ) hus an unbased estmator of 2 Use of restrcted resduals j s gven by sj K In ths approach to fnd an estmator of j, the resduals obtaned by takng nto account the restrctons on the coeffcents whch dstngush the SURE model from the multvarate regresson model are used as follows Regress y on, e, regress each equaton, =, 2,, M by OLS and obtan the resdual vector u = I y = H y A consstent estmator of j s obtaned as Usng * sj = uu j = yh H y j j H = I = j j j j j H I s * j, a consstent estmator of S can be constructed If n * s j s replaced by = + j j j j j j tr H H k k tr then s s an unbased estmator of * j j Econometrcs Chapter 8 SURE Models Shalabh, II Kanpur 7