Reading Assignment. Panel Data Cross-Sectional Time-Series Data. Chapter 16. Kennedy: Chapter 18. AREC-ECON 535 Lec H 1

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1 Readng Assgnment Panel Data Cross-Sectonal me-seres Data Chapter 6 Kennedy: Chapter 8 AREC-ECO 535 Lec H

2 Generally, a mxtre of cross-sectonal and tme seres data y t = β + β x t + β x t + + β k x kt + e t where =,, and t =,, Sample sze s hs s a Balanced Desgn Example of an Unbalanced Desgn: =,, t and t =,, Each has a dfferent nmber of Or? What do the data matrces look lke? y = y y y y y y X = x x x x x x AREC-ECO 535 Lec H

3 How to nterpret smlar/dentcal models wth dfferent data strctres? hnk abot the one nt change n X What s t changng across? And Y s smlar Ex) 5 states and 5 obseratons per state Ex) states and 5 obseratons per state Ex) Hosng prces for all US contes for 3 years Ex) Hosng prces for the Colorado Front Range (transactons) for the past years Ex) Hosng prces for major metropoltan areas n the US for past years AREC-ECO 535 Lec H 3

4 Fxed-Effects Models Sppose we want each th nddal to hae ts own mean y t = β + β x t + β x t + + β k x kt + α D + e t e t ~ (, σ ) where D = for obseraton on th nddal, and otherwse Sppose we want each tth tme perod to hae ts own mean y t = β + β x t + β x t + + β k x kt + θ t D t + e t e t ~ (, σ ) where D t = for obseraton on tth perod, and otherwse (We hae sed these arables before) AREC-ECO 535 Lec H 4

5 Motaton for Fxed-Effects y hree demand fnctons across three contres bt f the contry s not nclded as a dmmy arable f the fxed effects are mportant and yor model does not nclde them then yor reslts are lkely based (Heterogenety or nobsered heterogenety) x AREC-ECO 535 Lec H 5

6 Both? y t = β + β x t + β x t + + β k x kt + Σ α D + Σ θ t D t + e t e t ~ (, σ ) where D = for obseraton on th nddal and where D t = for obseraton on tth perod Carefl of degrees of freedom Freqently se the restrcton Σ α = Σ θ t = Ex) Bsness frm acton oer tme nestment n technology Want nestment (e, expendtre on ) to be dfferent across frms of dfferent sze and fncton (e, large erss small and manfactrng erss fnancal serces) And want t to be dfferent for dfferent years (e, dot com years or bsness cycle phases) AREC-ECO 535 Lec H 6

7 he crtcal sse and decson by the researcher wth respect to sng fxed effects s recognzng what yo are holdng constant n those ceters parbs condtons Mean shfts across nddals and/or across tme so then dfferences n means Example: = 3 & = α α α 3 θ θ Each? Means measred and then other ndependent arables measres aratons arond these dfferent (or shftng) means Dfference n means approach: how many means do yo hae and are they approprate? Yo measred them, now do they generalze or predct? AREC-ECO 535 Lec H 7

8 ext, we wll ntrodce aros forms of heteroskedastcty across the nddals and seral correlaton across the tme hen make se of EGLS β = (X Ω - X) - X Ω - y V(β) = (X Ω - X) - so we need to thnk abot and specfy what Ω looks lke AREC-ECO 535 Lec H 8

9 AREC-ECO 535 Lec H 9 Fxed-Effects Model wth heteroskedastcty y t = β + β x t + β x t + + β k x kt + (Σ α D ) + e t E(e t ) = σ (error arance s dfferent for each nddal )

10 AREC-ECO 535 Lec H Fxed-Effects Model wth heteroskedastcty and seral correlaton y t = β + β x t + β x t + + β k x kt + (Σ α D ) + e t e t = ρ e t- + t (there s seral correlaton wthn each nddal) E(e t ) = σ (error arance s dfferent for each nddal ) & 3 3 V V V V What f ρ = ρ j for all, j? (Remember the ewey-west Seral Correlaton Consstent Var-Co Estmator? Very approprate here and modfcatons needed for an nbalanced desgn)

11 AREC-ECO 535 Lec H Fxed-Effects Model wth heteroskedastcty, seral correlaton, and correlaton between nddals y t = β + β x t + β x t + + β k x kt + (Σ α D ) + e t e t = ρ e t- + t (there s seral correlaton wthn each nddal) E(e t ) = σ (error arance s dfferent for each nddal ) E(e t e jt ) = σ j (correlated errors across nddals and j) & 3 3 V V V V (Cont the parameters)

12 Random Effects Models Sppose we want the effects to be stochastc we want the errors to be combnatons of errors from dfferent sorces So we are pshng the mean effect nto the error y t = β + β x t + β x t + + β k x kt + e t where e t = + t + w t V(e t ) = σ = σ + σ + σ w Ex) Want nestment (e, expendtre on ) to be dfferent across frms of dfferent sze and fncton And want t to be dfferent for dfferent years bt not constraned to an estmated mean Bt mst be wllng to assme nddal or temporal effects are ncorrelated wth the regressors Random effects also saes potentally a large nmber of degrees of freedom AREC-ECO 535 Lec H

13 AREC-ECO 535 Lec H 3 & A A A A (Cont the parameters)

14 AREC-ECO 535 Lec H 4 Specal case, jst random effects across nddals y t = β + β x t + β x t + + β k x kt + e t where e t = + w t V(e t ) = σ = σ + σ w & A A A A

15 Specal case, fxed effects across nddals and random effects across tme y t = β + β x t + β x t + + β k x kt + (Σ α D ) + e t where e t = t + w t V(e t ) = σ = σ + σ w Ex) ransacton prces for each t there are t transactons he model explans market fndamentals expected prce here are systematc dfferences between frms prchasng de to locaton and shppng costs bt f any one transacton drng week t s away from the expected ale then t s lkely that all transactons are away from that ale f one frm pays too mch or too lttle based on market fndamentals then all play too mch or too lttle AREC-ECO 535 Lec H 5

16 Many software packages wll test for and estmate fxed and random effects Hasman est for fxed erss random effects Fxed effects are dmmy arables Random effects are a specfc form of heteroskedastcty combnatons of arances across sample grops Do yo want conclsons to be based on a mean effect? f not, then ncorporate arance effect Or are conclsons based on arance effect? Example prodcton decsons across farmers choce decsons across consmers Ex) f effect s from a contry n a mltple-contry model Any random draw s from a known contry so fxed effect may be better Howeer, f effect s tme perod n a mltple-tme perod model then what f the draw s from the next (ot-of-sample) perod? Random effects may be better Cannot hae fxed and random effects on the same aspect of the sample he qeston s wll the software handle nbalanced desgn? AREC-ECO 535 Lec H 6

17 Hasman est for Fxed Verss Random Effects H = (β FE β RE )' (Σ FE Σ RE ) - (β FE β RE ) hs s a Wald test and s dstrbted χ wth k- degrees of freedom ll hypothess s that the effects are ncorrelated wth the data or that random effects are acceptable to se he test s based on an nterestng dea Under the assmpton of the nll, the OLS and GLS reslts are consstent bt the GLS s effcent Bt nder the alternate, OLS s consstent bt GLS s not consstent herefore, nder the nll, OLS and GLS reslts shold not be dfferent Or H = (β β E )' (Σ Σ E ) - (β β E ) Under H : both β and β E are consstent bt β E s effcent whereas β s neffcent Bt nder H : β remans consstent bt β E s nconsstent AREC-ECO 535 Lec H 7