Testing for Unit Roots in Panel Data: An Exploration Using Real and Simulated Data

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1 Testng for Unt Roots n Panel Data: An Exploraton Usng Real and Smulated Data Bronwyn H. HALL UC Berkeley, Oxford Unversty, and NBER Jacques MAIRESSE INSEE-CREST, EHESS, and NBER

2 Introducton! Our Research Program:! Develop smple models that descrbe the tme seres behavor of key varables for a panel of frms: Sales, employment, profts, nvestment, R&D U.S., France, Japan! Substantve nterest: use of these varables for further modelng (productvty, nvestment, etc.) requres an understandng of ther unvarate behavor! Techncal nterest: explore the use of a number of estmators and tests that have been proposed n the lterature, usng real data.! Ths paper: a comparson of unt root tests for fxed T, large N panels, usng DGPs that mmc the behavor of our real data. 3/1/0 NSF Symposum - Berkeley

3 Outlne! Basc features of our data! Motvaton ssues n estmatng a smple dynamc panel model! Overvew of unt root tests for short panels! Smulaton results! Results for real data 3/1/0 NSF Symposum - Berkeley 3

4 Dataset Characterstcs Scentfc Sector, Country France Unted States Japan Data sources Enquete annuelle sur les Standard and Poor s Needs data; moyens consacres a la Compustat data Data from recherche et au dev. annual ndustral and OTC JDB (R&D dans les entreprses;enq. OTC, based on 10-K data from annuelle des entreprses flngs to SEC Toyo Keza survey) # frms # observatons 5,84 6,417 5,088 After cleanng 5,139 5,71 4,60 No jumps 5,108 5,31 4,15 Balanced (# obs.) 1,87,448,65 (# frms) Postve Cash Flow (# frms) The scentfc sector conssts of frms n Chemcals, Pharmaceutcals, Electrcal Machnery, Computng Equpment, Electroncs, and Scentfc Instruments. 3/1/0 NSF Symposum - Berkeley 4

5 Varables! Sales (mllons $)! Employment (1000s)! Investment (P&E, mllons $)! R&D (mllons $)! Cash flow (mllons $) All varables n logarthms, overall year means removed (so prce level changes common to all frms are removed Levn and Ln 1993). 3/1/0 NSF Symposum - Berkeley 5

6 Representatve data - sales 5 Log of deflated sales Year Selected U.S. Manufacturng Frms 3/1/0 NSF Symposum - Berkeley 6

7 Representatve data R&D 0 Log deflated R&D Year Selected U.S. Manufacturng Frms 3/1/0 NSF Symposum - Berkeley 7

8 Autocorrelaton Functon for Real Varables Unted States Autocorrelaton Lag Sales R&D Employment Investment Cash Flow 3/1/0 NSF Symposum - Berkeley 8

9 Autocorrelaton Functon for Dfferenced Logs of Real Varables Unted States Autocorrelaton Lag Sales R&D Employment Investment Cash Flow 3/1/0 NSF Symposum - Berkeley 9

10 Varance of Log Growth Rates σ () log σ () 5 0 Estmated Sgsq() for Dfferenced Log Sales - U.S Estmated Log(Sgsq()) Dstrbuton for Dfferenced Log Sales - U. S Number of obs Var(log growth rate) /1/0 NSF Symposum - Berkeley 10

11 Summary 1. Substantal heterogenety n levels and varances across frms.! However, frm-by-frm estmatons yeld trends wth dstrbutons smlar to those expected due to samplng error when T s small. (not shown)! The sgma-squared dstrbuton dffers from that predcted by samplng error, mplyng heteroskedastcty. (see graph). Hgh autocorrelaton n levels > fxed effects or autoregresson wth root near one? 3. Very slght autocorrelaton n dfferences; however, the wthn coeffcent s substantal and postve >heterogenety n growth rates? 3/1/0 NSF Symposum - Berkeley 11

12 3/1/0 NSF Symposum - Berkeley 1 A Smple Model 1 f ) : ( ) ( ) )( (1 ) : ( ) (1 or 0 ) ~ (0, Years 1,..., Frms; 1,..., nterest. the varable of logarthm of 1, 1, 1, 1 1 > > ρ ε δ ε δ ρ δ α ρ ε ρ ρδ δ ρ α σ ε ε ρ δ α t t t t t t t t t t t t t t js t t t t t t t t t y y RW y y FE y y j s,t ] ε E[ε T t N u u u y y

13 Estmaton wth a Frm Effect Drop δ t (means removed) and dfference out α : y ρ t y, t 1 ε OLS s nconsstent; use IV or GMM-IV for estmaton wth y,t-,,y 1 as nstruments. t Advantages: robust to heteroskedastcty and nonnormalty; consstent for β s; allows for some types of transtory measurement error n y. Dsadvantages: based n fnte samples; mprecse when nstruments are weakly correlated wth ndependent varables. 3/1/0 NSF Symposum - Berkeley 13

14 Three Data Generatng Processes ρ 1. 1 t, t 1 or y t y δ ε y t δ OLS s consstent; IV wth lagged nstruments not dentfed.. ρ 0 or y t y t δ α ε t δt ε t OLS s nconsstent; IV or GMM wth lag nst. s consstent ε t 3. ρ < 1, no effects or y t ρ y, t 1 y t δ α ε t ρy, t 1 δt ε t OLS s nconsstent; IV or GMM wth lag nst. s consstent 3/1/0 NSF Symposum - Berkeley 14

15 Results of Smulaton N00 T1 No. of draws1000 Estmated coeffcent for dy on dy(-1) Instruments are y(-)-y(-4) Truth OLS IV GMM1 GMM GMM CUE rho1.0 (RW) (.06) 0.79 (.690) (.175) (.8)** (.168) rho0.0 (FE) (0.019)** (.046) (0.04) (.333) (.041) rho0.9 (no effects) (.05)** (.089) ** Dfferent from truth at 5% level of sgnfcance. 3/1/0 NSF Symposum - Berkeley 15

16 Concluson from Smulatons! As wth ordnary tmes seres, t s essental to test frst for a unt root (even though asymptotcs n the panel data case are for N and not T).! Falure to do so may lead to the use of estmators that are very based and msleadng n fnte samples even though they are consstent.! If unt root > assume no fxed effect and then OLS level estmators approprate.! If no unt root > fxed effect (usually) and IV.! Near unt root > OLS bas can be large. 3/1/0 NSF Symposum - Berkeley 16

17 Unt Root Tests Consdered Note that these tests are generally vald for large N and fxed T.! IPS: Im, Pesaran, and Shm (1995) alternatve s ρ <1 for some. Based on an average of augmented Dckey-Fuller tests conducted frm by frm, wth or wthout trend. Normal dsturbances assumed.! HT: Harrs-Tzavals (JE 1999) alternatve s ρ<1. Based on the LSDV estmator, corrected for bas and normalzed by the theoretcal std. error under the null. Homoskedastc normal dsturbances assumed. 3/1/0 NSF Symposum - Berkeley 17

18 Unt Root Tests (contnued)! SUR: OLS wth no fxed effects and an equaton for each year (suggested by Bond et al 000) consstent under the null of a unt root. Has good power. Allows for heteroskedastcty and correlaton over tme easly.! CMLE:! Krunger (1998, 1999) CMLE s consstent for statonary model and for ρ1 (fxed T). Use an LR test based on ths fact. Homoskedastc normal dsturbances assumed, but not necessary.! Lancaster and Lndenhovus (1996); Lancaster (1999) smlar to Krunger. Bayesan estmaton wth flat pror on effects and 1/σ for the varance yelds estmates that are consstent when ρ1 (fxed T). σ s shrunk slghtly toward zero.! CMLE-HS: suggested n Krunger (1998) heteroskedastcty of the form σ σ t can be estmated consstently. 3/1/0 NSF Symposum - Berkeley 18

19 Condtonal ML Estmaton (HS) Model: Or y Stackng the model: Wth u t t y t α ρu E[ u u ρ y ε (1 ) α ρ, t 1 '] u t ε ε ~ N(0, σ, t 1 t t σ V ρ σ 1 ρ 1 ρ ρ... T ρ t ) y α ι 1 ρ ρ 1 ρ... T ρ ρ ρ u T 1 T T /1/0 NSF Symposum - Berkeley 19

20 Condtonal ML Estmaton (HS) Dfferenced: Dy Du where D Dy N(0, Σ) The log lkelhood functon: log L( ρ, Σ DV ~ wth ρ { } σ ) N( T N logφ 1 σ 1) log(π ) 1 N ( Dy )' Φ σ 3/1/0 NSF Symposum - Berkeley 0 D' 1 Dy ( T 1) N σ 1 Φ log( σ )

21 Condtonal ML Estmaton (HS) The σ can be concentrated out usng σ T 1 1 tr 1 ( Dy ( Dy )') Φ whch yelds log L( ρ) ( T 1) N 1 N( T 1) log(π 1) N log( σ ( ρ)) logφ( ρ) for estmaton. 3/1/0 NSF Symposum - Berkeley 1

22 Condtonal ML Estmaton (HS)! Krunger (1999) proves consstency of the CMLE-HS estmator for ρ!(-1,1].! However, the concentrated or profle lkelhood verson s problematc:! Nusance parameters (σ ) ncrease wth N standard error estmates based downward; not effcent (see B- N & Cox, ex. 4.3).! Non-orthogonal parameters (ρ, σ t, and σ )! Possble alternatves:! Modfed profle lkelhood - Barndorff-Nelsen and Cox (1994), but not clear how to do ths.! Integrated lkelhood (Woutersen 000). 3/1/0 NSF Symposum - Berkeley

23 Results of Smulatons! IPS! zero augmentng lags to be consstent wth other tests.! we found sze was too large f the data were allowed to choose the number of augmentng lags.! sze slghtly too large! power weak aganst large rho alternatves.! HT! sze correct f homoskedastc;! power weak aganst large rho alternatves, wth or wthout FE.! SUR! sze correct; slghtly too large f heteroskedastc! power weak aganst large rho alternatves, wth or wthout FE. 3/1/0 NSF Symposum - Berkeley 3

24 Results of Smulatons! CMLE! sze correct f homoskedastc! power weak aganst large rho alternatves, wth or wthout FE! CMLE-HS! sze wrong! power slghtly weak aganst large rho alternatves, wth or wthout FE! requres sandwch var-cov estmator; appears to have downward-based standard errors, so rejects too often. 3/1/0 NSF Symposum - Berkeley 4

25 Results of Smulaton - Homoskedastc DGP N00 T1 No. of draws1000 Emprcal sze or power (nomnal sze.05) Truth (DGP) IPS no trend IPS trend H-T CMLE t test CMLE-HS t test SUR rho1.0 (RW) rho0.0 (FE) rho0.99 (no effects) /1/0 NSF Symposum - Berkeley 5

26 Results of Smulaton - Heteroskedastc DGP N00 T1 No. of draws1000 Emprcal sze or power (nomnal sze.05) Truth (DGP) IPS no trend IPS trend H-T CMLE t test CMLE-HS t test SUR rho1.0 (RW) rho0.0 (FE) rho0.99 (no effects) /1/0 NSF Symposum - Berkeley 6

27 Results of Unt Root Tests Seres wth unt roots IPS no trend IPS wth trend HT CMLE CMLE wth HS SUR Sales US,J US,F,J US,F,J US,F,J US,F,J J only Employment US,F,J US,F,J US,F,J US,F US,F,J J only R&D US only US,F,J US only US only US,F,J -- Investment Cash flow US only US,J /1/0 NSF Symposum - Berkeley 7

28 Conclusons! A model wth a very large autoregressve coeffcent and no level fxed effect may be a good descrpton of these data the substantve mplcaton s that we use the ntal condton rather than a permanent effect to descrbe dfferences across frms.! CML estmaton s feasble and may be a useful estmator n the cases where we cannot use the SUR dea.! Next steps:! Heteroskedastc-consstent standard errors to correct sze n CMLE-HS, etc.! Further exploraton of heterogeneous trends.! Modelng a more complex AR process for our data wth heteroskedastcty but no fxed effects. 3/1/0 NSF Symposum - Berkeley 8

29 Trends real and smulated data Estmated tme trend 3/1/0 NSF Symposum - Berkeley 9

30 Intercepts real and smulated data Estmated Intercept 3/1/0 NSF Symposum - Berkeley 30