A Random Walk through Seasonal Adjustment: Noninvertible Moving Averages and Unit Root Tests

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1 CONFERENCE ON SEASONAIY, SEASONA ADJUSMEN AND HEIR IMPICAIONS FOR SHOR-ERM ANAYSIS AND FORECASING 0- MAY 006 A Random al hrogh Seaonal Admen: Nonnverble Movng Average and Un Roo e oma del Barro Caro Dene Oborn

2 A Random al hrogh Seaonal Admen: Nonnverble Movng Average and Un Roo e b oma del Barro Caro Unver of Barcelona And Dene R Oborn Unver of Mancheer Arl 006 Ke word: Un roo e, eaonal, eaonal admen, X-. JE code: C, C, C8

3 all (974 and Sm (974. del Barro Caro and Oborn (004, Ercon, Hen and ran (994, Frane (995, 996, Ghel (990, Ghel and Perron (993, 996, Ghel and ebermann (996, Maravall (993, Maa Mr and Oborn (004 and Oero and Smh (00. Seaonal admen fond o have no amoc mac on e nder he nll hohe of (zero freenc negraon and conegraon; ee, n arclar, Ghel and Perron (993 and Ercon e al. (994. Galbrah and Znde-alh (999 and Gonzalo and Para (998.- Inrodcon.- Seaonal admen and movng average comonen 3.- Nonnverble movng average 4.- Seaonall aded random wal

4 . Seaonal admen and movng average comonen Un roo e are baed on: ρ - ( H 0 :ρ or, α ρ - 0. n ( ma exhb emoral deendence and/or heeroedac. mng bon of UR e nder H 0 : Phll (987, heorem 3.. cal amon n UR anale beng ha he roce for aonar and nverble. (for examle, Ghel and Perron, 993, Ello, Rohenberg and Soc, 996, Galbrah and Znde-alh, 999. However, he nverbl of ma be eoned when he ere nder anal ha been eaonall aded b convenonal rocedre..e. X- or X- ARIMA rogram. Seaonal admen b X- can be rereened a he alcaon of a eence of lnear fler, ee aroe (977 arerl cae and Ghel and Perron (993 monhl cae. A hown b Brrdge and all (984, X- ame ha he naded ere generaed b: d ( S( w ( where d or (wh he be-fng model mlng d and w a movng average (MA roce. 3

5 Emrcal de of he roere of eaonal me ere fnd, n general, lle evdence for he reence of he fll e of eaonal n roo mled S( n (; ee, among oher Beale and Mron (993, Oborn (990, or he dcon n Ghel and Oborn (00 Alcaon of eaonal admen baed on an amon of a DGP of he form ( when he re DGP ha no eaonal n roo wll ndce he fll e of (eaonal n roo mled b S( no he MA comonen. Coneenl, he lzed fac ha macroeconomc me ere are no eaonall negraed mle ha he drbance n he n roo e regreon of (, when eaonall aded, ma be ancaed o be a nonnverble movng average roce. In Schwer (989 wa hown ha UR e are oorl zed n he reence of movng average comonen n (. Galbrah and Znde-alh (999 and Gonzalo and Para (998 analcall how wh ch doron occr. However, Galbrah and Znde-alh (999 and Gonzalo and Para (998 foc on nverble MA comonen n (. Neverhele, Galbrah and Znde-alh hn a he morance of h amon, 4

6 3. Nonnverble movng average -,,, (3 where - and ~ d(0, σ he movng average n roo of - n (3 mle a zero n he ecral den of a a freenc of π. 3. No correcon for aocorrelaon α -,,,, (4 α 0 and -. Prooon. e follow (3 wh - and ~ d(0, σ. he amoc bon of he normalzed ba e ac n (4 hen gven b: [ ( ] 0.5 ˆα. (5 [ ( r] and ha for he -rao e ac : [ ( ] 0.5 ˆ α. (6 / { } [ ( r] ˆ α [ ( [ ( r] ] [ ( ] ˆ α (7 { } [ ( r] / 5

7 6 3. Aoregreve agmenaon Now conder he al ADF regreon v φ α (8 where he DGP agan gven b (3. nder he H 0 :α 0 he aoregreve agmenaon of (8 rel n an MA( drbance roce, wh coeffcen e φ.. (... ( ( ( ( e φ φ φ φ φ φ φ φ...,,, ( (9 Noce ha, he coeffcen do no declne oward zero a ncreae. he AR aroxmaon doe no accon for he nonnverble MA eaonal n roo n roo -.

8 7 Prooon. e follow (3 wh - and ~ d(0, σ. he normalzed ba and -rao e ac e ac n regreon (8 hen af: [ ] [ ] even r odd r ( ( 4 ( ( 4 ˆ α (0 and [ ] [ ] [ ] [ ] even r odd r / / / / ˆ ( ( ( ( α (

9 able. he Nll Drbon of he DF -Sac n he Preence of a Nonnverble MA. Qanle Sze DF d Nonnverble MA Proce wh Agmenaon P P Noe: he anle of he emrcal bon of he ADF e -rao e are baed 5,000 relcaon and a amle ze of 4,000 obervaon. he DF bon obaned from a random wal where he nnovaon he whe noe roce ~ N(0,. he nonnverble MA an I( roce where he nnovaon gven b -, ~ N(0,, and he ADF regreon emaed wh no agmenaon and agmenaon of order,, 3, 4, 8,, 6, 0, 4. 8

10 9 3.3 PP aroach ( ( l Z ˆ αˆ α. ( and ( ( l l l Z ˆ ˆ α α (3 where ( l w ˆ ˆ, ˆ ˆ (4

11 Prooon 3. e follow (3 wh - and ~ d(0, σ. hen he amoc bon of he PP n roo e ac of ( and (3 are gven b: Z ( ( w(, [ w( r ] [ ( ] 0.5 ˆ α (5 [ ( r] 4 ( [ ( Z ˆ α /. (6 ( w(, / [ [ ( r] ] 0.5( w(, / w(, [ ( r] ( [ ] / ] h he Barle wndow (, [ /( ] w. ( Z ˆ α 0.5 / [ ( ] 0.5[ /( ]. (7 [ [ ( r] ] / 0

12 he Nll Drbon of he PP ( able. Z αˆ Sac n he Preence of a Nonnverble MA Qanle Sze DF d Nonnverble MA Proce wh Aocorrelaon Correcon Noe: he anle of he emrcal bon of he Z ( αˆ e are baed 5,000 relcaon and a amle ze of 4,000 obervaon. he DF bon obaned from a random wal where he nnovaon he whe noe roce ~ N(0,. he nonnverble MA an I( roce where he nnovaon gven b -, ~ N(0,, and he Z ( αˆ ac comed for aocorrelaon correcon o order,, 3, 4, 8,, 6, 0, 4 ng he Barle ecral wndow.

13 4.- Seaonall aded random wal...,,,, (8 where (agan for mlc 0 for 0. f f ( (9 Coeffcen are nown (ee aroe, 977, Ghel and Perron, 993. Qarerl cae 0 for 0,,, 7. Ung he Beverdge-Nelon (98 decomoon, he flered ere can be wren (ee he Aendx a: (9 4. No correcon for aocorrelaon f f α. (0 f 0

14 3 Prooon 4. e follow (. hen for e regreon (5, he normalzed ba ha amoc bon [ ]. ( ( ˆ r α ( whle he amoc bon for he -rao ac ( [ ] ( r ˆ α ( 4. Aoregreve agmenaon ( ( ( e φ φ ( (3 ( 0

15 4 Prooon 5. e nflered follow he DGP (, whle he ADF regreon (8 aled o he flered ere of (3. hen he normalzed ba e and -rao e ac af: ( ( [ ] ( ( A r A 0 0 ˆ α (4 and ( ( [ ] ( ( ( ( ( 0 ˆ. B B r A α (5 reecvel, where are defned n (3.

16 able 3. Scalng and Shf erm for he DF -Sac afer Seaonal Admen Rao of calng Scaled Shf Nmeraor Nmeraor Denomnaor Agmenaon hf calng calng Noe: he calng and hf erm relae o eaonal admen of a random wal; ee (9 and (0. he nmeraor hf erm defned b (, 0 0 he nmeraor calng gven b ( and he denomnaor calng b (. he rao of calng reen he rao of he nmeraor o he denomnaor calng, whle he caled hf he nmeraor hf dvded b he denomnaor calng. 5

17 able 4. Qanle and Sze of DF -Sac for Seaonall Aded Random al Qanle Sze DF d Seaonall Aded Random al wh Agmenaon P P P P P P P P P P P P Noe: he anle of he emrcal bon of he ADF e -rao e are baed 5,000 relcaon and a amle ze of 4,000 obervaon. he DF bon obaned from a random wal where he nnovaon he whe noe roce ~ N(0,. Seaonal admen erformed ng he lnear aroxmaon o he wo-ded arerl X- fler, wh 50 addonal obervaon generaed and dcarded from he begnnng and end of he amle. he nomnal ze 0.05 for all cae. 6

18 4.3 Phll-Perron aroach Snce, afer eaonal admen, n (9. [ ] [ ] E E 0 σ γ σ γ Prooon 6. For an naded ere followng he random wal roce of (8, he PP e ac of ( and (3, aled o he eaonall aded ere of (9, have amoc bon ( [ ] ( r w Z (, ( αˆ (7 ( ( ( ( ( ( ( w r w w r Z ˆ,,, ] [ α (8

19 able 5. Scalng and Shf erm for he Phll-Perron Z ( Sac aled o a Seaonall Aded Random al Shf Agmenaon facor Scalng facor Noe : he calng and hf erm relae o eaonal admen of a random wal; ee (7 and (8, ng an aocorrelaon correcon o order and he Barle wegh. he hf erm gven b ( σ [ / ] whle he calng facor ( [ / ]. αˆ 8

20 able 6. Z αˆ Sac for a Seaonall Aded Random al Qanle Sze DF Seaonall Aded Random al wh Aocorrelaon Correcon Noe: See able 4. he aocorrelaon correcon aled o order ng Barle wegh. Qanle and Sze of he PP (

21 4.3 Fne amle Mone Carlo anal 00 nomnal ze 5 %. Order of agmenaon 0,,, 3, 4, 8, n φ n n n 4 n n n n 4 (9 For NSA daa reaonable emrcal ze are generall obaned for 8 for he ADF e ( for DGP wh 0, ma be rered.e. -0.5, In conra he PP alwa banall overzed for NSA. ADF For DGP wh rong ove eaonal AR coeffcen (φ 0.5, 0.8, 0.9 and wh a ove noneaonal MA coeffcen ( 0.5, he SA daa ha good ze roere, even who agmenaon. Oherwe, he e alwa overzed when no agmened. h agmenaon he e can be banall nderzed or nderzed. hen he hf erm of able negave, ( 3 he emrcal ze for SA daa bgger han wh NSA daa. For ove hf, ( 8 or he ze lower ng SA han NSA daa (agmenaon accon for he aocorrelaon roere of he DGP, b canno accon for he nonnverble MA roo relng from SA.

22 Reaonable emrcal ze are generall obaned for SA daa wh and. h a rel from: (a he mall doron ndced b SA, (b roxm o he DGP mlcl amed b X-. (wh he exceon of he cae of negave. PP he PP e aled o SA daa doe no erform well. he onl e of eaonal me ere for whch he PP e ha aroxmael he correc ze afer SA are hoe wh where φ 0.5, 0.8, 0.9 combne he ove noneaonal MA wh 0.5, and hence (a menoned above he re DGP ha mlar emrcal roere o he DGP for whch X- he omal fler.

23 able 7. Sze of ADF for N00 ng Unflered ( and Flered (f Daa φ f f f f f f f f f f f f f f f f f f f

24 able 7 (conned φ f f f f f f f f n n n n n n n Noe: he DGP, φ 4 4. he ADF e regreon nclde an nerce and agmened o order. Rel are baed on 5,000 relcaon and a amle ze of 00 obervaon. Flerng ale he lnear aroxmaon o he wo-ded arerl X- eaonal admen fler, wh 50 addonal obervaon generaed and dcarded a he begnnng and end of he amle. he nomnal ze n all cae

25 able 8. Sze of PP Z ( αˆ for N00 ng Unflered ( and Flered (f Daa φ P f f f f f f f f f f f f f f f f f f f

26 able 8 (conned φ f f f f f f f f n n n n n n n Noe: he DGP, φ 4 4. he Z ( αˆ e regreon nclde an nerce and an aocorrelaon correcon aled o order. Rel are baed on 5,000 relcaon and a amle ze of 00 obervaon. Flerng ale he lnear aroxmaon o he wo-ded arerl X- eaonal admen fler, wh 50 addonal obervaon generaed and dcarded a he begnnng and end of he amle. he nomnal ze n all cae

A Random Walk through Seasonal Adjustment: Noninvertible Moving Averages and Unit Root Tests

A Random Walk through Seasonal Adjustment: Noninvertible Moving Averages and Unit Root Tests O R K I N G P A P E R S A N D S U D I E S ISSN 75-485 A Random al hrogh Seaonal Admen: Nonnverble Movng Average and Un Roo e Conerence on eaonal, eaonal admen and her mlcaon or hor-erm anal and orecang