University of Kansas, Department of Economics Economics 911: Applied Macroeconomics. Problem Set 2: Multivariate Time Series Analysis

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1 Univrsiy of Kansas, Dparmn of Economics Economics 9: Applid Macroconomics Problm S : Mulivaria Tim Sris Analysis Unlss sad ohrwis, assum ha shocks (.g. g and µ) ar whi nois in h following qusions.. Considr h following mulivaria procss. y = y +ε +θε x = φ +µ y i) Is his vcor procss saionary? ii) Is h vcor procss of firs diffrncs ( y, x) invribl?. Is h following procss invribl? y = ε +.5 ε.3ε,,,, y = ε +.6 ε +.7ε,,,, 3. Suppos ha h procss (y,x) has h following vcor auorgrssiv rprsnaion. y =.4y +.4x +ε x =.4y +.4x +µ i) Is his mulivaria procss saionary? ii) Calcula h univaria rprsnaions for y and x, rspcivly. 4. Prov ha if a mulivaria im sris procss has auorgrssiv roos on h uni circl, hn h corrsponding univaria procsss will also hav auorgrssiv roos on h uni circl. 5. Suppos h RBC modlrs ar corrc and ha xognous shocks o h mony supply hav no ffc on oupu. Assum ha oupu is a funcion of on lag of an xognous variabl ha conomricians do no obsrv and an indpndn whi nois shock. Also assum ha h cnral bank adjuss mony o h currn valu of his xognous variabl ha conomricians don obsrv and a diffrn whi nois shock ha is indpndn. Will mony fail o Grangr caus oupu in h bivaria VAR? 6. Suppos ha y is an n-vcor of im sris ha is scond diffrnc saionary and has h following 0

2 Wold rprsnaion: y =α+ C(L) ε wih α an n vcor, i C(L) = CiL i= 0 and ach C i an n n marix. Suppos ha h prsampl valus of g ar qual o zro. Assuming iniial condiions y 0, y -, y -,... for h vcor of im sris, driv h procss for y (in lvls)? Ar h drminisic and sochasic lmns for his procss fundamnally diffrn from h cas whr y is saionary afr firs diffrncing? If so, plas xplain. If hr ar som linar combinaions of y ha ar coingrad, wha kind of rsricions would ha impos on C(L)? If hr ar som linar combinaions of y ha ar coingrad, wha kind of rsricions would ha impos on C(L)? 7. Considr h following bivaria VAR modl: whr: y =.3 y + y m m =.9 m + (, ) y m = and.0.5 E =.5.0 and is no srially corrlad. Driv an alrnaiv rprsnaion which has shocks ha ar uncorrlad and h marix of long-run muliplirs for hs shocks is lowr riangular. Driv h impuls rspons funcion and varianc dcomposiion for y, m, y and m from his rprsnaion. 8. Considr h following bivaria VAR modl. whr: x =.7x + x y = + y.5y x y.0.5 = (, ) and E =.5.0 This modl can b viwd as a spcial cas of a gnral modl givn as h vcor Φ 0Z =Φ Z Φ = 0 Φ = Z = (x,y ) wih and in his paricular modl. (a) Driv an alrnaiv rprsnaion of h daa wih uncorrlad disurbancs and a lowr riangular Φ 0 marix. This rprsnaion implis ha x dpnds on laggd valus of h daa plus a whi nois innovaion, whil y dpnds on currn x, laggd valus of h daa plus a whi nois innovaion. (b) Driv h IRF and VDF of x and y wih rspc o h innovaions in h rprsnaion in and

3 par (a). β (L)Z = = Z (x,y ) 9. Considr h bivaria VAR modl: whr, β (L) = a L b L ii ii ii β = a L b L for i=,,, β = a L b L ( x, y) E =σ Ev = wτ 0 = v w vw β(l) is invribl. τ, for v = x or y and w = x or y, whn, and a. Wha condiions mus hold on β(l) paramrs, for a fini VAR in firs diffrncs o xis? b. Wha condiion mus hold on β(l) paramrs for hr o xis a Vcor Error Corrcions Modl wih only on coingraing vcor? c. Wha condiion mus hold on β(l) paramrs if h wo variabls in h modl ar saionary. For h rmaining pars of his qusion, assum h sris ar saionary in lvls. d. Paramriz h univaria rprsnaion for ach sris (assuming hr ar no common facors.). Consruc h non-orhogonalizd moving avrag rprsnaion associad wih ach innovaion. f. Consruc h moving avrag rprsnaion for ach shock obaind using a Cholsky ordring wih x placd firs. 0. Considr h bivaria VAR modl in firs diffrncs whr β(l) Z = Z = ( y, m ) and = ( y, m) E τ if = = 0 ohrwis. Assum ha v w vw β (L) = I β L β L L whr, and h invrs of B(L) xiss. β = α α α α δ δ β = δ δ a) Can an addiional condiion b imposd on β(l) o yild a Vcor Error Corrcions Modl wih saionary linar combinaions of Z undr hs assumpions? If so, wha is ha condiion? Ignor your answr o par (a) in h rmaindr of his qusion. b) Assuming no common facors, consruc h univaria rprsnaion for oupu (y) and mony (m). Wha ARIMA procss dos ach sris follow?

4 c) Consruc h moving avrag rprsnaion for Z assuming xognous shocks o mony ar uncorrlad wih xognous shocks o oupu (hrfor i is convnin o normaliz h covarianc marix for xognous shocks o b an idniy) and xognous shocks o mony hav no long-run ffc on h lvl of oupu. Wri h MAR as a funcion of h VAR paramrs and h variancs and covariancs of h VAR innovaions. (You nd o show prcisly how o calcula h MAR, bu you nd no muliply ou ach marix.). Suppos ha h raional xpcaions hory of h rm srucur of inrs ras is corrc and h -priod inrs ra (R) is an avrag of h currn and xpcd fuur on-priod inrs ra (r) and a risk prmium (γ):, R = ( )( r + E r ) + + γ whr h risk prmium is dynamic and a funcion of r -, R - and g which is h srially uncorrlad risk prmium shock which has consan sandard dviaion σ g : γ = α r + α R + ε. E indicas xpcaions basd on all informaion known a im. Suppos h rducd form for r is a funcion of on lag of ach variabl:. r = β r + β R + r A. Wri an quaion for R as a funcion of r, r -, R -, and g showing xacly how h cofficins from h rducd quaion for r and h conomic srucur drmin h paramrs in his quaion. B. Your rsul in par A can b rwrin as: R = τ r + τ r + τ R + ε * 0 whr g * is a linar funcion of g and ach τ is a consan cofficin. Suppos h shor ra is givn by h following cnral bank policy racion funcion: r = π 0R + π r + π R + µ, whr h cnral bank rsponds o currn valus of h -priod ra and laggd valus of and priod inrs ras and a monary policy shock (µ). Prov ha his srucural sysm can also b wrin as h following funcion of rsiduals from a bivaria VAR: = τ 0 + ε = π + µ * R r r 0 R, (). () C. Why can OLS b usd o sima h scond quaion? Givn ha you know τ 0 from h rm srucur hory and ha h varianc covarianc marix for h VAR s rsiduals is givn by: E R r R r [ ] σ σ R σ σ rr r =, how would you sima π 0? Wha would his sima convrg rr o asympoically (i.. in larg sampl)? [Hin: Whil you can us OLS o obain a consisn sima of E r R rr π 0, if you did sima quaion by OLS, π 0 would convrg asympoically o = σ. ] E σ R R 3

5 . Many conomiss hav found vidnc ha ach inrs ra has a uni roo and ha diffrn inrs ras ar coingrad. (Whil h hory of h rm srucur of inrs ras can b usd o jusify his finding, you can ignor ha hory in answring his qusion.) Considr h xampl from h prvious qusion. If ach ra has a uni roo and h inrs ra sprad, R-r, is saionary, h following VECM can b simad: b( L) R + b ( L) r = α ( R r ) + r b ( L) R + b ( L) r = α ( R r ) + R whr ach b ij (L) is a lag polynomial of a common lngh for all combinaions of i and j, b ii (0)= for all i (which mans h firs quaion prains o R and h scond o r ), b ij (0)=0 for i no qual o j, α and α ar cofficins and h las rm in ach quaion is h rsidual. A. Show how h VECM can b rwrin as a VAR in h lvls of R and r. Ar his VAR modl rsiduals h sam rsiduals as h rsiduals in VECM? B. Show how h VECM can b rwrin as a VAR in r and (R-r). Show whhr or no hs rsiduals ar h sam as h rsiduals in h VECM? C. Using h srucur in quaions and from h prvious qusion, driv impuls rsponss of R and r o h srucural shocks (g * and µ). Rpor h lag polynomials associad wih ach of hs rsponss. 3. Inflaion (π) oupu (y) and h inrs ra (R) ar variabls in h following hr rducd-form quaions: π π = β π + β y + β 3R + y y = β π + β y + β 3R + R = β π + β y + β R + R Th variancs and covariancs of h rsiduals ar givn by: i j E i j = σ ij E = 0 τ τ, for, wih i=π,y,r and j =π,y,r. For pars A, B and C, you may us lag polynomials o calcula IRFs: A) Calcula IRFs for h (π,y,r) Cholsky ordring B) Show ha h IRFs for h firs shock in h (π,r,y) Cholsky ordring ar prcisly h sam as IRFs for h firs shock in par A. ii) Show ha h IRFs for h hird shock in h (y,π,r) Cholsky ordring ar prcisly h sam as IRFs for h hird shock in par A. For pars D, E and F, assum β = β = β 3 = 0. D. Can OLS b usd o obain consisn and fficin simas of h β, β, β 3, β 3, β 3 and β 33 paramrs? Explain your rasoning. 4

6 ms E. Assum ha monary policy shocks ( ε ) immdialy affc R bu hav no immdia ffc on y or π. Also assum ha monary policy shocks ar indpndn of h srucural shocks o inflaion and oupu (call hm and ) a all poins in im. Assum hs ohr wo srucural ε y ε π shocks hav immdia and simulanous ffcs on inflaion and oupu. Using lag polynomials, calcula h impuls rspons funcion of ach variabl o a monary policy shock. In his calculaion, normaliz h srucural shocks o hav uni varianc. F. Suppos β 3 =0. Wha xacly is h impuls rspons of y o a shock o mony supply a an arbirary poin in im ( polynomials. dy dε + h ms )? Wri his answr in rms of paramrs, no in rms of h lag 4. In conras o h prvious qusion, suppos ha h conomic srucur is givn by π = α E π + α y + α π + α y + α R + ε as y = γ E y + γ R + γ π + γ y + γ R + ε ad R = φ π + φy + φ π + φy + φr + ε ms whr E dnos raional xpcaions basd on im informaion and i Eε ε i Eε ε j j τ = 0 if i j = σ if i = j i = 0 if τ (for all i and j). for i=as,ad,ms and j=as,ad,ms A. Carfully xplain why his modl dos no impos any ovr-idnifying (i.. sabl) rsricions on h rducd form VAR, which is givn by h unconsraind vrsion of h firs 3 quaions in h prvious qusion. B. Suppos you know h valus for paramrs φ, φ and γ. How would you sima all h ohr paramrs in h 3 srucural quaions? B spcific abou mhods and momn condiions. C. Show prcisly wha h paramrs α, α and γ ar qual o in rms of momns. (Bu don grind ou h rlaionship bwn ach momn and h covarianc marix of rsiduals.) 5. Assum x and y ar ach ingrad of ordr and ha h srucur is givn by: x = Θ ( L) ε + Θ ( L) ε y = Θ ( L) ε + Θ ( L) ε 5

7 whr ach srucural shock is uncorrlad wih all ohr srucural shocks (a all lags and lads) and ach srucural shock has uni varianc. Assum you hav simad a VAR for his bivaria sysm and obaind a varianc covarianc marix for rsiduals, G, and a marix associad wih h sum of cofficins in h VAR, β(). (Tchnically, h β() marix is qual o h idniy marix lss h sum of VAR cofficins marix). L h marix of long-run muliplirs b givn by ρ ρ ρ ρ. Assum ha x and y ar NOT coingrad in pars A and B. A. For convninc dfin M = r M M M = β() Σ [ β()'] wh M M Driv h valus of h ohr 3 long-run muliplirs for any givn valu of ρ. B. Now assum ha ρ =0 and ha you hav corrcly idnifid all h θ(l) paramrs. Driv h fracion of forcas rror varianc for y +h xplaind by shock. Show and xplain wha his varianc will convrg o as h gos o 4. (Imporan rmindr: This varianc prains o h lvl, no h firs diffrnc of y.) y λx C. Now assum x and y ar coingrad. L b h saionary linar combinaion (i.. h coingraing vcor) for any non-zro valu of λ. Spcifically wha non-linar rsricion dos coingraion impos on h long-run muliplirs? ε. 6. Bullard and Kaing (JME 995) sima bivaria VAR modls of h chang in inflaion ( π) and h chang in h log of oupu ( y). Using a variaion on h prmann and ransiory shock dcomposiion for oupu ha Blanchard and Quah usd o idnify aggrga supply and dmand ffcs, Bullard and Kaing idnify prmann and ransiory shocks o inflaion (no ha boh yps of shocks may hav a prmann ffc on h lvl of oupu in his framwork). Spcifically, hy dvlop h following saisical modl: π = R( L) υ + R ( L) υ y = R( L) υ + R ( L) υ Rk = R () = 0 Eυ υ which is idnifid by sing and by rsricing. For k= 0 = 0 convninc, assum ach shock in his saisical modl has uni varianc. On criicism of his approach is ha prmann changs in mony growh ar no xognous, bu insad obain bcaus h mony supply is ndognous o ral aciviy. I is argud ha incrass in inflaion wr primarily h rsul of a cnral bank incrasing mony growh in rspons o rcssions ha wr causd by advrs supply shocks. In ohr words, criics would modl h srucur as: 6

8 π = θ ( L) ε + θ ( L) ε m m y = θ ( L) ε + θ ( L) ε whr g as is an xognous aggrga supply shock, g m is an xognous mony growh shock wih θ θ 0 k = () < k= 0 bcaus advrs supply shocks caus h Fd o prmannly rais mony growh in an amp o parially offs h dclin in oupu. For convninc, assum h wo srucural shocks ar uncorrlad and ach shock has varianc qual o on (jus as was don wih h saisical modl). [Hin: Considr how h srucur and h saisical modl will map ino h rducd form.] A. Show prcisly how R () is biasd from θ (), assuming θ ()>0 and θ ()>0. Wha is h dircion of h bias (upward, downward, oward zro, away from zro, ambiguous)? B. Bullard and Kaing find for mos counris ha prmann shocks o inflaion (which virually vry hory lls us ar associad wih prmann changs in mony growh) ar NOT associad wih prmann changs in h lvl of oupu. Th primary xcpion is mos of h low inflaion counris for which hy find ha a prmann incras in inflaion lads o a prmann INCREASE in h lvl of oupu. Th criics argu ha xognous mony growh shocks (i.. xognous inflaion shocks) hav no long-run oupu ffc, and ha Bullard and Kaing g hir mpirical rsul bcaus of h ndognous rspons of mony growh o aggrga supply shocks. Basd on your analysis in par A, ar h criics righ or wrong? Why? C. Modify h horical framwork in quaions 3 and 4 by assuming θ θ 0 k = () = and θ (0)=0. Th ida bhind hs rsricions is ha h Fd dosn allow h inflaion ra in h shor run or h long run o rac o aggrga supply shocks (ignoring h implausibiliy for hs rsricions). Would hs horical rsricions impos any sabl rsricions on h rducd form s cofficins or on h covarianc marix for is rsiduals? If so wha xacly would hs rsricions b? as as k= 0 (3) (4) 7. Suppos h conomy can b dscribd by h following VARMA modl: Φ (L)y =Θ(L) whr y = (x,q), Φ ii (L) = aiil and Θ ii(l) = + biil for i=,, Φ (L) = al, Φ (L) = al, Θ (L) = bl, Θ (L) = bl, Also, = ( x, q), Ev w =σvw for v = x or q and w = x or q, and Ev = wτ 0 for all τ. A. Wha condiion(s) mus hold on h paramrs for h vcor procss o b saionary? 7

9 B. Wha condiion(s) mus hold on h paramrs for h vcor procss o b invribl? C. Assuming h vcor procss is saionary and invribl, driv h univaria rprsnaion for x. Assuming no common facors in h lag polynomials, wha yp of ARMA procss is i? D. Assuming h vcor procss is saionary and invribl, consruc h impuls rspons for ach variabl o h scond shock from a shor-run rcursiv modl in which x is placd firs in h ordring. E. Wha condiion(s) mus hold on h paramrs of h VARMA if boh variabls ar diffrnc saionary and hr is no coingraion? 8. In ach of h following srucural quaions no rsricions ar placd on dynamics. In addiion o unconsraind dynamics: Equaion (i) uss h raional xpcaions hory of h rm srucur of inrs ras o rla h -priod inrs ra (R) o an avrag of h currn and xpcd fuur on-priod inrs ra (r) as wll as a shock o h rm srucur; Equaion (ii) is an Old-Kynsian IS curv wih oupu as a funcion of h shor-rm ral ra of inrs and an IS shock; Equaion (iii) is a Taylor rul ha spcifis h shor-rm nominal ra as a funcion of inflaion, oupu and a monary policy shock; and Equaion (iv) wris inflaion as a funcion of shor and long rm inrs ras, oupu and an aggrga supply shock: (i) (ii) (iii) (iv) R = ( )(r + E r ) +φ Z +ε TS + R y =γ[r E π ] +φ Z +ε IS + y r =θπ +θ y +φ Z +ε π=α R +α y +α r +φ Z +ε MP r AS 3 π whr E dnos raional xpcaions basd on im informaion. L Z - b h s of k laggd ndognous variabls. In ohr words, w can wri X j R j π j = y j r j for j=0,,,..k and hn 8

10 Z X = j X k. Each φ is a vcor of cofficins for srucural quaions and j=r,π,y,r. For all i,j from h s {TS,IS,MP,AS}: E εε =0 if i j i j = σ if i = j i for i and j = TS, IS, MP, AP i j E εε τ =0 if τ (for any i and j). A. R-wri ach srucural quaion in rms of h rlaionship bwn VAR innovaions, srucural paramrs and srucural shocks. Assum h VAR is wrin as: β (L)X = for i=,,...k and k whr β (L) = I β L β L... β L, R π y r = (,,,) k 9 β β β β RR Rπ Ry Rr i i i i πr ππ πy πr βi βi βi βi i yr yπ yy yr βi βi βi βi rr rπ ry rr βi βi βi βi β = B. How would you sima h paramrs γ, θ, θ, α, α and α 3? For ach quaion, b spcific abou h lf hand sid variabl, h rgrssor(s) and, if ncssary, any variabls ha you ar using as insrumns. C. Suppos you ar NOT willing o impos rsricions from quaions (ii), (iii) and (iv). Howvr, you would sill lik o idnify h impuls rspons funcion of ach variabl o h rm prmium shock. I urns ou ha his can b don using a paricular Cholsky dcomposiion! Prov i. On asir way o show his is firs by rwriing h srucural quaions in rms of rsiduals, srucural paramrs and srucural shocks as: whr R TS δ δ ε V = V δ δ ε V π y r = (,, ) and ε V ( AS IS MP = ε, ε, ε ). I v placd bars ovr h δ cofficins ha ar known bcaus of h rm srucur hory. Th ohr δ cofficins ar unknown bcaus w now ar assuming ha w don wan o impos rsricions from any of h ohr quaions. Rwri his sysm of quaions for rsiduals such ha a Cholsky dcomposiion can b usd o

11 idnify h ffcs of h rm srucur shock. Show wha marix you would prform a Cholsky dcomposiion on and how i would b usd o idnify h rspons of ach variabl o h rm srucur shock. For convninc, you can wri h covarianc marix for rsiduals as: whr, for xampl, Σ RR is Σ Σ h varianc of h rsiduals o h quaion for R from h VAR. RR VR Σ Σ RV VV 9. Suppos h following riangular rprsnaion --- which is assumd o b srucural, no a rducd form --- characrizs h rlaionship bwn variabls x and y: y θ (L) θ (L) τ = s (L) (L) θ θ η whr ach lmn in τ and in η is an indpndn whi nois shock wih varianc normalizd o, τ ar h srucural shocks ha hav prmann ffcs on a las som of h variabls, η ar h srucural shocks ha hav no prmann ffcs on any variabls and s = x δy is 0 on way o rprsn all of h saionary linar combinaions of variabls. Assum ha θ() is full rank and ha θ(l) is invribl. W know ha whn η shocks hav mporary ffcs, θ () = 0, howvr, θ () and θ () ar unrsricd in his yp of modl. Assum hr ar n x (> ) variabls in x, n y (> ) variabls in y, and hrfor δ is an n x n y marix of paramrs. β = = [ ] A. Givn a VAR ha is wrin as: (L)X whr and a X y,s covarianc marix for rsiduals givn by Σ, show and xplain prcisly how you could sima h marix of paramrs θ () if his marix is lowr riangular (i.. h long-run rlaionship is rcursiv). In your answr, you migh find i convnin o l M =β() Σβ() and o M M M M M = wri wih h dimnsions of M ij conformabl o θ ij. (If you g suck on his qusion, Kaing (00, Macro Dynamics) may b a good plac o look for inspiraion). B. Show ha h impuls rsponss of y and s o h τ shocks ar idnifid, using marics from h VAR and h modl from par A. C. Th srucural riangular rprsnaion givn a h bginning of his qusion, can A (L) A (L) y τ = A (L) A (L) s η also b wrin as follows: whr h A(L) marix is simply qual o h invrs of θ(l). Using his vrsion of h riangular rprsnaion, driv h

12 srucural vcor rror corrcion modl wih prcisly h sam srucural rrors. [Hin: Considr h mulivaria vrsion of h Bvridg-Nlson dcomposiion.]