Proposed solution to the exam in STK4060 & STK9060 Spring Eivind Damsleth

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1 Proposed soluion o he exam in STK46 & STK96 Spring 6 Eivind Damsleh.5.6 NTE: Several of he quesions in he es have no unique answer; here will always be a subjecive elemen, in paricular in selecing he bes model. her alernaives han he ones presen here may be seen as equally good, if he argumenaion is solid.

2 Ex. Daa: KP from SSB from Jan. 986 o March 6, 43 monhly observaions. The daa are shown below. A clear, nearly linear rend is he main feaure of he daa. There seems o be some peculiar humps around observaion no Series a) This clear rend, combined wih he very persisen ACF shown below shows a clear need for differeniaion. The series afer a lag differeniaion is shown o he righ. The mos disinc feaure is he raher erraic behaviour beween observaions 3 and 9, i.e. November o July 3, and a slighly larger variabiliy owards he end of he series. Thus vaguely sugges a log-ransformaion of he daa, alhough he need for a ransformaion is by no means obvious. A log-ransformaion implies ha he KP varies on a relaive raher han an absolue scale, which is consisen wih economic heory. The. difference of he logransformed daa is shown o he righ, and show he same feaures.

3 The highly significan, slowly decaying peaks in he ACF a lags, 4 and 36 sand ou in he figure above, showing a need for differeniaion a lag as well. Below are shown he ACF and PACF afer differeniaion boh a lag and lag. There are a number of borderline significan values in he ACF and PACF, bu he negaive peak a lag sands ou. gnoring all lags bu, 4 and 36, he paerns in he ACF and PACF are consisen wih a SAMA(,,)x(,,) model wih seasonal MA-parameer around.5. Thus, he firs model proposal is his SAMA(,,)x(,,) model. Preliminary esimaion, using he innovaions mehod in TSM, gives Θ -.53, and he resuls from ML esimaion are shown o he righ: The Θ value of.77 is significan. A plo of he residuals (no shown) shows some raher exreme values in early 3, as expeced from he visual inspecion of he series. The mos exreme is almos 6

4 sandard deviaions, and way ouside he naural variaion. This value alone can easily resul in some spurious residual auocorrelaions. The residual ACF has wo (barely) significan values a lags and, a cluser of significan or near significan vales around lag, as well as some oher scaered near significan values. The Ljung- Box es for randomness rejecs he whie noise hypohesis a level.%. efining he model o a SAMA(,,)x(,,) model, where he non-seasonal MA-parameers are expeced o be fairly small, gives he model o he righ. Noe ha he non-seasonal parameers are significan, and ha he esimaes for he seasonal parameer has a much smaller sandard deviaion compared o he previous model. The ACC crierion has improved as well, while he BC has become slighly worse. For his model he residual ACF and PACF show no clear paerns. There are a few scaered nearsignifican values, bu he Ljung-Box saisic is no significan even a he 45% level. The only saisic o shed some doub on he whie noise hypohesis is he rank es, which rejecs he hypohesis a he % level. Conclusion: The SAMA(,,)x(,,) wih he parameers from above is an accepable model for he KP. b) Forecass for he period Jan. 5- March 6 based on daa hroughou Dec. 4: Using TSM wih he above model gives he forecass in he able o he righ: Afer he firs hree monhs, he predicions fall consisenly below he acual values. This is probably caused by he fall in he KP in he firs half of. This has resuled in a flaening of he local rend, and he model has no ye recovered and picked up he increase. The above forecasing and comparison is done wihin sample. This means ha he daa ha we forecas have also been used o idenify he model and 95% pred. bounds Predicion Lower Upper Acual Jan Feb March April May June July Aug Sep c Nov Dec Jan Feb March

5 esimae is parameers. Thus we will expec a beer fi o he acual observaions han we would ge from an ou of sample predicion. c) The KP values are given wih only one decimal. Thus a published value of X will acually be Xε, where ε is he rounding error, uniformly disribued over [-.5,.5>. Thus a lower limi for he - sep predicion error s.dev. (or any predicion error s.dev. for ha maer) is he s.dev. of ε. The s.dev. of a uniformly disribued variable is given by is range/. n his case he range is., and he s.dev. is./.9. d). Annual inflaion (in %) defined as (X -X - )/ X - %, where X is he KP value for monh and is he % change over he las monhs. We have from he exercise ex ha (-B )ln(x ). %, and assume he approximaion o be exac in he following. n par a) we found ha (-B)(-B ) ln(x )(θ Bθ B )(Θ B ). where is whie noise wih variance σ, wih he parameers given in par a). Bu since (-B )ln(x ) /, we have (- B) (θ Bθ B )(Θ B )U, where now U is whie noise wih variance σ. Thus, follows a SAMA(,,)x(,,) model wih parameers as found in par a). Using TSM wih his model, we find he following predicions, based on daa up o March 6. From he predicions we can find he sandard error, e.g. as (Upper Predicion)/.96. The probabiliy ha he acual value will exceed.5% is hen -Φ((.5-Predicion)/S.err.), where Φ is he sandard cumulaive normal disribuion. 95% pred. bounds Pred. Lower Upper S. err. Pr(>.5%) Dec % Dec % Noe ha in he analysis above we have assumed he mean of he differeniaed series o be. Subracing he small and insignifican, bu negaive mean will inroduce a deerminisic negaive rend, which will have a significan impac on he long-erm forecass for he inflaion rae.

6 Ex. ndusrial mixing process, ref. deails in he exercise ex Noaion: σ Toal acual (correc) inpu weigh o bach no. (unobserved) equired oal inpu weigh according o he recipe, so ha E Toal inpu weigh error, so ha (unobserved) Sandard deviaion for for he individual weighing processes) (known). (Known, calculaed as he square roo of he sum of he variances esidue in he blender afer discharging bach no. (unobserved) is assumed o follow a saionary A() process : ( ) ( where WN(, σ ) and he mean is known. Acual oupu weigh from bach no. (unobserved) ), σ Measured so ha E( oupu weigh from bach no. (bserved). The weighing equipmen is unbiased, ) Measuremen error on oupu, i.e. Sandard deviaion for : σ Sdv( (unobserved) ). (known) The dynamics of his indusrial process can hen be wrien: ( ) This indusrial process can be formulaed as a sae-space model X FX V GX W wih sae vecor X (,,, ). () () NB: Unforunaely here is yping error in he sign of he erm in he second line of eq. () in he exercise ex; i should be -, no! Also, in he second line of eq. () he subscrip of X was erroneously given as in he exercise ex.

7 a) epeaed use of eq. () gives: { V FX V X F ) ( ) ( ) ( ) ( ) ( X { { W W GX X G (,,,) 443 Thus we have he marices F and G, and he vecors X, V, and W as given in he ex. The form and shape of he marices and vecors in (), wih F and G independen of ime, are more han sufficien for a sae-space represenaion. n addiion, (i): {V } mus be whie noise, (ii): {W } mus be whie noise, and (iii): {V } and {W } independen of each oher. is reasonable o assume ha he measuremen errors, boh on he inpu and on he oupu, are independen over ime. Then { } and { } will be whie noise sequences. Furhermore, since { } is he noise in an A() model, i will be whie by definiion. Now, since V only conain elemens ha are linear combinaions of and from ime only, and { }and { }are whie noise, (i) is fulfilled. Similarly, {W } { }, which is whie noise by definiion, showing (ii). And assuming he measuremen error in he oupu o be independen of hose in he inpu as well as he residue noise we obain (iii). Finally, we mus assume ha he iniial sae X is independen of he noise processes {V } and {W }. b) The sae-space represenaion () is sable if all he eigenvalues are sricly wihin he uni circle. Sraighforward calculaions give ha F-λ λ (λ)(λ), so he eigenvalues are, and. Since one of he eigenvalues always, he represenaion is never sable, according o he sric definiion. However, he eigenvalue is relaed o he consan in he sae vecor. As long as <, i can be shown ha lim n n F, so ha he only informaion from he hisory ha persiss is he parameer values associaed wih he firs, consan elemen () in he sae vecor. f we define sable o mean ha he impac of previous observaion vanishes wih ime, he sae space represenaion is sable if <.

8 c) The parameer conrols he degree of persisence in he residue process. near implies a srong persisence, so ha high, or low, amouns of residue will end o cluser in ime. This could be he case if he amoun of residue is o some exen conrolled by emperaure, humidiy or oher persisen processes, or he echnical condiion of he blender. For, will be a simple random walk. near implies ha a high amoun of residue in one bach will end o be followed by a low amoun in he nex bach, and vice versa. This could be he case if he efficiency of he empying process depends on he volume in he bach, so ha he process becomes more efficien (giving less residue) if he amoun in he blender is large, and similarly he oher way around. f so, he hypohesis of independence beween he inpu process and he residue process becomes somewha dubious, hough. For -, will be an oscillaing random walk around he mean level. near means ha he residue process is near whie noise, where he residue afer each bach is nearly uncorrelaed wih previous and laer values. For, - is whie noise. d) AMA process for { }. From () we have: ( B)( ( B) ) ( B)( ) B (3) The lef hand side shows an A() srucure. The righ hand side is -correlaed, and hus equivalen o an MA()-srucure. Thus we have ha he { } process is an AMA(,): (-B)( - ) (θb). (4) The A parameer is he same as in he residue process. θ and he whie noise variance σ mus be found by equaing γ () and γ () for he righ hand sides of he wo represenaions. From (3) we find : γ () ( ) σ γ () σ σ σ while (4) gives : γ () ( θ ) σ γ () θσ θ and σ can hen be found by solving he equaion : θ σ σ θ ( ) σ σ for θ, using he inverible roo, and hen finding σ ( σ σ ) / θ θ As, he equaion for θ will approach, givingθ. Then he A and MA erms boh θ become (-B). They can hen be cancelled ou, and { - } becomes whie noise wih variance σ. σ θ σ As, he equaion for θ will approach, whereτ, givingθ ( τ τ 4τ ) and θ τ σ σ σ / θ. The negaive sign for he square roo mus be chosen so ha θ <, i.e.in he inverible region. When τ is small, so ha he residue variance is much smaller han he inpu measuremen variance, θ - and he process approaches a non-inverible MA() wih variance. When τ is σ

9 large, so ha he residue variance is much larger han he inpu measuremen variance, θ and he process approaches whie noise wih variance σ. e) AMA prosess for { }. We have from () ha. Using he resuls from d), we have ha: θ B B ( B)( ) θ (5) The parameer θ and he whie noise series are as found in d). Again he righ hand side is - correlaed, and we have an AMA(,) model for { }: (-B)( - ) (ωb)u, where U is W.N. wih variance ν (6) We use he same echnique as in d). From (5) we find : γ () γ () while (6) gives : ( θ ) σ ( ) σ (7) γ () ( ω ) ν (8) θσ σ γ () ων ω andν ω ω provided ω can hen, as in d), be found by solving he equaion :. f θσ σ for ω, using he inverible roo, and hen findingν ( θ ) σ ( ) σ ω we obain direcly ν Alernaively, we have from (3) and () ha ( B)( ( B)( ( B)( ) ) ) Thus, as an alernaive o (7), we have: γ () ( θ ) σ ( ) σ ( θσ σ ) / ω, γ () ( ) σ γ () σ σ σ σ ( ) σ (9) ω andν ω ω can hen, as before, be found by solving he equaion : σ σ σ for ω, using he inverible roo, ) ( σ σ ( ) σ and hen findingν ( σ σ σ ) / ω, provided ω. f ω we obain direcly ν γ () ( ) σ σ ( ) σ ()

10 f) observaions of. The plo shows a raher erraic behavior around a fairly sable mean value a appr.. The ACF and PACF are consisen wih an A() model wih parameer ca..5. There is no sign of any need for any MA parameer. Esimaion of an A() model gives he model o he righ: An analysis of he residuals shows no sign of model inadequacy, and here does no seem o be any room for furher model improvemen. Enforcing an MA() erm gives, as expeced, θ very close o, and far from significan. Thus we coninue wih he simple A() model, which is consisen wih he resuls from e) wih -.5, ω and ν 97.

11 Assuming ω, we ge from (): ν ( ) σ σ ( ) σ σ ( ν ( )( σ σ )) Wih -.5, ν 97 from he model esimaion, and wih σ obain σ 4.5 σ 6.5, as requesed. σ 4 from he ex, we However, insering hese values ino (9) we find γ () σ σ σ 365.5, which is far from consisen wih he lack of an MA erm in he idenified model for he daa. This may indicae ha here is a prining error in he exercise ex, so ha σ 4 or σ 4. Again, using (9) and requiring γ (), we obain: 97 ( (.5) ) σ σ ( (.5) ) σ σ σ 48..5σ σ.5σ σ 45 Thus he correc soluion, consisen wih he daa, is likely o be σ 5.7 and σ Ι / σ 9 or σ Ι 9 / σ.