THE TRANSFER FUNCTION MODEL

Transcription

1 THE TRANSFER FUNCTION MODEL Noaion Assume, for now, ha he arge variable is saionar i.e. has a consan mean, consan variance, and consan covariance and ha he proposed leading indicaor is saionar as well. The Transfer Funcion model is described b he following hree equaions. ω μ b ε δ θ ε a φ * θ μ u * 3 φ where μ is he inercep in equaion, μ is he mean of and a and u are whie noise error erms ha are uncorrelaed wih each oher a all forward and backward lags, and he "backshif" polnomials are defined as follows: ω ω ω ω ω r B B B L rb δ δ δ δ s B B B L sb θ θ θ θ q B B B L qb φ φ Bφ BL φ B p p θ θ Bθ B L θ B * * * * q* q* p* φ φ Bφ B L φ B. * * * * p* I is assumed for saionari and inveribili purposes ha he roos of he polnomials δ, θ, φ, θ *, and φ * are ouside he uni circle i.e. he roos are greaer han one in magniude if he are real and have a modulus greaer han one if he are comple. Thus, corresponding o his assumpion, i is quie imporan ha he

2 variables we are analzing, and, be in saionar form. If he are no and, sa, Δ and Δ are insead saionar, hen Δ and Δ should replace and in equaions and 3 above. Equaion is called he ssemaic dnamics equaion of he Transfer Funcion model because i describes he dnamic relaionship beween he leading indicaor,, and he arge variable. Wha is he naure of his relaionship? Obviousl, when changes i does no cause a change in unil b periods laer. b is called he dela parameer in he Transfer Funcion model. The naure of he numeraor and denominaor polnomials, ω and δ deermine wheher he change in has a finiel-lived effec on afer a dela of b periods or wheher he change in has an infiniel-lived bu diminishing effec on. Recall ha i is assumed ha he roos of he auoregressive denominaor polnomial δ are all ouside of he uni circle and his guaranees ha if δ is anhing oher han i.e. when s > hen he effec mus be diminishing. We will develop more of he inuiion on he relaionship beween and below. Equaion is called he error dnamics equaion. ε is he unobserved error in he ssemaic dnamics equaion. Thus, on average, he dnamic relaionship ha eiss beween and is described b ω μ b. δ The error ε represens he approimae naure of he relaionship. Then equaion sas ha, o he een ha he deerminisic par of he ssemaic dnamics beween and he righ-hand-side of he above equaion is approimae, wha is lef over o be eplained can be modeled b a Bo-Jenkins ARMAp,q model. Noice ha if he order of he numeraor polnomial, r, is equal o zero, ω reduces o ω. Likewise, if he order of he denominaor polnomial, s, is equal o zero, δ reduces o. Also noice ha if r, s, and ω, equaions and impl ha follows a Bo- Jenkins ARMAp,q model, θ θ μ a φ a φ φ and μ φ. Therefore, he Bo-Jenkins model for is us a special case of he Transfer Funcion model where he leading indicaor has no effec on he arge variable. We, of course, will be ver ineresed in disinguishing beween he cases

3 where has no effec on ω / δ and when has a ssemaic effec on ω / δ. In he former case, urns ou no o be a leading indicaor of while in he laer case is a viable leading indicaor of. Thus, one of he roles of he economerician is o deermine wheher or no he raional polnomial ω / δ is or is no equal o zero. If ω / δ is equal o zero, he economerician as compared o he saisician who doesn' know of or use should discard as an aid in forecasing he arge variable and r o come up wih anoher poenial leading indicaor, sa z, ha is useful in forecasing One wa he economerician can deermine wheher or no is a useful leading indicaor in forecasing over and above he special case Bo-Jenkins model for is o conduc an ou-of-sample forecasing eperimen and see if he forecasing accurac of he Transfer Funcion model using beer han he forecasing accurac of a simple Bo-Jenkins model for. Recall, if ω / δ, equaions and specialize o he Bo-Jenkins model for, namel,. θ θ μ a φ a φ φ where μ φ, of course. We will discuss he naure of he ou-of-sample forecasing eperimens in more deail laer. Equaion 3 is called he leading indicaor Bo-Jenkins equaion. In he Transfer Funcion model he leading indicaor is assumed o be purel eogenous in ha affecs bu curren and pas values of do no affec. This is someimes called one-wa Granger Causali. In oher words, follows a sochasic process of is own, namel, an independen Bo-Jenkins process ARMAp*,q*. We here use p* and q* as AR and MA orders o disinguish hem from he Bo-Jenkins orders p and q of he error dnamics equaion. The assumpion ha is purel eogenous is a ver imporan assumpion when adoping he Transfer Funcion model o characerize he relaionship beween he leading indicaor and he arge variable. If his eogenei assumpion for is no rue, we need o use some oher ime series model o characerize he relaionship beween and. One such model is called he Vecor Auoregressive model, VAR for shor, and given ime in his class will be discussed laer. One advanage of making a commimen o equaion 3 is ha, when forecasing more han b periods ahead, for eample, T b, T b, L, ec., we need fuure values of, namel T, T, L, ec. When we are forecasing, sa, T b he Bo-Jenkins model for equaion3 can be esimaed and used o produce he 3

4 forecas T, which in urn can be used in he esimaed equaion o produce a b forecas of, namel, T b. Wihou an esimaed version of equaion 3 we can' use an esimaed version of equaion o produce forecass beond b periods ahead. Thus, he Transfer Funcion model of equaions,, and 3 are dependen on he selecion of he backshif order b, he polnomial orders r, s, p, q, p*, and q* and implicil on he orders of differencing, sa d and d*, ha are required o make and saionar, respecivel. If a d-order difference is needed o make saionar and a d*-order difference is required o make saionar, d Δ should replace in equaion d* and Δ should replace in equaion 3 above and μ should be changed o be μ Δ d *, he mean of he d*-differenced series. From a noaional perspecive, we can represen equaions - 3 as TFb, r, s, p, q, p*, q*, d, d*. Before we go on, le s make some concree choices of he Transfer Funcion orders d, d*, b, r, s, p, q, p*, and q* so ha we can more full appreciae he naure of he Transfer Funcion model represened b equaions 3. Le d d* hus and are alread saionar, b, r, s, p, q, p* and q*. Also, for simplici le s assume ha he -inercep in equaion is zero μ and ha he mean of is zero μ. Then, he Transfer Funcion model for his specific case can be wrien as ω ω ε ε a u * φ. 3 In his case he ssemaic dnamics equaion is a wo-period disribued lag in wih a one-period dela, he error of he ssemaic dnamics equaion is whie noise a and he purel eogenous leading indicaor follows an AR Bo-Jenkins process. As anoher illusraion, le dd*, b, r, s, p, q, p*, q*, μ μ. Then he Transfer Funcion model akes he specific form Δ Δ ω ω Δ δ ε δ Δ ωδ ωδ 3 δδ ε δε Δ ω Δ ω Δ δ Δ ε δ ε 3 4

5 ε a Δ u θ u *. 3 In his case, he ssemaic dnamics equaion consiss of Δ being eplained b a woperiod disribued lag in Δ wih a wo-period dela and a one-period lag of he endogenous variable Δ. In his model, no onl does Δ have a wo-period dela effec on bu las period s change in also has an effec on his period s change in Δ. Also he error erm in he ssemaic dnamics equaion follows an MA process. From equaion we can see ha he error erm ε is whie noise, and from equaion 3 we see ha he change in he leading indicaor Δ follows an * MA process wih MA parameer θ. Thus wih he various choices of d, d*, b, r, s, p, q, p*, and q* we can have a ver sophisicaed descripion of he relaionship ha eiss beween he arge variable and he proposed leading indicaor. Impulse Response Funcion For he momen le us consider he deerminisic form i.e. wihou he error erm ε of he ssemaic dnamics equaion ω δ b ω ωb L ω B δ B L δ r r s sb b 4 where, for simplici, we le μ. Assuming ha he roos of he polnomial δ are all ouside of he uni circle, we can wrie 4 in he impulse response form b b b υ υ υ L, 5 an infinie disribued lag in b, b,, b,l. The coefficiens υ, υ, υ, L are called he impulse response coefficiens associaed wih he deerminisic ssemaic dnamics equaion 4. The inerpreaion of hese coefficiens is as follows: Consider increasing one uni a ime and in he ne period reurning i o is original value. υ is called he impac coefficien and represens he iniial impac ha he one- 5

6 period, one-uni increase in has on afer a dela of b periods. υ is he dela- coefficien ha represens he effec ha a one-period, one-period change in has on afer a dela of b periods. υ, υ3, L, have similar inerpreaions and are called he dela-, dela-3, ec. impulse response coefficiens. For eample, le b, r, and s. Then he deerminisic ssemaic dnamics equaion 4 becomes ω. 4 Furhermore, le be for all ime periods prior o and following, bu equal o a ime period. Now wha impac does his pe of change on have on? Well, is equal o zero ecep a ime and hen i is equal o ω. Therefore, he impulse response funcion for equaion 4 is υ w for for,,.. This can be ploed as Now consider he case where b, r, s. Therefore, he deerminisic ssemaic dnamics equaion becomes w w 4 6

7 Le have he one-period change of equaion 6. Wha is he impac of his change of? Well, is zero ecep a ime and hen period w w, Therefore, he impulse response funcion for 4 is. In he following v w for w for oherwise Tha is v w and v w, leing w < and w < w impulse responses:, we have he plo of he Then a one-period, one uni change on w of w unis and hen one more change in of w unis in period one and hereafer resumes he value of. gives rise o an immediae change in w, w Of course, if we had conemplaed he funcions w and w w 3. The corresponding impulse funcions would have been us like he ones above bu moved o he righ b wo periods, he amoun of he new dela of b insead of b. Now consider one more deerminisic, ssemaic dnamics equaion b, r, s : w w δ 4 where we assume < δ <, and in paricular, < δ <. Again le evolve as in equaion 6 and le s see wha happens o over ime for,-3,-,- w 7

8 w δ w w c, sa δ δ δ w δ w δ c 3 δ δ δ c ec. The impulse response funcion hen becomes v w for w δw c, for - δ c, oherwise Which, ploed is Where, for ploing purposes, we have assumed ha c > w, < δ <. Therefore, given he deerminisic ssemaic dnamics equaion of 4 we see ha a one period, one-uni change on a resuls in being w, c, δ c, δ,..., in ime periods c,,,3,.., respecivel. Tha is, from is original equilibrium,, he successive deviaions of from his original equilibrium beginning wih ime are unis and hen one more change in of w unis w, in period w w one and hereafer resumes he value of w, c, δc, δ c,..., ec. unil long enough ino he fuure of seles back down o is original equilibrium of. Of course, if he equaion 4 we had le b insead of b, we would have he same impulse responses as before bu he would be delaed wo periods and he impulse response graph immediael above would be shifed o he righ b wo periods. 8

9 We can, of course, generalize from his se of algebraic eercises. Again consider he general deerminisic ssemaic dnamics equaion w f b w w B... wr Br s δ B δ B... δ B s b 4 Is here anhing in general ha we can sa abou he impulse responses associaed wih such an equaion. The answer is es! Given b,,,.., or some ineger, we know ha he impulse responses are zero for lags,,,..,b-, unil b and hen v will equal o w. If s and he denominaor polnomial is he scalar δ, hen here will be r impulse responses afer b ha will be non-zero. In summar here will be r non-zero impulse responses beginning wih he lag b when he ssemaic dnamics follows he equaion w w w w 4 b b b... r b Bu now consider when s and is same posiive ineger. Then we can sa ha beginning wih lag b here will be a oal of r irregular impulse responses before he impulse responses begin a ssemaic deca o zero, eiher eponeniall or sinusoidal we don know eacl unil we know he signs and magniudes of he δ, δ,..., δ s coefficiens, The deca of he impulse responses is guaraneed b he assumpion ha he roos of δ, lies ouside he uni circle. In he case of he model 4 he impulse response funcion will be given b v { w,,,, In he case when s we have w w w w δ b b b... r br δ δ... s s 4 Here he impulse response funcion will be of he form v v, v,..., vr, " irregular" responsesr of hem vr, vr,...,eponeniall or sinusoidal declining responses beginning vr wih and coninuing. In summar hen, we can look a he number of periods dela before he firs nonzero impulse response occurs and we will be able o deermine he value of b. Thereafer, r will be deermined b he number of irregular no par of he deca 9

10 impulse responses before he impulse responses eiher because all zero. Thereafer or deca awa o zero Now wheher he impulse responses deca o zero or cu off o zero deermines wheher or no s or s and here is an auoregressive par o he deerminisic ssemaic dnamics equaion. If he impulse responses cu off o zero afer r irregular responses he s mus equal zero. Oherwise s,, or same posiive ineger. As ou can see, he impulse response funcion can help us idenif he b, r, and s orders in he ssemaic dnamics equaion. When we add back in he error erm of he ssemaic dnamics equaion u w f b ε Then he impulse response coefficiens need o be inerpreed as he epeced level of a various subsequen periods give a one-ime, one-uni change on. Of course, when he polnomials w and δ are esimaed from he daa resuling in r w w w B w... r B and δ B δ B s δ B... s we ge he esimaed impulse response polnomial w v v v B vb δ and he esimaed impulse response funcion v,,,,... is no alwas as... informaive as heoreical impulse response funcion v,,,,... The Cross-Correlaion Funcion One drawback of using he heoreical impulse response funcion and is empirical counerpar, he esimaed sample impulse response funcion is ha he choice of he scales of measuremen of and or alernaivel he scales of measuremen of Δ d and Δ d affecs he magniude bu no he paern of he impulse response coefficiens. Alernaivel, we can consruc a funcion called he cross correlaion funcion ha mimics he dela, irregular spike, and cuing off or declining behavior of he impulse response funcion e he correlaions are, b design, beween and and he invarian o he choice of he scales of measuremen of and. Le s urn o he definiion of he cross correlaion funcion. Le w and z be wo saionar ime series ha are poeniall relaed o each oher. Consider he following noaion: Le

11 E w μ z u 7 wz w z -3,-,-,,,,3 denoe he cross-correlaion beween w and z a lag. Noice ha he lags can be eiher posiive or negaive. For eample, if >, hen if is above below is mean z wz μ w now hen, more likel han no, will be above is mean wo periods from now. Of course is no invarian o he scales of measuremen one migh choose for w and z r wz s, s, s ec. bu he cross-correlaion a lag beween w and z is: ww wz ρ wz 8 var w var z ww w zz where E z μ z u zz z z and E w μ w u ww w w are he auocovariance funcions of w and z, respecivel, and hus w and z are he variance w and z, respecivel. B consrucion < ρ wz < and his is he case regardless of he choice of he scale of measuremen of w and z. For eample, if ρ wz.8 hen he correlaion of w now wih z wo periods from now is.8 and if, sa w is above is mean μ w, now hen, were likel han no, z will be above is mean, μ, wo periods from now. z Of course, if w is purel eogenous wih respec o z he ρ wz. 8 for., -3,-,-. Tha is, previous deviaions of z from is mean do no affec curren and fuure deviaions of w from is mean. However, if w does affec z eiher concurrenl or in he fuure as would be epeced of w is a leading indicaor of z hen measure of he ρ wz for,,,. Will be one-zero. Le us hen derive he cross-correlaion funcions for some simple ransfer funcion models: consider he case of * * * μ, μ, b b, r, s, p, q, p, q, d, d. We have w b ε 9 where ε a

12 is he whie noise process. Then w b and u [ w a ] E E b w E b E a w E b from Since, b assumpion, and are uncorrelaed a all leads and lags, following a E a E μ a for all. Therefore w for b oherwise is he auocovariance funcion for and given model 9-. The cross-correcion funcion for and given his model is ρ w w for b oherwise In summar, he cross-correlaion funcion for he model 9- is ploed as

13 where here we have assumed ha w >. There is one spike, afer a dela of b periods, and hen also, reflecing he eogenei of vis-a-vis he negaive i cus off logs of he cross-correlaion funcion are all zero ρ for...-3,-,-. Tha is he signaure vis-à-vis he cross-correlaion funcion of he model 9- where r,s,and b. Now consider a second model w b w b ε ε a 3 μ 4 * * b b, r, s, p, q, p, q, d, d * The covariance funcion is defined o be E Ew b E E-w [ w w a ] b -b- E a -b- w b w b where we have E a Eu a for all. Then w w for b for b oherwise This ranslaes ino he cross-correlaion funcion of ρ w w for b for b oherwise 3

14 In graphical form of he cross-correlaion funcion can be ploed as follows: In his graph we have assumed ha < w < w. * * * Finall, consider he model b b, r, s, p, q, p, q, d, d w b w b δ ε 5 ε a 6 a 7 E Ew w b E E-w b w [ w w δ a ] b -b- b δ -b- δ E E where we have E a Eu a for all. Then for <, because he covariance beween and previous lags of, acuel,,,, are all zero b he pure eogenei of he leading indicaor equaion. In he model 5-7,... b because E E μ for <b as well. When b however we have. b w w a Also for b we have b w δ b w δw m, 4

15 For b we have, m b b δ δ For b3 we have, 3 m b b δ δ In general hen our covariance funcion is,,3,4... s and s b for b for m b for m w w w s δ δ This implies ha he correlaion funcion for he model 5-7 is, s s, b or c c f b for w w b for w s δ δ ρ In graphical form he cross-correlaion funcion can be ploed as 5

16 where in graphing we have assumed ha < w, c > w > and < δ <. This is a signaure cross-correlaion funcion for a ransfer funcion model where bb here is a b period dela before he spikes begin, hen here arer- irregular spikes before an eponenial or sinusoidal deca begins and hen hereafer, insead of cuing off as when s, he cross-correlaion funcion decas awa when s>. The deviaion of he cross-correlaion funcion of a ransfer funcion model is somewha more complicaed when is no a whie noise series. However, we can calculae he cross-correlaion funcion of he pre-whiened and series and obain analogous resuls o hose we obained before. Assuming ha and are alread saionar, for eample, he pre-whiened series we need o cross-correlae are, and φ θ φ θ θ φ μ μ φ θ p where he pre-filer is φ / θ φ B φ B... φ B / θ B θ B L θ. p Cross-correlaing he pre-filered leading indicaor i.e he whie noise errors of he leading indicaor Bo-Jenkins model, μ, wih he pre-filered series,, produces a cross-correlaion funcion which provides a paern ha allows us o idenif he dela parameer, b and numeraor and denominaor polnomial orders, r and s, respecivel, ha correspond o he ransfer funcion for he original daa and. φ Appling he pre-filer o he ssemaic dnamics equaion. θ Summarizing, he cross-correlaion funcion of a ransfer funcion model wih a ssemaic dnamics equaion of w μ ε δ b should have no spikes unil b, hen have r more irregular spikes spikes no par of a ssemaic decaing paern, followed b eiher a cuing off behavior if s, or a decaing paern if s or some oher posiive ineger provides he pre-filer are ransfer funcion-model 6

17 φ θ b w φ φ ε δ θ θ w w δ δ ε μb ε b, where μ has been convenienlbu wihou loss of resul imposed. Analzing he cross-correlaion funcion beween and ie. μ clearl reveals he original raional polnomial srucure w δ Sample Cross-correlaion Funcion of he relaionship beween he original ' s and ' s. When we discussed he heoreical ACF and PACF funcions, i was noed ha we had o consruc sample esimaes of hem before proceeding o build a Bo-Jenkins model. Similarl, we need o consruc a sample cross-correlaion funcion which hopefull closel resembles he heoreical cross-correlaion funcion before we can build a ransfer funcion linking a leading indicaor,, wih a arge variable, sa c and c. Wha we need are he sample variances of he esimaed pre-filered series: and φ θ φ θ where he esimaed pre-filer is φ θ φ B φb θ B θ B... φ p B... θ B p p p and he φi and θi have been obained b esimaing an appropriae Bo-Jenkins model for we are implicil assuming in his discussion ha and are alread saionar A consisen esimae of he variance of is 7

18 T c 8 where ŷ he sample mean of he, namel ŷ T T A consisen esimae of he variance of is T c T μ 9 where are he Bo-Jenkins whie noise residuals for he leading indicaor equaion3. μ A consisen cross-covariance esimae a lag is given b T T T T c,-,-... for,,,3,... for μ μ Finall he sample cross-correlaion funcion is c c c..-3,-,-,,,,3.. This is a consisen esimae of he heoreical cross correlaion funcion ρ Under he assumpion ha and are oall correlaed wih each oher and hus ha he pre-filered and are unrelaed o each oher, he sandard error of he esimaes,, is approimael in large samples T, ha is, T SE when ρ. Then if an observed sample cross-correlaion 8

19 coefficien, sa, is ouside of he 95% confidence inerval.96,.96, one could conclude ha he heoreical cross-correlaion beween T T and a lag, ρ, is nonzero, using he above confidence inerval, hopefull, we can disinguish beween significan spikes in he sample cross-correlaion funcion and he zero values a cerain lags. Wha ma be difficul o discuss in he sample cross-correlaion funcion-is cuing off behavior and ailing off behavior which is he disinguishing characerisic beween ransfer funcion models wih s cuing off versus sor some oher posiive ineger when ailing off. Idenificaion of Transfer Funcion Models * * * The seps for idenif a TF b, r, s, p, q, p, q, d, d Model are as follows: Visuall inspec plos of he leading indicaor and arge variable and deermine he order of differencing needed o ransfer or possibl log o * saionari difference d and he order of differencing needed o ransfer or possibl log o saionari d. The saionar form of is hen Δ d* or d* possibl Δ log while he saionar form of is hen Δ d or possibl d Δ log. Fi a Bo-Jenkins model for he saionar form of he leading indicaor,, namel d* * * Δ. You will hen deermine he orders p and q for he ARIMA * * * p, d, q model of he leading indicaor. 3 Given esimaed Bo-Jenkins model for he leading indicaor form he d φ d d* φ d* esimaed pre-filered values Δ Δ and Δ Δ. θ θ Calculae he sample cross-correlaion funcion beween hese esimaed prefilered series. Use he 95% confidence inerval.96,.96 o T T deermine which sample cross-correlaions are significan and which ones are no. Choose b be he lag a which he firs significan cross-correlaion occurs, hen choose r based on he number of irregular spikes in he sample cross-correlaion minus one and hen choose s, if he sample cross-correlaion funcion cus off, and sor some oher posiive ineger if, afer he irregular spikes, he sample cross-correlaion funcion ssemaicall ails off. 9

20 4 Esimae he suggesed ransfer model using he b, r, and s values ou deermined in sep 3. Also for our chosen value of b, esimae addiional models, if an, suggesed b he sample cross-correlaion funcion. Beween he compeing models choose he model ha has he smalles goodness-of-fi measures, AIC and SRC, whie noise residuals and saisicall significan coefficien apar from possibl he μ. 5 In cerain insances ou ma no be able o find values of r and s, for ou given b value, which will produce whie noise residuals. If so, ou need o fi a Bo- Jenkins model o he residuals ε. Obain reasonable values for p and q for he Bo-Jenkins model of he residuals of our ssemaic dnamics equaion. This is called mapping up he auo correlaion in he residuals of he ssemaic dnamics equaion. In so doing will have enaive values of b, r, s, and p and q, ha produce he smalles goodness of fi measures AIC and SBC, whie residuals, and saisicall significan coefficiens. 6 Before making he enaive model of sep5 he final model of choice, ou need o eamine he -saisics of four overfiing models. Given a b value he dela parameer, here is one over-fiing model for each of he dimensions, r, s, p, and q incremening one order while holding he res of he orders fied a he enaive choice. If each he -saisics of he overfiing parameers of he four overfiing models are each saisic less han.96 or he absolue value of he saisic is less han.96 or he p-value of he -saisic is greaer han.5, hen fall back o he enaive model of sep 5 and make final choice. 7 Use our ransfer-funcion-model model in an ou of sample forecasing eperimen and compare he forecasing accurac of he model using eiher MAE, MSE, or he boss s loss funcion of choice wih he forecasing accurac of a properl chosen Bo-Jenkins model produces more accurae forecass using he leading indicaor hen he Bo-Jenkins model which ignores he leading indicaor hen he leading indicaor would appear o be useful and he ransfer funcion model should be used for fuure forecasing asks. On he oher hand, if he Bo-Jenkins model should prove o be more accurae han he Transfer Funcion model, we should drop consideraion of he leading indicaor and consider building a differen ransfer funcion using anoher proposed leading indicaor, sa,. z Esimaed ransfer funcion model for series M daa se. sales of carpe sore a ime housing permis issued in he coun a ime saionar form of : Δ d * saionar form of : Δ d

21 Bo-Jenkins model for he leading indicaor using he M sample daase of obs - Δ Therefore, p *, q *.