Time Varying Coefficient Models; A Proposal for selecting the Coefficient Driver Sets

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1 Time Varying Coefficien Models; A Proposal for selecing he Coefficien Driver Ses Sephen G. Hall, Universiy of Leiceser P. A. V. B. Swamy George S. Tavlas, Bank of Greece Working Paper No. 14/18 December 2014

2 Time Varying Coefficien Models; A Proposal for selecing he Coefficien Driver Ses Sephen G. Hall, P. A. V. B. Swamy and George S. Tavlas, Absrac Keywords Time-varying coefficien model Coefficien driver Specificaion Problem Correc inerpreaion of coefficiens JEL Classificaion Numbers C130 C190 C220 The views expressed in his paper are he auhors own and do no necessarily represen hose of heir respecive insiuions. P. A. V. B. Swamy G. S. Tavlas (Corresponding auhor) Reired from Federal Reserve Board, Member, Moneary Policy Council Washingon, DC, Bank of Greece, 21 El. Venizelos Ave Brockes Crossing Ahens, Greece Kingsowne, VA Sephen G. Hall Leiceser Universiy and Bank of Greece and NIESR, Room: Asley Clarke 116,. Universiy Road, Leiceser, LEI 7RH, UK,

3 1. Inroducion Time-varying-coefficien (TVC) esimaion is a way of esimaing consisen parameers of a model even when (i) he rue funcional form is unknown, (ii) here are missing imporan variables, and (iii) he included variables conain measuremen errors. 1 However, an imporan assumpion is needed o make he echnique operaional. This assumpion concerns he choice of wha are called coefficien drivers (formally defined below) and he separaion of hese drivers ino wo sub-ses. This separaion allows us o derive esimaes of bias-free coefficiens. Inuiively, coefficien drivers are a se of variables ha feed ino he TVC s and explain a leas par of heir movemen. As explained in wha follows, his se of drivers is spli ino wo sub-ses, one of which is correlaed wih he bias-free coefficien ha we wan o esimae and he oher of which is correlaed wih he misspecificaion in he model. This spli has always been somewha arbirary (much as in he case of choosing insrumenal variables). The objecive of his paper is o pu forward a mehod for producing his spli ha akes accoun of he non-lineariy ha may be in he original daa. As we argue below, his mehod provides a naural spli in he driver se. The remainder of his paper is divided ino four secions. Secion 2 presens a summary of he TVC approach and formally defines he concep of coefficien drivers and he need o spli he drivers ino wo ses. Secion 3 hen proposes a mehod for deermining his spli. Secion 4 provides an example of he pracical use of he echnique by applying i o modeling he effec of raing agencies raings on sovereign bond spreads. Secion 5 concludes. 2. TVC esimaion Here, we summarize he approach o TVC esimaion ha has been formalized in Swamy, Hall, Hondroyiannis and Tavlas (2010). TVC esimaion proceeds from an imporan heorem ha was firs esablished by Swamy and Meha (1975), and, which has subsequenly been confirmed by Granger (2008). This heorem saes ha any nonlinear funcional form can be exacly represened by a model ha is linear in variables, bu which 1 The developmen of TVC esimaion is due o Swamy (1969, 1974). See also, Swamy and Tavlas (2001, 2007) and Swamy, Hall, Hondroyiannis, and Tavlas (2010). 1

4 has ime-varying coefficiens. The implicaion of his resul is ha, even if we do no know he correc funcional form of a relaionship, we can always represen his relaionship as a TVC relaionship and, hus, esimae i. Hence, any nonlinear relaionship may be saed as; y = 0 + 1x1 + + K 1, xk 1, ( = 1,, T) (1) Where k-1 is he number of included variables in our model and L> [Seve: somehing is missing] Consequenly, his heorem leads o he resul ha, if we have he complee se of relevan variables wih no measuremen error, hen by esimaing a TVC model we will ge consisen esimaes of he rue parial derivaives of he dependen variable wih respec o each of he independen variables given he unknown, non-linear funcional form. If we hen allow for he fac ha we do no know he full se of independen variables and ha some, or perhaps all, of hem may be measured wih error, hen he TVCs become biased (for he usual reasons). Wha we would like o have is some way o decompose he full se of biased TVCs ino wo pars -- he biased componen and he remaining par; he laer would be a consisen esimae of he rue parameer. While his is asking a grea deal of an esimaion echnique, i is precisely wha TVC esimaion aims o provide (Swamy, Tavlas, Hall and Hondroyianis, 2010). This echnique builds from he Swamy and Meha heorem, menioned above, o produce such a decomposiion 2. Swamy, Tavlas Hall and Hondroyianis (2010) show wha happens o he TVCs as oher forms of misspecificaion are added o he model. If we omi some relevan variables from he model, hen he rue TVCs ge conaminaed by a erm ha involves he relaionship beween he omied and included variables. If we also allow for measuremen error, hen he TVCs ge furher conaminaed by a erm ha allows for he relaionship beween he exogenous variables and he error erms. Thus, as one migh expec, he esimaed TVCs are no longer consisen esimaes of he rue parial derivaives of he nonlinear funcion. Insead, hey are biased due o he effecs of omied variables and measuremen error. There are exac mahemaical proofs provided for our saemens up o his poin. 2 Mahemaically his model may appear o be a sae space one. However, he inerpreaion of he coefficiens is quie differen from he sandard sae space represenaion. Omied-variable biases, measuremen-error biases and he correc funcions of cerain sufficien ses of excluded variables are no considered pars of he coefficiens of he observaion equaions of sae-space models. This is he major difference beween (1) and he observaion equaions of a sandard sae-space model. 2

5 To make TVC esimaion fully operaional, we need o make wo key parameric assumpions; firs, we assume ha he ime-varying coefficiens hemselves are deermined by a se of sochasic linear equaions which makes hem a funcion of a se of variables we call driver (or coefficien-driver) variables. This is a relaively unconroversial assumpion. Second, we assume ha some of hese drivers are correlaed wih he misspecificaion in he model and some of hem are correlaed wih he ime-variaion coming from he nonlinear (rue) funcional form. Having made his assumpion we can hen simply remove he bias from he ime-varying coefficiens by removing he effec of he se of coefficien drivers which are correlaed wih he misspecificaion. This procedure, hen, yields a consisen se of esimaes of he rue parial derivaives of he unknown nonlinear funcion, which may hen be esed by consrucing ess in he usual way. An imporan difference beween coefficien drivers and insrumenal variables is ha for a valid insrumen we require a variable ha is uncorrelaed wih he misspecificaion. This ofen proves hard o find. For a valid driver we need variables which are correlaed wih he misspecificaion. We would argue ha his is much easier o achieve. To formalize he idea of he coefficien drivers, we assume ha each of he TVCs in (1) is generaed in he following way. Assumpion 1 (Auxiliary informaion) Each coefficien is linearly relaed o cerain drivers plus a random error, = j0 + j p1 jd zd + j d 1 (j = 0, 1,, K-1), (2) where he s are fixed parameers, he z d are wha we call he coefficien drivers; differen coefficiens of (2) can be funcions of differen ses of coefficien drivers. The regressors and he coefficiens of (2) are condiionally independen of each oher given he coefficien drivers. 3 These coefficien drivers are merely a se of variables ha, o a reasonable exen, joinly explain he movemen in j. Under our mehod, he coefficien drivers included in equaion (2) have wo uses. Inserion of equaion (2) ino equaion (1) parameerizes he laer equaion. This is he firs 3 The disribuional assumpions abou he errors in (12) are given in Swamy, Tavlas, Hall and Hondroyiannis (2010). 3

6 use of he coefficien drivers. Here, he issue of idenificaion of he parameerized model (1) is imporan. 4 The oher imporan use of he drivers allows us o separae he bias and bias-free componens of he coefficiens. Assumpion 2 The se of coefficien drivers and he consan erm in (2) divides ino hree differen subses A 1 j, A 2 j. and A 3 j such ha he firs se is correlaed wih any variaion in he rue parameer ha is due o he underlying relaionship being non-linear, he second se is correlaed wih bias in he parameer coming from any omied variables, and he final se is correlaed wih bias coming from measuremen error This assumpion allows us o idenify separaely he bias-free, omied-variables and measuremen-error bias componens of he coefficiens of (1). Assumpion 2 is he key o making our procedure operaional; i is he assumpion ha we can associae he various forms of specificaion biases wih ses means ha se A 2 j, and A 3 j, which A 1 j simply explains he ime-variaion in he coefficiens caused by he nonlineariy in he rue funcional form. If he rue model is linear, hen all ha would be required for se A 1 j would be o conain a consan. If he rue model is nonlinear, hen he bias-free componens should be ime-varying and he se of drivers belonging o explain he ime variaion in hese componens. A 1 j will 3. A Suggesion for he Choice of Coefficien Drivers Clearly, Assumpions 1 and 2 above are crucial for he successful implemenaion of he TVC approach. As noed above, he spli of coefficien drivers semming from hese assumpions has always been a problemaic par of he TVC-esimaion procedure. There are, however, cerain requiremens ha can help in selecing boh he variables ha make a good driver se and he spli ino he wo subses inheren under Assumpion 2. 4 To handle his issue, we use Lehmann and Casella s (1998, pp. 24 and 57) concep of idenificaion. 4

7 Consider, firs, he broad requiremens ha a complee se of drivers should fulfill; hese relae o predicive power and relevance. Consider, again, he driver equaions in equaion (2). j = j0z0 + p1 jd zd + j d 1 (j = 0, 1,, K-1) (2) Where z 1. For his se of drivers o be a good se, he drivers mus explain mos of he 0 variaion in j Hence, we can define an analog of he convenional R 2 for he esimaed counerpars o hese equaions as follows: R 2 SS j 1 SS j (3) (Where SS j and SS j are he sum of squared residuals and he oal variaion of he dependen variable, respecively). Here, we require he R 2 coefficien o be as close o 1 as possible, so ha he drivers explain a large amoun of he variaion in he TVC. This resul could, of course, be achieved simply by having a very large number of drivers. Therefore, we also require he drivers o be relevan in he sense ha he jd are significanly differen from zero. Esimaion of he full TVC model produces a covariance marix for he jd so convenional saisics and probabiliy levels may be produced in he sandard way. These wo condiions are closely analogous o he idea of relevance in insrumenal esimaion, where he insrumens mus have explanaory power in explaining he variables being insrumened. If he R 2 in (3) is low, hen we could infer ha we had a weak se of coefficien drivers. There is, however, no requiremen for he drivers o be independen of he coefficien j, as here is for insrumens o be independen of he error erm under an IV esimaion procedure. The more difficul issue, hen, is how o perform he spli in he coefficien drivers ino he wo ses oulined under Assumpion 2 -- ha is S 1, which conains he variables correlaed wih he unbiased coefficien, and S 2, which is he se of drivers correlaed wih he misspecificaion. Le s assume ha he rue, unknown model is given by 5

8 1 m y f ( x... x ) (4) We are, herefore, ineresed in esimaing y x i for i 1,..., K 1 (5) 1 m where he values of all he argumens of f ( x... x ) oher han x i are held consan. To undersand how he spli of drivers may be achieved, consider he following examples. Example 1 If (4) is linear, hen he S 1 se consiss of jus a consan, as he rue parameer is a consan, and all oher drivers explain he biases ha sem from missing variables and measuremen error. Example 2 Suppose ha (4) is a polynomial, such as a quadraic form. Consider, for simpliciy, he case of only 2 explanaory variables. Then, y 0 1x1 2 2x x x (6) Then, we are ineresed in esimaing y x i x1 (7) Thus, we esimae he TVC model y 0 1 x1 (8) And our driver equaions would be p iZi 0 i1 6

9 p p 1x1 1 izi 1 i1 5 (9) where here are p-1 coefficien drivers, Z, which are correlaed wih he measuremen errors and he missing variable x 2. Now we can see why his will give a consisen esimae of he rue coefficiens as when we remove he effec of he Z variables from (9) we ge p 1 0 0iZi 0 0 i1 (10) p1 1 izi p 1x1 i1 (11) Thus, by equaing he righ hand side of (11) wih (7) we can see ha hey are idenical; in oher words, he bias free componen of he TVC will hen be a consisen esimae of he rue parial derivaive. E 1 ( 10 11x1 ) 1 22x (12) I is also easy o see in his simple example exacly how he coefficien drivers correc for he omied variable bias. Wha is required for a good coefficien driver is somehing ha is well-correlaed wih he omied variables. Consider an exreme example in which he omied variables hemselves, in his case x 2 and x 2 2, are used as he drivers. Then he equaion for he ime-varying consan becomes x2 02x2 0 (13) The ime-varying consan, hen, represens exacly he omied variables and he final equaion is correcly specified and will, herefore, give consisen parameer esimaes. Of course, in general he coefficien drivers will no be perfecly correlaed wih he omied variables, bu as long as he correlaion is relaively high he esimae of he unbiased effec of x 1 will be consisen. Since we would no know wheher he rue model is quadraic, we could include higher order polynomial erms and es heir significance in he usual way o see how many 5 I is imporan o noe ha none of he variables included in he TVC equaion such as (8) can be included as a coefficien driver in he ime-varying consan equaion since he model is hen unidenified. 7

10 polynomial erms would be needed. If he non-lineariy was no, in fac, a polynomial, hen here are wo possible courses of acion. 1. We could include a number of polynomial erms and hink of his as a Taylor series approximaion o he rue unknown form. 2. We could ry a range of specific non-linear forms, again esing one form agains anoher. By using he second opion he sandard TVC model is able o nes a number of popular non-linear models wihin a single framework, which also allows for measuremen error and missing variables. This procedure is very differen from oher sandard procedures. For example, a popular non-linear model is he smooh ransiion auoregressive model (STAR). This allows a parameer o move smoohly beween wo values according o a funcion which responds o some hreshold variable. If we were esimaing a TVC such as (14) bu believed he rue non-lineariy followed a STAR form, hen we could specify he driver equaions as p izi 0 i1 (16) p G( z,, c) 12(1 G( z,, c)) 12 izi 1 i1 (17) Where G( z,, c) is he ransiion funcion -- ypically a logisic funcion for he LSTAR model, Z an exponenial funcion for he ESTAR model or a second order logisic funcion, z is he ransiion variable and and c are parameers. The model given by (17) is more general han he sandard STAR model as i includes he drivers associaed wih measuremen error and missing variables and will, hence, correc for hese misspecificaions. Model (17) also has a sochasic error erm and hus becomes a sochasic STAR model. The spli ino he wo sub ses is again obvious as he wo erms capuring he STAR effec are clearly he se which is appropriae for S 1. Anoher ineresing non-lineariy would be o use a se of combinaions of simple nonlinear funcions such as he log or exponen funcion. In his case he TVC model would begin o encompass a neural ne and he universal approximaion heorems of Hal Whie 8

11 would sugges ha wih sufficien complexiy he model could hen approximae any unknown funcional form o any degree of accuracy. Thus, o formalize he above approach for deriving he spli in he coefficien drivers, we propose o choose he iniial drivers in wo ways. The firs way involves specifically choosing nonlinear variables so as o capure any non-lineariy ha may be presen. The second se capures omied variables and measuremen error and would normally no involve nonlinear ransformaions of he basic variables in he model. Once he drivers are chosen in his way, he spli of he drivers so ha hey reveal he unbiased coefficien is sraighforward -- i simply follows he wo ses ha were iniially creaed. Equaion (1) and (2) are essenially a sae space form where (1) is he measuremen equaion and (2) are he sae equaions. This means ha all of hese models may be esimaed using he Kalman filer, which can esimae all of he parameers of he model by maximizing he likelihood funcion. The Kalman filer and he sae space form mus be linear in he sae variables, bu i can easily handle non-lineariies in he oher variables; all of he varians above may be esimaed in sandard sofware such as EVIEWS. An Appendix o his paper provides he EVIEWS code ha was used o esimae he example below. 4 An Applicaion: In his secion we invesigae he effecs of raings agencies decisions on he sovereign bond spread beween Greece and Germany. The underlying hypohesis is ha his relaionship is highly non-linear, a decline in raings by one caegory from say AA o A will have a relaively small effec on spreads while a decline from say B o CCC will have a much larger effec. So as he raing goes down he effec on spreads ge proporionally larger. We believe here is a highly non-linear relaionship beween spreads and raings. Of course, here are also many oher hings which migh affec spreads, deb, deficis relaive prices, poliics ec. Therefore, if we examine a simple relaionship beween spreads and raings here will be many omied variables, which would cause bias for a sandard OLS regression. 9

12 Our basic TVC model is hen sp 0 1 rae Where sp is he spread beween 2 year Greek governmen bonds and rae is he agencies raings, urned ino a numerical score where 1 is he bes raing and 20 is he worse. Our coefficien driver equaions ake he form: 0 1 = pol+ dgdp + cnewssq + relp + debogdp = + rae + pol+ dgdp + cnewssq + relp + debogdp + 14 In his driver se, rae gives a quadraic effec o he coefficien on raings and allows for a srong non-lineariy. We begin by esimaing his general model, o obain he following resuls 0 = pol+ (0.6) (0.4) 13.4 (2.9) dgdp cnewssq relp debogdp + (3.2) (1.6) = rae pol+ 0.27dgdp cnewssq - 3.5relp debogdp + (1.2) (17.4) (0.1) (0.18) (3.2) (3.35) (0.12) 2 saisics are in parenhesis. The R 2 1=0.80 and R 2 2=0.84, so boh ses of drivers are producing a reasonably high degree of explanaory power for he wo TVC s. A number of drivers in each equaion are insignifican, however, so we now simplify he model o exclude insignifican drivers and hen obain he following more parsimonious model (noe we do no exclude he consans even if insignifican) 0 = cnewssq +15.6relp debogdp + ( 2.0) (2.9) (3.9) (2.7) 1 = rae cnewssq - 2.3relp 1 (1.8) (20.9) (3.8) (4.2) 2 10

13 The coefficien of ineres is 1, The following figure shows he oal value of his TVC along wih he bias free effec which is given by subracing he error erm, and he effec from rae and cnewssq Bias free coefficien oal coefficien Because he scaling of he wo coefficiens is quie differen he following figure shows jus he bias free coefficien 11

14 Bias free coefficien This makes he srong non-lineariy clear, as raings sared o rise (ha is deeriorae). Afer 2008 he effec of raings on spreads became increasingly powerful. We, herefore, have found a very srong quadraic link beween raings and spreads. 5 Conclusions This paper has proposed a new way of selecing he spli beween coefficien drivers when deriving he bias free esimae of a coefficien wihin he TVC esimaion framework. We have argued ha if he rue model is linear hen only he consan in he coefficien driver se should be reained. If he rue model is non-linear hen an explici se of drivers should be chosen which capure he nonlineariy, in he absence of any specific informaion his can bes be done by using a se of polynomials in he explanaory variables. These drivers are hen he only ones which should be reained when deriving he bias free componen. We have also argued for wo condiions o be applied o he drivers which we call predicive power and relevance, ha is he drivers should explain a large proporion of he movemen in he TVC and hey should be saisically significan. 12

15 We illusrae his process by esimaing a nonlinear relaionship beween counry risk raing and sovereign bond spreads for Greece and show ha here is a highly non-linear effec here. Finally all his may be done wihin sandard sofware such as EVIEWS and he code for he esimaed model is given in appendix A. 13

16 Appendix A: EVIEWS code for he repored model in secion sp_gr = sv1 + sv1 = c(1) +c(6)pol_gr+c(7)dgdp_gr+c(8)cnewssq_gr+c(9)relp_gr+c(10)debogdp_gr+sv3(- sv2 = c(2)+c(3)rae_gr+c(11)pol_gr+c(12)dgdp_gr+c(13)cnewssq_gr+c(14)relp_gr+c(15)debogdp_gr +sv5(- sv3=[var = sv5= [var = sv6=sv5(-1) In his code SP_gr is he spread beween Greek and German 2 year governmen bond yields, Rae_gr is he raings on counry risk denoed as a variable scaled from 1 o 20 where 1 is equivalen o a AAA raing, pol_gr is an indicaor of poliical unres in Greece, cnewssq is an indicaor of news abou Greek GDP derived from ECB forecass., relp_gr is relaive process beween Greece and Germany, debogdp_gr is he deb o GDP raio for Greece. This model is also coded o have a firs order moving average error process on he wo ime varying coefficiens sv1 and sv2, which is usual in TVC models. The c(?) parameers are he usual EVIEWS noaion for parameers o be esimaed by maximum likelihood. 14

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