A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP SERIES *

Transcription

1 CUADERNOS DE ECONOMÍA, VOL. 43 (NOVIEMBRE), PP , 2006 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP SERIES * JUAN DE DIOS TENA Universidad de Concepción y Universidad Carlos III, España MIGUEL JEREZ Universidad Compluense, España SONIA SOTOCA Universidad Compluense, España NICOLE CARVALLO Universidad de Concepción, Chile An imporan limiaion in order o specify and esimae a macroeconomic model ha describes he Chilean economy resides in using variables wih sufficien number of observaions ha allow for a reliable economeric esimaion. Among hese variables, he GDP consiues a fundamenal magniude. Neverheless, for his variable here is no quarerly informaion before This paper compues quarerly GDP series for he period using he approach by Casals e al. (2000). As resul, he new series incorporaes he cyclical dynamic in he quarerly series laer o 1979 respecing, in addiion, all he annual exising informaion before he above menioned period. JEL: C22, E32 Keywords: Smoohing Algorihm, ARIMA Model, Transfer Funcion Model, Chilean GDP. 1. INTRODUCTION An imporan limiaion in order o specify and esimae a macroeconomic model ha describes he Chilean economy resides in using variables wih sufficien * We are indebed o an anonymous referee for helpful commens. The usual qualifier applies. Sonia Sooca and Miguel Jerez graefully acknowledge financial suppor from Miniserio de Educación y Ciencia, DGCICYT, ref. SEJ juande@udec.cl, jena@es-econ.uc3m.es

2 286 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 number of observaions ha allow for reliable economeric informaion. Among hese variables, he GDP consiues a fundamenal magniude. However, alhough here is annual informaion of his series from he XIX cenury, quarerly informaion is only available from In his paper we reconsruc quarerly series of he Chilean GDP for he period using he inerpolaion mehodology advocaed by De Jong (1989) and Casals e al. (2000) for he nonsaionary case. This procedure, explained in a simple way, consiss firsly of: 1) specifying and esimaing a saisical model of he series of ineres; 2) expressing he resuling model in sae space form; and hen 3) reconsrucing his series backward using a fixedinerval smoohing algorihm. As far as we are aware, in spie of is empirical relevance, here is only a previous aemp of reconsrucing he Chilean GDP by Haindl (1986). An imporan difference beween his paper and ours is ha he uses he mehodology proposed by Chow and Lin (1971). This mehodology imposes a very resricive funcional form ha canno represen accuraely he feaures of our daa. More specifically, his procedure assumes ha: 1) he dependen variable and he se of indicaors are fully coinegraed; 2) here is no feedback beween he variables in he model; and 3) he residual of he regression beween he dependen variable and he se of indicaors follows an ad-hoc funcional form. Here, insead of imposing any paricular funcional form on he variables, we specify models ha are plausible given he saisical properies of he ime series in our analysis. Two alernaive saisical specificaions are considered in our analysis: a univariae ARIMA model and a ransfer funcion model. These wo specificaions are robus and filer he daa o uncorrelaed and normal residuals. The quarerly series obained under boh procedures incorporae he cyclical dynamic in he quarerly series laer o 1979 respecing, addiionally, all he previous annual informaion. The remainder of he paper is organized as follows. In he nex secion, we specify and esimae an ARIMA model for he Chilean GDP. Secion 3 discusses how o wrie he univariae model in he sae space form and how o use his formulaion o reconsruc he quarerly series backward using a fixed inerval smoohing algorihm. In Secion 4, we exend he analysis o consider he use of ransfer funcion models. Quarerly GDP series obained from he univariae and he ransfer funcion model are shown and compared in Secion 5. Some concluding remarks follow in Secion UNIVARIATE ANALYSIS In his secion we describe some of he imporan feaures of he individual ime series used in he analysis and hen we specify and esimae an ARIMA model for he Chilean GDP ha will be used o inerpolae quarerly values of his series before Our main ineres is in Chilean GDP in real erms. This series is freely available from he Cenral Bank of Chile a he following URL:

3 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 287 hp:// The Cenral Bank of Chile publishes GDP series on a quarerly basis since However, i is possible o find GDP informaion on annual basis before his period. Addiionally, we also consider oher quarerly series ha will be used in he subsequen analysis o specify an economeric model. More specifically, our inenion is o sudy he properies of hese series o use hem as indicaors o inerpolae quarerly informaion of he Chilean GDP before Three series have received our aenion in his respec: 1) he moneary aggregae M1; 2) he price of copper; and 3) he erms of rade. Series of he moneary aggregae M1 were colleced from he Monhly Bullein of he Cenral Bank of Chile. Series of he price of copper was obained from Informe Económico y Financiero, also published by he Cenral Bank of Chile. Series of he erms of rade can be found in he paper by Benne and Valdés (2001). These hree series are available from In order o specify an univariae ARIMA model for he GDP series we choose no o consider he firs 10 quarerly observaions as hey are very erraic and generae an spurious AR(1) or AR(2) srucure. Tha is, we only consider he period 1982:3-2004:2. The evoluion of his series is shown in Figure 1. Inspecion of he figure reveals ha he GDP grows during he period under consideraion and his growh is affeced by he seasonal cycle. Figures of he moneary aggregae M1, he price of copper and he erms of rade are no shown for he sake of breviy. However, M1 grows during he period and is growh is also affeced by he seasonal cycle. Series of he erms of rade and he price of copper, on he oher hand, do no grow bu show lile endency o reurn o mean. This and he addiional informaion provided by correlograms suggess ha M1 and GDP series require wo differences o become saionary and one of hem could be a seasonal difference. Also, erms of rade and price of copper should be saionary afer one regular difference. FIGURE 1 REAL GDP IN MILLIONS OF CHILEAN PESOS. SERIES IN LEVELS

4 288 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 More formally, we employ he Augmened Dickey-Fuller (ADF) es for uni roos on he series in levels, firs and second differences. I is, of course, necessary o choose he number of augmenaion lags o accoun for serial correlaion and his is done using he sequenial approach in Ng and Perron (1995). The resuls are shown in Table 1. TABLE 1 AUGMENTED DICKEY-FULLER (ADF) TESTS Series Series in levels Series in firs differences Series in second differences Trend Inercep None Trend Inercep None Trend Inercep None GDP (**) 28.37(**) 28.52(**) M (**) 13.68(**) 13.69(**) Copper (**) 7.57(**) 7.58(**) 13.32(**) 13.39(**) 13.47(**) Terms of rade (**) 7.92(**) 7.91(**) 14.17(**) 14.24(**) 14.33(**) Trend, Inercep and None denoe he Augmened Dickey Fuller ess applied o a regression wih inercep and rend, inercep and no deerminisic parameers respecively. (**), (*) denoe rejecion of he null hypohesis a he 0.01, 0.05 significance levels respecively. Esimaion sample: 1982:III-2004:II. For series in levels, he uni roo null hypohesis canno be rejeced a convenional significance levels in any case. However, he uni roo null can be rejeced a he 0.01 level for firs differences of he price of copper and erms of rade. For M1 and GDP i is necessary o ake second differences in order o rejec he null a he 0.01 level. Once he number of uni roos has been deermined for each of he series, he nex sep is o specify and esimae an ARIMA model for he Chilean GDP. In order o do his, we use he Box-Jenkins mehodology based on he observaion and inerpreaion of correlograms; see Box e al. (1994). The simple correlogram for he GDP series wih one regular and one seasonal difference show ha he series is saionary and i does no have any known srucure in he regular par. However, he seasonal par suggess eiher a MA(1) or AR(2) specificaion. When he wo models are esimaed, he sandard deviaion of he esimaed residuals is 0.12 in boh cases. However, a MA(1) is more parsimonious han an AR(2) model and we choose he firs opion. This amouns o specifying he following model for he Chilean GDP (denoed by y ): (2.1) ΔΔy = ( 1 θl ) a 4 4 where L is he lag operaor; Δ and Δ 4 are operaors for he regular and seasonal differences respecively; θ is a consan parameer; and a ~N(0, σ ) a.

5 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 289 Esimaion of model (2.1) by exac maximum likelihood gives he following resuls (wih sandard errors beween brackes): 1 4 (2.2) ΔΔyx10 = ( L)ˆ a 6 4 ( 010. ) (2.3) σˆ a = 012. The correlaion beween he wo esimaed parameers ( ˆσ a and ˆθ ) is always below Besides, he Jarque-Bera saisic on he esimaed residuals is 1.80 indicaes ha hey can be considered as normal. Also, he correlogram of he residuals do no show any significan correlaion a any peak and is srucure could be regarded as a whie noise process. From model (2.1), i is sraighforward o describe a univariae model for he accumulaed yearly GDP, (denoed by Y ) as: 2 3 (2.4) Y = y + y + y + y = ( 1+ L+ L + L ) y Noice ha (2.5) ( 1 L 4 ) = ( 1 L)( 1+ L+ L 2 + L 3 ), hen, using expressions (2.2), (2.3) and (2.5) one can obain a model for Y as: (2.6) Δ 2 Yx10 6 = ( L 4 )ˆ a This las model is used for he backward inerpolaion of Y. An explanaion of his process follows in he nex secion. 3. AN ALGORITHM FOR THE INTERPOLATION OF THE CHILEAN GDP Once we have defined a model for Y, he quarerly series is consruced using a fixed inerval smoohing algorihm where for each four values one is fixed. For a brief descripion of his process, noice ha every linear economeric model wih fixed parameers can be represened in he sae space form as: 2 1 In he esimaion we use he economeric sofware E4 (hp:// One imporan advanage of his sofware compared wih oher more convenional ones is ha E4 esimaes parameers by exac maximum likelihood. This mehod is more efficien han he esimaion by condiional maximum likelihood. Moreover, exac maximum likelihood is especially convenien when iniial observaions in he sample are very erraic; see Hamilon (1994), Chaper 5. 2 See Terceiro (1990) for a more deailed and formal descripion of sae space models.

6 290 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 (3.1) Z = Hα + Du + cv (3.2) α = Φα + Γu + Ew + 1 where Z is a vecor of observed variables (in our case Z = Δ 2 Y ); u is a vecor of exogenous variables; α is a vecor of nonobservable sae variables; w and v are whie noise processes; and H, D, C, Φ, Γ and E are known parameer marices. Thus, for example, a possible represenaion of a seasonal MA(1) process in he sae space form could be: (3.3) α +1 = θa -3 (3.4) Δ 2 Y = α +a A fixed inerval smoohing algorihm consiss on obaining an esimaion of he sae variable and is variance from he available sample informaion. We denoe hese esimaions as α /N and P /N, P /N = E [(α α /N ) (α α /N ) /Ω N ].3 A deailed descripion of he compuaional procedure of inerpolaion proposed by De Jong (1989) and exended by Casals e al. (2000) is confined o he appendix. Inuiively, he algorihm depars from some iniial condiions α 1 and P 1 and hen esimaes he esae variable using a Kalman filer and reconsrucs he series backward using a smoohing algorihm. An imporan drawback in he algorihm advocaed by De Jong (1989) lies in he fac ha he iniial value of he covariance of he sae value, P 1, is arbirarily close o infinie for all he saes. Casals e al. (2000) exend his previous mehodology by proposing a fixed inerval algorihm ha allows for boh, saionary and uni roos, and also reas he case of exogenous inpus. This las procedure is used here o inerpolae nonexisen values of y. Thus, from 1980 backward, for each 4 values, 3 nonexisen values are inerpolaed using E4. 4 The algorihm gives he inerpolaed values of y and he esimaed sandard deviaions of he inerpolaions. 4. MULTIVARIATE ANALYSIS One poenial cavea in he previous analysis is ha a univariae ARIMA model only considers informaion referred o is own lagged values. Alhough our quarerly series respec he annual informaion provides by he Cenral Bank of Chile before 1980, however one could argue ha some exogenous shocks could marginally affec he GDP series a specific momens. For example, he 3 N denoes he oal number of observaions. 4 E4 codes can be obained from he auhors upon requess. Some pracical examples of inerpolaion mehods can be found in he E4 manual from is webpage (hp://

7 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 291 poliical crisis in he early sevenies, he coup d ea in Sepember 1973, he downfall in he price of copper in 1975 and he revaluaion of he Chilean peso in In order o ake ino accoun some of hese facs, we specify an economeric model o inerpolae he Chilean GDP before 1980 using as indicaors he quarerly series presened in Secion 2. The use of hese series can be jusified as follows. The moneary aggregae M1 series can be considered as a poenial indicaor o measure he relaion beween money and oupu suggesed by moneariss. The use of he price of copper can be jusified as here is a general agreemen among economiss on he influence of his mineral on he Chilean economy, see Meller (2002). The erms of rade is also an imporan variable o consider as i can ake ino accoun open marke consideraions in a small open counry as Chile. Typically, he mehodology proposed by Chow and Lin (1971) has been used o esimae quarerly series from heir annual values; see Pons and Surinach (1997). The procedure consiss on relaing he unknown quarerly series o a se of quarerly variables or indicaors. When he model is expressed in erms of annual accumulaed values, i can be wrien as: (4.1) Y = I β+ ε where I is a vecor of explanaory variables; β is a vecor of parameers; and ε follows an ad-hoc univariae process. For example, Chow and Lin (1971) assume o be a whie noise or a saionary AR(1) process. Fernández (1981), on he oher hand, assumes ha i follows a random walk. An obvious drawback of his procedure is ha i imposes a funcional form ha could be inaccurae given he feaures of he ime series in our analysis. More specifically, expression (4.1) imposes he following assumpions: Series Y and I are coinegraed, unless we specifically assume a funcional form similar o he one proposed by Fernández (1981). Y does no Granger cause any of he variables in I. Tha is, here is no reason o prefer a VAR sysem raher han a single ransfer funcion equaion. Even in he unlikely case ha he wo previous assumpions are rue, an addiional assumpion imposes ε o follow an ad-hoc specificaion. In our paricular case, given he analysis in Secion 2, i is clear ha GDP canno be coinegraed neiher wih he series of price of copper nor wih erms of rade as hey have differen inegraion orders. Also, Granger causaliy ess, no explicily repored here, clearly indicae ha he null of no causaliy beween he GDP and he wo series, he price of copper and he erms of rade, canno be rejeced in any direcion a convenional significance levels. In order o es coinegraion beween GDP and M1 we use he ess proposed by Engle and Granger (1997) as his procedure is simple, inuiive and i is especially suiable o be used wih only wo series. Thus, firs we run a regression beween he wo variables (GDP and M1). Then, in a second sep, we

8 292 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 es for he presence of a uni roo in he residuals of he regression using an ADF es. Resuls of he es do no indicaed he presence of a coinegraion relaionship beween hese wo variables. More specifically, he values of he ADF saisic for a regression wih inercep and rend, inercep and no deerminisic componens are 2.07, 0.65 and 0.75 respecively. Given hese resuls, we include in our mulivariae model he GDP and M1 series wih one regular and one seasonal difference in order o ensure ha hey are saionary. In a firs aemp, no repored here, we considered a VARMA model. However, no evidence of causaion from M1 o GDP was found in ha model. Then, we esimae he following model (sandard deviaions are beween brackes): (4.2) ( 1 + L+ L + L ) y x10 = ( 0. 43) + ( L)( 1+ L+ L + L ) i x10 + ( 015. ) ( 015. ) ( L ) ( 010. ) + 2 aˆ, Δ σ ˆ = 011., a where i is he quarerly indicaor (M1). In his regression, he correlaions among he esimaed parameers are always below Also, he residuals can be assumed o be normal wih a Jarque Bera saisic of 2.09 and correlograms of he esimaed residuals do no show any known srucure and hey are no serially correlaed. A his sage, i is imporan o emphasize ha model (4.2) is only a reduced form saisical model and i is beyond purpose of his paper o develop any srucural analysis on he relaion beween GDP and M1. Using his economeric model, we inerpolae he quarerly GDP series before 1980 using he smoohing algorihm described in he previous secion. 5. COMPARING ALTERNATIVES This secion compares he quarerly GDP series obained under boh he ARIMA and he ransfer funcion model. The main advanage of using a univariae model is based on is simpliciy. In general, parsimonious models are especially accurae o forecas values ou of he sample. Transfer funcion models, on he oher hand, are useful in his conex as hey can measure he impac of oher variables for which quarerly values before 1980 can be obained. However, one can argue ha here have been imporan srucural changes in he Chilean economy from 1967 and i is very risky o assume ha a se of economic indicaors may have a consan effec on Chilean GDP hrough he whole sample. Following his argumen, o es how robus our esimaions are, we esimae models (2.6) and (4.2) using annual daa for he whole sample ( ) bu resuls were very similar o hose obained from he sample ( ). Figure 2 shows he quarerly, y, and he accumulaed annual, Y, GDP series obained under boh procedures. The figure describes he differen economic

9 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP FIGURE 2 GDP SERIES IN MILLIONS OF CHILEAN PESOS Quarerly GDP series obained from an ARIMA model. Accumulaed annual GDP series obained from an ARIMA model Quarerly GDP series obained from a ransfer funcion model. Accumulaed annual GDP series obained from a ransfer funcion model

10 294 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 TABLE 2 QUARTERLY GDP SERIES IN MILLIONS OF CHILEAN PESOS OBTAINED FROM A UNIVARIATE AND A TRANSFER FUNCTION MODEL Year Univariae Transfer Year Univariae Transfer model funcion model model funcion model 1966-I I II II III III IV IV I I II II III III IV IV I I II II III III IV IV I I II II III III IV IV I I II II III III IV IV I I II II III III IV IV I I II II III III IV IV cycles in Chile during he period Thus, here is a moderae growh in he years moivaed by a srong public invesmen. Afer his period, here is sagnaion phase in characerized by igh moneary policies aiming o conrol a high rae of inflaion. There is a srong recovery in simulaed mainly by foreign rade. Finally, he Asian crisis affeced he Chilean economy a he end of he nineies slowing down economic growh from 1998 up o he presen dae. From his figure, an imporan feaure o remark is ha he wo new quarerly series describe he same economic rend as he accumulaed annual series from he Cenral Bank of Chile. Besides, he wo quarerly series reflec he seasonal behaviour inheren in he Chilean economy.

11 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 295 When he wo differen models are compared, i is clear ha he series inerpolaed under he univariae and he ransfer funcion model are almos idenical. This is also corroboraed by a Pearson s correlaion value of beween he quarerly series inerpolaed under boh procedures for he period 1967: :04. Given his resul, i is no clear which alernaive is superior. Therefore, we leave o he analys s crieria o choose beween he wo opions. This elecion will no have dramaic consequences in any case as boh series respec he annual GDP informaion provided by he Cenral Bank of Chile. The numerical values obained from he wo inerpolaions are shown in Table CONCLUSIONS In his paper we obain quarerly series of he Chilean GDP for he period using he fixed inerval smoohing algorihm proposed by De Jong (1989) and exended by Casals e al. (2000) for he nonsaionary case. We propose wo alernaives for he inerpolaion based on a univariae and a ransfer funcion model. Quarerly series obained in boh cases are consisen wih he annual informaion provided by he Cenral Bank of Chile before A new line of research opened by his paper is based on he use of he quarerly GDP series o specify and esimae more efficien economeric models ha could be considered o describe he relaion of he Chilean GDP wih oher economic (naional and inernaional) variables for which quarerly informaion is already available for long periods. REFERENCES Benne, H. and R. Valdés (2001). Series de Términos de Inercambio de Frecuencia Mensual para la Economía Chilena: 1965:1999, Working paper N 98, Cenral Bank of Chile. Box, G.E.P., G.M. Jenkins and G.C. Reinsel (1994). Time Series Analysis: Forecasing and Conrol. (Third Ediion), Englewood Cliffs, HJ: Prenice Hall. Casals, J., M. Jerez and S. Sooca (2000). Exac Smoohing for Saionary and Nonsaionary Time Series, Inernaional Journal of Forecasing, 16, Chow, G. and Lin, A.L. (1971). Bes Linear Unbiased Disribuion and Exrapolaion of Economic Time Series by Relaed Series, The Review of Economics and Saisics, 53, De Jong, P. (1989). Smoohing and Inerpolaion wih he Sae-Space Model, Journal of he American Saisical Associaion, 84, Engle, R.F. and Granger, C.WJ. (1987). Co-Inegraion and Error Correcion: Represenaion, Esimaion and Tesing, Economerica, 55, Fernández, R.B. (1981). Mehodological Noe on he Esimaion of Time Series, The Review of Economic and Saisics, 63, Haindl, R. (1986). Trimesralización del Produco Geográfico Bruo por Origen y Desino, Esudios de Economía, Universidad de Chile, 13,

12 296 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 Hamilon, J.D. (1994). Time Series Analysis. Princeon Universiy Press. Meller, P. (2002). Dilemas y Debaes en Torno al Cobre. Dolmen. Ng, S. and P. Perron (1995). Uni Roo Tess in ARMA Models wih Daa-Dependen Mehods for he Selecion of he Truncaion Lag, Journal of he American Saisical Associaion, 429, Pons, E. and J. Surinach (1997). Trimesralización y Conciliación de Magniudes Económicas: una Ampliación del Méodo de Chow-Lin, Working Paper N 20, Universidad de Barcelona. Terceiro, J. (1990). Esimaion of Dynamic Economeric Models wih Errors in Variables. Berlin: Springer-Verlag.

13 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 297 APPENDIX INTERPOLATION PROCEDURE This appendix describes he procedure used for inerpolaion of he Chilean GDP before The mehodology follows he lines saed by De Jong (1989) and exended by Casals e al. (2000) for he nonsaionary case. Le assume he following sae-space model: (A1) Z = Hα + Du + Cv (A2) α = Φα + Γu + Ew + 1 where (A1) is he observaion equaion ha generaes he (mx1) vecor of measures Z, = 1,2, N, u is a (rx1) vecor of inpus and he sae equaion (A2) describes he evoluion of he (nx1) sae vecor α. We make he following assumpions abou (A1)-(A2): i) w ~ IID ( 0, Q ), v ~ IID( 0, R), cov( w, v ) = S, for all = 12,,..., N. ii) The iniial sae is independen of w and v, and such ha α / u,..., u ~( α, P) 1 1 N 1 1. iii) The marices H, D, C, Φ, Γ, E, Q, R and S have been esimaed previously whereas α 1 and P 1 are unknown. We also denoe he informaion available up o = j by: Ω j = {Z 1,Z 2,,Z j,u 1,u 2,u j } and he firs and second-order condiional momens of he sae vecor by: α /j = E(α /Ω j ) and P /j = E[(α α /j ) (α α /j ) / Ω j ]. Following De Jong (1989), based on (A1) and (A2), one can inerpolae Z, u or α ono he space [Z 1,Z 2,,Z 1,Z +1,,Z N ] using a fixed inerval smoohing algorihm. This algorihm consiss of a forward sep given by a sandard Kalman filer and a backward recursion ha, for he case of fixed-inerval smoohing, akes he form: (A3) α = α + P N / / 1 / 1 1 (A4) P = P P R P N / / 1 / 1 1 / 1 1 T (A5) r = H ' B Z + Φ r, wih r = T (A6) R = H ' B H + Φ R Φ, wih R = 0 r 1 (A7) Φ = Φ KH N N

14 298 CUADERNOS DE ECONOMÍA Vol. 43 (Noviembre) 2006 where α / 1 and P / 1 were compued in a forward sep, Z = Z Z 1 is he sequence of Kalman filer innovaions corresponding o (A1)-(A2); B is he covariance marix of Z ; and K is he Kalman filer gain. Model (A1)-(A2) can be saionary, nonsaionary or parially saionary, depending on he eigenvalues of Φ. De Jong (1989) emphasize he imporance of iniializaion in he forward filering phase and propose adequae soluions for saionary sysems. However, in he nonsaionary case he iniial sae covariance, P 1, is arbirarily close o infiniy and, herefore, hese iniializaion crieria canno be used. In a pure nonsaionary framework, a common pracice consiss of approximaing he diffuse iniial condiions by P 1 = ki, where k is an arbirary big value. Frequen rule of humb values for k may vary beween k = 10 2 and k = Bu, alhough his iniializaion allows one o keep using sandard algorihms, however i commonly generaes biased resuls when k is no adequaely chosen and induces numerical error. Casals e al. (2000) derive an exac algorihm ha can be applied o saionary, nonsaionary and parially nonsaionary sysems. This algorihm is obained from he combinaion beween exac fixed-inerval smoohed momens and hose obained from an arbirarily iniialized smooher. To see his, consider he sae space model: * * (A8) Z = Hα + Du + cv * (A9) α = * Φα + Γu + Ew + 1 where he saes and measures correspond o (A1)-(A2) wih he iniial condiions α * 1 = 0 and P * 1 = 0. I is sraighforward o see ha propagaing he sae equaions (A2) and (A9), i follows ha * (A10) α = α + α Φ 1 / N 1/ N / N Also, from (A1)-(A2) and (A8)-(A9), Casals e al. (2000) prove ha (A11) Z = H α + Z * Φ 1 1 where Z * * * = Z Z / 1 are he innovaions resuling from a Kalman filer applied o (A8)-(A9), hereafer KF(0,0); and he marices Φ 1 are given by Φ = Φ Φ ( KH ) wih Φ 1 = I. Equaion (A11) can be wrien for all he sample as: (A12) Z = X + Z * α 1 where X is he block-diagonal marix whose -h block is HΦ 1 and he (mxn)x1 vecors Z and Z * conain he KF(0,0) innovaions Z and Z * respecively. Noe ha Z *, is independen of a 1.

15 A PROPOSAL TO OBTAIN A LONG QUARTERLY CHILEAN GDP 299 Then, he problem consiss of obaining he condiional expecaions in he righ-hand-side of (A10), aking ino accoun he relaionship (A12). The soluion is given by he following expressions: { } + * * * * (A13) α = E α ( Z )' Φ 1 B X α E α ( Z / N 1 1 )'B Z 1/ N (A14) P/ N= { 1 1 } P/ N + 1 Φ?? 1 Φ * E α * { ( Z * 1 )' B X E α * ( Z * 1 )' B X}+ P / N An imporan poin o be noed abou hese wo expressions is ha hey can be applied o saionary, nonsaionary and parially nonsaionary sysems, as he only erm affeced by P 1 are Z 1/N and P 1/N and his dependence occurs hrough P 1 1, which is finie. Algorihms for he compuaion of E α * ( Z * )' B X 1 and E α * ( Z * 1 )' B Z can be found in Casals e al. (2000), Secion 3.