DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Piotr Fiszeder Nicolaus Copernicus University in Toruń

Transcription

1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 006 Pior Fiszeder Nicolaus Copernicus Universiy in Toruń Consequences of Congruence for GARCH Modelling. Inroducion In 98 Granger formulaed e idea of congruence, as similariy of main properies of e endogenous variable and main properies of exogenous variables. Zielinski (984) inroduced e idea of dynamic congruen modelling. Congruence of e model in e meaning of Zieliński means a armonic srucure of e endogenous process is congruen wi e join armonic srucure of explanaory processes and e error erm, wic is independen of explanaory processes. Te idea of dynamic congruen modelling is based on capuring informaion abou e inernal srucure of processes a e model specificaion sage and building e congruen model on a basis of wie noise error erms equaion. Models wic were buil according o e procedure of dynamic congruen modelling are ofen beer models concerning saisical properies, if only e inernal srucures of analysed processes are correcly discovered and specified in e model. Te use of informaion abou e inernal srucure of economic processes was also a basis for e idea of congruen modelling inroduced by Granger (990). Te concep of congruence of condiional variances is no menioned in ose wo ideas of modelling. In is aricle e idea of congruence is exended o condiional variances. In is paper congruence of e model in condiional variances means equaliy of condiional variance of endogenous variable wi condiional variance of linear funcion of explanaory processes and error erm. I is assumed a all analysed processes ave consan and finie uncondiional variances. A model is congruen in condiional variance if e uncondiional Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House Financial suppor of e Polis Commiee for Scienific Researc for e projec H0B carried ou in is graefully acknowledged.

2 44 Pior Fiszeder variance of e endogenous variable is equal o e uncondiional variance of linear funcion of explanaory processes and e error erm and ere is congruence of armonic srucure of e endogenous process squared wi armonic srucure of e square of linear funcion of explanaory processes and e error erm. Te aricle is laid ou in four secions. Secion inroduces e models congruen in variance for GARCH processes. Secion 3 describes consequences of modelling wic is no congruen in variance and inroduces a new volailiy measures. Secion 4 concludes.. Models Wic Are Congruen in Condiional Variance Te GARCH ( p, process can be wrien as: ε ψ ~ D(0, ), () = α q p o + αiε-i + i = j = β j -j, () were ψ is e se of informaion available a ime and D ( 0, ) is probabiliy disribuion (usually e normal or Suden disribuion) wi zero mean and variance. Le ε y, ε x s ( s =,,..., k) and ε be wie noise GARCH processes (wi zero mean, consan uncondiional variance and no auocorrelaion). Te model: k y = ρ sε xs ε s= ε +, (3) x s were E( ε ε x ) = 0, is congruen, wen armonic srucures of e lef and s rig and side erms of e equaion are idenical. If ε y is an error process in e model describing inernal srucure of dependen variable, and ε are error processes in e models explaining inernal srucures of explanaory variables, en congruen model may be consruced in a radiional way using in equaion (3) inernal srucure of e processes (see Talaga and Zieliński, 986). Reurns of financial processes are ofen saionary and usually ey may be sufficienly described by auoregressive models wi low order of auoregression. Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House Te idea of congruence can be exended o iger condiional momens of a disribuion, owever in is case inerpreaion is no easy.

3 Consequences of Congruence for GARCH Modelling 45 Presened congruen models do no ave o be congruen in condiional variance. Firs le us consider an example wi one explanaory variable. Le ε x and ε be GARCH processes of orders: ( p, and ( p, q ). If, ε x and y ε are wie noise processes, en model: ε = ρε + ε (4) y x ε y is congruen. Model (4) is congruen in condiional variance, if uncondiional variance ε y is equal o uncondiional variance ρε + ε, and armonic srucures of e processes and ρε + ε ) are idenical, a is if e model: y ( x x ρε xε ε ε = ρ ε + + x is congruen. Te equaliy of uncondiional variances of ε and ρε x + ε is required in every congruen model of ype (4). Assuming ν = ε, GARCH ( p, model may be wrien as an ARMA (m,p) model for (were m = max {p,q}). I follows a and may be presened as: x ε = α + ε y = α ε q p p αiε x i + β jε x j β jν x j + i= j= j= q p p αiε y i + β jε y j β jν y j + i= j= j= Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House ν x ν y ε x ε y ε y (5), (6), (7) were ν y, ν x are wie noise processes (if ε y and ε x ave finie four order momens). Mos financial processes ave finie four order momens. Te excepions are mainly financial insrumens from money marke and less liquid insrumens from oer markes. Te srucure of ε will depend on inernal srucure of ε y and ε x processes. Condiional variance of e error erm ε in equaion (4) may en be described using GARCH, ) model. Te values of and q will depend on e ( p q 3 3 p3 3 ε y caracer of e relaionsip beween and ε x and e properies of ese processes (more abou e properies of processes wic are a sum of oer auoregressive processes may be found in Kufel, Piłaowska and Zieliński, 996, Sawicki and Górka, 996). If Y = A ( u ε and B )

4 46 Pior Fiszeder B Y A ε ( p, q ) = are independen ARMA processes of degree ( p, and respecively, en and Y = Y Y = B A ε B A ε B ) B Y = B A ε B A ( u ε. (9) Tus Y process is of ARMA ( p, ype, were p = p + p and q = max ( pq, pq ). If some parameers are close o zero, en e process may be idenified as a process wi fewer lags. In e case of financial ime series ere is usually a grea number of observaions, us will almos always be an idenifiable ARMA process. Similarly, wen and are independen en e variance ε in equaion (4) may be presened as a ARMA ( p, process. If equaion (4) is a model for error processes in e models describing inernal srucure of dependen and explanaory variables, en only correc specificaion of model (4) and GARCH model for ε will assure congruence in variance of model (4). In specificaion one sould properly describe inernal srucure of dependen and explanaory variables (in order o receive wie noise properies) and properly describe orders in GARCH model. One reaces similar conclusions from a general model for k explanaory variables (equaion (3)), assuming E( ε ε ) = 0. Wen explanaory vari- ' ables are correlaed en variance of ε will also depend on covariances of paricular explanaory variables. I is wor o menion a ere is no possibiliy o creae a classical congruen model beween GARCH processes ε y and ε x on e basis of sandardised processes. Le z y, z x and z be sandardised wie noise processes wi consan condiional variances: y y y xs ε y Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House xs z = ε /, z x = ε x / x, z = ε /. (0) Te model: z = ρ z + z () y x is congruen. Subsiuing (0) in equaion () and muliplying e equaion by y one obains: Y ε x (8) y y ε y = ρ ε x + ε. () x

5 Consequences of Congruence for GARCH Modelling 47 Subsiuing y y ρ = ρ and ε = ε one acquires a congruen model: x x ε = ρ ε + ε, (3) y ρ wi socasic parameer. Obviously ere exiss classical congruen model beween processes: ε y and ε x xy a is beween ε y and properly sandardized process ε x. 3. Congruen Models in Condiional Variance Proposiions of New Volailiy Measures Volailiy is an imporan quaniy in many financial analyses, eg. derivaives pricing, capial asse pricing, analysis of e flow of informaion beween markes and financial insrumens. Volailiy of condiional variance of regression model error erm lowers e efficiency of parameer esimaors obained wi leas squares meod, and covariance marix of esimaors σ ( X ' X ) is no valid if e regression involves lagged dependen variable. Sandard esing for parameers saisical significance may in is case lead o wrong resuls. Te alernaive, more suiable procedures include e use of oer esimaion meods wic are robus o canges of variance or describing e canges of error erm variance direcly in e model for example wi GARCH or SV specificaions. If e model wic serves as a ool for creaion of volailiy measure is no congruen in condiional variance, en e variance is usually underesimaed. Tus e congruence in variance of a model is imporan in every analysis using volailiy forecass. Le y describe AR (r) GARCH ( p, process: y r = 0 + i= φ φ y i i + ε, (4) ε ψ ~ N(0, ), (5) = q p αo + αiε-i + i = j = β j -j. (6) Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House Te esimaes of y variabiliy obained on a basis of e condiional variance of ε will be underesimaed, because ey do no accoun for canges of e auoregressive process. Variabiliy of is obviously greaer an variabiliy of y

6 48 Pior Fiszeder error erm ε. In empirical applicaions i may appear a inclusion of variabiliy resuling from auoregressive par of e model is imporan (for example in applicaions of Mone Carlo meods in pricing of derivaives or calculaing e Value a Risk). One can use condiions of model congruence in condiional variance and apply a new volailiy measure: s = r φ i= i, (7) Proposed volailiy measure is no a pure condiional variance, because i is rescaled up by a auoregressive variabiliy. Similarly e esimaes of variabiliy ε y based on condiional variance ε in equaion (3) will be underesimaed, because canges of explanaory variables ε x s are no accouned for. Le E( ε x, ε x ) = 0 s s '. Proposed volailiy measure for ε y may be wrien as: s y = k s= s ρ xs +, (8) were x s ( s =,,..., k) and are condiional variances of explanaory variables and e error erm respecively. If model (3) is no congruen in condiional variance, a is if ε does no accoun for inernal srucure of ε y and ε x s ( s =,,..., k ), en e esimaes of variabiliy generaed by is model will be incorrec (underesimaed or overesimaed). 4. Conclusions In e paper e concep of model congruence as been exended o condiional variances. Te implicaions of neglecing e informaion abou e inernal srucure of dependen variable or explanaory variables ave been indicaed. New measures of variabiliy ave been proposed. Empirical applicaion of proposed measures for financial analyses, in paricular for volailiy forecasing is lef for fuure researc. Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House References Granger, C. W. J. (98), Some Properies of Time Series Daa and eir Use in Economeric Model Specificaion, Journal of Economerics, 6, 30.

7 Consequences of Congruence for GARCH Modelling 49 Granger, C. W. J. (990), Were Are e Conroversies In Economeric Meodology?, in: C. W. J Granger (ed.), Modelling Economic Series, Clarendonpress, Oxford. Kufel, T., Piłaowska, M., Zieliński, Z., (996), Symulacyjna analiza poznawczyc własności dynamicznyc modeli zgodnyc (Simulaion Analysis of Cogniive Properies of Dynamic Congruen Models), Przesrzenno czasowe modelowanie i prognozowanie zjawisk gospodarczyc (Cross Secion Time Modelling and Forecasing of Economic Evens), AE, Kraków. Sawicki, J., Górka, J. (996), An ARMA Represenaion for a Sum of Auoregressive Processes, Dynamic Economeric Models, vol., UMK, Toruń. Talaga, L., Zieliński, Z. (986), Analiza spekralna w modelowaniu ekonomerycznym (Specral Analysis in Economeric Modelling), PWN, Warszawa. Zieliński, Z. (984), Zmienność w czasie srukuralnyc paramerów modelu ekonomerycznego (Time Varying Srucural Parameers of Economeric Model), Przegląd Saysyczny (Saisical Survey), 3, /, Copyrig by Te Nicolaus Copernicus Universiy Scienific Publising House