Algunas Técnicas Útiles Para La Identificación

1 Numerical Algorithms in Time Series Models. The case of Banana Production in Canary Islands González-Concepción, Concepción Gil-Fariña, María Candelaria Department of Applied Economics University of La Laguna Campus de Guajara, s/n 38701 La Laguna. Tenerife. Canary Islands

Abstract: In economics there exists a great interest for studying its raw material, the data and obtaining specific properties from data series. This can be useful to help to predict, with certain degree of certainty, the behaviour of relevant economic variables and beginning economic and social policy measures. This paper is concerned with illustrating the application of several numerical methods, in particular, the corner method, epsilon-algorithm, rs-algorithm, qd-algorithm, to the problem of model identification in time series analysis. These methods can be useful to identify some type of rational structure associated to economic data in different contexts (financial, marketing, farming…). Moreover, they incorporate the expectations of exogenous economic variables to improve the fit and forecasting of classic time series models. The empirical work is carried out with of a sensitivity analysis of the banana production in Canary Islands, considering different schemes for the compensatory aid expectations. Key-Words: Padé Approximation, economics, numerical algorithms, time series modelling, expectations, Canary economy, banana.

mentioning some numerical methods (corner method, 1 Introduction epsilon-algorithm, rs-algorithm and qd-algorithm). In the context of time series analysis, several authors Next, we show the theoretic characterization of these (Lii [20], Claverie et al. [9], Berlinet and Francq [5] techniques in TF models with a one output and one or …) have considered the rational theory of series in multiple inputs or explained variables in a causal way. econometric modelling and have proposed the use of Furthermore, we extend this type of models to include some techniques. expectations for the input variables and expose some The contribution of this paper is to show the way in proposals for their rational characterization. the one some techniques from the context of rational The empirical work is made in the context of the Box- approximation theory and closely related to the Padé Jenkins´s [7] approach. We illustrate the methods in Approximation, can be useful to identify some type of the context of the banana production in Canary rational structure associated to chronological data or Islands. time series. As the covariance structure of underlying The paper concludes with an outline of the results and processes exhibits features connected with the order of the most relevant conclusions. models, it is possible to use numerical algorithms linked with Hankel and Toeplitz determinants to 2 The Univariate Case: Some Methods estimate the unknown orders from observations and of Rational Characterization in ARMA expectations. Models In the univariate case, the identification of ARMA Let us consider a minimal stationary and invertible models has been extensively considered in last two Autoregressive-Moving Average (ARMA) model of decades (Beguin et al [2], Mareschal and Mélard [25], order (p,q) defined as Claverie et al [9], Berlinet and Francq [5]...). As for the multivariate case, some results have been given to p (L)Yt  q (L)a t , t  Z identify VARMA models (Tiao and Tsay [32], Reinsel where L is the backward-shift operator such that m [29], Lütkepohl and Poskitt [24], Pestano and L Yt=Yt-m,tZ, p(L), q(L) are polynomials of González [27]...) and Transfer-Function (TF) models degree p and q respectively and a t ;t  0,1,2,... is a (Liu and Hanssens [23], Lii [21], González et al sequence of independently and identically distributed [18]...). 2 random variables with mean zero and variance  a. It is Here, we begin making a briefly reference to the assumed that p(L) and q(L) have no common rational characterization for ARMA models, factors. 2 Various methods have been proposed to identify the of VARMA models and, as a particular case, TF orders p and q, starting on the autocorrelations of the models (Box and Jenkins [7]). process and the application of several criteria from the Next, we show this type of dynamic specification and PA’s theory. Among them, we can mention the C- the references, which contain the generalisation of the table method (Baker and Graves-Morris [1]), from its methods that have been exposed for the univariate properties it can be obtained the corner method in case. econometric literature (Beguin et al [2]). The characterization from the Hankel determinants in relation with the PA is basically the same given by 3 The Multivariate Case: Some Beguin et al for ARMA processes, although both Methods of Rational Characterization in approaches evolved in an independent way. Many Causal TF Models works later have considered the corner method in Here we refer to TF models for stationary processes ARMA modelling trying to get to the maximum their with one output yt and one or multiple inputs power (Mareschal and Mélard [25]). xit,i=1,...,n, that is The relation of this method with the Hankel n is (L) i q (L) determinants and PA has stimulated the study of other i b yt  L xit  a t algorithms in ARMA models. This is the case of the   (L)  (L) i1 iri p epsilon-algorithm (Wynn [34]), proposed by Berlinet s where  (L)     L  ...   L i and [3] for ARMA models. Its relation with PA and the is i i0 i1 is i corner method can be seen in Brezinski [8] and the ri ir (L)  i0  i1L  ...  ir L , bi is the lag in the characterization for an ARMA process in Berlinet and i i Francq [5]. response of yt to xit and at is a gaussian white noise We can also refer to the rs-algorithm (Pye and process. It is assumed a one-way causal dynamic Atchison [28]), proposed by Gray, Kelley and relation xit yt. McIntire [19] for ARMA models and whose relation With the purpose of identifying the values of bi, si and with PA can be seen in Brezinski [8]. The study of the ri and obtain a satisfactory response of yt for each statistical significance of the algorithm is given in input, several proposals have been considered on the González [13]. basis of algorithms related with the PA. It is also necessary to mention the qd-algorithm This technique allows us not only to identify the (Rutishauser [31]), which has been considered by polynomial orders but also offers consistent and Berlinet [3] to study the partial autocorrelation reliable initial parameters estimations without knowing function in an ARMA model and by González [13] to previously the noise structure. model identification. We can write the following compact relation: n   (L) The characterization for a stationary process is as j q y t   vi (L)x it  N t ; vi (L)   vijL ; N t  a t wh follows: i1 0  p (L) Theorem 1 (González [13]).- If a stationary process Xt ere vi(L) is the Impulse Response Function (IRF) has a minimal ARMA(p,q) representation, then one of which transforms xit into yt. the following statements is verified: First, the weights vij for each input and the matrix qi j ()  0, i  q  1, j > p covariance are computed using ordinary least squares  j  or maximising the likelihood function in accordance qip ()  0, i  q a)  p with the following expression:  q j q ()  0, j  p n ki  j j * y t    vˆ ijL x it  N t di j1()  0, i > q, j  p 1 i1j0  j1  ip1 where ki is a chosen finit number of terms, vˆ ij are the b) d p1 ()  0, i  q  estimated weights and N* is the reestimated noise dq j1()  0, j  p t  j1 term. We can define ˆ The study of the statistical significance of the elements vij vˆ i,max  max vˆ ij ; ˆ ij  can be also seen in González [13]. j vˆ i,max The methods explained here allow obtaining a where (ˆ )  (ˆ ) represents the sequence of tentative specification or even several possible models, i ij jN which will be discriminated by statistic methods relative weights for xit. The theoretical sequence of and/or in the estimation stage. Other techniques can be relative weight verifies the following linear difference found, for example, in Berlinet and Francq [5]. equation of order ri and rank bi+si PA and its properties can be also useful in the context  0 j > bi  si ij  i1i,j1  i2i,j2  ...  ir i,jr  of causal rational models to carry out the identification i i  0 j  bi  si 3 that constitutes a theoretic characterization for a TF modelling leads to formulations which, although valid model. from the perspective of fitting available data, they can Various methods have been proposed to obtain a not always provide a satisfactory response in terms of rational characterization in a TF model. Among them, their predictive performance. we can mention the corner method (Liu and Hanssens Faced with this situation, the consideration of [23], Tsay [33], Lii [22], Claverie et al [9]…), which expectations models for input variables (Gil and provides a generalisation of the one given in the González [11]) affords a more general framework that univariate case. The study of the statistical improves the short-term predictions in comparison significance can be also found in Tsay [33]. with traditional causal formulations. In the context of a TF model with multiple inputs we The methodological basis, which underlies this can also refer to the epsilon-algorithm (González and proposal, is the generalisation of the PA to formal Cano [14, 15], González et al [16,17], Gil and Laurent series (Bultheel [6]). This perspective in the González [11]...). The study of the statistical context of time series implies to generalise the significance can be seen in Berlinet and Francq [5] and assumptions and extend the identification procedures González et al [17]. proposed to date. Furthermore, we point up the rs-algorithm, proposed by González [13] for a TF model according to the next result: 4 The Multivariate Case: Some

Theorem 2.- vi(L) has a rational representation with Methods of Rational Characterization orders (bi,si,ri) if the following conditions are verified: for Expectations TF Models kr  i s r (i )  C1, k  bi  si  i Now we consider an expectations TF model (Gil and a)  bi si ri s (i )  C1 González [11]) for stationary process with yt  ri unidimensional and x* , x** of dimension n, r k j1( )  0, j, k < b t t  j1 i i n  n   kri 1 * * ** ** b) r (i )  0, k  bi + si y  v x  v x  N ; ri 1 t   ij it j   ij it j t  i1j0 i1j0 bi si ri r (i )  0  ri 1 q (L) 2 N t  a t ;a t ~ N(0, ) In certain cases, like it happens in the epsilon- p (L) algorithm, some transformations in the sequence of relative weights could be necessary to avoid The output yt is explained by two components: one * computational instability. systematic, described by input variables x t (real data) Lastly, the qd-algorithm has been also proposed by and ** (expectations) and a non-systematic one González [13] to identify a TF model according to the x t next characterization: described for an ARMA univariate process. We

Theorem 3.- If vi(L) has a specification with orders assume that a one-way causal dynamic relation exists * ** (bi,si,ri), then one of the following statements is x t , x t  yt. verified: A closer form for the expectations TF model can be k j expressed as: q ( )  0, j, k < b  j i i n q k j ( )  0, k > b + s , j > r y t   vi (L)x it  N t  j i i i i i1 a)  kri * ** q r (i )  0, k  bi  si where x i,t j  x i,t j if j  0, x i,t-j if j  0 and  i bi si  j  q j (i )  0, j  ri j  vi (L)   vijL represents the IRF, that is, the j d k j1 ( )  0, j, k < b series of weights evaluated in Z.  j1 i i We generalise the proposal given in Lii [22] to compute d k j1 ( )  0, k > b + s , j > r  j1 i i i i the estimated weights vˆ ij , approximating the lag and b)  kri 1 d (i )  0, k  bi  si lead structures, by means of a finite number of terms ri 1  (mi<0 and ki>0): bi si  j1 d j1 (i )  0, j  ri n k  i * yt    vˆ ijx i,t j + Nt i1 j=m The methods here exposed are useful for tackling the i Starting from the sequence of relative weights deterministic specification in a causal TF model. ˆ  (ˆ ) However, the presence of causal models in time series i ij jZ , some alternative techniques to 4 characterize an expectations TF model can be We identify an expectations TF model starting from obtained. the particular case 12 i 2 Two procedures proposed in this context are the y t  v(L)x t  N t ; v(L)   vi L x t ; N t  (1 1L  2 L )a t T-Table method and the generalisation of the epsilon- i12 algorithm (Gil and González [11]). These methods where yt=(1-L)lnPt and xt=(1-L)lnIt being Pt the generalise the corner method and the epsilon- production and It the farmer income in the period t. * algorithm respectively. Defining the Toeplitz In the four series, data x t (transformed data It till determinants 1992) are the same. They only differ in the generation ** gi of x t that includes the expectations from 1997. Tf ,g (ˆ i )  det(ˆ i,f ) i i j ihk h,k1 Scene 1: The farmer considers that the earnings will continue stable in the next decade. for the sequence of relative weights ˆ i , a finite order representation it can be obtained for the FRI according In this case, according to the T-table the final estimated model is to the next result (Gil and González [11]): -7 yt=(0.2001(4.07)+0.1296(2.69)L)L xt+ Theorem 4.- v (L) has a rational structure with orders 2 i +(1+0.8776(11.32)L-0.1153(1.48)L )at (pi,si,di,ri,ai,bi) if and only if the following conditions Scene 2: The farmer considers that the earnings are verified: diminish according to the European Union’s decision

to reduce costs. T ( )  0, f  b + s ; T ( )  0, g  r ; f ,r i i i i b s ,g i i i In this case, we obtain i i i i i T ( )  0, f  p + a ; T ( )  0, g  d 0.1661(3.00) 7 2 f ,d i i i i p a ,g i i i y  L x  (1 0.7589 L  0.2469 L )a i i i i i t 2 t (8.81) (2.89) t 1 0.5417(3.62) L T ( )  0, f  b + s + 1, f ,r i i i i i i Scene 3: The farmer considers that after a crisis period g  r  1  f  p + a - 1, g  d + 1 the earnings will be again in an acceptable level. i i i i i i i The chosen model is With respect to the epsilon-algorithm generalised, it is 0.1645(3.06) 7 2 possible to obtain the next characterization (Gil and yt  L x t  (1  0.7342(8.94) L  0.2770(3.41) L )a t 1  0.5543 L2 González [11]) for an expectations TF model: (3.86) Scene 4: The farmer doesn’t perceive when the crisis Theorem 5.- vi(L) has a rational representation with orders (pi,si,di,ri,ai,bi) if and only if the following will finish 0.1649 conditions are verified: y  (3.03) L7 x  (1  0.7959 L  0.2129 L2 )a s b r j t 2 t (10.29) (2.82) t i i i 1  0.5480 L  (i )  0,  (i )  0, j  si  bi  ri (3.60) 2ri 2ri

pi a i di j In the Fig. 2, we include these four models and two  2d (i )  0,  2d (i )  0, j  pi  a i  d i i i additional models in the classic context: The first one only considers the market price and the 5 Banana Production Models with Producer second one the producer earnings due to the market Incomes’ Expectations price and the compensatory aid. Both estimated Next, we model the evolution of the banana production models are respectively 4 and the producer earnings in Canary Islands yt=0.1707(5.39) L xt+(1+0.9318(27.78)L)at considering annual data for the period 1938-1997. An 4 yt=0.1658(4.98) L xt+(1+0.9285(-24.46)L)at important date is the year 1993 when the European We observe that all models provide a satisfactory Union gets going a mechanism of compensatory aid to response from the perspective of fitting available data. guarantee the subsistence of the banana sector. However, the expectations TF models offer better We carry out a sensitivity analysis of the banana predictions in the short-term. production considering different schemes for the In fact, without the compensatory aid, the subsistence compensatory aid expectations according to different of the sector would be threatened. scenes. Other aspects about this application can be seen in Gil and González [12]. We consider four theoretical hypotheses about the production incomes’ expectations from the 6 Conclusions introduction of the compensatory aid in 1993. This paper highlights the usefulness of several In the four models xt represents the transformed techniques closely related to the PA to identify some variables of the incomes according to the Fig. 1. types of rational structure associated with time series. The four series coincide till 1997. In this period the This is illustrated in the context of causal and compensatory aid didn’t exist (till 1992) or we had real expectations TF models, including the expectations for data. From 1997 we generate four types of the input variables in the last case. expectations for the compensatory aid. 5 The practical work points out the role of the statistical [13] C. González, Algunas técnicas útiles para la significance for the numerical values in the algorithms. identificación de estructuras racionaes en series In general, different possible models will be obtained temporales, Documento de Trabajo 9, Facultad de according to certain critical values. Ciencias Económicas y Empresariales, Universidad de Particularly, in the considered application we show the La Laguna (1997). effect of the farmer income’s expectations on the [14] C. González and V. Cano, Determinación de los banana production. órdenes de los polinomios de retardo en una función We observe the importance of maintaining the de transferencia: Comparación de algoritmos, Rev. compensatory aid for the subsistence of the banana Acad. Canaria de Ciencias 1 (1990a) 173-183. production in Canary Islands. If it were eliminated, the [15] C. González and V. Cano, Especificación de una half production would be disappeared with serious función de transferencia bajo limitación en el economic, social and ecological consequences. comportamiento de la variable dependiente, Anales de Economía Aplicada, Murcia (1990b) 337-344. [16] C. González, V. Cano and C. Gil, Comparación References: de algoritmos para la identificación de una función de [1] G. A. Baker Jr. and P. Graves-Morris, Padé transferencia: una generalización al caso de varios Approximants. Encyclopedia of Mathematics and its inputs, Rev. Española de Economía, Segunda Época Applications, 53, Cambridge University Press. 2th 10 (1993) 163-175. edition (1996). [17] C. González, V. Cano and C. Gil, The epsilon- [2] J. M. Beguin, C. Gourieroux and A. Monfort algorithm for the identification of a transfer-function (1980), Identification of a Mixed Autoregressive- model: some applications, Numerical Algorithms 9 Moving Average Process: The Corner Method in Time (1995) 379-395. Series, O. D. Anderson (ed.), North-Holland, [18] C. González and C. Gil, Padé Approximation in Amsterdam,(1980) 423-436. Economics, Numerical Algorithms (accepted). [3] A. Berlinet, Estimating the degrees of an ARMA [19] H.L. Gray, G.D. Kelley and D.D. Mac Intire, A model, Physica-Verlag, Compstat. Lectures 3 (1984) new approach to ARMA modeling, Commum. Stat. - 61-94. Simul. and Comp. B7 (1978) 1-77. [4] A. Berlinet, Sequence Transformations as [20] D. Hanssens and L. Liu, Lag specification in Statistical Tools, Applied Numerical Mathematics 1 rational distributed lag structural models, J. Bus. Econ. (1985). 531-544. Statist. 1 (1983) 316-325. [5] A. Berlinet and C. Francq, Identification of a [21] G.G. Judge, R.C. Hill, W.E. Griffiths, H. Univariate Arma Model, Computational Statistics 9 Lütkepohl T.-C., Introduction to the Theory and (1994) 117-133. Practice of Econometrics, (Second Edition, John [6] A. Bultheel, Laurent Series and their Padé Wiley and Sons, Inc. U.S.A, 1988). Approximations, Birkhaüser, Basel/Boston (1987) [22] K. Lii, Transfer Function Model Order and [7] G. E. P. Box and G.M. Jenkins, Time Series Parameter Estimation, Journal of Time Series Analysis Analysis: Forecasting and Control, Revised Edition, 6 (3) (1985) 153-169. (Holden Day, San Francisco, 1976). [23] L.M. Liu and D. M Hanssens, Identification of [8] C. Brezinski, Padé-Type Approximants and Multiple Inputs Transfer Function Models, Commun. General Orthogonal Polynomials (Birkhäuser, Basel, Statist. - Theor. Math. A11 (3) (1982) 297-314. 1980). [24] H. Lütkepohl and D.S. Poskitt, Specification of [9] P. Claverie, D. Szpiro and R. Topol, Echelon-Form VARMA Models, J. Bus. Econ. Statist. Identification des Modèles à Fonction de Transfert: La 14 (1) (1996) 69-79. Méthode Padé-Transformée en z, Annales d’Economie [25] B. Mareschal and G. Mélard, The Corner et de Statistique 17 (1990) 145-161. Method for Identifying Autoregressive Moving [10] C. Francq, Identification et Minimalité dans les Average Models, Applied Statistics 37(2) (1988) 301- Séries Chronologiques, Thèse, Université des Sciences 316. et Techniques du Languedoc, Montpellier II (1989). [26] H. Padé, Sur la Représentation Approchée d’une [11] C. Gil and C. González, Modelización racional Fonction par des Fractions Rationnelles, Ann. Ec. de series temporales no causales: algunas propuestas Norm. Sup. 9 (1982). de caracterización dinámica, Estudios de Economía [27] C. Pestano and C. González, C., Aplicada 8 (1997) 77-107. Characterization of the orders in VARMA Models, [12] C. Gil and C. González, La producción del Przeglad Statystyczny, Vol XLIII (3-4) (1996) 183- plátano en Canarias y las expectativas del agricultor 190. sobre la ayuda compensatoria, Revista Española de [28] W.C. Pye and T.A. Atchison, An algorithm for Estudios Agrosociales y Pesqueros, nº 194, 2002, pp. the computation of higher order G-transformation, 127-146. SIAM J. Numer. Anal. 10 (1973) 1-7. 6 [29] G.C. Reinsel, Elements of Multivariate Time [32] G. C. Tiao and R.S. Tsay, Model Specification Series Analysis (Springer Verlag, New York, 1993). in Multivariate Time Series, J. Roy. Statist. Soc., [30] P.M. Robinson, Distributed Lag Approximation Series B 51 (1989) 157-213. to Linear Time-Invariant Systems, The Annals of [33] R.S. Tsay, Model Identification in Dynamic Statistics 7 (3) (1979) 507-515. Regression (Distributed Lag) Models, J. Bus. Econ. [31] H. Rutishaüser, Der Quotienten-Differenzen Statist. 3(3) (1985) 228-237.

Algoritmus (Birkhäuser-Verlag, Basel, 1957). [34] P. Wynn, On a device for computing the em(sn) transformation, MTAC, 10 (1956) 91-96.

Figure 1

Figure 2 3 7 2 6 5 0 4 3 8 7 1 6 8 1 9 2 6 9 3 4 4 5 5 6 7 7 7 8 8 9 0 0 9 9 9 9 9 9 9 9 9 9 0 0 9 9 9 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

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Producción (* Datos 1998, 1999) M. Clásico - Escenario 2 M. Clásico - Escenario 1 M. Expectativas - Escenario 2 M. Expectativas -Escenario 3 M. Expectativas - Escenario 1 M. Expectativas - Escenario 4