Some Algorithms to Identify Rational Structures in Stochastic Processes with Expectations

This research was concerned with the identification of rational models in stochastic processes and we use the Padé-Laurent Approximation to identify a Transfer-Function Model with Expectations. We mention the T-table method and focus in studying the generalised epsilon-algorithm, emphasizing the main role of the statistical significance of their numerical entries. Empirical work is made for non-causal TF models using both simulated and economic real data.


INTRODUCTION
Over the last two decades, several studies have considered the use of rational approximation and related techniques in stochastic processes. Some of these techniques (e.g. [5,2,14,20,8] ), which are closely related to the Padé Approximation and Orthogonal Polynomials (e.g. [1,6] ), constitute useful tools for modelling time series and computing the orders of their rational structures.
Some interesting research provide reliable methods to specify the causal linear Transfer-Function (TF) model, a special case of VARMA models that constitutes one of the most frequent dynamic specification in time series modelling (e.g. [5,14,13,19] ).
The inclusion of expectations for the input variables in this type of models implies considering to non-causal models proposed [10] that we present in Section 2. This approach has provided not only a general framework to study dynamic relationships between variables with expected values but also extending the identification methods that have been proposed for classic causal models. The methodological basis which underlies this approach is the Padé-Laurent Approximation [7] . In this context we deal with the Toeplitz and the generalised epsilon methods. Section 3 is concerned with the generalised epsilon algorithm to identify a TF model with expectations. Also, the statistical significance of numerical entries in the algorithm is introduced analogously as in the classical case.
Empirical work is made using SCA package [15] , MATHEMATICA Software [20] and FORTRAN Programming. We contribute with numerical findings for non-causal TF models. Empirical results point out the role of the statistical significance for the epsilonalgorithm to get parsimonious formulations and reduce possible competing models to only a few for further testing.
In an economic context, we illustrate this proposal for studying some consequences in the banana production's evolution in the Canary Islands, the main banana producer area in European Union. The research concludes with the most relevant conclusions and some open questions of interest.

THE NON-CAUSAL TF MODEL
A TF model with expectations [10] for a bivariate and stationary process with y t onedimensional and ' * ** ' t t t x (x , x ) = of dimension m can be expressed as follows Now y t (output variable) is explained by two terms: a systematic one, which is described by one or more independent variables x it * (real data) and that includes x it ** (expected values) that can follow or not the same distribution and a non-systematic component which is described by an ARMA process.
The existence of a dynamic, one-way, causal relation x it * ,x it ** →y t is assumed. Therefore, a unidirectional relationship is given on y t by both contemporary and lagged effects of the input variables but also by the expected values described by x it ** . In the case of a single input variable and defining L j (x it ) = x it-j ∀j∈ Z, a closed form for the TF model with expectations can be expressed as denotes the IRF, now the double infinite series of weights in Z. A finite representation for v (L) with orders (p+a,s+b) and (-d,r) could be given by the expression In this finite structure for V(L), W p+a,s+b (L) and Ψ -d,r (L) are Laurent polynomial operators in L.
The lagged and lead effects on y t due to changes in x t appear expressed in W p+a,s+b (L) with a<0 and b≥0 respectively: It is considered that W p+a,s+b (L) and Ψ -d,r (L) have no common factors and model stability is guaranteed in terms of the roots for the characteristic polynomial Ψ -d,r (L).
In accordance with the guidelines proposed by the Padé-Laurent Approximation for studying doubly infinite series, a TF model with expectations can be identified by using the Toeplitz method [10] which generalises the corner method [2,14,20] given for causal models.
Therefore, a similar method to that one described for the causal model can now be used. For instance, given the sequence of weights j Z v ) ∈ ≡ and of relative ones Toeplitz determinants This theoretic characterization can be clearly displayed in a tabular form (T-table) in Table 1.
This proposal extends the well-known classic approach to carry out the identification of a causal TF model starting from the sequence x is not considered. In this particular case, another possible numerical method to provide a rational characterization is the epsilonalgorithm [12] . The study of the statistical significance in the epsilon-algorithm can be seen in [3] and [12] . Other methods have recently been proposed by [11] .
Some adequate transformations in the sequence of relative weights could be necessary to avoid computational instability in some specific cases.
Next, we refer to the generalised epsilon-algorithm in a non-causal context and contribute with by studying the statistical significance of their numerical entries. Table 1: T-table   g "0" denotes a significantly null element

THE GENERALIZED EPSILON-ALGORITHM AND ITS STATISTICAL SIGNIFICANCE
The epsilon algorithm constitutes an alternative method to identify a causal TF model. This iterative procedure can be derived from the cross rule between five adjacent Padé Approximants (see [6] ). If  Table 2: Epsilon-table  Col  - With all these values we can construct a new table, the Asymptotic Standard Deviation Table, for the epsilontable containing  Consequently, we can consider the value n 2k ε in Table 2 as a significantly null element if the n 2k n 2k ( )v ar( ( )) ε η ε η value associated is minor than a given critical value (1.28 or a greater one).

EMPIRICAL RESULTS
Empirical work is made using MATHEMATICA software to estimate weights j j 9...9v (v ) =− = by OLS method. Having computed the sequence of relative weights ( j j 9...9( ) =− η = η ) and the variance-covariance matrix, the generalised epsilon algorithm was implemented using FORTRAN programming language. In order to illustrate the generalized epsilon-algorithm and the role of the statistical significance to confirm the identified patterns, numerical results were obtained for the average weights.
with orders (-2,1,0,1,-1,2). In order to obtain asymptotic results, 200 dataset were simulated with 130 data each one. First data were not considered in order to avoid numerical instability. Therefore, a 113 effective data series was considered for each dataset. The epsilon and asymptotic standard deviations values for the average over 200 simulations with the statistical significance of their numerical entries are shown in Table 3 and 4 respectively.
It can be also concluded that as the number of simulations increases the results are clarify still more. From the perspective of real data modelling it frequently happens that according to certain critical values, the statistical significance table reduce possible competing patterns to only a few for further estimating and testing. Example 4.2 multivariate models to banana from canary islands: production, income and cost: A three-dimensional relationship between Production, Income and Average Production Cost is studied, taking data for the period 1938-2002 in the Canary Islands. Banana production and farmer income historical series were taken from [4,18,16] . We had to consider data about water and fertiliser collected from the accounting of three representative private properties in the North of Tenerife. Linear interpolation was necessary to obtain a reasonable sample size [17] . Data are shown in Fig. 1, 2  and 3.
Some necessary logarithmic and first-differences transformations have been achieved to obtain a more homogenous variance in data and guarantee both stationary series. Let us define at time t, z t = (1-L)lnProd t , where Prod t is banana Production in thousands of tons; x t = (1-L) lnI t where I t is the farmer indexed Income (ptas/kg) (Index 1996 = 100), including the market price and the compensatory aid; y t = (1-L)lnCost t , where Cost is the indexed Average Cost (ptas/kg) (Index 1996 = 100). "ln" represents the logarithm operator.

Fig. 1: Banana production data
A multivariate causal TF model for production with two inputs (income and cost): Previous results and the other aspects about the relationship between Production and Income can be found in [9] . The existence of a one-way causal dynamic relation x t ,y t →z t and the non-presence of autocorrelation between the input series can be confirmed. Having studied the noise term structure and estimated the IRF for each input variable, the corner method and epsilon-algorithm lead finally to the model Outlier detection only produced one abnormal observation in 1955 and the re-estimated model is basically the same. Fig. 4 shows fitted models until 1993 and forecasted values considering the last observation as origin of the forecasted interval (Prediction 1) and respectively the last forecasted value as origin of the next forecasted point (Prediction 2). So, we see four lines in the Fig.: two lines for initial model and the other two for the model including the outlier effect. However, although the causal model is suitable for fitting available data it fails to provide a satisfactory response in terms of its predictive performance from 1993.
Empirical results show the farmer income is a more relevant variable for the banana farmer than Production Costs. That is why we concern with a bivariate Production-Income TF model which suggests including income expectations.

Non-causal TF model with income expectations:
Next we consider the dataset for production, income and also income expectations. A main date is 1993, when the European Union (EU) introduced the compensatory aid to guarantee the banana sector subsistence, a sector actually hard conditioned by a tariff only system for the banana imports in the EU.
Specifically, four theoretical hypotheses on the income expectations are considered using 1993 as the starting date of the compensatory aid, ' . The four series differ in how ** t x is generated and that includes income expectations from 1997 (Fig. 5). A sensitivity analysis is carried out considering next possible schemes. We observe that all identified models provide a satisfactory response from the perspective of fitting to banana output data. However, TF models with expectations offer better predictions in the short-term. This behaviour could be explained by the fact that the farmer plans banana output not only considering his previous earnings but also taking into account other main variables, such as his future income expectation in the next three or four years. The case 3 can be considered the least realistic but it shows the optimism vision of the banana grower. The case 1 is the most desirable, although it implies reducing banana output. However, it could help to prevent the overproduction that occurred in the final years of our survey period. As it can be deduced from Case 4, without the compensatory aid, the subsistence of the sector will be threatened. An approximately half of current production would be lost and serious economic, social and ecological consequences would result.

CONCLUSIONS AND OPEN QUESTIONS
This research points up the goodness of several numerical methods closely related to the Padé Approximation for providing suitable rational structures in the context of time series modelling. Here we refer to the generalised epsilon algorithm as an alternative to the T-table method in a non-causal TF model, both proposals linked to the Padé-Laurent Approximation. Special emphasis is given to the statistical significance as a generalization of several previous results in the field of time series analysis. These related techniques can be considered "feasible alternatives" to contrast and/or confirm previously identified models.
Empirical findings emphasize the role of the epsilon table to pick up dynamic structures and also the table of asymptotic standard deviations to confirm the identified patterns and reduce possible competing models to only a few for further testing. Obtained results also stand out the goodness in terms of their asymptotic results.
For future research topics, we point out that the correct generalization of these results to VARMA models is not evident. For the corner method, for example, consideration has to be given to the rank of the matrices and not the determinants. The generalization of other numerical methods such as the rs and q-d algorithms to VARMA models has not yet been considered. It would also be of interest to consider the generalization of these numerical algorithms to noncausal contexts, considering the TF models with expectations.