In time series modelling, several authors have considered the use of rational approximation theory in econometric modelling. Consideration is given to both the univariate case (Beguin et al. [2]) and the multivariate case (Berlinet and Francq [3] and Reinsel [11]). Some interesting results have been given for the specific case of a transfer-function model (González et al. [7] and Lii [10])

In this context, several techniques closely related to the Padé approximation and orthogonal polynomials (Baker and Graves-Morris [1] and Brezinski [5]) have been proposed to identify possible rational structures. Since the covariance structure of the underlying processes exhibits features related to the order of the models, it is possible to use certain numerical algorithms (corner method, epsilon-algorithm, rs-algorithm, and qd-algorithm) to estimate the unknown orders from observations and expectations. In particular, these methods have been proposed for the study of economic data in different contexts (financial, marketing, farming) (González and Gil [8] and González et al. [9]).

This paper is the continuation of previous papers which are concerned with illustrating the application of certain numerical methods for identifying certain rational structures associated with data series.

Special emphasis is given here to the study of the statistical significance of two of these numerical methods, namely, the rs-algorithm and the qd-algorithm in terms of their asymptotic standard deviations.

Empirical work is carried out in the context of the Box-Jenkins [4] guidelines. Both proposals are illustrated for the univariate and multivariate case, considering a simulated ARMA model, a simulated transfer-function model with two inputs, and two economic applications, one for the series M given by Box and Jenkins [4] and Tsay [12] and the second one, the volatility series in the Spanish market modelled by Gil and Alegría [6] and González and Gil [8].

Empirical findings emphasize the role of the statistical significance for the numerical values in the aforementioned algorithms. In general, different possible models will be obtained depending on certain critical values.

1

G.A. Baker Jr. and P. Graves-Morris, "Padé Approximants", Encyclopaedia of Mathematics and its Applications, 53, Cambridge University Press. 2nd edition, 1996.

2

J.M. Beguin, C. Gourieroux and A. Monfort, "Identification of a Mixed Autoregressive-Moving Average Process: The Corner Method in Time Series", O.D. Anderson (ed.), North-Holland, Amsterdam 423-436, 1980.

3

A. Berlinet and C. Francq, "On the Identifiability of Minimal VARMA Representations", Statistical Inference for Stochastic Processes 1, 1-15, 1998. doi:10.1023/A:1009955223247

4

G.E.P. Box and G.M. Jenkins, "Time Series Analysis: Forecasting and Control", Revised Edition, Holden Day, San Francisco, 1976.

5

C. Brezinski, "Padé-Type Approximants and General Orthogonal Polynomials", Birkhäuser, Basel, 1980.

6

C. Gil and R. Alegría, "An Application of Padé Approximation to Volatility Modelling", International Advances in Economic Research 5(4), 446-465, 1999. doi:10.1007/BF02295543

7

C. González, V. Cano and C. Gil, "Comparación de algoritmos para la identificación de una función de transferencia: una generalización al caso de varios inputs", Rev. Española de Economía, Segunda época 10, 163-175, 1993.

8

C. González and M.C. Gil, "Padé Approximation in economics", Numerical Algorithms 33, 277-292, 2003. doi:10.1023/A:1025580409039

9

C. González, M.C. Gil and C. Pestano, "Some numerical methods of rational characterization in causal time series models", WSEAS Transactions on Mathematics, 1-4, 24-30, 2006. doi:10.1090/S0002-9947-06-04038-4

10

K. Lii, "Transfer Function Model Order and Parameter Estimation", Journal of Time Series Analysis 6 (3), 153-169, 1985. doi:10.1111/j.1467-9892.1985.tb00406.x

11

G.C. Reinsel, Elements of Multivariate Time Series Analysis, Springer Verlag, New York, 1993.

12

R.S. Tsay, "Model Identification in Dynamic Regression (Distributed Lag) Models", J. Bus. Econ. Statist. 3(3),228-237, 1985. doi:10.2307/1391593

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