Robustness of Forecasting for Autoregressive Time Series with Bilinear Distortions

There is a growing interest in the investigation of nonlinear time series models that is caused by nonlinearity of real processes. This paper contributes to the development of the particular nonlinear time series model analysis, the bilinear time series model. Bilinear time series model was proposed by Granger and Andersen in 1978 (Fan and Yao, 2003; Granger and Andersen, 1978) as an alternative to the linear time series model. The model gives a possibility to describe “sufficiently large stochastic outbursts” that often appear in applications to seismological and financial data analysis. Since the time of introduction, quite a lot of researches has been carried out in the investigation of the bilinear model; stationarity, casuality, invertibility conditions have been found, moreover, parameter estimation techniques, tests for bilinearity and other solutions have been presented and studied (Granger and Andersen, 1978; Rao and Gabr, 1984; Terdik, 1999). This paper points at the problem of the bilinear time series BL(p,0,1,1) prediction. The forecasting statistic of the well-known linear autoregressive model AR(p) is applied; its robustness under bilinear distortions is studied. The paper is organized according to the following structure. In Section 2 the bilinear model BL(p,0,1,1) and the corresponding linear model AR(p) are defined and some their properties are listed. Further, the linear forecasting for bilinear time series is considered in Section 3, and its robustness is studied in Section 4. Section 5 presents the results of numerical experiments.


Introduction
There is a growing interest in the investigation of nonlinear time series models that is caused by nonlinearity of real processes.This paper contributes to the development of the particular nonlinear time series model analysis, the bilinear time series model.
Bilinear time series model was proposed by Granger andAndersen in 1978 (Fan andYao, 2003;Granger and Andersen, 1978) as an alternative to the linear time series model.The model gives a possibility to describe "sufficiently large stochastic outbursts" that often appear in applications to seismological and financial data analysis.Since the time of introduction, quite a lot of researches has been carried out in the investigation of the bilinear model; stationarity, casuality, invertibility conditions have been found, moreover, parameter estimation techniques, tests for bilinearity and other solutions have been presented and studied (Granger and Andersen, 1978;Rao and Gabr, 1984;Terdik, 1999).
This paper points at the problem of the bilinear time series BL(p,0,1,1) prediction.The forecasting statistic of the well-known linear autoregressive model AR(p) is applied; its robustness under bilinear distortions is studied.
The paper is organized according to the following structure.In Section 2 the bilinear model BL(p,0,1,1) and the corresponding linear model AR(p) are defined and some their properties are listed.Further, the linear forecasting for bilinear time series is considered in Section 3, and its robustness is studied in Section 4. Section 5 presents the results of numerical experiments.
2 Models BL(p, 0, 1, 1), AR(p) and their Properties The bilinear model BL(p, 0, 1, 1) is a simple nonlinear modification of the well-known and intensively used linear autoregressive model AR(p) of the order p.Therefore, it is quite reasonable to consider these models together and apply some earlier obtained results of linear model AR(p) to nonlinear model BL(p, 0, 1, 1).
Further only stationary models AR(p) (1) and BL(p, 0, 1, 1) (2) will be considered.The conditions (Anderson, 1994;Rao and Gabr, 1984) provide their stationarity respectively.Here the matrices A and B are and ⊗ means the Kronecker product.Note that the model (1) defines zero-mean time series, while the model (2) has the mean equal to Let's define the second order moments of AR(p) and BL(p, 0, 1, 1) time series respectively: It is known (Anderson, 1994) that the moments of the AR(p) process c 0 (s), s ∈ Z, satisfy the Yule-Walker system of equations.The second order moments of bilinear time series can be found as a solution of the Yule-Walker-like linear system of equations as well.
Introduce the matrix notation: Lemma 1.The second order moments of the stationary bilinear time series BL(p, 0, 1, 1) satisfy the following equations: or in the matrix form: Proof.Calculate mathematical expectation of both sides of the difference equation ( 2) multiplied by x t−s : In particular, when s = 0, Because of the independence property for The proof of the last statement on E{x 2 t u t } is based on the equation By substituting the values E{x t−k x t u t }, k = 0, . . ., p, into (3), we obtain the statement of the lemma at first in scalar form and then represent it in matrix form.
Together with exact values of the second order moments, the asymptotic expansion at β → 0 is useful.
Austrian Journal of Statistics, Vol. 37 (2008) For further analysis, it is more convenient to use the matrix form of the models AR(p) and BL(p, 0, 1, 1).The following lemma gives this representation.
Introduce the notation: The stochastic difference equations ( 1) and ( 2) can be represented in the matrix form: (5) Proof.The equations ( 4) and ( 5) follow from the difference equations ( 1), ( 2) and the notations that has been introduced above.

Autoregressive Forecasting of Bilinear Time Series
The significant and complicated problem is to find a forecasting statistic xT+τ as an estimator of the future value x T +τ of the time series optimal w.r.t.some criterion by the observed history X = {x 1 , . . ., x T } of length T .Here τ , τ = 1, . . ., p, is a horizon of prediction.As an optimality criterion we will use a set of scalar risks For the time series corresponded to the model AR(p) (1) the problem of forecasting w.r.t. the mean-square risk optimality criterion is solved.The forecasted values, which provide minimum of defined earlier risks, can be found from the following system of equations (Anderson, 1994): where X0 T +p T +1 = (x T +1 , . . ., xT+p ) is the p-vector of forecasted values, X 0 T T −p+1 = (x T −p+1 , . . ., x T ) is the p-vector of the last p observed values, matrices S 1 , S 2 ∈ R p×p have been defined earlier.
Theorem 1.The forecasting statistic defined by ( 6) for the stationary model AR(p) (1) or ( 4) is unbiased; moreover, its matrix and scalar risks are defined by the following formulas (Anderson, 1994): Let us assume a low level of bilinear distortions in the model BL(p, 0, 1, 1) that means the coefficient β to be sufficiently small.So it is reasonable to use the autoregressive forecasting statistic (6) to predict future values of bilinear time series.Let us define the forecasting statistic from the equation equivalent to (6): Theorem 2. The forecasting statistic ( 7) for the stationary model BL(p, 0, 1, 1) (2) or ( 5) is biased: Proof.From ( 5) and ( 7) . Therefore, after calculation of mathematical expectation we obtain that the forecasting statistic ( 7) is biased.
However, it needs to be noted that for the low level of bilinear distortions β the forecasting statistic (7) tends to be unbiased.
Theorem 3.For the stationary model BL(p, 0, 1, 1) defined by ( 2) or ( 5) the autoregressive forecasting statistic (7) has the following matrix mean-square risk: where I −1,0,1 p ∈ R p×p is the matrix with identities only under and above diagonal, while other elements are zeros.
Proof.From ( 5) and ( 7) By calculating the mathematical expectation of every element in the right part of this expression and relying on Lemma 1, due to |S 1 | = 0, we get the required statement of the theorem.It is obvious from the Figures 1, 2 that the risk starts to increase very rapidly after β comes over the δ-critical level of distortions β + (δ, τ ) computed by Theorem 4 (Table 1).Figure 3 shows that for small bilinearity level β the main terms of asymptotic expansion provide us with quite accurate approximation of exact risks.

Conclusion
The paper contributes to the sensitivity analysis of the mean-square risk of the so-called "autoregressive forecasting statistic" (constructed for the hypothetical model AR(p)) in the situation where the observed time series satisfies really the model BL(p,0,1,1) with a small bilinearity coefficient β.Exact values and asymptotic expansions (at β → 0) of matrix and scalar risks allow to find quantitative estimates of robustness and supply a statistician with the δ-critical (for the given δ > 0) bilinearity level β + (δ, τ ) satisfying the δ • 100%-admissible increment of the risk.Numerical results are in a good coincidence with the approximations of the risk generated by main terms of asymptotic expansions.

Table 1 :
The δ-critical level of distortions