由於時間序列在不同範疇的廣泛應用，許多實證結果已明白指出時間序列

資料普遍地存在非線性(nonlinearity)，使得非線型方法在最近幾年受到

極大的重視。然而，對於某些特定的非線型模式，縱然現在已有學者提出

模式選取之檢定方法，但是它們的模式階數確認問題至今卻仍無法有效率

地解決，更遑論得到最佳的模式配適與預測結果了。所以，我們試圖利用

一已於其他科學領域成功應用之新技術──神經網路，來解決非線型時間

序列之預測問題，而我們之所以利用神經網路的原因是其多層前輸網路是

泛函數的近似器(functional approximator)，對任意函數均有極佳之逼

近能力，使我們免除對時間序列資料之屬性(線性或非線性)作事先檢定或

假設的必要。在本篇論文中，我們首先建構15組雙線型時間序列資料，然

後對於這些數據分別以神經網路與自我迴歸整合移動平均(ARIMA) 模式配

適。藉著比較兩者間的配適與預測結果，我們發現神經網路對於預測非線

型時間序列是較具有穩健性。最後，我們以台幣對美元之即期匯率作為我

們的實證資料，結果亦證實了神經網路對於預測一般經濟時間序列亦較具

穩健性。

With rapid development at the study of time series, the

nonlinear approaches have attracted great attention in recent

years. However, there are no efficient processes for the

problem of identification to many specifically nonlinear models

. Even if many testing methods have been proposed, we still

can not find the best fitted model and obtain the best forecast

performance. Hence, we try to solve the forecast problems

by a new technique -- neurocomputing, which has been

successfully applied in many scientific fields. The reason why

we apply the neural networks is that the multilayer feedforward

networks are functional approximators for the unknown function.

In this paper, we will first construct several sets of bilinear

time series and then fit these series by neural networks and

ARIMA models. In this simulation study, we have found that the

neural networks perform the robust forecast for some nonlinear

time series. Finally, forecasting performance with favorable

models will also be compared through the empirical realization

of Taiwan.

[1] Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis: Fore-casting and Control. 2nd ed. San Francisco : Holden-Day.
[2] Brockett,R.W.(1976).Volterra series and geometric control theory. Au-tomatica, 12. 167-176.
[3] Chan, W.S. and Tong, H. (1986). On test for non-linearity in time series analysis. J. Forecasting, 5, 217-28.
[4] Cynbento, G., (1989). Approximation by superposition of a sigmoidal function, Mathematics of Control, Signals and Systems, 2, 303-314.
[5] De Gooijer, J.G. and Kumar, K.(1992). Some recent developments in nonlinear time series modelling, testing and forecasting. International Journal of Forecasting, 8, 135-156.
[6] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987-1008.
[7] Funahashi, K. I., (1989). On the approximate of continuous mappings by neural networks, Neural Networks, 2, 183-192.
[8] Granger, C.W.J. and Anderson, A. P. (1978). An Introduction to Bi-linear Time Series Models. Vandenhoeck and Ruprech, Gottingen.
[9] Granger, C.W.J. (1991). Developments in the nonlinear analysis of economic series. Scand. J. Of Economics. 93(2), 263-276.
[10] Grosberg, S. (1988). Studics of Mind and Brain: Neural Principles of Learning, Perception, Development, Cognition and Motor Control. Boston, MA: Reidel.
[11] Guegan, D. and Pham, T.D. (1992). Power of the score test against bilinear time series models. Statistica Sinica, Vol. 2, 1, 157-169.
[12] Hecht-Nielsen, R., (1989). Neurocomputing, IEEE Spectrum, March, 36-41.
[13] Hinich, M. (1982). Testing for Gaussianity and linearity of a stationary time series. J. Time series Analysis, Vol.3, No.3, 169-76.
[14] Kolen, J. F. and Goel, A. K. (1991). Learning in parallel distributed processing networks: computational complexity and information con-tent. IEEE Transactions on Systems, Man, and Cybernetics, 21, 2, 359-367.
[15] Kosko, B. (1992). Neural Networks for Signal Processing, Prentice Hall, Englewood Cliffs, NJ.
[16] Lapedes, A., and Farber, R., (1988). How Neural Nets Work. The-oretical Division. Los Alamos National Laboratory Los Alamos, NM 87545.
[17] Luukkonen, R., Saikkonen P. and Terasvirta, T. (1988). Testing lin-earity against smooth transition autocorrelation models. Biometrica, 75, 491-500.
[18] McKenzie, E. (1985). Some simple models for discrete variate time series. In Time Series Analysis in Water Resources. (ed. K. W. Hipel), 645-650, AM. Water Res. Assoc.
[19] Priestley, M. B. (1980). State-dependent models: a general approach to nonlinear time series. J. Time Series Anal. 1, 47-71.
[20] Saikkonen, P. and Luukkonen, K. (1988). Lagrange multiplier test for testing non-linearities in time series models. Scand. J. of Statistics, 15, 55-68.
[21] Saikkonen, P. and Luukkonen, K. (1991). Power properties of a time series linearity test against some simple bilinear alternatives. Statistica Sinica, Vol. 1, 2, 453-464.
[22] Subba Rao, T. and Gabr, M. M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in statistics, Springer- Verlag, London.
[23] Tjoostheim, D.(1986). Some doubly stochastic time series models J. Time Ser. Analysis, 7, 51-72.
[24] Tong, H. And Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data. J. Roy. Statist. Soc. Ser. B, 42, 245-292.
[25] Tsay, R. S. (1989). Testing and modeling threshold autoregressive pro-cesses. Journal of the American Statistical Association, 84, 231-240.
[26] Tsay, R. S. (1991). Detecting and modeling nonlinearity in univariate time series analysis. Statistica Sinica, Vol. 1, 2, 431-51.
[27] Weiss, A. A. (1986). ARCH and bilinear time series models: compari-son and combination. J. Business Economic Statistics. Vol. 4, No. 1, 59-70.
[28] Wu, B., Liou, W. And Chen, Y. (1992). Robust forecasting for the stochastic models and chaotic models. J. Chinese Statist. Assoc. Vol.30, No. 2, 169-189.
[29] Wu, B. And Shih, N. (1992). On the identification problem for bilinear time series models. J. Statist. Comput. Simul. Vol.43. 129-161.