時間序列在過去二十年當中，受到熱列地討論，而絕大多數的文獻都是研

究線性時間序列模式。但在現實生活中，很多時間序列並不符合線性的假

設，因此近十年來很多學者致力研究非線性時間序列模式。其中有一種雙

線性模式，因其性質與線性模式類似，故引起了廣泛注意。在本篇文章中

我們是採用Subba Rao 和Gabr(1984)提出的迭代等式以及高斯－賽德迭代

法估計參數，再配合 Subba Rao(1981)提出的巢狀搜尋程序，來選取雙線

性模式的階數。將其選模結果與AIC、BIC以及修正後的 PKK 選模法比較

。

Bhaskara Rao, [vl., and Subba Rao, T ., and Walker, A.IVl.(1983), "On the
Existence of some Bilinear Time Series Models," Journal of Time Series
Analysis, 4,95-110.
Brockett, R.W.(1976), "Volterra Series and Geometric Control Theory," A•uto'matica,
12, 167-176.
Granger, C.W.J.,and Andersen, A.P.(1978),An Introduction to Bilinear Time
Series Analysis) Vandenhoeck and Ruprecht: Gottingen.
Haggan, V. and Ozaki) T. (1981) , " rvIodelling Non-linear RanclOln Vi brations
Using an Amplitude-dependent Autoregressive Tirne Series Model,"
Biometrika, 68, 189-196.
Koreisha, S., and Pukklia, T.(1990), "A Generalized Least -Squares Approach
for Estimation of Autoregressive Moving-Average Models," J ournal
of Time Series Analysis, 11, 139-151.
Eumar, K. (1986), "On the Identification of some Bilinear Time Series rdodels,"
Journal of Ti'me Series Analysis, 7, 117-122.
Li, W.K. (1984), "On the Autocorrelation Strnct ure and Identification of
some Bilinear T ime Series," Jo urnal of T i'me S eries Analysis, 5 172-
181.
Liu, J., and Brockwell, P,J.(1988), "On the General Bilinear Time Series
Ivlodels," Journal of Applied Probability, 25, 553-564.

Pham Dinh, T., and Tran, L.(1981), On the First Order Bilinear Time
Series Moclels," Journal of Applied Probability, 18, 617-627
Priestley, M.B.(1988),Non-Linear and Non-Stationary Time Series Analysis)
Academic Press: London.
Pukkila, T.M., and Koreisha, S., and Kallinen, A.(1990), :'The Identification
of ARMA Models," Biornetrika, 77, 537-548.
Pukkila, T,IvI., and Krishnaiah , P.R.(1988), "On the Use of Autorgressive
Order Determination Criteria in Univariate "White~ oise Tests," IEEE
Transaction, on Aco'lLstics) Speech, Signal Processing, 36, 764-774.
Quinn, B. G. (1982), " Stationarity and Invertibility of Simple Bilinear Mod-els, Stochastic Processes and their Applications, 12,225-229.
Subba Rao, T. (1981), "On the Theory of Bilinear Time Series Model," Jour-nal of The Royal Statistical Society Series B ; 43, 244-255.
Subba Rao, T., and Eduarda A. da Silva, M.(1992), "Identification of Bilinear
Time Series Models," Statistica Sinica, 2, 479-494.
Subba Rao, T., and Gabr, M.M.(1980), 'A Test for Linearity of Stationary
Tinle Series ," Journal of Time Series Analysis, 1, 145-158.
Subba Rao, T., and Gabr, }VLNI.(1984),An Introd'uction to Bispectral Analysis
and Bilinear Time Series NIodels) Springer-Verlag: Berlin.
Tong, H., and Lim, K.S.(1980), "Threshold Autoregression, Limit Cycles
and Cyclical Data," Journal of The Royal Statistical Society Series
B, 42,245-2 92.
Tsay, R.S .(1986), "Nonlinearity Tests for Time Series," Biometrika, 73, 461-
466.