Biometrika Trust A Portmanteau Test for Self-Exciting Threshold Autoregressive-Type Nonlinearity in Time Series Author(s): Joseph D. Petruccelli and Neville Davies Source: Biometrika, Vol. 73, No. 3 (Dec., 1986), pp. 687-694 Published by: Oxford University Press on behalf of Biometrika Trust Stable URL: https://www.jstor.org/stable/2336533 Accessed: 31-07-2019 20:19 UTC REFERENCES Linked references are available on JSTOR for this article: https://www.jstor.org/stable/2336533?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms Biometrika Trust, Oxford University Press are collaborating with JSTOR to digitize, preserve and extend access to Biometrika This content downloaded from 130.215.176.72 on Wed, 31 Jul 2019 20:19:11 UTC All use subject to https://about.jstor.org/terms Biometrika (1986), 73, 3, pp. 687-94 Printed in Great Britain A portmanteau test for self-exciting threshold autoregressive-type nonlinearity in time series BY JOSEPH D. PETRUCCELLI Department of Mathematical Sciences, Worcester Polytechnic Institute, Massachusetts 01609, U.S.A. AND NEVILLE DAVIES Department of Mathematics, Statistics and Operational Research, Trent Polytechnic, Nottingham NG1 4BU, U.K. SUMMARY A portmanteau test to detect self-exciting threshold autoregressive-type nonlinearity in time series data is proposed. The test is based on cumulative sums of standardized residuals from autoregressive fits to the data. Significance levels for the test under the hypothesis of linearity are obtain from the asymptotic distribution of the cumulative sums as Brownian motion. The performance of the test is evaluated for simulated data from linear, bilinear and self-exciting threshold autoregressive models. It is also compared with another test which has been suggested for detecting general nonlinearity. Features of the proposed test, which make it useful in identifying the autoregressive order and the lag in threshold models, are discussed. Some key words: Nonlinear time series; Portmanteau test; Self-exciting threshold autoregressive model. 1. INTRODUCTION Recently authors such as Tong (1978, 1983), Tong & Lim (1980), Granger & Anderson (1978), Priestley (1980) and Haggan & Ozaki (1981) have argued convincingly for the need for nonlinear models in time series analysis. The incidence of nonlinearity in a large sample of sets of real data is discussed by Davies & Petruccelli (1986). If, as appears to be the case, there is a need for nonlinear modelling in time series analysis, it certainly follows that there is a need for methods to ascertain when a series is nonlinear. One such method relies on the fact, inherent in the work of Granger & Newbold (1976), that, for a series Y, which is normal, Pk( Yt) = {Pk( Yt)}2, where Pk is the lag k autocorrela- tion. Granger & Anderson (1978) point out that departures from this relation could indicate nonlinearity. Maravall (1983) considers the possibility of examining.the sample autocorrelation functions of either the series itself and its squared values or the residuals from a fitted model and their squared values in order to detect nonlinearity. Davies & Petruccelli (1986) discuss this approach and some possible problems with it. McLeod & Li (1983) propose a portmanteau test, based on the autocorrelations of squared residuals from a linear fit, to detect nonlinearity in time series data. Their test statistic is analogous to the well-known statistics of Box & Pierce (1970) and Ljung & Box (1978). Significance levels are based on the asymptotic x2 distribution of the test statistic when the process is linear. This content downloaded from 130.215.176.72 on Wed, 31 Jul 2019 20:19:11 UTC All use subject to https://about.jstor.org/terms 688 JOSEPH D. PETRUCCELLI AND NEVILLE DAVIES Keenan (1985) takes a novel approach in devising a test for nonlinearity. He first assumes the series can be adequately approximated by a second-order Volterra expansion (Wiener, 1958). 00 00 00 Yt = O+ E Ouat-u + E E 0UVat_uat_v5 (1 1 U=-00 v=-00 u=-00 where {at} is a strictly stationary process. The second term on the right-hand side of (1-1) is analogous to an interaction term in regression analysis, and linearity of (1-1) is equivalent to this term being zero. To test whether this is the case, Keenan obtains an order M autoregressive approximation to the process (1.1) and calculates an F statistic which emulates Tukey's one degree of freedom for nonadditivity test in regression analysis. Spectral tests for nonlinearity (Subba Rao & Gabr, 1980; H-inich, 1982) are not considered here. The state dependent models described by Priestley (1980) contain many standard nonlinear models, such as threshold autoregressive, bilinear and amplitude dependent, as subclasses. The fitting procedure for state dependent models proposed by Haggan, Heravi & Priestley (1984) appear to indicate at the same time, using graphical methods, to which subclass the model belongs. In the present paper we introduce a portmanteau test designed to detect the specific class of self-exciting threshold autoregressive models. In ? 2 we describe the theory and implementation of the test, while in ? 3 we summarize results obtained in applying the test to simulated data from linear, bilinear and threshold autoregressive models. 2. A PORTMEANTEAU TEST FOR NONLINEARITY The proposed test is based on cumulative sums of standardized one-step forecast errors from autoregressive fits to the data. The idea for such a statistic in the piecewise linear regression context was developed by Ertel & Fowlkes (1976). We assume that, rather than following a piecewise linear regression model, the data follow the self-exciting threshold autoregressive model of order p, p Yt = +o)+5 I ( Yt-i + at (2.1) i=l if rj-1 < Yt-d d rj (=j =1, . ..., l), t=p+1, ..., n and d is a positive integer. Here the thresholds are -oo = ro < r, < ... < r, = 0o, and {at} is a sequence of identically indepen- dently distributed zero mean random noise terms with variance a2. In the discussion that follows we assume = 2, so that there is one nontrivial threshold. Let Y(i) (i = 1, ... n -p) be the ith smallest observation among { Yp+l-d, ..., Yn-d}. The expression in (2-1) can then be formulated as a finite autoregression in the Y(1). If the threshold lies between the m and (m + 1)th ordered Yt values, the complete pth order autoregression implied by (2.1) can be written as P 0o) ( 1)Y(i)+d-1+a(i)+d (i= 1, ... ., m), Y(i)+d = I- (2-2) This content downloaded from 130.215.176.72 on Wed, 31 Jul 2019 20:19:11 UTC All use subject to https://about.jstor.org/terms Test for nonlinearity in time series 689 If we proceed with successive autoregressions, of fixed order p, that use the first r values of the Y(i), for 1 < rmin - r - m, then the standardized one-step-ahead forecast errors should be roughly identically independently distributed with zero mean and unit variance. However, from (2.2), as r begins to exceed m, the nonlinearity of the process should cause systematic deviations in the forecast errors. Let the ordered autoregression implied by (2. 1) be represented by Y = X4 + a, where Y is a column vector that contain the Y(i)+d (i = 1,..., n -p), X is an (n -p) x (p + 1) matrix with first column containing unity and the remaining columns appropriate lagged Y(i)+d values, 4 is a column vector of the Oyi) (i = 0, 1, ... , p; j= 1, ..., 1) and a is a column vector of noise terms. In general our procedure is as follows. Step 1. Choose the order of autoregression, p, the lag d and rmin. Step 2. For rmin < r - n -p, find the multiple regression of the first r rows of Y on the first r rows of X and compute the successive one-step-ahead standardized forecast errors, Zr+i, say. Step 3. Form the cumulative sums r Zr= E zi (r=rmin+1 ..., n-p). i =rmin+ 1 The zi's may be calculated very efficiently using regression updating methods (Ertel & Fowlkes, 1976). The cumulative sums may be plotted sequentially to give a graphical method for detecting not only nonlinearity, but also, for threshold autoregressive models, the approximate location of thresholds. Standard tests may be employed to detect when the cumulative sums go out of control. The scheme of Ertel & Fowlkes (1976), for example requires a run in the sign of the zi's of sufficient length and a minimum increase in the cumulative sum over this span in order to conclude the process is no longer linear. To develop a portmanteau test for nonlinearity, we use an invariance principle for random walks (Feller, 1970, pp. 342-3). Let Tn= max lZrl. rmin+l1rGn-p Then as n -> oo, 00 pr{Tn/(n-p-rmin)1<t}->4,-1 Z (-)k(2k+ 1)-1 exp {-(2k+1) r2/(8t2)}. (2 3) k=O We consider the hypothesis that model (2.1) is linear Ho: +'l '-'(i = 0x 19 . .. 9 p) versus the alternative hypothesis that Ho does not hold. Under H0, (2.3) should hold approximately for moderate samples sizes. For a given series let 1 -p* denote the value computed from the right-hand side of (2 3) with t given by max IZrl/(n-p-rmin)2.