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A Portmanteau Test for Self-Exciting Threshold Autoregressive-Type Nonlinearity in 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

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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, 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 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 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.

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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 , 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)

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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)14,-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 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. rmin+1 G rG n-p

That is, p* is the observed significance level of the test. The test rejects Ho at the a level of significance if p* < a. Step 1 of our procedure could allow for different choices for p. Although it is tempting to use an automatic procedure such as Akaike's information criterion, AIC, to choose an

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'optimal' order, such model fitting techniques would have to presume a linear model structure for the data. In fact varying p and d coupled with examination of the graphical displays of the cumulative sums can help identify these parameters.

3. SIMULATION RESULTS 3 1. Empirical significance levels In this section we check the empirical significance levels of our proposed identification portmanteau, P, statistic based on cumulative sums for finite samples. We simulated 1000 realizations of linear autoregressive processes of order 1, for values of the autoregressive parameter, 4 in the range -1 - 4 - 1 for sample size 50 and 100. We chose rmin= 6 with values of the autoregressive order, p and the lag d required in the calculation of P, to be 1 or 2. We found a tendency for significance levels to be too high for negative 4 values and too low for positive 4. Davies & Petruccelli (1986) show that there is a similar effect with empirical significance levels for Keenan's F statistic although overall, P is closer to nominal levels than is F. Further details are available from the authors in an unpublished report.

3 2. Bilinear processes We considered the bilinear model Y, = (4 + Pa,) Yt-1 + a,, where a, - N(0, 1) and simulated this process 1000 times each, for values of 4 of 0 5 and 0 9, with -1 <,B < 1. The power curves for both F and P are plotted in Fig. 1. We used sample sizes 50 and 100 at three levels of significance, 10%, 5% and 1%. In line with the suggestions of Keenan (1985), we chose an autoregressive approximation of order M = 6 to calculate F. For 4 = 0.5, and at sample size 50 and 100, P marginally outperforms F for moderate to large bilinearity, as reflected by the , parameter being above approximately Qs4. For ,B < 0 there appears to be little to choose between the two tests at any significance level. For 4 = 0*9, P outperforms F for 3 > 0 7 at both sample sizes. For 0 < ,B < 0 7, F is rather better at detecting bilinearity. For ,3 < 0 and sample size 50 there is little to choose between the statistics. For sample size 100, P is outperformed by F for ,B < 0. From this limited study, it appears P, which was designed specifically for another type of non- linearity, can be better than F at detecting the more extreme cases of bilinearity as measured by the size of the parameter ,B. We did not consider the diagnostic statistic Qaa of McLeod & Li (1983), since Davies & Petruccelli (1986) show that F is better at detecting both bilinear and threshold autoregressive time series.

3-3. Self-exciting threshold autoregressive processes We restrict our consideration to the process (2 1) in which p = 1, 4(1) = 4(P2) = 0 and r, = 0. For brevity we write ()(1) = (), and 4()2) = 42. Hence model (24) becomes

={4)Yt-l+at (Yt_<0), (3.1)

The stationarity region of (3.1) was shown by Petruccelli & Woolford (1984) to be 41 < 1, 42 < 1 and 'p142 <1. We simulated processes from within that region and investi- gated the power of the P statistic to reject linearity of each process. The graphs in Fig. 2

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(a) n =50, b = 0*5 (b) n =50,= 0*9

Power Power

50 5

30 30~ ~~~~~~~~~~6 -1-0 0 1-0 -1*0 0 1.0

(c) n= 100,+=0.5 (d) n= 100,4=0.9 \ ~~~~Power , a Power\'0

X 0 1''' f\ Sof~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\I

-140 0 1, 0 4-1r 0 1-0

pormaneaustaistc, dote30 line, forw bilnea f noesY=+,3tY_+ 30 th 100 0 ra0iuaios 9o o 50 /_..*R, - 50R

e stati i20 a 20

ii, ~ ~ ~~~~~~~ ~10 ;/ thresold utorgresivePower prdcsltl6hng0ntepwro tets.Hwvr

The~~~~~~5 reaiedces npwrwihi beval nFg 0 o 1 05ad+ /1 ~ ~ ~~~1 ls -1.0 0 1.0 -1.0 0 1*0 Fig. 1. Empirical power curves ()for Keenan's F statistic, shown by continuous line, and cumulative sum portmanteau statistic, dotted line, for bilinear models Y'I = (q5 + fla,) Y, -1 + a, ; 1 000 simulations; from top to bottom the three sets of lines correspond to 10%, 5 %, 1% significance levels respectively. are plots of the power curves for rmin = 6, different values of p and d, and sample size 100. We see that misspecifying a higher autoregressive order p than that of the true threshold autoregressive model produces little change in the power of the test. However, misspecifying the lag, d, in the calculation of the P statistic causes a great loss of power. The relative decrease in power which is observable in Fig. 2 for k1 = -0-5 and 42 close to -2 -O and p = 2, rmin = 6, d = 2 is explained by the fact that the process (3.-1) is near the nonstationary boundary 'k1& = 1. 'A The graphs in Fig. 3 show some power curves for both the P statistic and the Keenan F statistic for processes within the stationarity region of process (3.-1). From these graphs it is clear that for some processes the Keenan tF" statistic marginally outperforms the P

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(a) Power (b) Power (c) Power

W"~~~~~~~~~~' "-" I + ~~ ~~~"" I II

-2-0 -1 0 0 i-O -20 - -10 0 1.0 -20 -1 0 0 1.0

(d) Power (e) Power ()Power 4100 t s ' 100 1 00 - - -*4~0 -b '2 - 4)2

50- +" 50 5"0~] b

-2-0 -1 0 0 1.0 -2-0 -1.0 0 1.0 -2-0 -1P0 0 1.0

50 .~' 50 50 2 (g) Power (h) Power (i Power - z - - .100 ----100 .100

l_ 02 - +2----2 2 -2-0 -1*0 0 1.0 -2-0 -1*0 0 1i0 -2-0 -1o0 0 1.0

Fig. 2. Empirical power curves )for cumulative sum portmanteau statistic for self-exciting threshold autoregressive model (3=1); n = 100; 1000 simulations; from top to bottom the three dotted lines correspond to 10%,) 5%, 1% significance levels respectively; (a) 01 = -0 5 (p = 1, rnin= 6, d = 1); (b) 01 = -0 5 (p =2, rmin = 61, d = 1); (c) 0, = -0 5 (p =2, rmin = 6, d = 2); (d) '01 = 0 (p = 1, rmin = 6, d = 1); (e) 0, = 0 (p =2, rmin = 63, d = 1); (f) 010 (p = 2,- rmin= 63, d =2); (g) 0 I= 0 5 (p = 1, rmin = 6, d = 1); (h) 0 =? 5 (p= 23, rmin = 63, d = 1); (i) +1 =0X5 (P =2, rmin =6, d =2).

There appears to be a definite region where the P statistic is inferior but it is our experience, as Fig. 3 illustrates, that where the Keenan statistic outperforms P the difference in performance is marginal whereas P very substantially outperforms the Keenan statistic over a wide range of parameter values. In an unpublished report we have applied P to 234 time series, the majority of which were real data collected from textbooks, journals, etc. Our objective was to determine the incidence of autoregressive threshold type nonlinearity in these series. There is at least some evidence to suggest that some real series are nonlinear and it may well be beneficial to reanalyse these data using self-exciting threshold autoregressive models. Also we found that nonstationarity can severely distort all tests for nonlinearity, so that we recommend appropriately differencing a series before proceeding to test for non- beaneficialtionooulinearity. The evidence reanlyeths is also data that usring other selfexcitimngea forms of transformation sthreshold bautoegressuuaive to cure problems models like heteroscedasticity may be necessary, although it is not clear how filtering and/or transfor- wetrectdaommnd appropritelye diseriencn ao seroninesr before proinfiaceigt testeorlon mation techniques affect types of nonlinearity in data. After suitable diff erencing and/or

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(a) 41 -0 5, n 50 (b) 4+=0, n= 50 (c) 41=05, n= 50 Power Power Power 100 100 100

80 280 2 - -80

60 ',\ 060

40 (f40 + 40

\

20

(d) 41 -0-5, n 100 (e) &=0, il=100 (f) o,=0-5, n=100

Power Power Power 100 100 0

.80 80 80

40404

\\ 20 20vX2

-2*0 -1*0 -2 0 10 -2-0 -1.0 -2 0 1.0 -2*0 -1P0 -2 0 2 1.0

Fig. 3. Empirical power curves (%) for Keenan's F statistic, shown by continuous line, and cumulative sum portmanteau statistic, dotted line, for self-exciting threshold autoregressive model (3 1); (p, rmin, d) = (1, 6, 1); 1000 simulations; from top to bottom the three sets of lines correspond to 10%, 5%, 1% significance levels respectively.

4. CONCLUSIONS In this paper we have proposed a portmanteau-type statistic to detect and identify self-exciting threshold autoregressive-type nonlinear time series. We compared, via simu- lation studies, the performance of the proposed statistic with a statistic proposed by Keenan (1985) on both bilinear and SETAR time series. For the bilinear series we found, in some cases, the Keenan statistic to be marginally more powerful, but for a broad range of stationary threshold autoregressive processes the proposed statistic outperformed Keenan's statistic on most occasions. Given that the relative strengths of these two statistics complement each other, it may be beneficial for an analyst looking for non- linearity in time series data to make use of both the Keenen F statistic and our proposed statistic. The main strengths of P is that for threshold autoregressive time series (i) it is an identification statistic and so does not depend on the 'optimal' model, (ii) plots and tests based on the cumulative sums could identify thresholds, (iii) the ability to vary the parameters p and d could help identify the autoregressive order and lag in the self- exciting threshold autoregressive process.

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We are now analysing 1001 series (Makridakis et al., 1984) to investigate further evidence of nonlinearity and hope to report our findings later.

REFENCES

Box, G. E. P. & PIERCE, D. A. (1970). Distribution of residual autocorrelation in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 70, 70-9. DAVIES, N. & PETRUCCELLI, J. D. (1986). Detecting nonlinearity in time series. Statistician 35, 271-80. ERTEL, J. E. & FOWLKES, E. B. (1976). Some algorithms for linear spline and piecewise multiple linear regression. J. Am. Statist. Assoc. 71, 640-8. FELLER, W. (1970). An Introduction to Probability Theory and its Applications, 2. New York: Wiley. GRANGER, C. W. J. & ANDERSON, A. P. (1978). An Introduction to Bilinear Time Series Models. Gottingen: Vandenhoeck and Ruprecht. GRANGER, C. W. J. & NEWBOLD, P. (1976). Forecasting transformed series. J.R. Statist. Soc. B 38, 189-203. HAGGAN, V., HERAVI, S. M. & PRIESTLEY, M. B. (1984). A study of the application of state-dependent models in nonlinear time series analysis. J. Time Series Anal. 5, 69-102. HAGGAN V. & OZAKI, T. (1981). Modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model. Biometrika 68, 189-96. HINICH, M. J. (1982). Testing for Gaussianity and linearity of a stationary time series. J. Time Series Anal. 3, 169-76. KEENAN, D. M. (1985). A Tukey nonadditivity-type test for time series nonlinearity. Biometrika 72, 39-44. LJUNG, G. M. & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 65,297-303. MAKRIDAKIS, S., ANDERSON, A., CARBONE, R., FILDES, R., HIBON, M., LEWANDOWSKI, R., NEWTON, J., PARZEN, E. & WINKLER, R. (1984). The Forecasting Accuracy of Major Time Series Methods. New York: Wiley. MARAVALL, A. (1983). An application of nonlinear time series forecasting. J. Bus. Econ. Statist. 1, 66-74. McLEOD, A. I. & LI, W. K. (1983). Diagnostic checking ARMA time series models using squared-residual autocorrelations. J. Time Series Anal. 4, 269-73. PETRUCELLI, J. D. & WOOLFORD, S. W. (1984). A threshold AR(1) model. J. Appl. Prob. 21, 270-86. PRIESTLEY, M. B. (1980). State-dependent models: A general approach to nonlinar time series analysis. J. Time Series Anal. 1, 47-71. SUBBA RAO, T. & GABR, M.M. (1980). A test for linearity of stationary time series. J. Time Series Anal. 2, 145-58. TONG, H. (1978). On a Threshold Model in Pattern Recognition and Signal Processing, Ed. C. H. Chen. Amsterdam: Sijthoff and Noordhoff. TONG, H. (1983). Threshold Models in Nonlinear Time Series Analysis, Lecture Notes in Statistics, 21, Ed. D. Brillinger, S. Fienberg, J. Gani, J. Hartigan and K. Krickeberry. New York: Springer-Verlag. TONG, H. & LIM, K. S. (1980). Threshold autoregression, limit cycles and cyclical data (with discussion). J.R. Statist. Soc. B 42, 245-92. WIENER, N. (1958). Nonlinear Problems in Random Theory. Cambridge, Mass: M.I.T. Press.

[Received October 1985. Revised January 1986]

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