On the Asymptotic Standard Errors of Residual Autocorrelations in Nonlinear Time Series Modelling W. K. Li Biometrika, Vol. 79, No. 2. (Jun., 1992), pp. 435-437. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28199206%2979%3A2%3C435%3AOTASEO%3E2.0.CO%3B2-O Biometrika is currently published by Biometrika Trust.
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Biometrika (1992), 79, 2, pp. 435-7 Printed in Great Britain
On the asymptotic standard errors of residual autocorrelations
in nonlinear time series modelling BY W. K. LI
Department of Statistics, University of Hong Kong, Hong Kong
The asymptotic distribution of residual autocorrelations for some very general nonlinear time series models is derived. This includes the important class of threshold models. Consequently more accurate standard errors for residual autocorrelations can be obtained facilitating model diagnostic checkings in many situations. A small simulation result applying the methodology to threshold models is reported. Some key words: Asymptotic distribution; Nonlinear models; Residual autocorrelations; Standard errors; Threshold models.
Nonlinear time series analysis has in recent years attracted a lot of attention in the literature. Many classes of models have been proposed. Among these, the two most popular classes of models are the bilinear models (Granger & Andersen, 1978) and the threshold models (Tong, 1978; Tong & Lim, 1980). See the recent book by Tong (1990) for a more complete introduction to nonlinear time series modelling. However, as is well known, for residuals obtained from a fitted autoregressive moving average model the actual standard errors could be much less than l / J n (Box & Pierce, 1970; Ansley & Newbold, 1979). In nonlinear time series, the asymptotic distribution of the residual autocorrelations for all the major models has been unknown and consequently the value l / J n is only a rough guide in diagnostic checking. For example, Tong (1990, p. 324) suggested the use of l / J n but cautioned that this could be conservative especially for the first few lags. In the present paper, the distribution of the residual autocorrelations from a general nonlinear model is derived. This includes automatically the threshold autoregressive models as special cases. The asymptotic covariance matrix could be evaluated using sample averages instead of expected values. A similar approach has also been considered by Li (1991) for extensions of generalized linear models. A small simulation applied to threshold models shows that the standard errors thus obtained are much closer to the actual ones. The main result is given in 9 2. Application to threshold type nonlinear models is given in § 3. 2. THE M A I N RESULT Let {Xi)be a stationary and ergodic time series, with F, the a-field generated by {X,, Xi-, ,. . .). Let {Xi) satisfy the nonlinear model
where f is a known nonlinear function of past Xi's and 4 is a p x 1 vector of parameters. The function f is assumed to have continuous second order derivatives almost surely. The noise process {a,) is assumed to be independent, Gaussian with mean zero and variance a?.It is further assumed that (1) is invertible or equivalently {a,) is measurable with respect to F,.
Let the length of realization ke n. Let the lag k white noise autocovariance be Ck = 'C ata,-k/n ( k = 1,. .. , M ) and denote by Ck the corresponding residual autocovariances. The residuals {a^,) are asiumed to be from a least squares fit of (1) to {X,). Define the lag k residual autocorrelations to be ?k = e k / e o . Using a Taylor series expansion of ?k it can be shown that the asymptotic and therefore we can ignore in deriving the asymptotic distribution of ?k does not depend on by scaling. distribution of ?k.The result for F, will follow from that of The residual variance 62 is estimated Lt! rk = Ck/CO,r = (r, ,... , rM)' and ? = (PI, . .. , h)'. by Co. If {a,) have finite fourth order moments then it is well known that r J n is asymptotically normally distributed with mean zero and variance I,, where IMis the M x M identity matrix. Under regularity conditions as given by Klimko & Nelson (1978) the least squares estimator $ of 4 can be shown to be asymptotically normally distributed with mean 4 and covariance matrix at V-'/ n, where
eo
ek
eo
V = ~ { n -C ' (aa,/a4)(dar/a4)'). Denote f ( F , - , , 4 ) by f,-, . Suppose that E (af,-,/a4ar-,) exist for j = 1, .. ., M, and that corresponding sample averages converge in probability to the respective expected values. A sufficient condition for the latter would be the covariance between a,-,aft-l/a4 and a,,-,af,,-,/a4 goes to zero as It - t'l +a.This seems to be a reasonable assumption in practice. The foll~wingtwo lemmas follow (McLeod, 1978; Li, 1991) using Taylor series expansion of 'C a: and Ck. LEMMA1. The asymptotic cross covariance between J n ( $ - 4 ) and J n C = ( C , ,. . . , CM)' is equal to a; V-' J, where J = E{C af;-,/a~a,-,,. . . ,C af,-l/a4ar-M)n-1. Proof: This follows from the standard result
$ - 4 - ( C af,-l/a4af:-l/a4)-'(C LEMMA2. For large n,
2- C - J'($
-4).
Proof: This follows from a Taylor series expansion of
af,-,/a@,).
ekabout 4 and evaluated at $.
From these two lemmas and the martingale central limit theorem (Billingsley, 1961) we have the following theorem. THEOREM. The large sample distribution of u n is normal with mean zero and covariance matrix IM- u;~J'v-'J. Note that, for autoregressive moving average models, V and J can be evaluated in terms of 4. For nonlinear models closed form expressions for these quantities are usually unavailable. Our proposal here is to use observed quantities instead of the expectations. This is in some sense analogous to the use of observed rather than expected Fisher information (Efron & Hinkley, 1978). Simulation results reported in 9 3 suggest that this is a viable approach. RESULTS 3. SIMULATION A small simulation experiment was conducted to compare the asymptotic and the empirical standard errors of tkin threshold models. The design of the experiment is as follows. We considered a simple SETAR (2; 1 , l ) model X, = &X,-, + a, if X,-, > 0; and X, = 4;X,-, + a, otherwise, where {a,) were normally distributed with mean 0 and variance 1. Then it can be easily shown that Vfi n-'(X'X), where X is given by Tong (1983, p. 140). Similarly, elements of J can be shown to be the limits in probability of the quantities 'C X,-la,-kl,/n, where k = 1, . . . , M, j = 1,2. Here I, indicates X,-, > 0 and I, = 1- I , . For each pair ( 4 , , 4;), 1000 independent realizations each of length 200 were generated. The values of ( 4 , , 4:) considered were (0.5, -0.5), (-0.8,0.8), (0.95, -0.95), (0.8,0.3), (-0.8, -0.3). The series were generated and fitted using IMSL subroutines. The sample variances V(Lk) of Fk over the 1000 replications were computed for each model. Denote JV(?k) by Sdk. These were taken to be the 'true' standard errors of Pk. The
Miscellanea Table 1. Empirical resultsfor residual autocorre~ationsin S E T A R ( ~ ;1, 1) models, n = 200, 1000 replications
asymptotic variances C ( t k ) were also estimated for each realization using the Theorem in § 2. and The sample averages of C ( t k ) were obtained and were denoted as c k . The results for Sdk ( k = 1, ... , 6 ) are reported in Table 1. As in the linear autoregressive situation the results in Table 1 showed that the 'true' standard errors for tkcould be smaller than the value 1/ 4200 = 0.0707. This discrepancy is more prominent if the values of k are small. Consequently, using 1.96/dn as critical value would give a very conservative confidence limit for the first few residual autocorrelations. Note also the much closer and Sdk. This suggests that the result in Q 2 could be usefully applied to give match between more accurate standard errors in practice. Our result will give a more stringent criterion in diagnostic checking for threshold models. This is consistent with the comments by Tong (1990, p. 324) that the first few Fk should be given more careful scrutiny. Note that as k becomes larger approach the value l / J n . both Sdk and
dck
Jck
Jck
ACKNOWLEDGEMENT
I would like to thank Professor Howell Tong for helpful comments.
REFERENCES ANSLEY,C . F. & NEWBOLD,P. (1979). On the finite sample distribution of residual autocorrelations in autoregressive-moving average models. Biometrika 70, 275-8. BILLINGSLEY,P. (1961). The Lindeberg-Levy theorem for martingales. Proc. Am. Math. Soc. 12, 788-92. BOX, G. E. P. & PIERCE, D. A. (1970). Distribution of the residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65, 1509-26. EFRON, B. & HINKLEY,D. V. Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika 65, 457-87. GRANGER,C. W. J. & ANDERSEN,A. P. (1978). An Introduction to Bilinear Time Series Models. Gottingen: Vandenhack and Ruprecht. KLIMKO,L. A. & NELSON, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6 , 629-42. LI, W. K. (1991). Testing model adequacy for some Markov regression models for time series. Biometrika 78, 83-9. MCLEOD, A. I. (1978). On the distribution of residual autocorrelations in Box-Jenkins model. J. R. Statist. Soc. B 40, 296-302. TONG, H. (1978). On a threshold model. In Pattern Recognition and Signal Processing, Ed. C. H. Chen, pp. 575-86. Amsterdam: Sijthoff and Noordhoff. TONG, H. (1983). Threshold Models in Nonlinear Time Series Analysis, Springer Lecture Notes in Statistics, 21. New York: Springer-Verlag. TONG, H. (1990). Non-linear Time Series: A Dynamical System Approach. Oxford University Press. TONG, H. & LIM, K. S. (1980). Threshold autoregression, limit cycles and cyclical data (with discussion). J. R. Statist. Soc. B 42, 245-92.
[Received February 1991. Revised January 19921
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References Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models G. E. P. Box; David A. Pierce Journal of the American Statistical Association, Vol. 65, No. 332. (Dec., 1970), pp. 1509-1526. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197012%2965%3A332%3C1509%3ADORAIA%3E2.0.CO%3B2-A
Assessing the Accuracy of the Maximum Likelihood Estimator: Observed Versus Expected Fisher Information Bradley Efron; David V. Hinkley Biometrika, Vol. 65, No. 3. (Dec., 1978), pp. 457-482. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28197812%2965%3A3%3C457%3AATAOTM%3E2.0.CO%3B2-Y
On Conditional Least Squares Estimation for Stochastic Processes Lawrence A. Klimko; Paul I. Nelson The Annals of Statistics, Vol. 6, No. 3. (May, 1978), pp. 629-642. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28197805%296%3A3%3C629%3AOCLSEF%3E2.0.CO%3B2-5
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Testing Model Adequacy for Some Markov Regression Models for Time Series W. K. Li Biometrika, Vol. 78, No. 1. (Mar., 1991), pp. 83-89. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28199103%2978%3A1%3C83%3ATMAFSM%3E2.0.CO%3B2-C
On the Distribution of Residual Autocorrelations in Box-Jenkins Models A. I. McLeod Journal of the Royal Statistical Society. Series B (Methodological), Vol. 40, No. 3. (1978), pp. 296-302. Stable URL: http://links.jstor.org/sici?sici=0035-9246%281978%2940%3A3%3C296%3AOTDORA%3E2.0.CO%3B2-9
Threshold Autoregression, Limit Cycles and Cyclical Data H. Tong; K. S. Lim Journal of the Royal Statistical Society. Series B (Methodological), Vol. 42, No. 3. (1980), pp. 245-292. Stable URL: http://links.jstor.org/sici?sici=0035-9246%281980%2942%3A3%3C245%3ATALCAC%3E2.0.CO%3B2-Q
