{'fraction_non_alphanumeric': 0.037940379403794036, 'fraction_numerical': 0.03387533875338753, 'mean_word_length': 5.2747875354107645, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
DIAGNOSTIC CHECKING FOR THE ADEQUACY OF NONLINEAR TIME SERIES MODELS

We propose a new diagnostic test for linear and nonlinear time series models, using a generalized spectral approach. Under a wide class of time series models that includes autoregressive conditional heteroskedasticity (ARCH) and autoregressive conditional duration (ACD) models, the proposed test enjoys the appealing “nuisance-parameter-free” property in the sense that model parameter estimation uncertainty has no impact on the limit distribution of the test statistic. It is consistent against any type of pairwise serial dependence in the model standardized residuals and allows the choice of a proper lag order via data-driven methods. Moreover, the new test is asymptotically more efficient than the correlation integral–based test of Brock, Hsieh, and LeBaron (1991, Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence) and Brock, Dechert, Scheinkman, and LeBaron (1996, Econometric Reviews 15, 197–235), the well-known BDS test, against a class of plausible local alternatives (not including ARCH). A simulation study compares the finite-sample performance of the proposed test and the tests of BDS, Box and Pierce (1970, Journal of the American Statistical Association 65, 1509–1527), Ljung and Box (1978, Biometrika 65, 297–303), McLeod and Li (1983, Journal of Time Series Analysis 4, 269–273), and Li and Mak (1994, Journal of Time Series Analysis 15, 627–636). The new test has good power against a wide variety of stochastic and chaotic alternatives to the null models for conditional mean and conditional variance. It can play a valuable role in evaluating adequacy of linear and nonlinear time series models. An empirical application to the daily S&P 500 price index highlights the merits of our approach.We thank the co-editor (Don Andrews) and two referees for careful and constructive comments that have lead to significant improvement over an earlier version. We also thank C.W.J. Granger, D. Tjøstheim, and Z. Xiao for helpful comments. Hong's participation is supported by the National Science Foundation via NSF grant SES–0111769. Lee thanks the UCR Academic Senate for research support.