A New Nonlinearity Test to Circumvent the Limitation of Volterra Expansion with Applications

Hui, Yongchang and Wong, Wing-Keung and Bai, Zhidong and Zhu, Zhenzhen (2016): A New Nonlinearity Test to Circumvent the Limitation of Volterra Expansion with Applications.

Abstract

In this paper, we propose a quick, efficient, and easy method to examine whether a time series Yt possesses any nonlinear feature. The advantage of our proposed nonlinearity test is that it is not required to know the exact nonlinear features and the detailed nonlinear forms of Yt. We find that our proposed test can be used to detect any nonlinearity for the variable being examined and detect GARCH models in the innovations. It can also be used to test whether the hypothesized model, including linear and nonlinear, to the variable being examined is appropriate as long as the residuals of the model being used can be estimated. Our simulation study shows that our proposed test is stable and powerful. We apply our proposed statistic to test whether there is any nonlinear feature in the sunspot data and whether the S&P 500 index follows a random walk model. The conclusion drawn from our proposed test is consistent those from other tests.

• Item Type: MPRA Paper
• Language: English
• Keywords: Nonlinearity, U-statistics, Volterra expansion, sunspots, efficient market
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; G - Financial Economics > G1 - General Financial Markets > G10 - General

References

[1] Box, G.E.P., Jenkins, G.M., Reinsel, G.C., 1994. Time Series Analysis Forecasting and Control. Prentice-Hall.

[2] Carlin, B.P., Polson, N.G., Stoffer, D.S., 1992. A Monte Carlo approach to nonnormal and nonlinear state space modeling. Journal of the American Statistical Association 87, 493-500.

[3] Chan, K.S., Tong, H., 1990. On likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society B 52, 469-476.

[4] Chen, R., Tsay, R.S., 1993a. Functional-coefficient autoregressive models. Journal of the American Statistical Association 88, 298-308.

[5] Chen, R., Tsay, R.S., 1993b. Nonlinear additive ARX models. Journal of the American Statistical Association 88, 955-967.

[6] Cheng, Q., Chen, R., Li, T., 1996. Simultaneous wavelet estimation and deconvolution of reflection Seismic signals via Gibbs sampler. IEEE Transactions on Geoscience and Remote Sensing 34, 377-384.

[7] Cox, D.R., 1981. Statistical analysis of time series: Some recent developments (with discussion). Scandinavian Journal of Statistics 8, 93-115.

[8] Denker, M., Keller, G., 1983. On U-statistics and v. Mises’ Statistics for weakly Dependent Processes. Zeitschrift f¨ur Wahrscheinlichkeitstheorie und verwandte Gebiete 64, 505-522.

[9] Dom´ınguez, M.A., Lobato, I.N., 2003. Testing the martingale difference hypothesis. Econometric Reviews 22(4), 351-377.

[10] Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987-1008.

[11] Fan, J.Q., Yao, Q.W., 2003. Nonlinear time series: nonparametric and parametric methods. Springer-Verlag, New York.

[12] Granger, C.W.J., Andersen, A.P., 1978. An Introduction to Bilinear Time Series Models. Vandenhoek and Ruprecht, Gottingen.

[13] Haggan, V., Ozaki, T., 1981. Modeling Nonlinear Vibrations Using an Amplitude- Dependent Autoregressive Time Series Model. Biometrika 68, 189-196.

[14] Hamilton, J.D., 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.

[15] Hansen, B.E., 1996. Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413-430.

[16] Hiemstra, C., Jones, J.D., 1994. Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance 49(5), 1639-1664.

[17] Izenman, A.J., 1983. J.R.Wolf and H.A.Wolfer: An historical note on Zurich sunspot relative numbers. Journal of the Royal Statistical Society Series A 146, 311-318.

[18] Keenan, D.M., 1985. A Tukey non-additivity-type test for time series nonlinearity. Biometrika 72, 39-44.

[19] Kowalski, J., Tu, X.M., 2007. Modern Applied U-Statistics. John Wiley & Sons, New York.

[20] Li, G.D., Li, W.K., 2011. Testing for Linear Time Series Models against Its Threshold Extension. Biometrika 98(1), 243-250.

[21] Li, W.K., 1992. On the asymptotic standard errors of residual autocorrelations in nonlinear time series modeling. Biometrika 79(2), 435-7.

[22] Ljung, G.M., Box, G.E.P., 1978. On a Measure of a Lack of Fit in Time Series Models. Biometrika 65(2), 297-303.

[23] Moran, P.A.P., 1954. Some experiments on the prediction of sunspot numbers. Journal of the Royal Statistical Society Series B 116, 112-117.

[24] Newey, W.K., West, K.D., 1987. A simple, positive semi-definite, heteroskedasticity and autocorrelation constistent covariance matrix. Econometrica 55(3), 703-708.

[25] Petruccelli, J.D, Davies, N., 1986. A portmanteau test for self-exciting threshold autoregressive-type nonlinearity in time series. Biometrika 73, 687-694.

[26] Priestley, M.B., 1980. State-dependent models: a general approach to nonlinear time series analysis. Journal of Time Series Analysis 1, 47-71.

[27] Serfling, R., 1980. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.

[28] Tiao, G.C., Tsay, R.S., 1994. Some advances in nonlinear and adaptive modeling in time series. Journal of Forecasting 13, 109-131.

[29] Tjϕstheim, D., 1994. Non-linear time series: A selective review. Scandinavian Journal of Statistics 21, 97-130.

[30] Tong, H., 1978. On a threshold model. In C. H. Chen (ed.), Pattern Recognition and Signal Processing. Sijhoff & Noordhoff, Amsterdam.

[31] Tong, H., 1983. Threshold Models in Non-linear Time Series Analysis. Springer- Verlag, New York.

[32] Tong, H., 1990. Non-linear Time Series: A Dynamical Systems Approach. Oxford University Press, Oxford.

[33] Tong, H., Lim, K.S., 1980. Threshold autoregression, limit cycles and cyclical data (with discussion). Journal of the Royal Statistical Society Series B 42, 245-292.

[34] Tsay, R.S., 1986. Nonlinearity tests for time series. Biometrika 73, 461-466.

[35] Tukey, J.W., 1949. One degree of freedom for non-additivity. Biometrics 5, 232-242.

[36] Wiener, N., 1958. Non-linear Problems in Random Theory. Cambridge, Mass: M.I.T. Press.

[37] Yule, G.U., 1927. On a method of investigating periodicities in disturbed series with special reference to Wolfer’s sunspot numbers. Philosophical Transactions of the Royal Society (London) A 226, 267-298.