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The spurious regression of fractionally integrated processes. (English) Zbl 1054.62586

Summary: This paper extends the theoretical analysis of the spurious regression and spurious detrending from the usual \(I(1)\) processes to the long memory fractionally integrated processes. It is found that when we regress a long memory fractionally integrated process on another unrelated long memory fractionally integrated process, no matter whether these processes are stationary or not, as long as their orders of integration sum up to a value greater than 0.5, the \(t\) ratios become divergent and spurious effects occur. Our finding suggests that it is the long memory, instead of nonstationarity or lack of ergodicity, that causes such spurious effects. As a result, spurious effects might happen more often than we previously believed as they can arise even between stationary series while the usual first-differencing procedure may not completely eliminate spurious effects when data possess strong long memory.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
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[1] Baillie, R. T.: Long memory processes and fractional integration in econometrics. Journal of econometrics 73, 6-59 (1996) · Zbl 0854.62099
[2] Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods, 2nd Edition. Springer, New York. · Zbl 0709.62080
[3] Cheung, Y. -W.; Lai, K. S.: A fractional cointegration analysis of purchasing power parity. Journal of business and economic statistics 11, 103-112 (1993)
[4] Chung, C.-F., 1994. Calculating and analyzing impulse responses and their asymptotic distributions for the ARFIMA and VARMA models. Econometrics and Economic Theory Paper No. 9402, Michigan State University.
[5] Chung, C.-F., 1995. Sample variances, sample covariances, and linear regression of stationary multivariate long memory processes. Preprint, Michigan State University.
[6] De Jong, R., Davidson, J., 1997. The functional central limit theorem and weak convergence to stochastic integrals: results for weakly dependent and fractionally integrated processes. Preprint.
[7] Davydov, Y. A.: The invariance principle for stationary processes. Theory of probability and its applications 15, 487-489 (1970)
[8] Durlauf, S. T.; Phillips, P. C. B.: Trends versus random walks in time series analysis. Econometrica 56, 1333-1354 (1988) · Zbl 0653.62068
[9] Fox, R.; Taqqu, M. S.: Multiple stochastic integrals with dependent integrators. Journal of multivariate analysis 21, 105-127 (1987) · Zbl 0649.60064
[10] Granger, C. W. J.: Long memory relationships and the aggregation of dynamic models. Journal of econometrics 14, 227-238 (1980) · Zbl 0466.62108
[11] Granger, C. W. J.: Some properties of time series data and their use in econometric model specification. Journal of econometrics 16, 121-130 (1981)
[12] Granger, C. W. J.; Newbold, P.: Spurious regression in econometrics. Journal of econometrics 2, 111-120 (1974) · Zbl 0319.62072
[13] Granger, C. W. J.; Joyeux, R.: An introduction to long-memory time series models and fractionally differencing. Journal of time series analysis 1, 15-29 (1980) · Zbl 0503.62079
[14] Hosking, J. R. M.: Fractional differencing. Biometrika 68, 165-176 (1981) · Zbl 0464.62088
[15] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noise and applications. SIAM review 10, 422-437 (1968) · Zbl 0179.47801
[16] Marmol, F.: Spurious regressions between \(I(d)\) processes. Journal of time series analysis 16, 313-321 (1995) · Zbl 0819.62075
[17] Nelson, C. R.; Kang, H.: Spurious periodicity in inappropriately detrended time series. Econometrica 49, 741-751 (1981) · Zbl 0468.62099
[18] Nelson, C. R.; Kang, H.: Pitfalls in the use of time as an explanatory variable in regression. Journal of business and economic statistics 2, 73-82 (1984)
[19] Nelson, C. R.; Plosser, C.: Trends and random walks in macro-economic time seriessome evidence and implications. Journal of monetary economics 10, 139-162 (1982)
[20] Park, J. Y.; Phillips, P. C. B.: Statistical inference in regressions with integrated processespart 1. Econometric theory 4, 468-497 (1988)
[21] Phillips, P. C. B.: Understanding spurious regressions in econometrics. Journal of econometrics 33, 311-340 (1986) · Zbl 0602.62098
[22] Phillips, P. C. B.: Nonstationary time series and cointegration. Journal of applied econometrics 10, 87-94 (1995)
[23] Sowell, F.: The fractional unit root distribution. Econometrica 58, 495-505 (1990) · Zbl 0727.62025
[24] Tsay, W. J.: On the power of durbin–Watson statistic against fractionally integrated processes. Econometric reviews 17, 361-386 (1998) · Zbl 0913.62087
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