Testing for Long Memory Using Penalized Splines and Adaptive Neyman Methods

Testing procedures for the null hypothesis of short memory against long memory alternatives are investigated. Our new test statistic is constructed using penalized splines method and Fan’s (1996) canonical multivariate normal hypothesis testing procedure. Using penalized splines method, we are able to eliminate the effects of nuisance parameters typically induced by short memory autocorrelation. Therefore, under the null hypothesis of any short memory processes, our new test statistic has a known asymptotic distribution. The proposed test statistic is completely data-driven or adaptive, which avoids the need to select any smoothing parameters. Since the convergence of our test statistic toward its asymptotic distribution is relatively slow, Monte Carlo methods are investigated to determine the corresponding critical value. The finite-sample properties of our procedure are compared to other well-known tests in the literature. These show that the empirical size properties of the new statistic can be very robust compared to existing tests and also that it competes well in terms of power.

[1]  Katsuto Tanaka,et al.  THE NONSTATIONARY FRACTIONAL UNIT ROOT , 1999, Econometric Theory.

[2]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[3]  P. Erdös,et al.  A limit theorem for the maximum of normalized sums of independent random variables , 1956 .

[4]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[5]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[6]  P. Robinson.  Efficient Tests of Nonstationary Hypotheses , 1994 .

[7]  Liudas Giraitis,et al.  Large Sample Inference for Long Memory Processes , 2012 .

[8]  Brendan McCabe,et al.  TESTING FOR LONG MEMORY , 2006, Econometric Theory.

[9]  P. W. Strike,et al.  Forecasting and control , 1991 .

[10]  J. Neyman.  »Smooth test» for goodness of fit , 1937 .

[11]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[12]  P. Phillips,et al.  Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? , 1992 .

[13]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[14]  Young K. Truong,et al.  RATE OF CONVERGENCE FOR LOGSPLINE SPECTRAL DENSITY ESTIMATION , 1995 .

[15]  David Ruppert,et al.  Semiparametric regression during 2003-2007. , 2009, Electronic journal of statistics.

[16]  David Ruppert,et al.  On the asymptotics of penalized spline smoothing , 2011 .

[17]  Jianqing Fan,et al.  Generalised likelihood ratio tests for spectral density , 2004 .

[18]  Jianqing Fan,et al.  Goodness-of-Fit Tests for Parametric Regression Models , 2001 .

[19]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[20]  D. Ruppert,et al.  On the asymptotics of penalized splines , 2008 .

[21]  P. Schmidt,et al.  On the Power of the KPSS Test of Stationarity Against Fractionally-Integrated Alternatives , 1996 .

[22]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[23]  Jianqing Fan.  Test of Significance Based on Wavelet Thresholding and Neyman's Truncation , 1996 .

[24]  Jianqing Fan,et al.  Nonlinear Time Series : Nonparametric and Parametric Methods , 2005 .