本文主要以Lee and Shie (2003) 的文章為基石,來推導出Augmented Dickey-Fuller (ADF Test) 單根檢定,在一般化的分數單根資料下 (例如: ARFIMA(p,1+d,q) (|d|<1/2; p,q 為正整數時) 的t檢定統計量的極限分配,進而探討其檢定力隨著所選定延滯期數增加而產生變化的原因;在文章的最後,我們提供了一些圖表來說明與蒙地卡羅模擬的證據來佐證我們的推導結論

In this paper, we derive the asymptotic distribution of the Augmented Dickey-Fuller t Test statistics, t_{ADF}, against a generalized fractional integrated process (for example: ARFIMA(p,1+d,q) ,|d|<1/2,and p, q be positive integer) by using the propositions of Lee and Shie (2003).
Then we discuss why the power decreases with the increasing lags in the same and large enough sample size T when d is unequal to 0. We also get that the estimator of the disturbance's variance, S^2, has slightly increasing bias with increasing k. Finally, we support the conclusion by the Monte Carlo experiments.

Catalog:
1 Introduction p.7
2 Model Setting and Denotations p.9
2.1 Population Process p.9
2.1.1 The Binomial Expansion of Fractional
Diference p.10
2.1.2 Assumptions For t p.10
2.1.3 Data Generating Process p.10
2.2 Regression Model p.11
2.3 Denotations p.11
2.4 Estimation p.13
3 The Functional Central Limit Theorem p.14
4 The Propositions of Integrated Process p.15
5 Lemmas p.16
6 Theorems and Discussions p.17
6.1 Theorems p.17
6.2 Discussions p.18
6.2.1 Estimators p.18
6.2.2 Test Statistics p.19
6.3 Figures p.20
6.4 Monte Carlo Evidences p. 23
7 Conclusions p.28
8 Appendix p.29
8.1 Proof of Lemma 1-9 p.29
8.2 Proof of Theorem 1-4 p.38
*References p.44

[1]Anderson, T.W., 1951, Estimating Linear Restrictions on
Regression Coefficient for Multivariate Normal
Distributions.Annals of Mathematical Statistics 22, 327-
351.
[2]Anderson, T.W., 1971, The Statistical Analysis of Time
Series, Wiley, New York.
[3]Avram, F. and M.S. Taqqu, 1987, Noncentral limit
theorems and appell polynomials. Annals of Probability
15, 767-775.
[4]Baillie, R.T., 1996, Long memory processes and
fractional integration in econometrics. Journal of
Econometrics 73, 5-59.
[5]Beran, J., 1994, Statistics for long-memory processes.
Chapman Hall, New York.
[6]Box, G.E.P., and G.M. Jenkins, 1976, Time Series
Analysis : Forecasting and Control, 2nd. edn., Holden-
Day, San Francisco.
[7]Chung, C.F., 1994, A note on calculating the
autocovariances of the fractionally integrated ARMA
models. Economics Letters 45, 293-297.
[8]Chung, C.F., 2002. Sample means, sample
autocovariances, and linear regression of stationary
multivariate long memory processes. Econometric Theory
18, 51-78.
[9]Davidson, J., 1994, Stochastic limit theory, Oxford:
Oxford University Press.
[10]Davidson, J. and R.M. De Jong, 2000, The functional
central limit theorem and weak convergence to
stochastic integrals II: Fractionally Integrated
Processes, Econometric Theory 16, 643-666.
[11]Davidson, J., 2002, Establishing conditions for the
functional central limit theorem in nonlinear and
semiparametric time series processes. Journal of
Econometrics 106, 243-269.
[12]Davydov, Y.A., 1970. The invariance principle for
stationary processes. Theory of Probability and Its
Applications 15, 487-489.
[13]Dickey, D.A. and W.A. Fuller, 1979, Distribution of
the estimator for autoregressive time series with a
unit root, Journal of the American Statistical
Association 74, 427-431.
[14]Diebold, F.X. and G.D. Rudebusch, 1991, On the power
of Dickey-Fuller tests against fractional alternatives.
Economics Letters 35, 155-160.
[15]Dittmann, I., 2000, Residual-based tests for
fractional cointegration:$;;$ A Monte Carlo study.
Journal of Time Series Analysis 21, 6, 615-647.
[16]Dolado, J., J. Gonzalo, and L. Mayoral, 2002, A
fractional Dickey-Fuller test for unit root.
Econometrica 70, 5, 1963-2006. Economics Letters 35,
155-160.
[17]Geweke, J.F. and S. Porter-Hudak, 1983, The estimation
and application of long memory time series models.
Journal of Time Series Analysis 4, 221-238.
[18]Granger, C.W.J. and R. Joyeux, 1980, An introduction
to long-memory time series models and fractional
differencing. Journal of Time Series Analysis 1, 15-39.
[19]Hassler, U. and J. Wolters, 1994, On the power of unit
root tests against fractional alternatives. Economics
Letters 45, 1-5.
[20]Helson, J. and Y. Sarason, 1967, Past and future.
Mathematica Scandanavia 21, 5-16.
[21]Hurst , H.E., 1951, Long-term storage capacity of
reserviors,Transactions of the American Society of
Civil Engineers 116, 770-779.
[22]Hurst , H.E., 1956, Methods of using long term storage
in reservoirs, Proceedings of the Institute of Civil
Engineer 1, 519-543.
[23]Hurst , H.E., 1957, A suggested statistical model of
some time series that occur in nature. Nature 180, 494.
[24]Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, and Shin
Y., 1992, Testing the null hypothesis of stationarity
against the alternative of a unit root :
How sure are we the economic time series have a unit
root ? . Journal of Econometrics 54, 159-178.
[25]Kramer, W., 1998, Fractional integration and the
augmented Dickey-Fuller Test. Economics Letters 61,
269-272.
[26]Lee, C.N. and F.S. Shie, 2003, Fractional Integration
and Phillips-Perron Test. Working paper, National Sun
Yat-Sen University.
[27]Lee, D., and P. Schmidt, 1996, On the power of the
KPSS test of stationarity against fractionally-
integrated alternatives. Journal of Econometrics 73,
285-302.
[28]Mandelbrot, B.B. , 1972, Statistical Methodlogy for
non periodic cycles : From the covariance to R/S
analysis. Annals of Economic and Social Measurement 1,
259-290.
[29]Mandelbrot, B.B. and J. Wallis, 1968, Noah, Joseph and
operational hydrology. Water Resources Research 4, 909-
918.
[30]McLeod, A.I. and K.W. Hipel, 1978, Preservation of the
rescaled adjusted range :A reassessment of the Hurst
phenomenon. Water Resources Research 14, 491-508.
[31]Marinucci, D., and P.M. Robinson, 1999, Alternative
forms of fractional Brownian motion. Journal of
Statistical Planning and Inference 80, 111-122.
[32]Mielniczuk, J., 1997, Long and short-range dependent
sums of infinite-order moving averages and regression
estimation. Acta Sci. Math. (Szeged) 63, 301-316.
[33]Nelson, C.R. and C.I. Plosser, 1982, Trends and random
walks in macroeconomic time series : Some evidence and
implications. Journal of Monetary Economics 10, 139-
162.
[34]Newey, W. and K. West, 1987, A simple positive semi-
definite, heteroscedasticity and autocorrelation
consistent covariance matrix. Econometrica 55, 703-708.
[35]Perron, P. and S. Ng, 1996, Useful modifications to
some unit root tests with dependent errors and their
local asymptotic properties. Reviews of Economic
Studies 63, 435-463.
[36]Phillips, P.C.B., 1987, Time series regression with a
unit root. Econometrica 55, 277-301.
[37]Phillips, P.C.B. and S. Ouliaris, 1990, Asymptotic
properties of residual based tests for cointegration.
Econometrica, 58, 1, 165-193.
[38]Phillips, P.C.B. and P. Perron, 1988, Testing for a
unit root in time series regression. Biometrika, 75,
2, 335-346.
[39]Said, E.S. and David A. Dickey, Testing for unit root
in autoregressive-moving average models of unknown
order. Biometrika 71, 3, 599-607.
[40]Schwert, G.W., 1987, Effects of model specification on
tests for unit roots in macroeconomic data. Journal of
Monetary Economics 20, 73-103.
[41]Sowell, F.B., 1990, The fractional unit root
distribution. Econometrica 58, 2, 495-505.
[42]Stout, W.F., 1974, Almost sure convergence. Academic
Press.
[43]Tanaka, K., 1999, The nonstationary fractional unit
root. Econometric Theory 15, 549-582.
[44]Wang, Q., Y. Lin, and C. Gulati, 2003, Asymptotics for
general fractionally integrated processes with
applications to unit root tests. Econometric Theory,
19, 143-164.
[45]White, P., 1951, Hypothesis testing in time series
analysis. Almquist and Wiksells, Uppsala.
[46]White, P., 1956, Variation of yield variance with plot
size. Biometrika 43, 337-343.
[47]White, H., 2001, Asymptotic theory for
econometricians. Academic Press.