Lag Order and Critical Values for the RMA Based Augmented Dickey-Fuller Test
by
Zhixiao Liu
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 9, 2010
Keywords: Finite and asymptotic critical value; Recursive mean adjustment;
Monte Carlo; Response surface.
Copyright 2010 by Zhixiao Liu
Approved by
Hyeongwoo Kim, Chair, Assistant Professor, Department of Economics
John Jackson, Professor, Department of Economics
Randy Beard, Professor, Department of Economics
ii
Abstract
This thesis examines the validity of asymptotic critical values for a Recursive Mean
Adjustment (RMA) based Augmented Dickey-Fuller (ADF) unit root test. Cheung and Lai show
that critical values for the Ordinary least square (OLS) based ADF test depend substantially on
the lag order in finite samples. The present article extends their work to a newly proposed RMA-
based unit root test, which is more powerful than the OLS-based test. Our Monte Carlo
simulation results show that asymptotic critical values for the test with the deterministic terms
are valid only when the lag order is one. When lag order is greater than one, the RMA based test
with asymptotic critical values tends to be overall over-sized. I also provide finite sample critical
values for an array of lag-order and sample size pairs.
iii
Acknowledgments
First and foremost, I would like to thank my advisor, Dr. Hyeongwoo Kim. Without his
direction, his patience and encouragement, this thesis would not have been possible. I sincerely
appreciate invaluable academic and personal support I have received from him throughout this
thesis.
I would also thank the rest of my thesis committee members: Dr. John Jackson and Dr.
Randy Beard for their valuable feedbacks and suggestions that helped me to improve the thesis.
I also respectfully acknowledge Dr. Barry Burkhart, Chair, Department of Economics at
AU for giving me his valuable time. I thank Dr. Jackson for admitting me to Master of Science
program to pursue higher education.
I thank you all for teaching me and guiding me.
Finally, my further gratitude goes to my family: my husband Xueyi, my son Kaishuo, My
parents, and my sister Zhixin. Thanks for your love, support, encouragement and patience.
iv
Table of Contents
Abstract.........................................................................................................................................ii
Acknowledgments........................................................................................................................iii
List of Tables ...............................................................................................................................vi
List of Figures.............................................................................................................................vii
List of Abbreviations .............................................................................................................?viii
Chapter 1 Introduction ................................................................................................................. 1
Chapter 2 ADF
ols
Root Test and ADF
RMA
Test........................................................................... 4
2.1 ADF unit root test ..................................................................................................... 4
2.1.1 Autoregressive Unit Root Test ........................................................................... 4
2.1.2 Three cases under alternative hypothesis............................................................. 6
2.1.3 Dickey-Fuller ( ADF) unit root test ..................................................................... 8
2.2 Recursive Mean Adjusted ADF unit root test (ADF
RMA
) ......................................... 9
2.2.1 Recursive mean adjusted based ADF unit root test (ADF
RMA
) ......................... 9
2.2.2 Recursive trend adjusted based ADF unit root tests (ADF
RTA
) ........................ 11
Chapter 3 Experimental Designs for Response Surface Methodology....................................... 13
3.1 Response surface literature review ........................................................................... 13
3.2 Response surface method and experimental design ................................................. 13
Chapter 4 Simulation Results and analysis................................................................................. 17
Chapter 5 Conclusion.................................................................................................................. 27
v
References ................................................................................................................................. 28
Appendix?????????????????????????????????. 31
vi
List of Tables
Table 1 Response surface estimation of Critical values for the ADF
RMA
statistic ..................... 18
Table 2 Lag Order and Finite-sample Critical Values................................................................ 19
Table 3 Lag Order and Finite-sample Critical Values (for constant) ....................................... 20
Table 4 Lag Order and Finite-sample Critical Values (for time trend)??????????21
vii
List of Figures
Figure 1 Distribution of ADF t statistics..................................................................................... 14
Figure 2 Plots of Monte Carolo-Estimated critical values for various RMA based ADF test.... 22
Figure 3 RMA based critical value as a function of (a) T and (b) K in the case of constant
no trend for 1%, 5%, and 10% test .............................................................................. 25
Figure 4 RMA based critical value as a function of (a) T and (b) K in the case of constant
and trend for 1%, 5%, and 10% test............................................................................ 26
viii
List of Abbreviations
ADF Augmented Dick-Fuller
AR Autoregressive
CV Critical Value
DF Dickey -Fuller
DGP Data Generating Process
GDP gross domestic product
OLS Ordinary Least Squares
RMA Recursive Mean Adjustment
RSM Response Surface Methodology
RTA Recursive Trend Adjustment
1
Chapter 1
Introduction
The analysis of unit root nonstationarity has been one of the major areas of research in time
series econometrics over the last two decades. Stationararity or nonstationarity of macro-
economic time series is quite important to investigate statistically because (a) macro economic
time series are known to exhibit persistence in their intertemporal behavior, (b) Spurious
regression problems can lead to misleading inference, (c) conventional statistical analyses may
be invalid when applied to regressions with nonstationary variables. Early motivation for a unit
root test was to help determine whether to use forecasting models expressed in differences or
levels in particular applications (e.g. Dickey, Bell, and Miller, 1986). Nowadays, unit root tests
are useful to test certain hypotheses such as purchasing power parity (e.g. Rogoff, 1996), the
efficient market hypothesis (e.g. Balvers et al ,2000), and the natural rate of unemployment or
hysteresis hypothesis (e.g. Blanchard and Summers,1987), just to name a few. Generally, the
major problem when working with nonstationarity results from the breakdown of conventional
asymptotic distribution theory under nonstationarity. Standard statistical inferences become
invalid, and many test statistics developed for nonstationarity converge to nonstandard
distributions. Therefore, unit root tests are important.
Many methods for unit root tests have been developed. Among them, the Augmented Dick-
Fuller (ADF) test is by far the most popular. This test examines the null hypothesis of
nonstationarity against stationary alternatives. Asymptotic critical values for the test were
2
tabulated by Dickey-Fuller (1976). Despite of its popularity, it is well known that the ADF test
has a low power to find stationarity, especially when the sample size is small.
In order to improve the power of unit root tests, many new methods have been put forward.
For example, Zilliott, Rothenberg and Stock (1992) proposed a simple modification of the ADF
test, referred to as the DF-GLS test, which is shown to have higher power by Cheung and Lai
(1995b). Recently, an ADF unit root test based on recursive mean adjustment (RMA) has been
put forth by So and Shin (1999) and Shin and So (2001), which showed significant power
improvement according to their Monte Carlo studies.
1
Shin and So (2001) derived the limiting
distribution of the test with a constant. Their asymptotic critical values for the test with a
constant are tabulated for some sample sizes based solely on AR(1) processes.
Cheung and Lai (1995a) showed that finite sample critical values are determined by lag order
in addition to sample size. It is crucial to correcting for the lag order impact in implementing a
RMA based ADF test (ADF
RMA
), for critical values that ignore the dependence of lag order can
be misleading. Kim et al (2009) showed that the RMA based ADF test outperformed the DF-
GLS and standard ADF tests in their study for G7 stock markets. Despite its power and
convenience to implement, this method is largely overlooked in the financial literature.
The purpose of this study is to examine the validity of asymptotic critical values for a
Recursive Mean Adjustment based Augmented Dickey-Fuller test. Our Monte Carlo simulation
results suggest that asymptotic critical values (e.g. Shin and So(2001)) computed based on k=0 )
for the test with the deterministic terms are valid only when the lag order is one .When the lag
1
The logic behind RMA method is to use partial mean instead of global mean:
t
t
t
t
t
YYYY ?? +?=? ?
?
? )( 1
1
1
,
t
? is
uncorrelated to the recursive mean adjusted regressor
11 ??
?
tt
YY
,which results in biased reduction RMA estimator, while LS
estimator is to estimate ,)(
1 ttt
YYYY ?? +?=?
?
,
t
? is correlated to regressor YY
t
?
?1
,which is biased. See chapter 2
for details.
3
order is greater than one, the test with asymptotic critical values tends to be overall over-sized
even when the sample size is fairly big.
2
.
Response surface analysis has been used by Mackinnon (1991) to obtain approximate finite
sample critical values for the traditional ADF unit root test. In his method, lag order is assumed
to be fixed and equal to 1 for ADF test. Cheung and Lai (1995a) extended the response surface
analysis and showed that although the asymptotical ADF test may not depend on the lag
parameter, lag order can be important in finite samples. Employing their ideas by properly
accounting for the effect of lag order, our study provides improved estimation of lag-adjusted
critical values for the ADF
RMA
test. Our experimental design generalizes Mackinnon?s method
(1991) by including lag order but still omits those other nuisance parameters as in Cheung and
Lai (1995a). Finite-sample correction for the nuisance parameter, although is desirable, is hard to
make, given the potential size of the parameter space of these unknown parameters, it is
plausible to omit them.
This thesis is organized as follows: In chapter 2, conventional ordinary least square (OLS)
DF and ADF unit root tests (ADF
ols
) are described, and compared to the RMA based ADF test
(ADF
RMA
). Chapter 3 discusses the methodology of response surface analysis and our
experimental design. Chapter 4 reports and analyzes response surface estimation of the critical
values of ADF
RMA ,
and provides finite sample critical value Tables for the ADF
RMA
test. Finally
in Chapter 5 we offer conclusions.
2
A test is oversized when the actual size with asymptotic critical value is greater than the nominal size. That is, such
tests tend to reject the null hypothesis too often.
4
Chapter 2
ADF Unit Root Tests
2.1 OLS-based ADF unit root test
Why people worry about unit root? Most macroeconomic time series are known to exhibit
high persistence, possibly nonstationarity, in their intertemporal behavior. Conventional
statistical inferences may be invalid when the true data generating process is nonstationary
Therefore, unit root tests are important. A widely used unit root is the Augmented Dickey-fuller
or ADF test (Dickey and Fuller, 1979). The test typically examines the null hypothesis (random
walk without a drift) of nonstationarity against three stationary forms of alternatives.
2.1.1 Autoregressive Unit Root Test
To illustrate the important statistical issues associated with an autoregressive unit root test,
we considered the following simple AR (1) model
ttt
YY ?? +=
?1
(1)
Where
t
? is white noise. The hypotheses of interest are
H
0
: 1=? (unit root in ?=0)
3
, )1(IY
t
?
H
1:
,1|| 1) when the error term
t
? is serially correlated. Consequently,
Said and Dickey (1984) developed a test, known as augmented Dickey-Fuller (ADF) test. This
test is conducted by ?augmenting? the preceding three equations with the lagged values of the
differenced dependent variable
t
Y . To be specific, we use form (4). The ADF test here consists of
estimating the following regression:
t
K
j
jtjtt
YYtCY ???? +?+++=
?
=
??
1
1
5
(5)
The specification of deterministic terms depends on the assumed behavior of Y
t
under the
alternative hypothesis of trend stationarity as describe in the previous section. Under the null
hypothesis, y
t
is I(1), which implies that ,1=? The test statistics are based on the least square
estimate of (5) and are given by
)?(
1?
?
?
se
ADF
t
?
=
k
T
ADF
??
?
?
??
1
)1?(
1
K??
?
=
5
Alternatively
t
K
j
jtjtt
YYtCY ???? +?+++=?
?
=
??
1
1
can be used, where 1?=??
k
t
T
ADF
se
ADF
??
?
?
?
?
??
1
)
?
(
,
)
?
(
?
1
K??
==
9
ADF
t
and
?
ADF follow the same asymptotic distribution as the Dickey-Fuller tests with
white noise error when lag order P is selected appropriately.
It is well-known that LS for autogressive (AR) suffers from serious downward bias in the
persistence coefficient when the process includes deterministic. To see the bias, assume that the
regression equations follow (3). By the Frisch-Lowell-Waugh theorem, estimating ?? by OLS is
equivalent to estimating the following regression with de-meaned terms.
ttt
YYYY ?? +?=?
?
)(
1
where
?
=
?
=
T
j
j
YTY
1
1
. We see that
t
? is correlated with
j
Y , for j=t, t+1,...,T, thus it is also
correlated with Y . Therefore, the OLS estimator for the AR(1) process with an intercept creates
a mean-bias. The bias has an analytical representation, and as Kendall (1954) shows, the OLS
estimator is biased downward. It is known that correcting for bias may help enhancing the power
of the test. In what follows, we demonstrate that this is also the case for the recursive mean and
recursive trend adjusted versions of the ADF unit root tests.
2.2 Recursive mean adjusted (RMA) based ADF test (ADF
RMA
)
The RMA-based unit root test possesses greater power than an ADF
ols
test. Due to reduced-
bias estimation, the left percentile of the null distribution (of the test) shifts to the right, while the
asymptotic distribution of RMA and the OLS estimator are identical under the alternative. This
leads to an improvement in power over the ADF
ols
(Shin and So, 2001). We will examine the
principle behind the ADF
RMA
test by reviewing recursive demeaning and detrending procedures.
2.2.1 Recursive mean adjusted based ADF unit root test (ADF
RMA
)
10
So and Shin (1999) originally introduced recursive mean adjustment in univariate
autogression to reduce the small sample bias of the least square estimator, and Shin and So
(2001) extended their recursive mean adjustment to a unit root test for the case of an unknown
mean .
Shin and So (2001) introduced the concept of recursive mean adjustment by considering the
following AR(1) model.
TtYY
ttt
,,2,1,)(
1
LL=+?=?
?
???? (6)
where
t
? is zero mean stationary process. Shin and So (2001) note that when the absolute value
of ?
is less than 1, because? is unknown, therefore, ? can be replaced by the mean of
t
Y
?
=
=
T
j
j
Y
T
Y
1
1
(7)
Application of the ADF or DF test to the mean-adjusted observation ( YY
t
? ) is achieved
using the following regression
ttt
YYYY ?? +?=?
?
)(
1
(8)
However, as Shin and So further note, replacing ? with Y in (6) leads to correlation
between the regressor ( YY
t
?
?1
) and
t
?
.
Denoting the OLS estimator as
0
?? , the resulting bias of
0
??
has been derived by inter alia, Kendall(1954), Tanaka (1984) and Shaman and Stine (1988)
as
t
ToTE ???? +++?=?
??
)()31()?(
11
00
(9)
To overcome the problem of correlation between the error term and regressor, Shin and So
(2001) propose the use of recursive mean, Y
t-1
, using the partial mean instead of global mean.
11
TtY
t
Y
t
i
it
,,3,2
1
1
1
1
1
L=
?
=
?
?
=
?
(10)
Define 1
~
??= t
tt
YYY , and 1
11
~
?
??
?= t
tt
YYY . The recursive mean-adjusted version of (6) and (8) is
then given as
ttt
YY ?? +=
?1
~~
(11)
In a nutshell, the logic behind RMA estimator can be seen by defining,
?
?
=
?
?
=
1
1
1
1
1
t
i
it
Y
t
Y
,
so
t
? is uncorrelated with the recursive mean adjusted regressor
11 ??
?
tt
YY
, which results in
substantial biased reduction for RMA estimator.
?
?
=
?
?
=
?
?
?
?
??
=
T
t
t
t
T
t
t
t
t
t
RMA
YY
YYYY
2
2
1
1
2
1
1
1
)(
))((
?? (12)
Similarly, the extending the RMA estimation to higher order autoregressive process AR(p)
(where p is greater than 1) is as:
t
k
j
jtjtt
YYY ??? +?+=
?
=
??
1
1
~~
(13)
RMA
?? can be obtained by regression (13). We control for nuisance parameters (
j
? ) by a method
described in Kim et al (2010).
2.2.2 Recursive trend adjusted based ADF unit root tests(ADF
RTA
)
Consider the following model:
ttt
YTY ???? +++=
?110
(14)
where
t
? is white noise, null hypothesis to be tested is
0
H : ? =1
The model of interest includes a constant and time trend so that the vector of deterministic
variables considered is Z
t
=(1,t)?, with corresponding vector of parameters to be estimated,
12
( ).,
10
?? In order to consider the recursive trend adjustment, Shin and So (2001) took an OLS
based approach whereby the vector of estimators of the deterministic component at time t is
given by:
k
t
k
kk
t
k
kt
yZZZ
??
=
?
=
=
1
1
1
)'(
~
? (15)
Thus, once the T by 2 vector of parameters of the deterministic component is estimated as in
equation (16), following Shin and So (2001), the test regression can be set up using the following
recursively adjusted variable,
11
~
'
~
??
?=
tttt
ZyY ? (16)
1111
~
'
~
????
?=
tttt
ZyY ? (17)
As equations (17), (18) show, only the sample mean of the observations up to time t-1 is
considered. Where
11
~
'
?? tt
Z ? is the mean value of recursively trend variable.
We have,
ttt
YY ?? +=
?1
~~
(18)
And the relevant test statistic given as )
?(/1? ??? se?=
, where )
?(?se
is a standard error.
Remark: In order to account for potential autocorrelation, equation model (18) can be
augmented with lags of depended variable as in the conventional Augmented DF (ADF) test as
t
k
j
jtjtt
YYY ??? +?+=
?
=
??
1
1
~~
(19)
see inter, alia, Shin and So (2001) and Taylor (2002). We also control for nuisance parameter
following Kim et al (2010).
13
Chapter 3
Experimental Designs for Response Surface Methodology
3.1 Response surface literature review
The response surface methodology (RSM) is important in designing, formulating,
developing and analyzing new scientific studies. It is also efficient in improving existing studies.
In statistic, the response surface methodology explores the relationship between several
explanatory variables and one or more response variables. The method was introduced by Box
and Wilson (1951). Their main idea of RSM is to use a sequence of designed experiment to
obtain an optimal response. Box and Wilson (1951) suggested using the second degree
polynomial model to approximate the response variable. They acknowledged that this model is
only an approximation, not exact, but such a model is easy to estimate and apply even when little
is known about the process. Response surface methodology has been used in many fields of
applied statistics (Myers, Khuri and Cater, 1989) since this method was introduced by Box and
Wilson . Researchers have applied the RSM in econometrics fields in 1970.
Early studies that use the response surface methodology in econometrics include Hendry
(1979), Hendry and Harrison (1974), and Hendry and Srba (1977); the references to later work
were reviewed by Hendry(1984). Cheung and Lai (1993a) estimated finite-sample critical values
for reduced-rank integration tests, Cheung and Lai (1995) estimated finite sample critical values
for ADF tests by taking into account dependence on the lag order in addition to sample size.
3.2 The response surface methods and experimental design
14
Figure 1 ADF distribution of ADF t-statistic
Response surface analysis applies to a system where the response of some variables depends
on a set of other variables that can be controlled and measured in experiment. Simulations are
conducted to evaluate the effect on the response variable of designed change in control variables.
A response surface describing the response variables as a function of control variables is then
estimated. When there are constraints on the design data, the experimental design has to meet the
requirements of the constraints. In general, the response surface changes can be visualized
graphically. The graph is helpful to see the shape of the response surface, hence, the function
f(x
1
,x
2
), where x
1
,x
2
are control variables can be plotted versus the level of x
1
and x2. The three
Note:: itr stands for number of iterations. T is the sample size and k is the lag order parameter. This
distribution was obtained from case of T=100, K=0, with iterations are 1000, 5000, 10000 and 50000.
The vertical line gives 5% critical values at different itr.
15
dimensional graph shows the response from the side is called response surface plot. Sometimes,
however, it is easier to see the response surface in two-dimensional graphs, which our study will
provide in addition to three dimensional graphs to show how response variables are affected by
control variables.
In our analysis, the response variable is the finite sample critical value of the RMA based
ADF (ADF
RMA
), and the control variables are the sample size (T) and the lag order parameter
(K). Our design covers 168 different pairing of (T,K), for which T varies from 50 to 700 with an
increment 50, and K={0,1,2,3,4,5,6,7,8,9,10,11}. In each experiment for given (T,K), the 1
percent, 5 percent, and 10 percent critical value are computed as corresponding percentiles of the
empirical finite-sample distribution based on a same number of iterations. (Figure 1 shows the
distribution in the case of T=100 and K=0 with iteration numbers 1000, 5000, 10000, 50000. The
vertical lines give the 5% critical value at different iterations. It is found that the distributions are
almost saturated at iteration 10000? . Therefore, in the following simulation study, the iteration
number is chosen 10000.
The data generating process considered in the simulation is a conventional random walk.
ttt
eXX +=
?1
( 20)
where e
t
is an independently distributed standard normal innovation. Sample series of X
t
are
generated by setting an initial value x
0
equal to zero, and creating T+50 observations, of which
the first 50 observations are discarded to avoid the problem of initialization. The GAUSS
programming language and subroutine RNDN are used to generate random normal innovations.
The regression model given by the equations in section 2.2 is more general than the DGP
considered. Higher-order DGP?s , for which e
t
can be autocorrelated, is allowed for in our tests
provided that the lag order parameter K is large enough to capture the dependence. Because if
16
the lag order is too small relative to the true lag order, the error term e
t
in the regression will no
longer be white noise. In this case, RMA based ADF test can be seriously biased, making
estimates of critical values inaccurate.
Selecting the functional form for response surface is not entirely arbitrary and need to be
satisfied some restrictions. In our case here, intuitively, with a given sample size T, the choice of
lag order parameter can effect on RMA base ADF test by determining the effective number of
observation available and number of parameters to be estimated in the test. As the sample size
increase to infinity, the effect of K on critical value may be diminished to zero. When sample
size goes to infinity, the effect of sample size on critical value should also be diminished to zero.
Taking these restrictions into consideration, we adopted the response surface polynomial
equation by Cheng and Lai (1995a). This experimental design generalizes Mackinnon?s (1991)
by including lag order, but omits that nuisance parameter that e
t
contains due to autocorrelation.
The polynomial equation is the following:
t
j
s
j
j
i
r
i
iKT
T
K
T
CV ???? +++=
??
==
)()
1
(
11
0,
(21)
where
KT
CV
,
is the critical value estimate for sample size of T and lag order parameter K,
t
? is
error term. r and s are respective polynomial orders for variables 1/T and K/T. The second
summation term capture the incremental contribution from the lag order. It is obvious that K/T
variable will be diminished to zero as value of T goes to infinity. Since both 1/T and K/T 0? as
,??T the intercept term gives an estimate of asymptotic critical value.
In order to find the response surface equation that fits the data well, a range of different value
of r and s have been considered, the test values to be found in next chapter.
17
Chapter 4
Monte Carlo Simulation Results
Considering different values of r and s (r = 1,2,1/2, s = 1,2,1/2) in estimating equation (21).
For the critical value in the tests of the constant with a trend and without a trend model, It was
found that data fits well at r=1 and s=1, the higher orders of polynomial term do not add much
power to the explanatory variables. So, the response surface equation can be written as:
tKT
T
K
T
CV ???? +++= )()
1
(
110,
(22)
Table 1 shows the results of response surface regression from equation (22). The tests with
and without a trend are conducted at 1%, 5%, and 10% significance levels. (6 response surface
regressions were run).
0
? gives intercepts at three different significance levels ,which are very
close to the asymptotic critical values that computed by Shin and So (2001) when the sample size
is large.
1
? is the coefficient of variable 1/T,
1
? is the coefficient of variable K/T. Note that in
both cases, variable 1/T showed to be statistically significant in all regressions at all three levels.
K/T variable showed up to be even more statistically significant in all regressions at all three
levels than the variable 1/T. This implies that the effect of lag order on critical values can be
more sensitive than that of the sample size in the finite sample. In other words, lag order in
addition to the sample size has a strong effect on finite sample critical values for RMA based test.
Various measures of data fit are also computed, including goodness of fit, the standard error of
regression and mean absolute error. The results in Table 1 show the ability of the response
surface equation (22) to fit the data, not only the intercepts are close to asymptotic critical value,
but also in the view of goodness of fit (R squares are high at all three significance level in all
18
regressions), and in the views of standard error and mean absolute error(both measures of
standard error and the mean absolute error are fairly small in all six regressions).
Table 1 Response surface estimation of Critical values for the ADF
RMA
statistic
Constant no trend Constant and trend
Coefficient
& statistics
1% 5% 10% 1% 5% 10%
-2.51033 -1.86617 -1.53426 -2.51148 -1.8583 -1.5118
?
0
0.00877 0.00602 0.00476 0.00497 0.00387 0.00337
4.41123 4.24757 4.13655 7.05625 6.7304 6.17495
?
1
1.99228 1.36615 1.08037 1.12764 0.8787 0.76494
-21.902 -13.7101 -10.4743 -32.536 -22.3816 -18.2898
?
1
0.27332 0.18742 0.14822 0.1547 0.12055 0.10494
R
2
0.98859 0.98604 0.98479 0.99832 0.99781 0.9975
? 0.08206 0.05627 0.0445 0.04645 0.03619 0.03151
Mean(|?|) 0.05901 0.0362 0.02924 0.03335 0.02393 0.01971
Max(|?|) 0.58802 0.41714 0.34164 0.15864 0.23551 0.22572
notes: The response surface regression is given by equation (22). The ADF
RMA
, Corresponding heterorskedasticity-consistent
standard errors for coefficient estimates are put in parentheses. ? represents the standard error of the regression. Mean gives
the mean absolute error of the response surface prediction against estimated critical value from simulations.
Some finite sample critical values were estimated by Shin and So (2001) for RMA based
ADF unit root tests based on K=0. It is interesting to compare those estimates directly with the
response estimate of critical values obtained here as displayed in Table 2. The estimates provided
by Shin and So (2001) are given in the third column. The first column is sample size, The second
column is the significance level (1%, 5%, and 10%). The last four columns contain response
surface estimates for K=0, K=4, K=7, K=10. Not unexpectedly, when K equals zero, the two
estimates are matched very closely. However, if we look at K=4, K=7 and K=10, it is evident
that critical values obtained (5-7 column) are different from those obtained by Shin and So
(2001). Note that differences in those estimates decrease as sample size increase. Therefore, if
19
lag order is greater than one, using the asymptotic critical values that tabulated based on K=0 can
be misleading, which causes one to reject nonstationarity too often.
Table 2a Lag Order and Finite-sample Critical Values
Constant no trend
Sample
Size
Sig.
Level
SS
Estimate
K=0 K=4 K=7 K=10
10% -1.54 -1.52061 -2.2064 -2.87791 -3.70069
5% -1.88 -1.87178 -2.8269 -3.57357 -4.71707 50
1% -2.57 -2.53395 -4.06249 -5.43865 -6.9005
10% -1.54 -1.52744 -1.8864 -2.17908 -2.49887
5% -1.88 -1.89109 -2.3313 -2.76876 -3.11599 100
1% -2.54 -2.55762 -3.32239 -3.8541 -4.52167
10% -1.54 -1.53696 -1.70023 -1.81685 -1.90924
5% -1.88 -1.85298 -2.0581 -2.24729 -2.35806 250
1% -2.53 -2.47415 -2.82888 -3.07332 -3.2957
10% -1.54 -1.54083 -1.61355 -1.66143 -1.71053
5% -1.88 -1.86064 -1.95929 -2.04771 -2.07821 500
1% -2.53 -2.55082 -2.67263 -2.78071 -2.93049
note: The finite-sample critical values tabulated for the RMA based ADF test. The third column gives the estimated of critical
values provided by Shin and so, their critical values are tabulated based on K=0.
By comparing, we note that when lag order is greater than 1 in finite samples, the RMA
based test with asymptotic critical values can be oversized even when the sample size is fairly
large (e.g., T=500). Table3 and Table 4 contain the size samples for 50, 100, 150, 200, 250, 300,
350, 400, 500 and 700. Lag order parameter for k=0, k=1,k=4, k=7, and k=10, for the constant
with a trend and the constant without a trend case respectively.
20
Table 3 Lag Order and Finite-sample Critical Values (with constant only)
Constant no trend
Sample
Size
Sig.
Level
K=0 K=1 K=4 K=7 K=10
10% -1.52061 -1.71129 -2.2064 -2.87791 -3.70069
5% -1.87178 -2.09758 -2.8269 -3.57357 -4.71707
50
1% -2.53395 -2.91018 -4.06249 -5.43865 -6.9005
10% -1.52744 -1.61213 -1.8864 -2.17908 -2.49887
5% -1.89109 -1.97353 -2.3313 -2.76876 -3.11599 100
1% -2.55762 -2.74225 -3.32239 -3.8541 -4.52167
10% -1.53581 -1.59643 -1.73442 -1.93768 -2.13946
5% -1.88011 -1.95934 -2.14825 -2.40002 -2.63186 150
1% -2.51324 -2.71848 -3.01869 -3.37777 -3.80271
10% -1.56443 -1.61965 -1.72762 -1.85436 -1.96858
5% -1.92177 -1.98652 -2.15246 -2.31493 -2.47407 200
1% -2.52997 -2.66364 -2.94306 -3.2297 -3.52086
10% -1.53696 -1.58198 -1.70023 -1.81685 -1.90924
5% -1.85298 -1.91337 -2.0581 -2.24729 -2.35806 250
1% -2.47415 -2.56988 -2.82888 -3.07332 -3.2957
10% -1.54128 -1.57486 -1.67971 -1.75265 -1.81263
5% -1.87465 -1.92166 -2.05973 -2.12496 -2.26618 300
1% -2.58249 -2.62431 -2.80629 -2.96733 -3.1465
10% -1.52718 -1.5624 -1.64502 -1.73164 -1.81761
5% -1.87848 -1.91812 -2.03097 -2.14781 -2.23945 350
1% -2.54502 -2.65563 -2.80415 -3.0574 -3.17389
10% -1.57436 -1.59626 -1.67024 -1.71469 -1.78129
5% -1.90003 -1.93776 -2.04227 -2.11551 -2.19669 400
1% -2.56224 -2.62814 -2.76469 -2.93333 -3.03889
10% -1.54083 -1.5592 -1.61355 -1.66143 -1.71053
5% -1.86064 -1.89132 -1.95929 -2.04771 -2.07821 500
1% -2.55082 -2.60103 -2.67263 -2.78071 -2.93049
10% -1.54923 -1.57325 -1.60717 -1.63943 -1.67771
5% -1.88772 -1.90553 -1.96961 -1.99999 -2.06051 700
1% -2.49949 -2.4997 -2.5837 -2.64767 -2.76336
21
Table 4 Lag Order and Finite-sample Critical Values (with constant and trend)
Constant no trend
Sample
Size
Sig.
Level
K=0 K=1 K=4 K=7 K=10
10% -1.47205 -1.80006 -2.77986 -3.86395 -5.13153
5% -1.84173 -2.2253 -3.39948 -4.79138 -6.25507
50
1% -2.52754 -3.10357 -4.81799 -6.86265 -8.92842
10% -1.49502 -1.66452 -2.15989 -2.71169 -3.27288
5% -1.84282 -2.06463 -2.65397 -3.34345 -4.01644 100
1% -2.5349 -2.86154 -3.73827 -4.79021 -5.74805
10% -1.47004 -1.60452 -1.92896 -2.30694 -2.66329
5% -1.82663 -1.97507 -2.41763 -2.8466 -3.27923 150
1% -2.49265 -2.69174 -3.31183 -3.91968 -4.71777
10% -1.53278 -1.62822 -1.8635 -2.1257 -2.37419
5% -1.84629 -1.99704 -2.30306 -2.61474 -2.9213 200
1% -2.4987 -2.66193 -3.11449 -3.68513 -4.04091
10% -1.50982 -1.5751 -1.78759 -1.99093 -2.17857
5% -1.83116 -1.92892 -2.21428 -2.43789 -2.69948 250
1% -2.49963 -2.64609 -2.97453 -3.43644 -3.76817
10% -1.51862 -1.5623 -1.73839 -1.9168 -2.07993
5% -1.85901 -1.93811 -2.13809 -2.34395 -2.53138 300
1% -2.47546 -2.58012 -2.88916 -3.25563 -3.53404
10% -1.49331 -1.54098 -1.67846 -1.84522 -1.99903
5% -1.83554 -1.90016 -2.06524 -2.26431 -2.472 350
1% -2.473 -2.57033 -2.85746 -3.13251 -3.41309
10% -1.52667 -1.56711 -1.69544 -1.84959 -1.95867
5% -1.85861 -1.92469 -2.09782 -2.26432 -2.41766 400
1% -2.55818 -2.61636 -2.84022 -3.11207 -3.33235
10% -1.52286 -1.55684 -1.6599 -1.76309 -1.85708
5% -1.88443 -1.92506 -2.02402 -2.16568 -2.25709 500
1% -2.54 -2.63973 -2.75189 -2.94264 -3.17992
10% -1.51037 -1.54111 -1.61852 -1.67488 -1.74792
5% -1.84951 -1.88258 -1.95933 -2.04298 -2.13594 700
1% -2.4927 -2.5482 -2.69819 -2.77213 -2.93323
22
Figure2 Plots of Monte Carlo-Estimated critical values
Note: figure 2 is estimated value for various RMA based ADF test, where T is the sample size, and K is
the lag order parameter. In each graph, the vertical axis gives the Monte-Carlo estimated values
corresponding to different combination of T and K
23
Finally, the Monte Carlo simulation critical values are plotted as Figure 2 for various
ADF
RMA
tests. Three dimensional graphs provide a sense of the numerical fluctuation in the
critical values as a function of the lag order parameter K and sample size T. The 6 graphs are
arranged in a 3 by 2 matrix to allow efficient comparison across types of tests and across the test
sizes. To see how critical values are affected by sample size T and lag order K more clearly, we
also provide two dimensional graphs (Figures 3 and 4).
Three dimensional graphs of Figure 2 show that the presence of a signed the correction to the
asymptotic critical value at k=0 and ??T . This is consistent with the fact that
1
?
,
which
determines the effect of pure sample size, are positively signed in response surface. The effect of
lags is unambiguously signed in all response surfaces. The graphs also show that critical values
decrease (while absolute values increase) when k increases. Therefore, the test with asymptotic
critical values is overall oversized. A comparison between graphs shows that the similar speeds
at which finite-sample critical values approach the asymptotic levels.
Figure 3a and 3b are two dimensional graphs for the constant only case. The color bar in
Figure 3a represents different number of k, which varies from 0 to 11. The color bar in Figure 3b
represents different sample sizes, which varies from 50 to 700. By observing these graphs, we
have found that (1) given a k, critical values increase as sample sizes increase and the signs are
consistent with
1
?
in response surface. Finite-sample critical values converge gradually to their
asymptotic critical value; (2) given a relatively small sample size, critical values linearly
decrease (absolute critical values increase) as k increases, which causes the test with asymptotic
critical values to be oversized. When sample size (T) goes to infinity, critical values with k and
without k converge to the asymptotic critical value.
24
Figure 4a and 4b are 2-D plots for the constant with a trend case. It is clear that the pattern of
critical value as a function of T and K is very similar to that of the constant only case. The
asymptotic critical values for both time trend and no trend case are also nearly the same.
However, the variations of critical value in time trend case is greater than that of no trend case,
which means the test with asymptotic critical values in constant with time trend becomes more
oversized.
In summery, based on those simulation results, we found that asymptotic critical values are
valid only when lag order is one.
25
Figure 3 RMA based critical value as a function of (a) T and (b) K in the case of constant
without time trend.
(a)
(b)
26
Figure 4 RMA based critical value as a function of (a) T and (b) K in the case of constant and
time trend.
(b)
(a)
27
Chapter 5
Conclusion
Usually the practice of applying the RMA based unit root test has largely ignored the
sensitivity of the lag order, which is often justified by asymptotic results that the limiting
distribution of the test is free of the lag order. Even though the lag order may not affect the
critical values when T goes to infinity, this practice may not be valid. Cheung and Lai (1995a)
showed that critical values for the ordinary least square (OLS) based ADF test depend on the lag
order in finite samples. We extend their work here by examining a more powerful RMA based
ADF unit root test. Our Monte Carlo simulation results show that asymptotic critical values for
the test are valid only when the lag order is one (k=0). When the lag order is greater than one, the
RMA based unit root test with asymptotic critical values tends to be overall over-sized when the
deterministic terms are allowed.
28
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31
Appendix: Outline for Generating Critical Values
1. Given T, generate N sets of random walk observations, where N=10000. Each series is
generated by setting initial value equals zero, and creating T+50 observations, of which the first
50 observations are discarded to avoid problem of initialization. The GAUSS programing
language and subrouinte are used to generate random normal innovations.
2. In the case of the constant with no trend, obtain recursively adjusted mean value by equation
(10). For the case of constant with trend, equation (15) is calculated and the recursively adjusted
trend mean value
11
~
'
?? tt
Z ? is thus obtained.
3. For each parameter K, test regression of equation (11) or (18) for k=0 and equation (13) or (19)
for k>0 to estimate
RMA
?? . Then RMA based ADF t statistics are calculated.
4. From N statistic for each K and for RMA based ADF test statistic, obtain ? % percentile,
? =1, 5, 10, which gives ? % critical value.