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This is to certify that the

dissertation entitled

Testing Economic Time Series
For Stationarity and Nonstationarity

presented by

Yongcheol Shin

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Economics

936.2%

Major professor

Date 9.! 351/ 6'2.

MS U is an Affirmative Action/Equal Opportunity Institution 0— 12771

 

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MSU lo An Afflrmetlvo Action/Equal Opportunity Institution

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TESTING ECONOMIC TIME SERIES
FOR STATIONARIT Y AND NONSTATIONARITY

By

Yongcheol Shin

A DISSERTATION
submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Economics

1992

ABSTRACT
TESTING ECONOMIC TIME SERIES
FOR STATIONARITY AND NONS'I‘ATIONARITY
By

Yongcheol Shin

It is a well-established empirical fact that standard unit root tests fail to reject
the unit root hypothesis for many economic time series. However, these results do not
indicate strong evidence against relevant trend stationarity alternatives, because it is
well-known that unit root tests are not very powerful.

Recently, various attempts, including a Bayesian approach, have been made to
reconsider the important problem of distinguishing trend stationary and unit root
processes. However, there have been very few previous attempts to test the null
hypothesis Of stationarity my. Kwiatkowski, Phillips, Schmidt, and Shin (1992,
KPSS) propose an LM test Of the null hypothesis that an observable series is
stationary around a deterministic trend, using the components representation in which
the series is decomposed into the sum of deterministic trend, random walk, and
stationary error.

This dissertation extends the KPSS test statistic for stationarity in two ways.
First, finite sample size and power of the KPSS statistic for stationarity are extensively
studied in a Monte Carlo experiment. Next the use of the KPSS statistic as a unit root

test is suggested, because the KPSS statistic is consistent and a different limning

 

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distribution is Obtained under the hypothesis that the series is difference stationary.

Both tests are applied to the Nelson-Flosser data, and for many of these series
it is not very clear whether they contain a unit root or are trend stationary. These
results are quite consistent with recent (inconclusive) empirical findings.

One implication of the above empirical findings is that many economic time
series may be in the region of "near stationarity." A lot of Monte Carlo studies have
shown that standard unit root tests have severe size distortions when the process is
nearly stationary. This dissertation also considers the asymptotics of standard unit root
tests in this case using generalized "nearly stationary model." It is found that the
above size distortion problem is well predicted by our asymptotics. It is also argued
that the superiority of the augmented Dickey-Fuller statistic is not established and that

more efficient estimation techniques will be needed to improve the tradeoff between

size distortions and low power.

Dedicated to my family

iv

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,|

Ann.“

ACKNOWLEDGEMENTS

My first thanks should be devoted to Professor Peter Schmidt, my dissertation
chairman for his incomparable guidance of this dissertation. I appreciate his kindness,
sincerity, and wit all the time. He has supported me in many ways and suggested
many ideas developed in this dissertation. I have really enjoyed working with him.

I also thank the other dissertation committee members, Richard T. Baillie,
Robert H. Rasche, and Jefferey Wooldridge for their helpful comments. Finally, I
appreciate assistance by many staffs and faculties in department of economics.

My best thanks must go to my parents and my wife, Malhee for their
invaluable support. I also wish to express my greatest delight to my beloved son,

Daniel. Finally, I am grateful to my two younger brothers for their encouragement.

 

 

.>.

'ne

TABLE OF CONTENTS

LIST OF TABLES .....................................
CHAPTER 1 : INTRODUCTION ...................................
1. GENERAL INTRODUCTION ...........................
2. UNIT ROOT TESTS AND ERROR AUTOCORRELATION ......
3. UNIT ROOT TESTS UNDER NEAR STATIONARITY ........
4. TESTING THE NULL HYPOTHESIS OF STATIONARITY ...
5. THE KPSS TEST AS A UNIT ROOT TEST ..............
6. PLAN OF THE DISSERTATION .......................
CHAPTER 2 ° FINITE SAMPLE PERFORMANCE OF THE

STATIONARITY TEST

1. INTRODUCTION ...................................
2. THE KPSS TEST FOR STATIONARITY .................
3. FINITE SAMPLE PERFORMANCE ......................
1. SIZE .......................................
2. POWER ......................................
3. COMPARISON TO THE SAIKKONEN AND LUUKKONEN
TEST .......................................
4. APPLICATIONS TO THE NELSON—PLOSSER DATA ........
5 SUGGESTIONS AND CONCLUDING REMARKS .............
APPENDIX A .........................................
APPENDIX B .........................................
TABLES .............................................
CHAPTER 3 ' TESTING FOR A UNIT ROOT: A DUAL APPROACH
1. INTRODUCTION ...................................
2. THE KPSS TEST AS A UNIT ROOT TEST ..............
3. COMPARISON TO OTHER SIMILAR TESTS ..............
4. FINITE SAMPLE BEHAVIOR .........................
1. SIZE ........................................
2. POWER .......................................
3. COMPARISON WITH THE DICKEY AND FULLER
UNIT ROOT TESTS: SIZE ......................

4. COMPARISON WITH THE DICKEY AND FULLER

vi

17
19
25

28

33
35
36

39
41

43

59
60
65
67
68
72

74

 

 

 

UNIT ROOT TESTS: POWER ..................... 76

5. APPLICATIONS TO THE NELSON-PLOSSER DATA ........ 78
6. CONCLUDING REMARKS ............................. 81
TABLES ............................................. 85

CHAPTER 4 : ASYMPTOTIC DISTRIBUTION OF UNIT ROOT TESTS
WHEN THE PROCESS IS NEARLY STATIONARY

1. INTRODUCTION ................................... 103
2. MODEL AND THE ASYMPTOTIC RESULTS ............... 106
3. SIMULATION RESULTS ............................. 114
4. DISCUSSIONS AND CONCLUDING REMARKS ............. 120
APPENDIX ........................................... 124
TABLES ............................................. 141
CHAPTER 5 : CONCLUSION ....................................... 161
LIST OF REFERENCES ................................. 168

vii

o
A"

 

CHAPTER 2

Table
Table
Table
Table
Table
Table

Table
Table
Table
Table

Table

2—7
2-8
2—9

2-10

CHAPTER 3

Table
Table
Table
Table
Table
Table
Table
Table

Table

Table

Table

3-0
3-1
3-2
3-3
3-4
3-5
3-6
3—7

LIST OF TABLES

Critical Values for Stationarity Test ... 43
Size for iid Errors (A - 0) ............. 43
Size for AR(l) Errors (A - O) ........... 44
Size for MA(1) Errors (A - 0) ........... 45
Power for iid Errors .................... 46
Comparison of the Limiting Power with
Predictions Based on Asymptotics ........ 47
Comparison of the Power of the nu test to
Tanaka's Limiting Power ................. 48
Power for AR(1) Errors .................. 49
Power for MA(1) Errors A .................. 53
Size Comparison of the nu test to

The Saikkonen and Luukkonen Test ........ 57
Power Comparison of the "u test to

The Saikkonen and Luukkonen Test ........ 58
Critical Values for Unit Root Tests ..... 85
Size with iid Errors .................... 86
Size with AR(1) Errors .................. 87
Size with MA(1) Errors .................. 91
Power with iid Errors (A - 0) ........... 95
Power with AR(l) Errors (A - O) ......... 96
Power with MA(1) Errors (A - 0) ......... 97

Size Comparison of the KPSS Unit Root Tests

To The Dickey—Fuller Tests

Under MA(l) Errors ...................... 98
Power Comparison of the KPSS Unit Root Tests

To The Dickey-Fuller Tests .............. 99
The KPSS Unit Root Tests Applied To

The Nelson-Plosser Data ................. 101
Augmented Dickey-Fuller Unit Root Tests
Applied To The Nelson—Flosser Data ...... 102

viii

 

CHAPTER 4
Table 4—1 (a)
Table 4-1 (b)
Table 4-2
Table 4-3
Table 4—4
Table 4—5
Table 4-6

Table 4-7 (a)

Table 4-7 (b)

5 % Fractiles of The Actual Sampling
Distribution ...........................
95% Fractiles of The Actual Sampling
Distribution ...........................
Fractiles of The Predicted Distribution
For 6 - 1/4 ............................
Fractiles of The Predicted Distribution
For 6 - 1/2 ............................
Fractiles of The Predicted Distribution
For 6 - 5/8 ............................
Fractiles of The Predicted Distribution
For 6 - 3/4 ............................
Fractiles of The Predicted Distribution
For 6 - 7/8 ............................
Comparison of the Predicted Distribution
With the Actual Sampling Distribution

(p test) ...............................
Comparison of the Predicted Distribution
With the Actual Sampling Distribution

(1 test) ...............................

ix

CHAPTER 1

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CHAPTER 1

INTRODUCTION

1.1 General Introduction

Many economic time series are clearly nonstationary, and one important
statistical issue is the appropriate representation of this nonstationarity. For simplicity,

assume that any deterministic trend is linear, so that we can write

(1) y. = \v +5.1 + X. t = 1.°~.T.

where y, is the observed series, (\V + it) represents deterministic trend, and X, is the
unobserved stochastic deviation of y, from deterministic trend. If X, is stationary, then
y, is often said to be "trend stationary", and the long-run behavior of y, is essentially
determined by its deterministic trend component. However, if X, is 1(1), so that AX, is
stationary, y, is said to be "difference stationary", and Ay, = g + AX, so that changes in
y, contain a component that is fundamentally unpredictable even in the long run. The
trend stationary (TS) and difference stationary (DS) time series have very different
long run properties, and this has important economic and statistical implications.
There has been extensive interest in the use of the autoregressive integrated
moving average (ARIMA) process for modelling nonstationary time series. Ignoring

deterministic trend, for the moment, suppose that the time series is represented by the

ARV.

ARMA process

(2) y. = 13y” + 2,, E. = n. + Oun-

If B = 1, y, has a "unit root" in its AR representation, and y, is difference stationary so
long as 6 is not equal to -1. In fact y, is a random walk if B = 1 and 6 = 0. Unit root
tests typically test the hypothesis B = 1, and 9 is a nuisance parameter. However,
these roles can be reversed. If B = 1 and 9 = -1, y, is white noise. More generally,
we can test the stationarity hypothesis 6 = -1, in which case B is a nuisance parameter.
If a series is generated by a member of the linear TS class we should fail to reject the
hypothesis of a unit MA root in the ARMA model for its first difference, and if it is
generated by a member of the DS subclass we should fail to reject the hypothesis of a
unit AR root in the ARMA model for its levels.

As noted above, the difference between DS and TS series may be economically
important, since a unit AR root implies long run persistence, in the sense that at least
part of the effects of random shocks on macroeconomic variables are permanent.
Correct treatment of the stationary or nonstationary nature of the data is also necessary
for meaningful statistical inference, owing to the spurious regression phenomenon
pointed out by Granger and Newbold (1974) and Phillips (1986). Thus, it is important
to be able to distinguish a series with a unit root from a stationary series. General

surveys of the unit root literature are given by Diebold and Nerlove (1990) and

Campbell and Perron (1991).

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1.2 Unit Root Tests and Error Autocorrelation

Attempts to distinguish the DS series from the TS series have generally taken
the form of a test of the null hypothesis of a unit AR root against the alternative of
stationarity. Most of the existing unit roots tests are variants of the Dickey and Fuller
(DF) tests provided by Fuller (1976) and Dickey and Fuller (1979). The BF unit root

tests are based on the regressions:

(3) Yr = BYt-l + 8t
(4) Yr = (I. + BYt-l + 8t
(5) Yz=a+BYr1+5t+€r

for t = 1,...,T. In each case, the unit root hypothesis is B = 1. Two types of test
statistics are used: one is the normalized coefficient test statistic T(B — 1), where B is
the OLS estimate of B; this yields the DF statistics B, 6,, and 6, from regressions (3),
(4) and (5), respectively. The other is the usual t—statistic for testing the hypothesis B
= 1, which yields the DF statistics 9, ‘9, and‘f, corresponding to the same three
regressions.

The DF regressions differ in the way that they handle level and deterministic
trend. The regression (3) does not allow for non-zero level or trend under the
alternative. The equation (4) allows for linear deterministic trend under the null, but it

allows only for non-zero level under the alternative. Finally, regression (5) allows for

'6'

 

Nor:

4

non-zero level and linear and quadratic trends under the null, but implies level and
linear trend under the alternative. The a, ands, tests based on regression (5) are the
most commonly used in econometric work, because inclusion of the trend term (St) in
(5) is necessary for the tests to be consistent against TS alternatives.

Schmidt and Phillips (1991, SP) suggest an alternative set of unit root tests,

based on the parameterization:

(6) Yr=V+§i+Xv Xi=BXt-1+Ev

Note that this parameterization is also used by Bhargava (1986), and it mimics the
form of equation (1) above. The SP test statistics are based on the LM (score)

principle for the null hypothesis B = 1 in equation (6). The test statistics are derived

from the regression:
(7) Ay, = constant + ¢S,_, + error, t = 2,°--,T

where S'H = y,,l - {Bx - E(t - 1) and the restricted MLE’s of § and wx = w + Xo are
given by: E = (yT - y1)/(T - 1), and {5, = y1 - E. Let 6 be the OLS estimate of (1) for

(5). The test statistics are given by:
(8) 5 = 1’43
(9) T = t-statistic for the hypothesis 4: = 0.

SP show that 5 and "E are monotonic transformations of each other, so the tests are

identical in the absence of corrections for error autocorrelation. Also, 5 is almost

5
identical to the R2 statistic of Bhargava (1986).

The main difference between the DF and SP parameterizations is that the
meanings of the parameters or and 8 in (4) and (5) are different under the null and
under the alternative, while in the SP parameterization (6), \V and E, represent level and
linear trend respectively under both the null and the alternative. The distribution of
the SP test statistics 5 and "t", and of the DF test statistics 6, and 6,, are independent of
‘1’ and § in (6); this is the further evidence of the usefulness of equation (6) to
represent the data generating process.

Since the tabulated distributions for the DF and SP test statistics are obtained
under the assumption that the errors in the model (i.e, a, in equations (3), (4), (5), and
(6)) are iid, they are not expected to be robust to more general error structures, in
particular to the presence of autocorrelated errors. Furthermore, it is known from
Phillips (1987), Phillips and Perron (1988) and Schmidt and Phillips (1991) that error
autocorrelation affects the distributions of the test statistics even asymptotically.
Therefore, modifications of the basic DF and the SP test statistics have been developed
that allow for error autocorrelation. These modifications can be put into four groups.

First, augmented Dickey-Fuller (ADF) tests are proposed to accommodate error
autocorrelation by adding lagged differences of y, to the regressions (3) - (5). Said
and Dickey (1984, 1985) show that, if the number of lagged differences is suitably
chosen, the ADF test statistics have the same asymptotic distribution as the original
DF test statistics would have under iid errors. If the errors are AR(p), the number of

lagged differences must be at least as large as p. If the errors have an MA

in

If:

6

component, the number of lagged differences is allowed to increase with sample size,
though at a slower rate (e.g., at the rate T‘”). Lee (1990) proposes an analogous
augmented version of the SP tests.

Second, Phillips (1987) and Phillips and Perron (1988, PP) use semiparametric
corrections to the DF statistics to develop general tests to allow for a wide class of
weakly dependent and heterogeneous errors. The limiting distributions of the corrected
test statistics are the same as the original DF test statistics would have under iid
errors. SP provide similar semiparametric corrections for their tests.

Third, Hall (1989) has considered unit root tests based on instrumental
variables (IV) estimation of the DF regressions. He assumes MA(q) errors, with q
treated as known and y,.,,, with k > q, is used as the instrument for y,,,. Lee and
Schmidt (1991) also propose similar 1V versions of the SP test statistics, where the
instrument for SH in (7) is S,,.,, with k > q.

Finally, Choi (1990) develops tests based on GLS estimation. This requires a
parametric (ARMA) form for the autocorrelation.

A large body of simulation evidence has shown that these methods of
accommodating error autocorrelation can perform poorly in finite samples. Phillips
and Perron (1988), Schwert (1989), Kim and Schmidt (1990) and Lee (1990) have
shown that the uncorrected DF and SP tests reject the true null hypothesis too often in
the presence of negative autocorrelation and too seldom in the presence of positive
autocorrelation. These size distortions can be quite considerable. For example, with

MA(l) errors with 0 = -0.8, the size of the tests (the probability of rejecting the null

7
when it is true) is almost unity, even for T as large as 500 or 1000. The Phillips-

Perron corrected DF and SP tests perform somewhat better than the uncorrected tests,
but still suffer from considerable size distortions even for surprisingly large sample
sizes. The augmented DF and SP tests also perform somewhat better than the
uncorrected tests, with the extent of the improvement depending on the number of
augmentations. With a sufficiently large number of augmentations, the size of the
augmented tests becomes more or less correct, even for cases in which the errors are
strongly autocorrelated. However, this result is less optimistic than it might first seem,
because the tests with many augmentations have almost no power.

Hall finds that his IV tests are a significant improvement over the uncorrected
DF or the PP tests, when the errors are MA(l). Lee and Schmidt (1991) also provide
fairly optimistic results for the IV versions of the SP tests, which have surprisingly
smaller size distortions and greater power than other tests of similar size. Pantula and
Hall (1991) provide some moderately optimistic results for the Hall’s IV tests when
the errors are ARMA. Similarly, Choi’s results for his GLS-based tests are fairly
good for most parameter values. However, no testing procedures seem to work well

in finite samples with strongly negatively correlated errors.

1.3 Unit Root Tests Under Near Stationarity

To interpret the Monte Carlo evidence summarized in the last section, we

return to the ARMA representation of y, given in equation (2). Suppose that B = 1 so

8

that there is a unit AR root. As 9 —> -1, y, approaches stationarity. Correspondingly,
the process can be called "nearly stationary" when 9 is close to but not equal to minus
one; for example, when 0 = -0.8.

It is not surprising that most unit root tests perform very poorly when the
process is nearly stationary. For example, Blough (1989) argues that there is no
discontinuity between unit root and stationary processes. For a given sample size,
every stationary process can be arbitrarily well approximated by a unit root process
which is nearly stationary. Correspondingly, a true level a test of theth root null
cannot have power greater than (1 against stationary alternatives. Therefore, the tests
with small size distortions in the presence of strongly negatively correlated errors
might be expected to be essentially without power. For example, Blough’s Monte
Carlo simulations show that, for T = 100, the ADF unit root test with 12 lags has
power of only 20% against white noise. Power is even lower when the stationary
process is serially correlated. Therefore, it seems that no tests may survive in the
presence of strongly autocorrelated errors in terms of bo_th_ size and power of the tests.
The tests with correct size have poor power and the tests with high power have serious
size distortions.

The preceding discussion reflects the fact that, in the nearly stationary case, the
unit root asymptotic distributions are poor guides to the finite sample distributions of
the test statistics, even for fairly large sample sizes. Thus it may be useful to find
other types of asymptotic distributions to approximate finite sample distributions in

this case. The important step in this direction has been taken by Pantula (1991), who

9

investigates the behavior of some unit root test statistics under the null of a unit root

when the process is nearly stationary, using the local approximation for 9 = -1:
(10) 9=-1+1/I'5, 8>0.

Combining (2) and (10), y, becomes a random walk as 8 approaches zero; however,
for fixed 5, y, approaches white noise as T —> oo. Pantula shows that the asymptotic
distribution of the unit root test statistics depends on the speed with which 6
approaches minus one (i.e., the value of 5) as well as the sample size, T. Pantula uses
his asymptotic distributions to infer differences among tests in their sensitivity to near
stationarity, and, based on these differences, he suggests the use of the augmented
Dickey-Fuller (ADF) test. However, note that he does not consider the power of the
tests.

In Chapter 4, we will investigate the behavior of the DF and the SP unit root
test statistics under the null of a unit root when the process is nearly stationary by

using the more general local approximation for 9 = -1:
(11) e=-1+cn‘,C>o,5>o.

With two parameters (C and 8) instead of only one, we can hope to find more accurate
asymptotic approximations to the finite sample distributions of the test statistics.
Furthermore, our concern is somewhat different from Pantula‘s. We will make
detailed comparisons of our asymptotic approximations and the true finite sample

distributions (calculated by a Monte Carlo simulation), to see under what conditions

10

these asymptotic distributions are accurate enough to be useful. This is actually a
logical prior step to Pantula’s type of analysis, since there is no point in choosing tests

based on inaccurate asymptotic approximations.

1.4 Testing the Null Hypothesis of Stationarity

Nelson and Plosser (1982) failed to reject the hypothesis that long historical
time series for the US. are difference stationary, using the DF tests. Similar results
have been obtained using the SP tests and other types of unit root tests, which have
generally failed to reject the null hypothesis of a unit root in many macroeconomic
time series.

However, it is important to note that in this empirical work the unit root is set
up as the null hypothesis to be tested, and the way in which classical hypothesis
testing is canied out ensures that the null hypothesis is accepted unless there is strong
evidence against it. Therefore, an alternative explanation for the common failure to
reject a unit root is simply that standard unit root tests are not very powerful against
relevant alternatives. For example, see Dejong ga_l. (1989). This point is also
discussed in the recent survey paper by Campbell and Perron (1991).

Therefore, in trying to decide whether economic data are stationary or
integrated, it would be useful to have available tests of the null hypothesis of
stationarity as well as tests of the null hypothesis of a unit root. There have been

relatively few previous attempts to test the null hypothesis of stationarity. See Park

11
and Choi (1988), Rudebush (1990), Dejong et a1. (1989), for examples. These are

 

reasonable first attempts to test stationarity, but they suffer from the lack of a
plausible model in which the null of stationarity is naturally framed as a parametric
restriction.

Recently, Kwiatkowski, Phillips, Schmidt, and Shin (1992, KPSS) propose a
test of stationarity based on the decomposition of the series into deterministic trend,

random walk, and stationary errors:
(u) m=¢+m+w
(13) Y. = 7m + Ur

Here §t represents linear deterministic trend, 7, represents random walk (so the u, are
iid), and v, is the stationary error. This parameterization provides a plausible
representation of both stationary and nonstationary variables, and leads naturally to a
test of the hypothesis of stationarity. Note that the decomposition into stationary and
random walk components is a popular way of thinking about the properties of a time
series in macroeconomics applications. See Nelson and Plosser (1982), Watson
(1986), Clark (1987), and Cochrane (1988). KPSS derive the statistic for stationarity
as the LM test of the null hypothesis 0,2 = 0; i.e., the variance of the random walk
component of y, equals zero. Thus the null hypothesis implies that the series is trend
stationary.

We can note that the model of (12) and (13) implies

12
(14) Ay, = g + u, + Av,.

Define w, = u, + Av, as the error in this expression for Ay,. If u, and v, are iid and
mutually independent, then w, has a non-zero one period autocovariance, with all other
autocovariances equal to zero, and accordingly it can be expressed as an MA( 1)

process. Thus the KPSS model is equivalent to the ARIMA model
(15) Y: = BYt-l + W” Wt = 8, + eel-1’ ‘3 = 1’ 8! iid

This is of the same form as equation (2) above. The connection between 6 and the
variances of u and v is straightforward. Let A = (Sf/6,2. Then we can get the

relationship between 0 and A. as
(16) 9 = -{(7l-+2) - IMAMHWVZ. or 7» = -(1 + 9)2/9.

where A. 2 0, |9| S 1. Thus 2. = 0 corresponds to 9 = -1 (stationarity) while A = 00
corresponds to 9 = 0 (a pure random walk). Equation (16) shows an interesting
connection between the KPSS tests and the usual DF tests. The BF tests test B = 1
assuming 6 = 0; 9 is a nuisance parameter. KPSS effectively test 6 = -1 assuming B =
1; now B is a nuisance parameter.

Since the reduced form of the KPSS model is ARIMA(0,1,1), a test of A. = 0
corresponds to a test of 9 = -1. The model is strictly noninvertible under the null and
it follows from the results of Sargan and Bhargava (1983) that classical procedures
cannot be applied in this case. An LM test statistic can be constructed but, as noted in

Tanaka (1983), its asymptotic distribution is nonstandard. A locally best invariant

Ii.’

.’

k

E)

13

(LED test of the hypothesis that 0,2 is zero can also be constructed and is in fact the
same as the LM test. See Nabeya and Tanaka (1988).

KPSS derive the LM statistic as a special case of the statistic developed by
Nabeya and Tanaka (1988) to test for random coefficients. Let e,, t = 1,2,...,T, be the
OLS residuals from the regression of y, on an intercept and time trend. Let 6,2 be the
usual estimate of the error variance from this regression. The LM statistic for

stationarity is derived as:
T A
(17) LM = 2183/63
t:
where S, is the partial sum process of the residuals
t
(18) S, =.21ei , t = 1,2,...,T
1:

However, the LM derivation assumes that the stationary enors v, in (12) are normal
white noise. If they are not white noise, but satisfy the regularity conditions of
Phillips (1987), the asymptotic distribution of the statistic is the same as under white
noise errors if we divide by an estimate of the "long run variance" 02 rather than the
innovation variance of. Let 02 be any consistent estimate of the long run variance.

Then KPSS define the statistic
A T A
(19) n, = T2331 S,2 / 0'2
Under the null hypothesis of = 0, they establish the asymptotic distribution

(20) 1i. —> 1: V.(r>dr

14

where V2(r) is the second-level Brownian bridge given by
(21) V2(r) = W(r) + (2r - 313)W(1) + (-6r + 6155(1)l W(r)dr

and W(r) is Standard Brownian motion.

The same statistic may also arise in other contexts. Saikkonen and Luukkonen
(1990) derive the statistic as the locally best unbiased invariant test of the hypothesis 6
= -l in the model Ay, = e, + 68,, with the e, iid normal. Based on the discussion
above, this is not a surprising result.

In Chapter 3 we will consider the finite sample properties of the KPSS
stationarity test, which uses semiparametric corrections for error autocorrelation in the
presence of the autocorrelated errors. These results will be compared with the results
for the Saikkonen and Luukkonen test, which uses parametric corrections based on an
assumed ARMA model for the stationary error. We will consider both the size and

the power of the tests.
1.5 The KPSS Test As a Unit Root Test

Many economic time series have sample autoconelations for their first
difference that are positive and significant only at lag one or two, but are
insignificantly negative at longer lags. Conventional model selection procedures
choose a low order ARIMA model in order to parsimoniously capture the short run

dynamics. Then only the role of the random walk component is important, and

15

possible trend reversion over long horizons is ignored. Variance ratio tests have been
suggested to handle this situation, but unfortunately their confidence bounds are very
wide, because there are very few independent observations on the long run behavior
for most macroeconomic time series. See Cochrane (1988) and Lo and MacKinlay
(1989). More generally, standard unit root tests, such as the DF or the SP tests,
propose the null hypothesis that a random walk component exists, whereas the tests of
the random walk hypothesis (e.g., variance ratio tests) have as their null hypothesis
that stationary components do not exist.

Stock (1990) has recently developed a "generic" class of unit root tests, based
on the fact that an 1(1) process grows at rate Tm, while an 1(0) process does not. If
we let e, be the detrended series, and let S, be the partial sum process of the e’s, as in
equation (18), then the 8, process is 0,,(T1’2) if the original series is stationary, while it
is 0,.(T’n) if the original series is 1(1). In either case, suitably normalized functions of
S, will converge to corresponding functionals of detrended Brownian motion. As a
result, may different functions of S, could be used to test the unit root null.
Furthermore, the same statistics (with a different asymptotic distribution) can be used
to test the null of stationarity.

A difficulty with such generic tests is that they may have low power compared
to tests derived more explicitly as unit root tests or stationarity tests. For example, the
KPSS test is derived from the LM (score) principle as a stationarity test, and can
therefore be expected to have desirable power properties near the null of stationarity.

However, it is a function of the partial sum process and can be fit into the class of

16

Stock’s generic unit root tests, despite the fact that there is no reason to expect it to
have good power properties as a unit root test.

In Chapter 3, we will consider the KPSS test as a unit root test. We provide
Monte Carlo evidence that shows that it is generally less powerful than traditional unit
root tests, like the DF or the SP tests. We also use the KPSS statistic to test the

hypothesis of a unit root in the Nelson-Plosser data.

1.6 Plan of the Dissertation

The structure of the dissertation is as follows. Chapter 2 investigates the finite
sample performance of the KPSS stationarity tests in the presence of autocorrelation.
Chapter 3 considers unit root tests based on the KPSS statistics. Chapter 4 provides
asymptotic results for the DF and SP unit root tests when the process is "nearly

stationary", and shows the possible source for the size distortion problem in this case.

Chapter 5 gives our concluding remarks.

CHAPTER 2

CHAPTER 2

THE FINITE SAMPLE PERFORMANCE OF THE STATIONARITY TEST

2.1 Introduction

There has been considerable interest in the use of autoregressive processes for
modelling nonstationary time series. Nonstationarity is implied by the presence of unit
roots in the autoregressive polynomial, and therefore the unit root hypothesis has
recently attracted a lot of attention. Furthermore, the standard conclusion that is
drawn fiom the empirical evidence is that most aggregate economic time series contain
a unit root. See Nelson and Plosser (1982). However, it is imponant to note that in
this empirical work the unit root is set up as the null hypothesis to be tested, and the
way in which classical hypothesis testing is carried out ensures that the null hypothesis
is accepted unless there is strong evidence against it. Therefore, an alternative
explanation for the common failure to reject a unit root is simply that standard unit
root tests are not very powerful against relevant alternatives. For further discussion

see Dejong et a1. (1989).

 

Therefore, it would be useful to have available tests of the null hypothesis of
stationarity as well as tests of the null hypothesis of a unit root. There have been

relatively few previous attempts to test the null hypothesis of stationarity. See Park

17

18
and Choi (1988), Rudebush (1990), and Dejong et a1. (1989). However, they all suffer

from the lack of a plausible model in which the null of stationarity is naturally framed
as a parametric restriction.

Nonstationary series can be decomposed into an integrated part and a weakly
stationary part. A well-known decomposition of time series into random walk with
drift and weak stationary components is proposed by Beveridge and Nelson (1981).
However, we note that, in the absence of some additional theory or assumption on the
data generating process, such a decomposition may not be unique. See Aoki (1990).

Recently, Kwiatkowski, Phillips, Schmidt, and Shin (1992) use a
parameterization which provides a plausible representation of both stationary and
nonstationary variables to derive a test for the null hypothesis of stationarity. They
choose an unobserved components representation in which the time series under study

is written as the sum of a deterministic trend, a random walk and a stationary error

process:
Yt=§t+7t+vt9 Yt=Yt-l+ut V

They wish to test the hypothesis 0,2 = 0, which implies that y, is stationary around a
deterministic trend (trend stationarity). Since the reduced form of the components
model is also ARIMA(0,1,1), a test for 0,2 = 0 corresponds to a test for a unit moving
average root. The model is therefore strictly noninvertible under the null and it
follows from the results of Sargan and Bhargava (1983) that the classical procedures

cannot be applied in this case.

19

A univariate time series model can be regarded as a special case of a time
varying parameter regression model. Very little attention has been paid to the way in
which time variation should be introduced into the coefficients of explanatory
variables. Testing for time variation in the coefficients of the explanatory variables is
also subject to the same problems encountered in carrying out tests of 0,2 = 0. The
difficulties stem from the random walk nature of the time varying parameters. Testing
the null hypothesis 0,2 = 0 has also been considered in context of time varying
coefficients model. See Nicholls and Pagan (1985), Nyblom (1986), and Nabeya and
Tanaka (1988).

The purpose of this chapter is to examine the finite sample performance of the
KPSS stationarity test. We explain and compare various models in section 2.2. The
main results of the simulations are given in section 2.3. The results of applying the
KPSS statistics for stationarity to the Nelson-Plosser data are briefly discussed in

section 2.4. Some suggestions and concluding remarks are given in 2.5.

2.2 The KPSS Test for Stationarity

Suppose that the economic time series y, can be decomposed into a
deterministic trend, random walk, and stationary enor process:
(1) yt=§t+h+w

(2) Yr = 7H + ut

20

where the u, are iid (0,0,2) and v, is stationary. KPSS derive the LM statistic for the
null hypothesis of stationarity, 6,2 = 0, under the assumption that the v, are iid
N(0,0’,,2). Their model is a special case of the model developed by Nabeya and
Tanaka (1988) to test for random regression coefficients. Nabeya and Tanaka consider

the regression

(3) Yr = xtBt + 2:7 + V,,

in which B, is a normal random walk and the errors v, are iid N(0,o,2). Therefore, the
KPSS model is the special case of (3) in which x, = 1 for all t, z, = t, and B, = 7,.

Let e,, t = 1,2,...,T, be the OLS residuals from the regression of y, on an
intercept and time trend (or intercept only for the test of level stationarity). Let 6,2 be
the usual estimate of the error variance from this regression. Then the LM statistic is

given by
T A
(4) LM = t218,2 lo,2
where S, is the partial sum process of the residuals
r
(5) S, =_231ei , t = 1,2,...,T
1:

The same statistic may arise in other contexts. Saikkonen and Luukkonen
(1990, SL) derive the locally best unbiased invariant (LBUI) test of the hypothesis 9 =

-1 in the model

(6) Ay, = w, + 9w,1

21
with E(yo) unknown and playing the role of intercept and w, iid normal. Note that y,

is stationary under the null hypothesis of 0 = -1.
Comparing both parameterizations carefully, we find that they are the same.
Consider starting with equations (1) and (2). After some algebra, we get the following

equation:

(7) y. = a + [Syn + C»

withB= l,a=§,'yo=yo, and

(8) c, = A(y, + v,) = u, + Av,

Therefore, 0,2 = 0 corresponds to c, = Av,. If we rewrite model (6) as
(9) y. = a + y... + 8,.

(10) 8, = w, + 0w,l

then testing for 6,,2 = 0 in equations (1) and (2) is exactly the same as testing for 9 =
-1 in the equations (9) and (10). We can also derive the exact relationship between A.

= (sf/cf in (1) and (2) and e in (9) and (10):
(1 1) 9 = -(1/2)[0~ + 2) - {m + 4)}"21
(12) 7. = -(1 + ewe

forkZO,|9|Sl.AsA—)0,0—>-1whileask—nme—afl

22

Tanaka (1990) also analyzes testing for a moving average unit root when the

data follow a simple MA(l) process given by

(13) x. = 6. - 92,,

and derives the score test statistic for 9 = 1:

 

T T
2: [(t-1)x, + (t-2)x2 +...+ x,,, - {t/(T+1)}Z(T - s + 1)x,]2
= =1
(14) ST = t 1 s
T
T 2 [(x1 + 2x2 +...+ tx,)2/ {t(t + 1)}]
t=1

This test can be extended to a more general model such as (9) and (10). Using (4),

we obtain the KPSS score test statistic for the level stationarity as follow:
T T _

(15) LM = 21 SLR/T1 21 (y. -y)2
t: t:

where 8,, =j 51 (yj - )7). Define x, = Ay, and S, =j§t1yj° Then, Tanaka’s score test
statistic (14) is the same as the statistic for level stationarity (15). See appendix A.
However, the assumption that the stationary error (v, in equation (1)) is iid is
unrealistic in time series modelling. There are two possibilities to generalize the
testing procedures to allow for stationary but not iid v,.
One possibility is a parametric correction. In the special case of a Gaussian
MA(l) model of the form of (6), SL derive the LBUI test statistic (R in their

notation), which is the same as the KPSS LM statistic for level stationarity under the

23
assumption that stationary errors are iid. The general ARMA(p,q) model for w, in (6)

is given by
(16) Ay, = w, + 6w,_,, p(L)w, = or(L)£,
where p(L) = 1 - p,L - - ppL" and or(L) = 1 + alL + + aqL“, and 8, are iid

normal. The roots of both the lag polynomials p(L) and or(L) are outside the unit
circle so that the stationarity and invertibility conditions are satisfied. They use a
parametric approach to generalize test statistic R, which is based on the appropriate
residuals of the general ARMA(p,q) null model first fitted to the original mean
corrected series. When p > 0, the idea of the test procedures is to replace the original

null model by an MA approximation. Define
(17) p(L>"a<L) = w.) =j§o ij’
Then w, can be decomposed as

(18) w, = w, + A2,

2% v, To derive modified

i=j+1

w, = W(1)£,; z, = 1t(L)e,; and 1t(L) 131m where n, = -
test statistics in the presence of ARMA errors, they approximate the rational transfer
function “(0-) by a polynomial of a finite but sufficiently large order, say m.

However, one caution is that unnecessarily large values of m have an adverse effect on

the power of the tests. Finally, the modification of test statistic R becomes

T ..
(19) Rm = (T - p)‘2 2 s3 / w2, where
t=p+l

24

.. T -
(20) W = (T - P)’1 E (h. - h)2
t=p+l
t
(21) s,=,-2‘. (hi-1i), t= p+1,---,T-1.
J=p+l

h, .=. y, - AZ and Z, = $19,051,, where R, = 124% ands, are the residuals. Then as T
—> co, the Ru, statistic has the same asymptotic distribution as R.

The other possibility, followed by KPSS, is to use a semiparametric correction
of the type suggested by Phillips and Perron (1988). An advantage of this approach is
that no knowledge of the parametric form of autocorrelation is required. The correction
only amounts to replacing the denominator of the LM statistic by an appropriate

variance estimator. Define the long run variance as
‘ T
(22) 02 = lim T‘E(ST2), ST = 2v,
T—-)oo i=1

which will enter into the asymptotic distribution of the test statistic, when the v, are
stationary but not iid. In this case the appropriate denominator of the LM statistic is

an estimate of 0'2 instead of of. A consistent estimator of 0'2, 32(0) can be constructed

as
T l T
(23) 52(0) = '1‘1 Zle,2 + 21"2‘. w(s,0) 2 e,e,_,
r=1 s=1 r=s+1

where w(s,f) = 1 - s/(l + 1) is the Bartlett window which guarantees the nonnegativity
of $202). For consistency of SW), it is necessary that the lag truncation parameter 1
-) on at an appropriate rate as T —-) 0°. The generalized KPSS LM statistics for level

stationarity and for trend stationarity are defined as follow:

25

(24) (i, = TZESuz/szfl) —+ g V(r)2dr
(25) fl, = TZZSnzlsza) —> g V2(r)2dr

SL, and ST, are the partial sum processes of the OLS residuals from the regression of y,
on [1] and on [1,t] respectively. The subscript ’u’ indicates that we have extracted
only a mean from y, and the subscript ’1’ indicates that we have extracted mean and
trend from y. W(r) is a standard Brownian motion; V(r) = W(r) - rW(1) is a standard
Brownian bridge; and V2(r) = W(r) + (2r - 3r2)W(1) + (~6r + 2?); W(r)dr is the
second-level Brownian bridge.

The critical values of IV(r)2dr and of IV,(r)2dr are given in Table 2-0, which
are calculated via a direct simulation using a sample size of 2,000 with 5,0000

replications, and the random number generator GASDEV/RAN3 of Press et a1. (1986).

 

2.3 Finite Sample Performance

The finite sample distribution of the test statistics 6,, and fi, will be tabulated by
simulation. The distribution of both statistics under the null hypothesis depends only
on the sample size T, while the distribution under the alternative depends on A as well
as T. See appendix B. The simulation results using 20,000 replications are given in
Tables 2-1 through 2—10. We consider three lag specifications in calculating the
denominator of the LM statistics, the long run variance of the residuals, 52(0) in (23):

r, = o, r, = mum/100)“), and e,, = int{12(T/100)““}.

26
2.3.1 Size

1) iid errors

We can see in Table 2-1 that the tests have approximately correct size except
when T is small and I is large. For I = 00, the tests have correct size even for T =
30, so that the asymptotic validity of the tests holds even for fairly small samples.
Using 1 = t4, the tests are slightly less accurate, and the improvement as T increases
is slow. For 2 = 1,2, there are considerable size distortions for T = 30, and moderate
distortions (too few rejections) even for T = 100 or 200, though the tests are quite
accurate for T = 500. Unsurprisingly, the larger 0 is, the larger is the sample size
required for the asymptotic results to be relevant.
_2_) AR(I) errors

We next consider the size of the tests in the presence of autocorrelated errors.
In particular, we will consider AR(l) errors, of the form v, = pv,l + a, with the e, iid.
The AR(l) parameter p is a convenient parameter to consider, since it naturally
measures the distance of the null from the alternative. In particular, under the null
that 6,2 = 0, y, approaches a random walk as p —> 1. As a result, we expect a
problem of over-rejection for p > 0, with its severity depending on how close p is to
unity. Table 2-2 presents our simulation results giving the size of the tests for p = 0,
21:02, $0.5, and 1:08, and for T between 30 and 500. As expected, the tests reject too
often for p > 0 and too seldom for p < 0. The over-rejection problem is very severe
for 1 = to, which is not surprising since the test is not valid even asymptotically in

this case. However, the 04 and 0,2 versions of the tests do not improve very rapidly

27

with the sample size. The tests using 14 have moderate size distortions for p = 0.5

and considerable size distortions for p = 0.8, while the test using 1,2 are fairly good
for T .>. 30 and p S 0.5, but not so good for p = 0.8. Unfortunately, p = 0.8 is a
plausible parameter value since, if we take most series to be stationary, their first-order
autocorrelations will often be in this range.

3) MA( 1) errors

We next consider MA(l) errors, of the form
(26) vr = 8r + (18,4

These results are given in Table 2-3. In the presence of MA(l) errors, the size
distortions of the test are not as severe as in the AR(l) case. Therefore, the use of
long lags (large value of 1) is not necessary for test statistics to have approximately
correct size. For example, for or = 0.8, the sizes of the fi, test using to, 04, 012 are
.206, .057 and .033 and the sizes of the fl, test using 2,, r, and e,, are .302, .061 and
.038, both of which are far less than for the AR(l) case with p = 0.8.

For positive or, the size of the tests using 0,2 is converging around the nominal
level as T increases; e.g., when T = 500, all sizes are very close to .05. However,
when T is small, the use of In still gives unsatisfactory results; e.g., when T = 30 and
a = 0.8, the size of 0.6912) is .004 but the size of fi,(0,,) is .217.

We also note that when T is in the range of 70 to 120, which are plausible
sample sizes encountered in economic data, the size performance of the tests using 04

is slightly better than when 0 = 2,2 and better than when 0 = 00.

28

For negative or, the sizes are too low for all cases, and these results are
consistent with the asymptotic results that the size —> 0 as or —> -1.

To sum up, in the presence of positively autocorrelated errors, the tests using
to have an over-rejection problem and the tests using 0,2 generally have the least size
distortion. The size distortions are more severe in the AR(l) case. We also note that
the tests using 24 perform better than the tests using 0,, in some cases. On the other

hand, we have an under-rejection problem for negatively autocorrelated errors.

2.3.2 Power

Results for the power of the tests in the presence of iid errors are given in
Table 2-4. We will discuss these results before going on to power in the presence of
AR(l) or MA(l) errors. Note that both test statistics, fl, and 13,, are consistent.
However, for fixed T, the power of the test approaches a limit (as A —> oo), which is
usually less than one. We can represent as the limit power the power of test for A =
00. For example, when T = 100, the limit powers of the fi, test using to, 04, and 0,2
are .998, .827, and .582, and they are .999, .820, and .410 for the fi, test, as can be
seen for A. = 10,000 in Table 24. Although the limit power of the tests using 0,, or
0,2 is close to one in large samples, it is generally far less than the power of the tests
using to in finite samples. This is especially so for the tests using 012. Even for T =
500, the limit power is about .901 for the man) test and .911 for the fi,(2,,) test.

However, the fact that the limit power of test is not equal to one can be

explained well by the asymptotics under the alternative. The asymptotic distribution

29

for the stationarity test under the alternative hypothesis 0,2 > O is given in KPSS. For

the test of level stationarity, we have the following results:
(27) (err) it, a {)1 1§W(s>dsrda / K§W<srds

whereW(s) = W(s) -(1: W(b)db is a demeaned Wiener process, and the

constant K is defined by
(28) K = I: k(s)ds

where k(s) represents the weighting function used in calculating s(0)in equation (23);
w(s,t) = k(s/Q) in the notation above. For the Newey-West estimator, k(s) = 1 - |s|
and therefore K = 1.

The analysis for the trend stationarity case (i.e., for the statistic fl,) is only
slightly more complicated. We just need to replace the demeaned Wiener process '

W(s) above with the demeaned and detrended Wiener process W‘(s):
(29) W’(s) = W(s) + (63 - 4) (I: W(r)dr + (-123 + 6) g rW(r)dr

defined by Park and Phillips (1988, equation (16), p.474). The rest of our analysis
then follows without further change.

Note that the asymptotic distribution (27) does not depend on the variance of
the stationary error. This is so, because, under the alternative hypothesis, the random
walk component dominates the stationary component. In that sense these asymptotics

correspond to 6,2 = 0, or A. = 00, and can be used to predict the limit power of the

30

stationarity test.

Percentiles of the asymptotic distribution in (27), and of the corresponding
asymptotic distribution for fi,, are calculated in Chapter 3, where we consider these
statistics as unit root test statistics. These can be converted to percentiles for fin and {1,
(under A = co) by multiplying by (T If), and we can therefore predict the limit power
of both tests. These results are given in Table 2-5. We find that the limit power of
the stationarity test in finite samples is generally consistent with the asymptotic results.
For example, for fi, with T = 100 and 2 = 12 the actual power of .410 compares to
.417 predicted by the asymptotic distribution under the alternative.

As described in section 2.2, Tanaka (1990) tabulates the limiting power of the
ST test in equation (14) under local alternatives of the form 9 = -1 + Cfl‘ and C fixed,
and therefore we can use his table for the limiting power as a good approximations to
the power against nearly stationary alternatives in finite samples. However, the
Tanaka’s test is not most powerful at higher values of A in finite samples. For
example, when T = 100 and 9 = -0.8 (A = 0.011), the power of ST is .863, which is
less than the power of the MPI test, .954. We also note that he does not consider
cases withT> 100 and0>6>-0.4(oo>A> 1).

We compare the actual (simulation) power of the *1, test with Tanaka’s limiting
power and these results are given in Table 2-6. When A < 1, the power of the 19,00)
test is close to Tanaka’s limiting power for all sample sizes, while the use of more
lags loses power in finite samples. For example, when T = 100 and A = 0.01, the

powers of the fi, test using to, 04, and 1,2 are .587, .508, and .376, and Tanaka’s

31

limiting power is .584. When A 2 1, the powers of the 1‘}, test are generally less than
Tanaka’s limiting power when T S 50. For example, when T = 50, the powers of the
*1, tests using to, 24, and 0,2 are .959, .704, and .343 respectively for A = 10,000,
which is less than Tanaka’s limiting power of .991.

The above results show that the power of the 1‘}, test for 2 = 00 and for small A
is well predicted by the asymptotics of Tanaka. More generally, its power
performance is satisfactory unless both A is large and T is small. However, note that
the final) test is not generally powerful at all even against the pure random walk
alternative of A = oo in finite samples.

For more detailed power investigations, we choose 4 different values of A
(0.0001, 0.01, 1 and 10,000). We first analyze the case of iid errors, for which the
results are given in Table 2-4. When A = 0.0001, the powers of the tests are not much
different from nominal level in finite samples. This implies that it is almost
impossible to distinguish between the distribution under the null and under a local
alternative in finite samples. When A = 0.01, the powers of the tests using to
approach one as T increases, but in finite samples they are not very large. As
expected, both tests are not very powerful against alternatives with small values of A,
say A < 1. Note that the if, test is generally less powerful than the fin test. This
finding is also consistent with previous studies. For A 2 1, the tests using 0,, and 04
are reasonably powerful even in finite samples but the tests using 0,2 are not powerful,
as mentioned, even against the pure random walk alternative. It should be noted that

the loss in power from using a large value of 0 persists for large sample sizes. This is

32

also expected from the asymptotics under the alternative; (UI‘) enters the expression
(27) above.

We now consider the power in the presence of AR(l) errors. These results are
given in Table 2-7. In this case the power generally depends on the value of p as well
as the value of A. As p —-) 1, the test is more powerful, as expected. For p = 0.8,
even when A = 0.0001 and T = 100, the powers of the fin tests using 20, 24, and 9,2
are .799, .252 and .083 and the powers of the fi, tests using to, 1,, and 0,2 are .953,
.340 and .092, all of which are far greater than the powers for iid errors. This is a
reflection of the size distortion caused by AR(l) errors with positive p. On the other
hand, for negative p, power is small unless T is very large.

In the presence of AR(l) errors, the use of longer lags (a larger value of 0) is
needed for the test to avoid size distortions, while the use of shorter lags is needed for
the test to be more powerful. This implies an inevitable tradeoff between power and
size in finite samples. Therefore, we have to weigh size against power of the test in
determining the choice of the number of lags to be used. Based on the above
simulation results and considering the fact that p = 0.8 is a plausible parameter value,
the use of l = 18 may be a compromise between the large size distortions under the
null that we would expect for 0 S 04 and the very low power under the alternative that
we would expect for l = 1,2.

We now consider power in the presence of MA(l) errors. These results are
given in Table 2-8. In this case the power also depends on the value of or as well as

the value of A, but the influence of or is not as important as the AR( 1) parameter p.

33

Generally, the test using to is more powerful for positive or than for negative or. For
A = 0.0001, when or = 0.8 and T = 100, the powers of the if, test using 20, 94, and 1,,
are .218, .062 and .037 and the powers of the fi, test are .307, .063 and .039
respectively. On the other hand, for negative or, the powers are less than nominal
level unless T is very large. Once again these levels of power reflect the size
distortions caused by or at 0.

In the presence of MA(l) errors, size distortion is not as severe as in the AR(l)
case, so that the use of very long lags is not required for the test statistics to be more
accurate. For positive values of the moving average parameter, there is still a tradeoff
between power and size in finite samples, but the size distortions with 1 = l, are not
as severe as for the AR(l) case. Based on the above simulation results, the use of I =
1., may be a compromise between some size distortions under the null that we would

expect for 1 = 0,, and decreased power under the alternative that we would expect for

(>94.

2.3.3 Comparison to The Saikkonen and Luukkonen Test

The simulation results for the ii, and fi, tests can be compared with those for
the test of Saikkonen and Luukkonen. They suggest the use of their R, statistic in the
presence of MA(l) errors and of their R,5 statistic in the presence of AR(l) errors.
These comparisons are given in Table 2-9 and 2-10. Note that for iid errors, size is

relatively correct in all cases.

With MA(l) errors, the R, statistic can be compared with the fip(f4) statistic.

34

For positive or, when T = 100, the sizes of both tests are correct and similar but R, is
always more powerful than 13,04). For example, for 0 = -0.95 (A = .0026), the power
offi,(r,) is .201 and the power of R, is .335 when o = 0.8. For negative or the sizes
of both tests are low but the size of the mm) test is almost zero, but interestingly,
mm) is always more powerful than R,.

With AR(l) errors, we compare the if, test using 0,, or 9,, with R,,. For p < 0,
the sizes of 13,04) or 11.6912) are too small but the size of R,, is relatively conect. On
the other hand, for p > 0, some size distortions occur for 13,04), but R,, and man)
show almost the correct size performance. However, the power of man) is always
less than the power of R,, except when 6 = -0.95 and p is negative. For example, for
0 = -0.90 (A = 0.0111) the power of R,, is .447 and the power of fipan) is .219 when
p = 0.8 and T = 100. However, no general conclusion can be drawn from the
comparison of the power of R15 with the power of mm). For example, for p = 0.5
and T = 100, the powers offi,(r,) and R,, are .209 and .265 for e = -095, but they
are .648 and .567 for 0 = -0.8. This ordering is reversed when p = -0.5. However, it
is generally true that the power of R,, exceeds the power of 13,04) except where mm)
suffers from considerable size distortions.

To sum up, it is very difficult to draw any clear conclusion from the above
comparison. The Saikkonen and Luukkonen testing procedure is more complicated
than KPSS’s, because they have to estimate a general ARMA process and then derive
the approximate MA(m) representation. There is also the problem of choice of the

appropriate number of m. However, interestingly, the size of their test is relatively

35

correct in most cases with autocorrelated errors, especially when the errors are AR(l).

2.4 Applications to the Nelson-Plosser Data

In this section we briefly discuss the results of application of the stationarity
tests to actual data. KPSS apply their tests for stationarity to the data analyzed by
Nelson and Plosser in order to check whether their approach to testing stationarity
corroborates the main findings of Nelson and Plosser. They consider values of the lag
truncation parameter 0 from 0 to 8. The values of the test statistics are fairly sensitive
to the choice of f, and in fact for every series the value of the test statistic decreases
as 2 increases. This is a reflection of large and persistent positive autoconelations in
the series. For all series except the unemployment rate and the interest rate, they can
reject the hypothesis of level stationarity, but this is not very surprising in light of the
obvious deterministic trends present in these series.

Based on the observations that for most of the series the value of the long run
variance estimate has settled down reasonably by the time 2 = 8 is reached, they use
the results for l = 8 for the trend stationarity test. The choice of 2 = 8 is relatively
consistent with the above simulation results. Their results have very different
implications for many of series considered from Nelson and Plosser. Nelson and
Plosser can reject the null hypothesis of a unit root at the 5 % significance level for
only two series (unemployment rate and industrial production) of the 14 series

considered, while KPSS can reject the null hypothesis of trend stationarity at the 5%

36

level only for five series (industrial production, consumer prices, real wages, velocity
and stock prices). Therefore, KPSS conclude that most economic time series are not

very informative about whether or not they contain a unit root.
2.5 Suggestions and Concluding Remarks

We have investigated the finite sample performance of the stationarity tests of
Kwiatkowski, Phillips, Schmidt, and Shin (1991). In the process we have shown the
close relationship between the KPSS stationarity test using a components
parameterization and a test for the moving average unit root using the traditional
ARIMA parameterization, and discussed some intrinsic testing problems involved. We
summarize the main findings as follows.

First, when the stationary errors are iid, the size of the tests using to is correct.
Given positive autocorrelation, the tests using 0,, show some size distortions but the
tests using 1,, show very small size distortions in most cases. The use of 04 shows
the intermediate behavior. The size distortion problem is more severe with AR(l)
errors than with MA(l) errors. In particular, when p is large and positive, the use of
longer lags (e.g., 912) is required to avoid severe size distortions. On the other hand,
when the errors are negatively autocorrelated, we have a under-rejection problem.

Second, with iid errors, the tests using 2,, are most powerful. The power
performance of the tests in finite samples depends on the value of A, as expected.

Power is low when A is small and increases as A increases. Power does not usually

37

approach one as A —-) 00 with T fixed, however.

Third, using 2, or 1,, instead of to loses power, so that there is an inevitable
trade-off between size and power of the test. Our simulation results suggest the use of
shorter lags (e.g., 24) unless it appears that the stationary errors follow an AR(l)
process with large positive parameter.

Fourth, 1‘}, is less powerful than fir This confirms that it is difficult to
distinguish between a unit root and trend stationary series in finite samples.

We have compared the finite sample performance of the KPSS stationarity test,
which uses semiparametric corrections for error autocorrelation, with that of the
Saikkonen and Luukkonen (1990) test, which uses a parametric correction instead.
Although it is difficult to draw any clear conclusion about their relative performance
in terms of size and power, it is important to mention‘the relatively good finite sample
performance of the Saikkonen and Luukkonen test when errors are the AR(l) with
large positive parameter. A possible combination of both approaches to tackle the
problem of autocorrelation is a further research topic.

It should be noted that we estimate the long run variance from residuals from a
fit of the model with the stationarity hypothesis imposed, and so if the null hypothesis
is not true we should expect s2(f) to diverge as 0 increases. This implies the need for
further research to find an estimate of the long run variance that is consistent under
the null and that increases the rate of divergence of the LM statistic under the

alternative.

We can possibly modify the stationarity test to make it into a cointegration test.

38

One of advantages of this approach is that we can set up the null of cointegration
directly, instead of the null of no-cointegration which is a direct extension of the unit
root test and has been mainly used in the literature. The basic idea is simple.
Suppose that the n x 1 vector X is 1(1). Then variables in X are cointegrated if there
exists an n x 1 vector r such that r’X is 1(0). r’X is called the long run relationship.
If we know r, or estimate it efficiently, we can set y = r’X and apply the stationarity
test to y. We expect this test of cointegration to give further light on the true
relationships among important economic variables. We are currently working on this
topic. In particular, the asymptotic theory for the case that r is estimated must be

derived.

39
Appendix A

We will now demonstrate that the extended version of Tanaka’s score test is
the same as the fin test. Tanaka’s simple model is given in (13) in the text and the
score test statistic for a moving average unit root is given in (14). We extend this to

the more general model
(A1) 43'. = 8t - 98st

where Ay, = x,, and y, = x,. As shown in the text, the null hypothesis 9 = 1 in (A1)
corresponds to the null of 0,2 = 0 in equations (1) and (2) in the text.

r l
Define S, =,Zl yj and SL, =2“.1 (yj -9). Then it is straightforward to show that
.1= 1"
(A2) SLr = St ' t); = Sr ' (mST
T
where S,- = 21y,
t:

Lemma A1 (t - 1)xl + (t - 2)x2 + + x,_, = S,,,
(proof) (t - 1)x, + (t - 2)x2 + + x,_,

= (t - 1)yr + (t - 2)(y2 - yr) + + (y.-r - Yr?)

= 1310 - 1) - (t - 2)] + y,[(t - 2) - (t - 3)]+ + yt-l

= yr + Y2 + + ysr = 81-1

. T
Lemma A2 21(T - s + 1)x, = 22y, = Ty
s:

(Proof) 1.8('1‘-s+ l)x,
s=1

40

= Ty, + (T - 1)(y2 - Yr) + (T - 2)(ys - Y2) +...+ (yr - yr.)
= y,[T - (T - 1)] + y2[(T - 1) - (T - 2)] +...+ yr

= g)“ = T);

Using Lemma A1 and A2 and (A2) we can show that
T _ T
(A3) Numerator of ( 14) = 231(8t - ty)2 = 121 SL3
I: =

We now show that the denominator of(14) is the same as the denominator of the 1‘},
test using Lemma A3. The denominator of (14) can be expressed as (see Tanaka

(1990))

T T T T
(A4) x’Q"x = [ 213x, 223 x, x,] [I - ee’fF] Dix, 52.1x, x,]’
where e is the unit column vector.

T
Lemma A3 x’Q“x = 21 (y, - y)2
t:
(proof) x’Q'1x
= [YT (YT ' Yr)“ (YT ' yt-1)][I ' ce’lTHyT (YT ‘ Yr) (YT ' yt-1)]’
= yr2 + (yr-yr)2 +--+ (YT'YT-1)2 - (T+1)"[yr + (yr-yr) +"~+ (yr-yr.r)l2
T T
= Elly? + (T+1)yr2 - 2yr(yr+y2+-°+yr) - (T+1)"[(T+1)yr - {Syn}2
T _ T _ 2
= )1;er ' Tyz = 21:0“ '3')

Here we use the fact that y, = Ex, with y0 = 0.

Therefore, the extended version of Tanaka’s score test should reduce to the

KPSS level stationarity test.

41
Appendix B

We will demonstrate that the distribution of the fin and fl, tests under the null
hypothesis of stationarity ((5,,2 = 0) depends only on the sample size T, while the
distribution under the alternative depends on A (the ratio of the random walk error
variance to the stationary error variance) as well as T.

Consider the level stationarity test first. Under the null ( 6,,2 = 0), y, can be

expressed as

(B1) Yr = 70 + Vt

Then the partial sum process 8,, is given by
t _ t _

(32) 8L! =j-Zl (Yj 'Y) =j§1 (Vj ' V)

8,, does not depend on yo and the fi, test is a function of 8,, (t = 1,~-,T), so that its
distribution is invariant to y,,. The scale factor 0,, cancels out of the expression for the
fin test, therefore, the distribution of the statistic under the null is independent of

nuisance parameters ((3" and 70).

Under the alternative ((5,,2 > 0), y,, e,, and 8,, are expressed as

I
(BB) Yr = 70 + Yr + Vt, Yr =j§1ui
(B4) C1 = (Yr 'Y) = (Yr '7) + (Vt '6)

1.
(BS) 8.51.3310, -9) =1; {0. ~73 + (v, 47)}

42

Since SL, does not depend on y,, and the fi, test can be written as a function of S,,, t =
1,---,T, its distribution under the alternative is still invariant to 7,. However, the
distribution for the *1, test under the alternative depends on A = (Sf/6,2 as well as T.

To show this, we rescale (B3) by dividing by 0,. Accordingly, we get the following

results

(B6) y,’ = 70" + y,’ + v,‘, y,‘ 13-51“;

(B7) Ct. = (Yr. '37.) = (Yr. '7.) + (V: 47')

(Ba) 5.: 1:510; -9‘) =3 ((7: 4‘) + (v; -m

Note that v,‘ (v, /o,) now follow iid N(0,1) and that the u,’ are iid N(0, A) where A =
(Sf/6,2. Then the *1, test can be written as a function of e,' and S,,', because e,' = e/o,
and S“. = 8,, /0,. Since e,° and SL,‘ depend on A but in a different way, the
distribution under the alternative clearly depends on A as well as T.

Following the same logic as for the case of level stationarity, it is
straightforward to show that the distribution for the *1, test under the null is
independent of nuisance parameters 0,, y,, and F, and that its distribution under the

alternative depends on A as well as T.

Table 2—0

43

Critical Values for Stationarity Test

30
50
80
90
100
120
200
500

OOOOOOOOOOOOOOO

.010
.025
.050
.100
.200
.300
.400
.500
.600
.700
.800
.900
.950
.975
.990

Table 2—1

OOOOOOOOOOOOOOO

.0248
.0302
.0367
.0460
.0624
.0788
.0970
.1193
.1473
.1853
.2435
.3493
.4648
.5826
.7444

OOOOOOOOOOOOOOO

.0174
.0204
.0235
.0280
.0349
.0413
.0481
.0557
.0645
.0757
.0915
.1203
.1488
.1787
.2193

Size for iid Errors (A=O)

.049
.050
.049
.048
.049
.051
.051
.050

.004
.012
.029
.030
.029
.034
.041
.046

.054
.052
.049
.051
.049
.052
.052
.052

.248
.043
.032
.034
.033
.038
.040
.049

0.

0.

44

Table 2-2 Size for AR(l) Errors (A = 0)

8

.5

2

.8

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

£0

.654
.725
.779
.796
.807
.833
.852

.321
.331
.350
.352
.359
.367
.370

.118
.118
.122
.123
.125
.128
.129

.017
.015
.014
.014
.014
.014
.013

.002
.001
.001
.001
.001
.001
.001

.000
.000
.000
.000
.000
.000
.000

.301
.264
.300
.250
.256
.271
.239

.114
.098
.108
.090
.092
.099
.090

.055
.053
.060
.054
.057
.061
.059

.025
.029
.034
.033
.036
.038
.042

.010
.015
.018
.019
.020
.021
.026

.002
.007
.008
.007
.003
.008
.010

"a
2:.

£12

.007
.039
.080
.081
.091
.094
.092

.005
.021
.042
.043
.047
.053
.058

.004
.015
.033
.033
.038
.045
.049

.003
.011
.024
.026
.029
.037
.045

.002
.007
.017
.020
.023
.030
.036

.001
.002
.007
.007
.010
.015
.022

30

.769
.886
.936
.952
.960
.977
.989

.425
.486
.521
.538
.542
.559
.586

.147
.156
.157
.159
.166
.168
.170

.016
.012
.011
.010
.013
.011
.010

.001
.001
.001
.000
.000
.000
.000

.000
.000
.000
.000
.000
.000
.000

’71
'24

.317
.319
.401
.337
.354
.396
.361

.129
.113
.124
.107
.114
.121
.110

.062
.059
.060
.057
.064
.065
.065

.027
.029
.031
.031
.035
.036
.039

.010
.016
.015
.016
.019
.020
.024

.001
.013
.008
.002
.002
.002
.008

£12

.124
.057
.084
.092
.104
.108
.111

.178
.047
.046
.047
.054
.054
.062

.227
.045
.036
.038
.042
.043
.052

.268
.039
.028
.029
.034
.036
.042

.301
.032
.021
.021
.025
.029
.036

.319
.028
.013
.009
.011
.015
.020

45

Table 2—3 Size for MA(l) Errors (A = 0)

m. n,
a T £0 34 £12 £0 14 £12
0.8 30 .202 .062 .004 .287 .068 .217
50 .201 .057 .016 .295 .064 .046
80 .205 .063 .033 .299 .064 .037
100 .206 .057 .033 .302 .061 .038
120 .208 .061 .039 .305 .068 .043
200 .209 .065 .046 .306 .070 .044
500 .208 .060 .050 .316 .067 .053
0.5 30 .169 .058 .004 .236 .065 .222
50 .169 .055 .016 .242 .061 .046
80 .174 .062 .033 .244 .062 .036
100 .176 .056 .033 .251 .059 .038
120 .176 .058 .038 .025 .066 .043
200 .179 .063 .045 .254 .067 .044
500 .181 .059 .049 .263 .066 .053
0.2 30 .102 .050 .004 .129 .055 .233
50 .102 .049 .014 .133 .052 .045
80 .106 .055 .031 .131 .053 .035
100 .105 .051 .031 .132 .052 .036
120 .011 .052 .036 .138 .058 .041
200 .110 .057 .044 .139 .059 .042
500 .110 .055 .048 .141 .060 .051
-0.2 30 .015 .021 .002 .012 .024 .274
50 .012 .025 .010 .008 .025 .038
80 .011 .029 .023 .008 .026 .027
100 .011 .030 .025 .007 .027 .027
120 .011 .032 .028 .010 .031 .032
200 .010 .034 .035 .008 .032 .035
500 .009 .036 .041 .007 .035 .043
-0.5 30 .000 .003 .001 .000 .004 .360
50 .000 .004 .003 .000 .004 .028
80 .000 .000 .003 .000 .003 .013
100 .000 .006 .010 .000 .004 .010
120 .000 .006 .012 .000 .005 .013
200 .000 .006 .016 .000 .004 .017
500 .000 .007 .023 .000 .006 .021
-0.8 30 .000 .000 .000 .000 .000 .481
50 .000 .000 .000 .000 .000 .019
80 .000 .000 .000 .000 .000 .000
100 .000 .000 .000 .000 .000 .000
120 .000 .000 .000 .000 .000 .000
200 .000 .000 .000 .000 .000 .000

500 .000 .000 .000 .000 .000 .000

Table 2—4 Power for iid Errors

A T

.0001 30
50
80
100
120
200
500

.001 30
50
100
200
500

.01 30
50

80

100

120

200

500

0.1 30
50

100

200

500

1.0 30
50

80

100

120

200

500

10000 30
50
80
100
120
200
500

£0

.050
.051
.056
.063
.066
.097
.307

.058
.075
.168
.399
.788

.146
.287
.489
.587
.667
.846
.997

.514
.721
.927
.990
1.00

.806
.924
.977
.989
.994
.999
1.00

.887
.959
.988
.998
.998
1.00
1.00

A

"u

£4

.038
.041
.052
.055
.060
.092
.295

.046
.060
.147
.372
.757

.110
.232
.429
.508
.587
.776
.962

.403
.566
.762
.924
.989

.600
.683
.810
.818
.865
.943
.992

.641
.704
.822
.827
.871
.947
.992

£12

.004
.013
.032
.038
.044
.078
.275

.004
.020
.100
.314
.682

.009
.089
.288
.376
.459
.626
.865

.034
.267
.551
.713
.897

.053
.332
.532
.579
.633
.725
.901

.059
.343
.536
.582
.635
.725
.901

.053
.054
.052
.054
.059
.065
.137

.054
.060
.084
.193
.621

.080
.129
.249
.352
.444
.729
.983

.287
.547
.878
.990
1.00

.725
.914
.982
.993
.996
1.00
1.00

.888
.974
.995
.999
.999
1.00
1.00

.243
.045
.034
.036
.041
.051
.118

.244
.048
.053
.132
.503

.240
.065
.117
.172
.235
.448
.843

.200
.129
.357
.637
.903

.152
.171
.344
.411
.492
.667
.911

.141
.176
.353
.410
.496
.675
.911

47

Table 2-5

Comparison of the Limiting Power

With Predictions Based on Asymptotics

T 30 50 100 200 500

2, n, Actual .887 .959 .995 1.00 1.00
Predicted .888 .961 .995 1.00 1.00

0, Actual .888 .974 .999 1.00 1.00
Predicted .880 .971 .999 1.00 1.00

2, n“ Actual .640 .704 .830 .947 .992
Predicted .641 .701 .826 .947 .992

n, Actual .508 .627 .820 .966 .998
Predicted .507 .623 .828 .966 .997

2,2 n“ Actual .059 .343 .580 .725 .901
Predicted .055 .348 .583 .728 .900

0, Actual .141 .176 .410 .675 .911
Predicted .137 .177 .417 .671 .909

1 Actual power is obtained as the proportion of rejection of the

null of stationarity when we apply the 95 % critical value for
the stationary test to data generated under the alternative of A
- 10,000.

2 Predicted power is calculate as the probability (for the
asymptotic distribution under the alternative) that the statistic
exceeds its 95% critical value.

48

Table 2—6

Comparison of the Power of the "a Test to

Tanaka's Limiting Power

T A 0 C £0 i, 2,2 Tanaka1
30 .0001 -0.990 0.299 .050 .038 .004
.001 -0.969 0.934 .058 .046 .004 .061
.01 -0.905 2.854 .146 .110 .009 .147
.1 -0.730 8.105 .514 .403 .034 .505
1 0 -0.382 18.541 .806 .600 .053 .839

100 -0.010 29.706 .883 .639 .059 .949
10000 -0.0001 29.997 .887 .641 .059 .949

50 .0001 -0.990 0.498 .051 .041 .013

.001 —0.969 1.556 .075 .060 .020 .079
.01 —0.905 4.756 .287 .232 .089 .293
.1 -0.730 13.508 .721 .566 .267 .733
1.0 —O.382 30.902 .924 .683 .332 .952
100 -0.0lO 49.510 .958 .703 .342 .990

10000 -0.0001 49.995 .959 .704 .343 .991

100 .0001 -0.990 0.995 .063 .055 '.038 .062

.001 -0.969 3.113 .168 .147 .100 .172
.01 -0.905 9.512 .587 .508 .376 .584
.1 -0.730 27.016 .927 .762 .551 .931
1.0 -0.382 61.803 .989 .818 .579 .9962
100 -0.010 99.020 .994 .826 .582

10000 -0.0001 99.990 .998

200 .0001 -0.990 1.990 .097 .092 .078 .099

.001 -0.969 6.225 .399 .372 .314 .400
.01 -0.905 19.025 .846 .776 .626 .848
.l -O.730 54.031 .990 .924 .713 .993
1.0 -0.382 123.607 .999 .943 .725 .9962
100 -0.010 198.039 1.00 .945 .726

10000 -0.0001 199.980 1.00 .947 .725

500 .0001 -0.990 4.975 .307 .295 .275 .309

.001 -0.969 15.563 .788 .757 .682 .786
.01 -0.905 47.562 .997 .962 .865 .988
.1 -0.730 135.078 1.00 .989 .897 .9962
1.0 -0.382 309.017 1.00 .992 .901
100 -0.010 495.098 1.00 .992 .901

10000 -0.0001 499.950 1.00 .992 .901

1. These powers are interpolated from Tanaka's table of the limiting
power for different C values.
2. Tanaka's results are available only up to C=60.

49

Table 2—7 Power for AR(l) Errors (A = 0.0001)

0.8

0.5

0.2

-0.2

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

10

.653
.728
.779
.799
.809
.839
.873

.322
.334
.353
.363
.368
.397
.531

.118
.122
.129
.134
.140
.176
.373

.018
.016
.020
.021
.027
.050
.280

.002
.001
.002
.003
.004
.017
.241

.000
.000
.000
.000
.000
.005
.217

A

"u

£4 £12
.303 .007
.264 .039
.300 .081
.252 .083
.258 .091
.282 .099
.304 .137
.115 .005
.097 .021
.111 .043
.095 .045
.098 .051
.121 .067
.218 .159
.055 .004
.055 .016
.065 .035
.061 .038
.067 .045
.094 .067
.260 .229
.026 .003
.031 .012
.042 .029
.045 .035
.054 .045
.097 .089
.365 .351
.011 .002
.019 .008
.029 .025
.038 .035
.051 .049
.118 .129
.470 .467
.002 .001
.014 .004
.031 .025
.026 .041
.040 .069
.149 .222
.649 .634

£0

.769
.886
.936
.953
.960
.979
.988

.424
.487
.520
.540
.544
.565
.634

.148
.157
.158
.165
.171
.186
.267

.016
.013
.012
.011
.015
.019
.073

.001
.001
.000
.000
.001
.001
.025

.000
.000
.000
.000
.000
.000
.008

'71
it

.316
.318
.401
.340
.356
.396
.383

.130
.114
.125
.108
.116
.130
.150

.061
.059
.060
.060
.067
.079
.124

.028
.031
.032
.034
.041
.054
.151

.011
.016
.018
.020
.026
.041
.210

.001
.014
.014
.005
.005
.021
.378

112

.123
.057
.084
.092
.104
.112
.123

.178
.047
.046
.049
.055
.061
.086

.227
.045
.038
.039
.045
.055
.101

.267
.039
.030
.031
.038
.054
.149

.300
.032
.024
.026
.033
.057
.225

.318
.029
.017
.017
.022
.065
.396

50

Table 2-7 (Continued)

"a
P T £0 £4 £12
0.8 30 .679 .335 .009
50 .772 .332 .066
80 .846 .434 .137
100 .881 .431 .207
120 .902 .487 .272

200 .960 .668 .431
500 .996 .891 .747

0.5 30 .389 .163 .009
50 .488 .207 .056

80 .625 .352 .189

100 .700 .400 .254

120 .755 .475 .336

200 .893 .686 .527

500 .991 .925 .818

0.2 30 .214 .118 .009
50 .341 .209 .072

80 .525 .388 .246

100 .617 .461 .328

120 .689 .540 .414

200 .864 .749 .596

500 .989 .952 .850

—O.2 30 .106 .116 .009
50 .251 .263 .108

80 .465 .477 .329

100 .568 .555 .422

120 .652 .632 .498

200 .846 .819 .652

500 .987 .973 .876

—0.5 30 .065 .131 .011
50 .215 .334 .148

80 .440 .553 .393

100 .551 .629 .478

120 .637 .699 .548

200 .837 .863 .684

500 .986 .981 .888

-O.8 30 .062 .156 .013
50 .209 .469 .205
80 .432 .666 .459
100 .547 .684 .530
120 .633 .750 .593
200 .831 .890 .706
500 .985 .987 .896

(A - 0.01)
”7

£0 24
.771 .327
.891 .345
.945 .442
.964 .411
.973 .448
.991 .612
.999 .876
.444 .140
.542 .151
.634 .214
.697 .237
.737 .295
.880 .524
.994 .901
.176 .077
.248 .108
.359 .188
.452 .245
.531 .320
.775 .590
.987 .940
.031 .047
.077 .105
.184 .229
.286 .321
.386 .421
.699 .707
.984 .973
.006 .032
.033 .126
.127 .298
.225 .405
.328 .514
.660 .792
.982 .986
.002 .023
.020 .234
.108 .471
.200 .483
.304 .599
.643 .851

.979

.994

£12

.122
.065
.105
.128
.157
.260
.630

.179
.058
.085
.116
.156
.320
.754

.223
.062
.100
.147
.199
.407
.823

.256
.067
.136
.204
.277
.507
.868

.279
.076
.177
.259
.341
.565
.889

.258
.088
.238
.322
.406
.620
.904

0.8

0.2

51

Table 2-7 (Continued) (A - 1)

m. n.
T 12,, 2, 12,, 12,, 2,

30 .863 .609 .049 .864 .472
50 .949 .674 .318 .964 .592
80 .983 .802 .517 .993 .789
100 .992 .807 .569 .998 .796
120 .996 .858 .624 .999 .848
200 1.00 .942 .720 1.00 .956
500 1.00 .989 .894 1.00 .996

30 .834 .592 .050 .799 .426
50 .936 .671 .326 .937 .564
80 .979 .802 .525 .988 .777
100 .990 .810 .574 .995 .795
120 .994 .860 .628 .997 .848
200 1.00 .943 .724 1.00 .956
500 1.00 .990 .900 1.00 .998

30 .817 .598 .052 .746 .421
50 .928 .676 .334 .920 .571
80 .978 .806 .530 .984 .787
100 .989 .815 .579 .994 .803
120 .994 .863 .631 .997 .855
200 1.00 .944 .725 1.00 .959
500 1.00 .991 .901 1.00 .998

30 .802 .612 .053 .705 .441
50 .924 .684 .340 .902 .591
80 .977 .812 .534 .981 .800
100 .988 .820 .582 .993 .813
120 .994 .867 .633 .996 .862
200 .999 .946 .725 1.00 .962
500 1.00 .991 .901 1.00 .998

30 .798 .619 .053 .687 .454
50 .922 .691 .342 .895 .605
80 .976 .816 .535 .980 .807
100 .988 .822 .582 .992 .817
120 .993 .869 .634 .996 .866
200 .999 .946 .726 1.00 .963
500 1.00 .991 .902 1.00 .998

30 .800 .624 .053 .686 .458
50 .922 .698 .345 .892 .617
80 .976 .819 .535 .980 .817
100 .988 .823 .584 .992 .820
120 .993 .870 .634 .996 .868
200 .999 .946 .725 1.00 .964
500 1.00 .992 .902 1.00 .998

.135
.162
.329
.394
.475
.656
.903

.152
.163
.335
.403
.485
.663
.911

.155
.166
.341
.409
.490
.667
.913

.152
.171
.346
.413
.494
.670
.914

.147
.174
.349
.415
.494
.670
.913

.140
.174
.350
.416
.496
.670
.914

0.8

0.5

0.2

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

Table 2—7

£0

.889
.961
.988
.995
.997
1.00
1.00

.889
.961
.988
.994
.997
1.00
1.00

.889
.961
.988
.994
.996
1.00
1.00

.889
.961
.988
.995
.997
1.00
1.00

.889
.961
.988
.994
.997
1.00
1.00

.889
.961
.988
.994
.997
1.00
1.00

£12

.057
.348
.536
.583
.635
.726
.903

.057
.348
.536
.584
.635
.726
.903

.057
.348
.536
.584
.635
.726
.903

.057
.348
.536
.584
.635
.726
.903

.057
.348
.536
.584
.635
.726
.903

.057
.348
.536
.583
.635
.726
.903

£0

.889
.973
.996
.999
.999
1.00
1.00

.889
.973
.995
.999
.999
1.00
1.00

.889
.973
.995
.999
.999
1.00
1.00

.889
.973
.995
.999
.999
1.00
1.00

.889
.973
.995
.999
.999
1.00
1.00

.889
.973
.995
.999
.999
1.00
1.00

(Continued) (A - 10,000)

’71
£4

.512
.627
.822
.826
.871
.965
.998

.512
.627
.822
.825
.871
.965
.998

.512
.627
.822
.826
.871
.965
.998

.512
.627
.822
.825
.871
.965
.998

.512
.627
.822
.825
.871
.965
.998

.512
.627
.822
.825
.871
.965
.998

£12

.141
.177
.353
.417
.496
.671
.914

.141
.177
.353
.417
.496
.671
.914

.141
.177
.353
.417
.496
.671
.914

.141
.177
.353
.416
.496
.671
.914

.141
.177
.354
.416
.496
.671
.914

.141
.177
.353
.416
.496
.671
.914

53

Table 2—8 Power for MA(l) Errors (A - 0.0001)

0.8

0.2

-0.2

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

10

.203
.202
.213
.218
.223
.256
.426

.170
.172
.181
.187
.193
.226
.409

.103
.105
.112 '.
.117
.124
.159
.360

.015
.013
.015
.017
.021
.044
.275

.000
.000
.000
.000
.000
.007
.226

.000
.000
.000
.000
.000
.002
.204

.004
.016
.035
.037
.044
.062
.191

.004
.016
.035
.037
.045
.063
.203

.004
.015
.034
.036
.044
.068
.238

.002
.011
.028
.034
.044
.091
.360

.001
.004
.018
.028
.045
.145
.533

.000
.000
.006
.019
.049
.252
.696

£0

.287
.296
.299
.307
.309
.320
.400

.236
.244
.245
.253
.256
.271
.351

.128
.135
.133
.138
.144
.156
.237

.012
.010
.009
.008
.011
.014
.065

.000
.000
.000
.000
.000
.000
.014

.000
.000
.000
.000
.000
.000
.003

.217
.046
.037
.039
.046
.054
.089

.223
.046
.037
.040
.045
.054
.092

.234
.045
.036
.038
.044
.054
.104

.275
.039
.028
.030
.036
.053
.153

.360
.029
.015
.015
.020
.047
.271

.480
.019
.001
.002
.003
.037
.462

0.8

0.2

54

Table 2-8 (Continued) (A - 0.01)

"a ”1
T 20 2, 3,, £0 1,

30 .290 .114 .007 .312 .081
50 .401 .182 .060 .380 .101
80 .564 .347 .214 .471 .168
100 .648 .416 .294 .552 .215
120 .713 .496 .379 .615 .281
200 .875 .712 .567 .818 .536
500 .990 .939 .836 .989 .920

30 .262 .113 .008 .263 .078
50 .379 .189 .063 .333 .103
80 .550 .359 .224 .431 .173
100 .636 .430 .304 .513 .223
120 .703 .510 .391 .586 .292
200 .873 .724 .578 .802 .553
500 .990 .944 .840 .989 .928

30 .201 .112 .009 .158 .070
50 .328 .208 .075 .224 .103
80 .518 .390 .252 .335 .185
100 .610 .468 .335 .431 .246
120 .685 .548 .422 .512 .323
200 .863 .754 .602 .765 .597
500 .989 .955 .853 .987 .944

30 .099 .112 .009 .027 .043
50 .244 .264 .110 .070 .101
80 .461 .481 .334 .176 .228
100 .566 .560 .427 .277 .324
120 .650 .637 .504 .377 .424
200 .845 .822 .657 .692 .711
500 .987 .974 .877 .984 .975

30 .050 .113 .010 .002 .019
50 .198 .323 .158 .020 .100
80 .428 .554 .414 .108 .279
100 .543 .641 .498 .204 .408
120 .631 .713 .566 .306 .527
200 .834 .869 .694 .650 .807
500 .986 .982 .892 .980 .988

30 .033 .116 .010 .000 .008
50 .181 .369 .201 .009 .098
80 .416 .600 .461 .083 .318
100 .533 .684 .537 .176 .468
120 .624 .750 .600 .280 .590
200 .830 .891 .712 .631 .852
500 .985 .986 .899 .978 .993

.215
.058
.088
.127
.172
.363
.795

.220
.060
.092
.132
.181
.379
.805

.229
.062
.102
.150
.204
.418
.830

.262
.068
.139
.209
.284
.512
.870

.318
.078
.190
.278
.360
.586
.896

.373
.085
.234
.324
.414
.629
.909

0.8

0.5

0.2

Table 2-8

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

£0

.825
.931
.978
.989
.994
1.00
1.00

.822
.930
.978
.989
.994
1.00
1.00

.815
.927
.977
.988
.994
1.00
1.00

.802
.924
.976
.988
.994
.999
1.00

.796
.921
.976
.988
.993
.999
1.00

.795
.920
.976
.988
.993
.999
1.00

(Continued) (A - 1)

"u

12,,

.588
.671
.802
.812
.861
.944
.991

.590
.672
.803
.813
.862
.944
.991

.598
.677
.807
.815
.864
.945
.991

.612
.685
.812
.820
.867
.946
.991

.619
.691
.816
.822
.869
.946
.992

.621
.695
.818
.823
.869
.946
.992

£12

.051
.330
.528
.578
.630
.725
.901

.052
.332
.528
.578
.631
.725
.901

.052
.335
.531
.580
.631
.725
.901

.052
.341
.534
.582
.634
.726
.902

.053
.343
.534
.583
.634
.725
.902

.052
.344
.535
.584
.635
.726
.902

£0

.777
.930
.985
.994
.997
1.00
1.00

.766
.926
.985
.994
.997
1.00
1.00

.742
.918
.984
.993
.997
1.00
1.00

.704
.902
.981
.993
.996
1.00
1.00

.684
.894
.979
.992
.996
1.00
1.00

.678
.889
.979
.991
.996
1.00
1.00

'71
£4.

.408
.560
.778
.797
.851
.958
.998

.412
.564
.781
.799
.852
.959
.998

.421
.572
.788
.803
.856
.959
.998

.440
.591
.800
.813
.863
.962
.998

.453
.602
.808
.818
.866
.963
.998

.458
.606
.811
.820
.868
.964
.998

.158
.165
.338
.406
.488
.665
.914

.157
.165
.340
.408
.489
.666
.914

.156
.166
.342
.409
.490
.667
.914

.151
.172
.346
.413
.494
.670
.914

.146
.175
.350
.416
.495
.670
.914

.147
.175
.352
.415
.495
.671
.914

56

Table 2-8 (Continued) (A - 10,000)

m. n.
a T 20 2, 2,2 20 2,,

0.8 30 .889 .643 .057 .889 .512
50 .961 .701 .348 .973 .627
80 .988 .822 .536 .995 .822
100 .995 .826 .583 .999 .825
120 .997 .870 .635 .999 .871
200 1.00 .947 .726 1.00 .965
500 1.00 .992 .903 1.00 .998

0.5 30 .889 .642 .057 .889 .512
50 .961 .701 .348 .973 .627

80 .988 .822 .536 .995 .822

100 .995 .826 .583 .999 .825

120 .997 .870 .635 .999 .871

200 1.00 .947 .726 1.00 .965

500 1.00 .992 .903 1.00 .998

0.2 30 .889 .642 .057 .889 .512
50 .961 .701 .348 .973 .627

80 .988 .822 .536 .995 .822

100 .995 .826 .583 .999 .825

120 .996 .870 .635 .999 .871

200 1.00 .947 .726 1.00 .965

500 1.00 .992 .903 1.00 .998

—0.2 30 .889 .642 .057 .889 .512
50 .961 .702 .348 .973 .627

80 .988 .822 .536 .995 .822

100 .995 .826 .583 .999 .825

120 .997 .870 .635 .999 .871

200 1.00 .947 .726 1.00 .965

500 1.00 .992 .903 1.00 .998

-0.5 30 .889 .642 .057 .889 .512
50 .961 .702 .348 .973 .627

80 .988 .822 .536 .995 .822

100 .995 .826 .583 .999 .825

120 .997 .970 .635 .999 .871

200 1.00 .947 .726 1.00 .965

500 1.00 .992 .903 1.00 .998

-0.8 30 .889 .642 .057 .889 .512
50 .961 .702 .348 .973 .627

80 .988 .822 .536 .995 .822

100 .995 .826 .583 .999 .825

120 .997 .870 .635 .999 .871

200 1.00 .947 .726 1.00 .965

500 1.00 .992 .903 1.00 .998

.141
.177
.354
.416
.496
.671
.914

.141
.177
.354
.416
.496
.671
.914

.141
.177
.354
.416
.496
.671
.914

.141
.177
.354
.416
.496
.671
.914

.140
.177
.354
.416
.496
.671
.914

.140
.177
.354
.416
.496
.671
.914

57

Table 2-9
Size Comparison of the gu'Test to the Saikkonen and Luukkonen Test

MA(l) errors

a
T Test —O.8 -0.5 0.0 0.5 0.8
100 R, .023 .041 .053 .056 .060
£0 .000 .000 .049 .176 .206
2, .000 .006 .043 .056 .057
2,2 .000 .010 .029 .033 .033
200 R, .030 .047 .052 .057 .060
20 .000 .000 .051 .176 .209
2, .000 .006 .049 .063 .065
2,2 .000 .016 .041 .045 .046
AR(l) errors
p
T Test -0.8 -O.5 0.0 0.5 0.8
100 R,0 .028 .051 .053 .052 .081
R,5 .041 .051 .053 .052 .066
R30 .039 .051 .053 .052 .059
20 .000 .001 .049 .352 .796
2, .007 .019 .043 .090 .250
2,2 .007 .020 .029 .043 .081
200 R,o .027 .046 .047 .045 .074
R,5 .044 .046 .047 .045 .056
R30 .041 .046 .047 .045 .050
£0 .000 .001 .051 .363 .833
2, .008 .021 .049 .099 .271

2,2 .015 .030 .041 .047 .094

58

Table 2-10
Power Comparison of the 6, Test to the Saikkonen and Luukkonen Test

MA(l) errors

a
T 0 Test -0.8 -0.5 0.0 0.5 0.8
100 —0.95 R, .122 .243 .304 .324 .335
£0 .197 .216 .301 .401 .419

2, .483 .416 .268 .209 .201

2,2 .439 .351 .193 .137 .130

-0.90 R, .309 .505 .587 .616 .631

£0 .557 .568 .609 .654 .666

2, .694 .652 .525 .450 .436

2,2 .541 .505 .392 .319 .308

-0.80 R, .534 .749 .831 .858 .867

20 .841 .843 .858 .874 .877

2, .785 .770 .715 .673 .665

2,2 .574 .564 .521 .491 .484

AR(l) errors
p

T 0 Test —0.8 —0.5 0.0 0.5 0.8
100 —0.95 R,5 .273 .292 .288 .265 .266
R30 .269 .292 .288 .265 .226

£0 .218 .233 .309 .513 .827

2, .492 .401 .268 .209 .312

2,2 .422 .317 .193 .115 .121

-0.90 R,5 .563 .581 .555 .481 .447
R30 .562 .581 .555 .480 .377

20 .573 .575 .609 .715 .885

2, .695 .643 .525 .417 .444

2,2 .535 .487 .392 .270 .219

-0.80 R,5 .822 .812 .743 .567 .605
R30 .823 .818 .743 .534 .457

£0 .844 .845 .858 .894 .950

2, .785 .766 .715 .648 .635

2,2 .572 .558 .524 .461 .399

Using equation (11) or (12) in text we find that if 0 - -0.95, —0.9,
and -O.8, then A - .0026, .0111, and .05.

CHAPTER 3

CHAPT ER 3

TESTING FOR A UNIT ROOT: A DUAL APPROACH

3.1 Introduction

This chapter considers using the KPSS test statistic for stationarity to test for
integration in time series data. Many of the existing tests for a unit root are motivated
by considering the problem of testing whether an autoregressive root equals one
against the alternative that it is not equal to one. In contrast, the statistics proposed in
this chapter can be derived from a very different specification, but have almost the
same implication.

The decomposition into stationary and random walk components is a popular
way of thinking about the properties of macroeconomic time series. Since a series
with a unit root is equivalent to a series that is composed of a random walk and a
stationary component, tests for a unit root are attempts to distinguish between series
that have no random walk component and series that have a random walk component.

KPSS propose a statistic (6,, given in equation (25) of the last chapter) to test
the null hypothesis of stationarity (no random walk component) against the alternative
of a unit root (a non-zero random walk component). In this chapter we reverse this
process and consider using the KPSS statistic to test the null hypothesis of a unit root

against the alternative of stationarity. The basic idea behind this procedure has been

59

60

suggested by Stock (1990). Suppose that y, is the series in question and S, =j=2t1 yJ is
the partial sum process of y,. If y, is 1(0), then y, is 0,0) and S, is 0,.(Tm); while if
y, is 1(1), then y, is 0,.(Tm) and S, is 0,.(T3n). Thus Stock suggests that functions of S,
can be used to test H,: y, is I(l) vs H,: y, is 1(0), or vice versa. The idea is that one
statistic can be used to test both hypotheses. The KPSS statistic is of this form (it
depends on 8,) and KPSS use it to test lrl0 vs H,. In this chapter we consider using it
to test H, vs H,. The question of interest is whether a test desigr_red as a stationarity
test will perform well as a unit root test As will be shown, the answer turns out to be

no.

We will derive the asymptotic distribution of the KPSS statistic as a unit root
test and discuss its characteristics in section 3.2. This test is compared to other similar
kinds of unit root tests in section 3.3. The finite sample performance of the test is
investigated via a Monte Carlo simulation in section 3.4. We apply our unit root test

statistics to the Nelson-Plosser data in section 3.5. The concluding remarks are given

in section 3.6.
3.2 The KPSS Test As a Unit Root Test

We will use a components representation of an economic time series to derive
test statistics for the unit root hypothesis. The series of interest y,, can be decomposed

into the sum of a deterministic trend, a random walk and a stationary error:

61
(1) Yr = at + ’Yt + V, 9 t = 1,2,...,T

(2) Yr = 7M + ur

where u, are iid (0,0,2) and v, are stationary errors.
The null hypothesis is simply that 0,2 > 0, so that y, is an 1(1) process under

the null. Under the null (1) can be expressed as:

(3) (1 ' ”K = S ‘1' “t + AVn 01'
I
(4) Yr = 70 + at +j=zluj + V,.

On the other hand, (1) can be expressed in the form of an 1(0) process under the

alternative of 0,2 = 0:
(5) Yr=70+§t+vr

The fundamental difference is that the deviations from trend in (5) are stationary while
in (4) they are an integrated process whose variance increases without bound as t gets
large.

Note that (3) or (4) is a generalization of the first order difference stationary

process which has been used as the counterpart of (5) in most of the unit root

literature:
(6) (1 - L)y, = § + u,, or

t
(7) Y. = To + fit +1333.

62
Comparison shows that (6) or (7) is a special case of (3) or (4); (4) reduces to (7)

when 0,2 = 0. Therefore, our model of (1) and (2) is more general than the Dickey-
Fuller type model. In fact, as shown in Chapter 1 (equations (15) and (16)), our
model (3) or (4) is equivalent to a Dickey-Fuller model with MA(l) errors.

KPSS derive the asymptotic distributions of their test statistics under of > O,

which is our null hypothesis. First, we consider the statistic used by KPSS to test for

level stationarity:
T
(8) fl. = “I" 318.2620)

1
where S, =21 e,, ej = y1 - y, and s2(2) is the Newey-West estimate of the long run
J:

variance of v,.
T l T

(9) 82(9) = (1m[ 26.2 + 22 W02) 23 6.6.-.)
t=1 s=1 t=s+1

where the Bartlett window, w(s,2) = l - 2/(s + l) is used for nonnegativeness of 82(2).
For consistency of $20) under stationarity, it is necessary that the lag truncation
parameter I -) oo as T —> oo. The rate 2 = 0,,(T1’2) will usually be satisfactory (see,
e.g., Andrews (1991)).

We now use the invariance principles (10) and (11) to derive the asymptotic

results for the unit root test statistics.
172,, m Rm
J:

W] W] -
(11) Tmsm, = Ti"2 1'5in -'y) + '1‘”2 13'1“] -v)

63
[a'l‘l -
= T” 13:10,. -7) + 0,.(1) —s 6,]:W(s)ds

where a,b e [0.1], [aT] and [bT] are integer parts of aT and bT, and W(s) = W(s) -

0J'1W(b)db is the demeaned Wiener process. Therefore,
T T -
(12) T“ 23,2 = T1 204/2392 —> 6,211 [I‘W(s)ds]2 da
t=l {=1 0 0
From Phillips (1991) we can also show that
(13) (21‘)1 52(2) —+ K631: W(s)2ds

provided Tm! --> 0 as T —> 00. K is defined by K = £1 k(s)ds where k(s) represents
the weighting function used in calculating s20). Note that if w(s,l) = 1 - t/(s + 1) is
used, then k(s) = 1 - Is! and therefore, K = 1. However, if 2 = 0 is used, the

following holds instead of (13):
(14) T‘s2(2) —> o,2£W(s)2ds

Combining (12) with (13), it is straightforward to see that the unit root test with level,

defined as 7],,(2), has the following asymptotic distribution:

(15) fire) = am fine) —» {)1 [£W(s)ds]2da / K§W(s)2ds
If I = 0 is used,
(16) fine) = (1(1) 11(0) 45 [£W(S)d812da /§W(s>2ds

The analysis for the unit root test statistic in the presence of trend is only

64
slightly more complicated. Now e, is the residual from an OLS regression of y, on

intercept and trend. Correspondingly, we just need to replace the demeaned Wiener

process W(s) above with the demeaned and detrended Wiener process W'(s):
(17) W(s) = W(s) + (6s - 4)£1 W(r)dr + (-12s + 6)!)" rW(r)dr

Therefore, the unit root test statistic with level and trend, defined as '71,, has the

following asymptotic distribution:

(18) rm) = (cm 11,0) —) {,1 [1:W'(s)ds]2da/ K§W(s)2ds
If 1 = 0 is used,
(19) fire) = (1m firm -+ (I: ll: W‘(s>ds12da lg Words

The stationary errors v,. do not have any effect on the asymptotic distribution
of the test statistics under the null hypothesis 0,2 > 0. This implies that under the unit
root null the statistic has the same limiting distribution as that for a pure random walk
process. Thus, although the unit root null can be stated as A = (Sf/0,2 > 0, in fact our
null is effectively A = 00 ((5,,2 = 0). fin and ii, are free of nuisance parameters because
the scale effect from the variance 0,2 > O in the numerator and the denominator of the
limiting distribution cancels out. It is important to note that this is so regardless of the
choice of the lag truncation parameter 2. From the point of view of the KPSS
statistics 1?, and 1‘], as unit root statistics, 2 = 0 is the obvious choice. Other choices of

l (e.g., l proportional to T'") are necessary to correct for error autocorrelation in

given
replic

No.1: 1

33

"resent
PTOCCSS

”511d p11
M an
‘ktendg.

humor],

65

testing the null hypothesis of stationarity, but are not necessary for the unit root test.
In Table 3-0, the critical values of the r.h.s of equations (16) and (19) are
given, calculated via a direct simulation, using a sample size of 2,000, 50,000
replications, and the random number generator GASDEV/RAN3 of Press 91.31. (1986).
Note that the unit root test is a lower tail test. Therefore, we compare fin“) =
(amine) and fi.(0 = (Wm!) [firm = (mire) and fire) = (lffli.(0) when 2 =

0] to the lower 5% critical values to test the unit root null at the 5% level.
3.3 Comparison to Other Similar Tests

As mentioned above, Stock (1990) develops a unifying framework for so—called
"generic" unit root tests based on the fact that an 1(1) process is 0,,(Tm) but an I(0)
process is 0,,(1). Under the null hypothesis that the time series contains deterministic
trend plus an integrated process, Stock works with functionals of the detrended series
itself and shows that those converge weakly to the corresponding functionals of a
detrended Brownian motion. Our unit root tests, 171,, and fi,, also converge weakly to
functionals of a detrended Brownian motion under the null.

According to Stock’s simulation results, the modified Sargan and Bhargava
tests perform relatively well, and they are similar in form to our tests, fl, and fi,. Note

that under the null of a unit root, Stock’s specification is the same as that of Schmidt

and Phillips (1990) or Bhargava (1986):

(17) y. = 4(9) +1.;1 u,

66

where d,(B) represents deterministic trend and the second term is an 1(1) process.
Therefore, (17) is basically the same as (7), though (7) assumes linear deterministic

trend. Modified Sargan-Bhargava tests are
2 T 2 Il' 2
(18) sews) = T t33107.") l82 —> ths)
B 2 T B 2 J’] B 2
(19) saw-r > = T 310. >/62 —> 0 w (s)

where y." = y. -i. y.” = y. - B}. - 6.0m. W“ = W(r) - (r - 1/2)W(1) - Iw<s). 13'. and B}
are the maximum likelihood estimators of B0 and B, assuming d,(B) = B0 «1» B,t. Stock
uses the detrended series itself while we use the partial sum process of detrended
series to derive the test statistics.

Furthermore, Stock estimates the long run variance of the residual as follows:

T l T
(20) 62 =(1/I‘)[Z 6‘3 + 2}: r. {1,111,}
ml j=lt=j+l

where (1‘, is the residual from the regression of detrended (or demeaned) series on
lagged detrended (or demeaned) series, used for consistency. That is, 62 replaces T‘l
$2 (Ay,)2 in the original Sargan-Bhargava tests to make the test nuisance parameters
free. This treatment of the long run variance is different from ours. The form of the
denominator of the KPSS statistic is chosen so as to estimate the long run variance of
the stationary error in the ab_seg_c_e_ of a unit root. Under the unit root null, the limiting
distribution of the denominator of the KPSS statistic involves a functional of Brownian
motion, basically because our residual e, is 1(1), while Stock’s residual is 1(0), under

the unit root null. However, the KPSS statistic is free of nuisance parameters under

67

the unit root null without the need to estimate a long run variance consistently. It is
reasonable to guess that this treatment of nuisance parameters might minimize size
distortions, but also entail some loss of power.

Blough (1989) also represents a time series as a convex combination of a
random walk and a general stationary component following the Beveridge-Nelson

(1981) decomposition:
(21) y, = 5y,, + (l — 5)y2,, 5 = [0,1]

where y,, = u + y,,_, + e,, and y,, = m, + a(L)£,,,. Here m, is deterministic trend, a(0)
= 1, a(1) < co and the e,, are iid. Blough then considers the problem of testing the null
of a unit root 5 = (0.1] against the stationary alternative of 5 = 0. Note that a
sequence of processes {y,(5)} with 5 approaching zero is equivalent to a sequence of
ARIMA process with MA root approaching minus one. For any 5 > 0, the asymptotic
properties will be dominated by the random walk component {y,,}. However, for
finite samples, {y,(5)} will behave like {y,,} for small 5 and therefore the finite sample
distribution of statistics will be dominated by {y,,}. This implies that some unit root
processes behave almost like white noise for a given sample size, which raises

questions about both the possibility of and the need for generic unit root tests.

3.4 Finite Sample Behavior

The finite sample distribution of the unit root test statistics '71,, and fi, will be

tabulated by a Monte Carlo simulation. The results of these simulations (using 20,000

68
replications when T S 100 and 10,000 replications when T > 100) are given in Tables

3-1 through 3-8.
We will use three different specifications of the stationary errors v,: iid, AR(l)

and MA(l) errors.
(22) v, = pv,_, + e, for AR(l) errors
(23) v, = e, + 012,, for MA(l) errors

where e, are iid and values of 21:02, $0.5, and :l:0.8 are used for p and or. When the
errors are iid, the size of the test statistics under the null depends on the variance ratio
A and the sample size T, whereas the power under the alternative depends only on T,
since the alternative specifies A = 0. We will consider three different choices of the
lag truncation parameter 2. These are to = O, 2, = integer[4(T/100)"‘], and 2,, =
integer[12('I‘/100)"‘]. As noted above, we expect 2 = to to be the best choice, but this

may depend on how one weighs the tradeoff between size distortions and low power.

3.4.1 Size

For investigation of size we will choose four different values of A (0.0001,
0.01, 1 and 10,000) and seven values of T (30, 50, 80, 100, 120, 250, 500). We
expect better size performance for large A and worse size performance for small A,
because our null is effectively A = 00 as discussed above. Furthermore, because size

distortions disappear asymptotically, we expect better size performance (for any given

A) when T is large than when T is small.

69

1) iid Errors

The sizes of the tests under iid errors are given in Table 3-1. The tests using
1,2 perform very poorly. They reject too seldom (except when A is very small), and
this problem persists even for rather large values of A and T, such as A = 10,000 and
T = 500.

The tests using 10 and 1, perform very poorly when A = 0.0001. Since A =
0.0001 represents near stationarity, it is not surprising that the tests overreject
substantially. When A = 0.01 the tests still perform poorly, but there is some evidence
of improvement as T increases. The tests using 1,, and 1, perform reasonably well
underthenullforAz 1 ande 100.

The fi, test does not do as well as the fi, test. This is unfortunate, since most
economic time series appear to contain deterministic trends, and'thus the if, test is the
one needed in practice.

2) AR(l) Errors

We next consider the size of the test in the presence of AR(l) errors. Table 3-
2 presents our simulation results. It consists of four pages, corresponding to the four
values of A that we consider.

For A = 10,000, the sizes of the tests are almost independent of the AR(l)
coefficient p, and therefore almost identical to the results for p = 0, as presented
previously in Table 3-1. This is so because, with A = 10,000, the stationary error is
negligible.

When A = l, the tests using 1,, and 1, show similar size performance for

posit

How
small

using

sever:
fort:
impro‘

Endor.

70
positive p, especially for T 2 100. For example, when p = 0.8 and T = 100, the sizes

of fi,(1,,) and fi,(1,) are .058, and the sizes of fi,(1,) and fi,(1,) are .056 and .047.
However, for negative p, the tests using 1,, show small size distortions when T is
small, but the sizes of the tests using 1, are relatively correct in most cases. The tests
using 1,, have low size and in fact, size is zero for fi,(1,,) when T < 200.

For A = 0.01, size distortion occurs for the tests using 1,, and 1,, but this is less
severe as p —> 1; e.g., when T = 100 and p = 0.8, sizes are .371 for fi,(1,,) and .345
for fi,(1,), which are far less than the sizes of the tests using 10 with iid errors. The
improvement of the tests as p —> 1 is expected, since the stationary error approaches a
random walk as p —> 1. The tests using 1, show mixed and intermediate behavior.
Comparing with the results for iid errors, for positive p there is a decrease in size
distortion for small T but an increase in size distortion for large T, while for negative
p we find a slight improvement of size performance. As in other cases, the tests using
1,, do not perform well.

For A = 0.0001, the sizes of the tests using 10 are almost one unless T is small
and p —-) 1. This is reasonable because, as A -—> 0, the series becomes stationary and
the test should reject the unit root null.

Comparing the above results with the results for iid errors, we find that size
distortions are generally less severe as p —> 1, as expected. However, this pattern is
more clear for tests using 1,, than for tests using 1, or 1,,.

3) MA(l) Errors

We now consider the size performance of the test in the presence of MA(l)

71

errors, the results of which are given in Table 3-3.

For A = 10,000, the results are essentially the same as the results for iid and
AR(l) errors. As A -> co, the size performance of tests does not depend upon the
autocorrelation of the stationary errors.

For A = l, the tests using 1, perform somewhat better than the tests using 1,,
and much better than the tests using 1,,. A similar pattern occurs for A = 0.01, except
that the size distortions are larger for the tests using 1,, and 1, (especially 1,). The
tests using 1,, perform well if T is large enough (T 2 200, say).

For A = 0.0001, the sizes of the tests using 10 approach one as T increases, and
the sizes of the tests using 1, are also quite large. Generally, the size distortion of the
tests using 1,, is slightly more severe as or —> -1. The tests using 1, show less size
distortion for small T, but more size distortion for large T, when or is positive
compared to when or is negative. The tests using 1,, also show size distortions for
large T, but their sizes are nearly zero in small
samples, especially for the 71’, test

To sum up, the main determinant of the size performance of the test in finite
samples is the relative variance ratio A. For small values of A, the tests are not
expected to be very exact in finite samples, and therefore the use of longer lags is
needed to avoid severe size distortion. However, when A is large so that random walk
components dominate stationary components, the sizes of the tests using 1,, and 1, are
relatively correct, but the sizes of the tests using 1,, are too low unless T is very

large. Therefore, the use of longer lags (e.g., 1,,) is not necessary or desirable in this

72

case. Comparing the results for MA(l) errors with the results for AR(l) errors, the
size distortions are more severe in the MA(l) case, especially when A is small and the
stationary errors are positively autocorrelated.

Generally, the location of the null (the value of A) is important for the accuracy
of inference since our null is simply A > 0. As our simulation shows, the test
distribution is close to the asymptotic null distribution in finite samples only when A is

sufficiently large.

3.4.2 Power
1) iid Errors

Results for the power performance of the tests in the presence of iid errors are
given in Table 3-4. The power of the tests using 1,, is almost 1 for both it, and fi,
when T > 50. The power of the tests using 1, is close to one in large samples, but it
is less than that of the tests using 1,, in finite samples. The power of the tests using
1,, is very small unless T is very large, and, in particular, power is almost zero for
fi,(1,,) when T S 50 and for 71, when T < 200.
2) AR(l) Errors

We now consider power in the presence of AR(l) errors. These results are
given in Table 3-5.

For positive p, the tests using 1,,, 1,, and 1,, all suffer from low power. This
is as expected, because even under the alternative of stationarity ((3,,2 = 0), y,

approaches an 1(1) process as p —> 1. The tests using 1,, show the poorest power

73

performance. The tests using 1,, show relatively good power performance when p is
away from 1. For example, when p = 0.5 and T = 100, powers are .904 and .908 for
fi,(1,,) and fi,(1,). The tests using 1, show reasonable power unless p is close to 1,
e.g., when T = 100 and p = 0.5, powers are .700 for fi,(1,) and .551 for fi,(1,).

For negative p, the powers of the tests using 1,, are almost one in most cases,
while the tests using 1, show reasonable power unless T is too small. However, the
tests using 1,, are not powerful unless T is large.

3) MA(l) Errors

We now consider the power performance of the test in the presence of MA(l)
errors. These results are given in Table 3-6.

The tests lose power as the MA(l) parameter or —) l, but the power loss is not
as large as it is in the AR(l) case. For example, when T = lOOand or s 0.8, power is
.966 for fi,(1,,) and .976 for fi,(1,,), both of which are far greater than the
corresponding powers in the presence of AR(l) errors with p = 0.8. The powers of
the tests using 1, are less than the powers of the tests using 1,,, but are reasonable.
The tests using 1,, are the least powerful, and power is nearly zero for small T.

The tests are also more powerful for negative or than for positive or. The
powers of the tests using 1, are almost one in most cases, and powers of using 1, are
above .9 when T > 50. However, the tests using 1,, are not powerful in finite
samples.

To sum up, the tests using 1,, are most powerful unless the stationary errors

follow an AR(l) process with p close to one. The tests using 1, have reasonable

74

power in most cases, but are not powerful either when p is close to one. The use of

longer lags (e.g., 1,,), especially for the fi, test, leads to a large loss of power.

3.4.3 Comparison with the Dickey-Fuller Unit Root Tests: Size

Consider the components representation (1), and A = 63/6,}. The series is
stationary if A = 0, and any A > 0 indicates the presence of a unit root. However, it is
important to note that the value of 6,,2 does not appear in the asymptotic distribution
theory for the KPSS statistic under the null hypothesis of a unit root. That is, the 1(1)
component of y, asymptotically dominates the 1(0) component, and asymptotically the
distribution is the same as if 6,,2 = 0 (i.e., A = 00). Thus, under the unit root null, we
should expect finite sample (but not asymptotic) size distortions when 6,2 > 0 (i.e., A
< co).

These size distortions can be related to the size distortions suffered by the
Dickey-Fuller tests or other similar unit root tests when the errors in the Dickey-Fuller
representation are autocorrelated. KPSS show that the model (1) is equivalent to the

ARIMA model (Dickey-Fuller regression with MA(l) enors):

(24) y, = d + By” + w,, w, = a, + 66,,,, B = 1, e, iid.

Indeed, the connection between 0 and A is straightforward (see Harvey (1989), p. 68):
(25) 6 = -[(A+2) - [A(A+4)]m}/2, A = ~(1 + 0)2/0; A 2 0, I 0| S 1

Thus, for a given A in (1), it is reasonable to compare the size distortions of the KPSS

75

unit root tests to the size distortions of the Dickey-Fuller tests in the presence of
MA(l) errors, with parameter 0 given by (25) so as to correspond to the given value
of A.

Table 3-7 gives some results on the sizes of the KPSS unit root test, the
Dickey-Pullers, and 19, tests, and the augmented Dickey-Fuller tests. Other tests that
are similar to the '8, rest, such as the Dickey-Fuller 6, test or the Schmidt and Phillips
test, give similar results. We choose A = oo (9 = O), A = 0.5 (0 = -0.5), and A = 0.05
(9 = -0.8); and we consider T = 25, 50, 100, 250 and 500. The results are based on a
simulation using 20,000 replications when T S 100, and 10,000 replications when T >
100.

We analyze first the tests that allow for level but not trend. These are the fi,
andfi, tests, and their augmented versions.

For 0 = O (A = co) most tests have relatively correct size. However, the size of
fi,(1,,) test is almost zero except when T is large. As 0 —-) -l (A —> 0), positive size
distortions occur (except for the fi,(1,,) test) and 611,) shows the worst size
distortions. The size of the fi(1,) test is also quite considerably distorted, but the size
of fi,(1,) is relatively correct. For example, when T = 100 and 0 = -0.8, the sizes of
6,0,) and £11,) are .997 and .434, but the sizes of fi,(1,,) and fi,(1,) are .311 and .118.
Finally, the sizes offi(1,,) are relatively correct but the sizes of fi,(1,,) are too low.

The results for the tests that allow for trend are similar. When 9 = 0, most test
statistics have relatively correct size, but the sizes of fi,(1,) and fi,(1,,) are low [in

small samples. The problem is much worse for 7],(1,,) than for fi,(1,). Again 641,)

76

shows the worst size distortions as 0 -) -1. When 9 = -0.8 and T = 100, for example,
the sizes ort‘,(r,) and €41.) are 1 and .568, but the sizes of fi,(r,) and fi,(r,) are .6
and .141. On the other hand, f,(1,,) has relatively correct size but the size of fi,(1,,)
is nearly zero unless T is large.

Generally. fire). fire). 12.0.). 22(6). he). and 1.0.) are not reliable because
of poor size, when 9 is close to minus one. This result is consistent with previous
empirical findings that the Dickey-Fuller tests show considerable size distortions when
the errors are MA(l) with negative MA(l) parameter. See Schwert (1987, 1989) and
Lee and Schmidt (1991). The situation where 6 is close to minus one is called the
"nearly stationary case". The asymptotic distribution of the Dickey-Fuller tests when
the process is ’nearly stationary’ is derived in Chapter 4, and shows possible sources
for the size distortions in this situation. As will be seen in Chapter 4, the above
results are consistent with the predictions of the asymptotic theory.

One thing to note is that for A = .5 and .05, the size distortions for fi,(1o) are
much smaller than those of the Dickey-Fullera test. The size distortions for fi,(1o)
are also smaller than those of the 6,0,) test for T 2 250, but not for T S 100.

Asymptotic theory appears to be relevant for smaller values of T for the fi,(1,,) test

than for the 6, test or its augmented versions.

3.4.4 Comparison with the Dickey-Fuller Unit Root Tests: Power
We now turn to the power comparison of the test. Here the null hypothesis of

a unit root is false, so that A = 0 in (l) and B < 1 in the Dickey-Fuller model (24). In

77
order to match the data generating process of the KPSS model (1) with that of the

Dickey-Fuller model, for a given value B < 1, we assume that the errors in the
Dickey-Fuller regression are iid (so 9 = 0 in (24) above), and let the stationary error in
(1) be AR(l) with parameter B: v, = Bv,,, + 6,, with e, iid. Thus the data generating
process implied by both parameterizations is the same; deviations from level and/or
linear trend are AR(l) with parameter B. Table 3—8 presents results for the same tests
as in Table 3-7, for B = 0, .2, .5, .8, .9, .95 and .99.

We consider first the case of level but no trend. When B = 0, the power of the
fi,(1,,) and 6,00) tests is close to one for most cases. Generally, the KPSS unit root
test is less powerful than the Dickey-Fuller test. The difference in power is sometimes
substantial. For example, for T = 100 and B = 0.8, arguably an empirically relevant
set of parameter values, the power of £11,) is .875 while the power of fi,(1,,) is .546.
The power of fi,(1,) is roughly comparable to that of 8,11,). It is generally the case
that the KPSS unit root test is slightly more powerful than the augmentedfi test for T
S 50 and slightly less powerful for T 2 100. The use of longer lags generally loses
power unless T is large, e.g., when T = 100, the powers of fi,(1,,) and 6,0”) are only
.150 and .448.

We now examine the case that allows for liner trend. When B = 0, the powers
of 11,00) and 19,0,) are almost one unless T is small. The comparison of the KPSS
unit root test to the Dickey-Fullera test is easy to summarize. Again the fi, test is
less powerful than the 8, test. The difference in power is also substantial. For

example, for T = 100 and B = .8, the power of 1].},(10) is .65 while the power of fi,(1,,)

78

is .41. The power of fi,(10) is roughly comparable to that of “9424)- It is generally the
case that fi,(1,,) is slightly more powerful than £04) for T S 100 and slightly less
powerful for T 2 250.

Interestingly, comparing these results to the results above for size distortions in
the presence of autocorrelated errors in the Dickey-Fuller representation, we find
support for the general supposition that the tests that are less susceptible to positive
size distortions are also less powerful. It is noted that the use of longer lags loses a
lot of power, and this is more severe for the KPSS unit root test than for the Dickey-

Fuller tests. For example, fi,(1,,) has no power at all when T S 100.

3.5 Applications to the Nelson-Plosser Data

In this section we apply our unit root tests, fin and fig to the data analyzed by
Nelson and Plosser (1982). They find that the unit root hypothesis is rejected at the
5% level for only the unemployment rate series, and it is rejected at about the 10%
level for the industrial production series. These results are typically interpreted as
indicating the presence of a unit root in most of the Nelson-Plosser series.

In Table 3-9 we first present the results for the if, test which we use to test the
null hypothesis of a unit root with level. We consider values of the lag truncation
parameter 1 from 0 to 8. The values of the test statistics are sensitive to the choice of
1, and in fact for every series the value of the test statistic fust decreases and then

increases as 1 increases. This is different from the results for the stationarity test,

tcsr t
and j
a un
trend
not 1
trend

Then

trend.
declir.
10 the

CODES.

wOtllcl

Consist

79

because in this case test statistic depends on 1 in a different way. Nevertheless, the
test outcome is not in very much doubt: for all series except the unemployment rate,
and possibly the nominal wage and the interest rate, we cannot reject the hypothesis of
a unit root with level. Because the Nelson-Plosser series contain obvious deterministic
trends, and because the fi, test does not allow for deterministic trend, these results may
not be reliable. It is well-known that the DF 6,, andf‘,I tests are inconsistent against
trend stationary alternatives. The 17],, test also suffers from this inconsistency problem.
Therefore, for data with linear deterministic trend, the '71, test should be used instead.
We therefore proceed to test the null hypothesis of a unit root with level and
trend, for which if, is the apprOpriate statistic. Once again the test statistics first
decline and then increase as 1 increases. In this case the choice of 1 is also important
to the conclusions. If we do not correct for residual autocorrelation at all, which
corresponds to picking 1 = 0, we would not reject the null hypothesis of the unit root
for any series except for the unemployment rate. Also, if we choose 1 2 4, then we
would not reject the null hypothesis of the unit root in any case, which is very
consistent with our simulation findings that the tests using longer lags are not
powerful. However, if we choose 1 = 1, we find that we can reject the null of a unit
root at the 5% level for three series: the unemployment rate, GNP deflator, and
money. We cannot reject the null of a unit root at the 5% level for the remaining
series, but we can reject a unit root at 10% level for the industrial production series.
We can compare these results to the results from the augmented Dickey-Fuller

‘9, test, assuming an AR(p) model with p = 1 to 9. (That is, p is the number of

80

augmentations of the regression leading to the test statistic). Table 3-10 shows that
the augmented Dickey-Fuller t-test cannot reject the unit root hypothesis at the 5%
level in almost all cases. The rare exceptions are the unemployment rate for the
AR(2) and AR(4) specifications and the money stock for the AR(8) specification.
These findings are quite consistent with those of Nelson and Plosser (1982), Rudebush
(1990), and others. Using the bootstrapping method, Rudebush derives the p-value
(the marginal significance level) for rejection of the unit root null hypothesis and finds
that, among all of the variables, there is only one for which the unit root hypothesis
can be rejected at the 5% level: the unemployment rate.

Combining the results of our unit root tests with the results of the KPSS
stationarity tests, the following picture emerges. Three series (unemployment rate,
GNP deflator, and money) appear to be trend stationary, since we can reject the unit
root hypothesis and cannot reject the trend stationarity hypothesis. Five series
(consumer prices, real wages, velocity, and stock prices and possibly industrial
production) appear to have unit roots, since we can reject the trend stationarity
hypothesis and cannot reject the unit root hypothesis. Three more series (real GNP,
nominal GNP, and the interest rate) probably have unit roots, though the evidence
against the trend stationarity hypothesis is only marginally significant. Employment
and real per capita GNP are probably trend stationary, which is consistent with
Cochrane (1988), though the evidence against the unit root is only marginally
significant. For the nominal wage we cannot reject either the unit root or the trend

stationarity hypothesis, and the appropriate conclusion is presumably just that the data

81

are not sufficiently informative.

There are two interesting cases: industrial production and real GNP. For
industrial production we can reject the trend stationarity hypothesis at the 5% level
and the unit root hypothesis at the 10% level, while for real GNP we can reject the
trend stationarity hypothesis at the 10% level and the unit root hypothesis at about the
20% level. It seems that the data are not sufficiently informative to distinguish clearly
between these hypotheses. These results may be indicative of "near stationarity" of
the series. These results are also consistent with the Clark’s (1987) finding that the
data allocate a substantial fraction of the short run variation in real GNP and industrial
production to a persistent business cycle, with less variation allocated to a stochastic
trend that evolves smoothly over time. However, the results could also indicate the
necessity to consider other, different models, such as fractional integration.

Finally, our result indicates that stock prices seem to behave as a pure random
walk process. This is contrary to the weak evidence of a slowly mean reverting
characteristics of stock prices over a long horizon (probably, 3 - 5 years) suggested in
the finance literature. See Fama and French (1988) and Poterba and Summers (1987).
One possible reason is that we deal with yearly data only and further research is

needed for more general analysis.

3.6 Concluding Remarks

The KPSS stationary test is based on a components model in which an

economic time series is expressed as the sum of a deterministic trend, a random walk,

82

and a stationary error. We have considered the use of this statistic to test the null
hypothesis of a unit root. We have derived its asymptotic distribution under the unit
root null, considered its finite sample performance by a Monte Carlo simulation, and
applied it to the Nelson-Plosser data series.

The asymptotic distribution of the test has been shown to be free of nuisance
parameters and is the same as the distribution for the pure random walk process. In
finite samples, the main determinant of the size performance of the test is the relative
variance ratio A rather than the autocorrelation of the stationary errors. Therefore,
since our null is simply A > 0, the location of the null is important for the quality of
inference in finite samples. Simulation results show that the distribution of the test
statistics in finite samples is close to the asymptotic null distribution when A and T are
sufficiently large. In this case the use of shorter lags (1) is preferred. On the other
hand, when A is small, the test is not expected to be exact. In this case, the use of
longer lags is needed to avoid size distortions. Comparing the results for AR(l) errors
with those for MA(l) errors, the size distortion is more severe in the MA(l) case,
especially when A is small and the errors are positively autocorrelated. This is not
surprising because the process with AR(l) errors approaches an 1(1) process as p -) 1.

Our results on power performance can be summarized as follows: the test using
1 = 0 is more powerful than the test using lags (1 > 0) in most cases. Especially,
fi,(1,,) does not have any power even against the white noise alternative (A = 0 and
stationary error is iid) in finite samples. The test using 1, has reasonable power in

most cases. However, when the stationary errors follow an AR(l) process with AR(l)

83

parameter close to one, even the tests using 1,, and 1, are not powerful.

To sum up, the use of longer lags is prefened in terms of size performance in
the "nearly stationary case" while the use of shorter lags is preferred in terms of power
performance in the "nearly integrated case."

We have also compared the size and power performance of our test statistics
with those of the Dickey-Fuller test statistics. Our results are not very encouraging for
dual-use statistics. The KPSS statistic, designed for use as a test of stationarity, does
not make a particularly powerful unit root test. In particular, its power is noticeably
less than the power of the Dickey-Fuller 8, test (or other similar tests) against trend
stationary alternatives. This is intuitively reasonable. We would expect a similar lack
of power if standard unit root test statistics were used to test the null hypothesis of
stationarity.

We have applied our unit root tests to the data analyzed by Nelson and Plosser
(1982). Based on simulation results on their finite sample performance and based on
the sample autocorrelations of the series in first differences, we choose the lag
truncation parameter as 1 = l (or 1 =0) for the fi, test. Using the results for 1 = l,
we find that we can reject the null of a unit root at the 5% level for only three series:
the unemployment rate, GNP deflator, and money.

Combining the above results with the results of the KPSS stationarity tests, the
following picture emerges. Three series (unemployment rate, GNP deflator, and
money) appear to be trend stationary. Five series (consumer prices, real wages,

velocity, stock prices, and industrial production) appear to have unit roots. Three

84

more series (real GNP, nominal GNP, and the interest rate) probably have unit roots,
while two more series (employment and real per capita GNP) are probably trend
stationary. For the nominal wage we have no clear conclusion.

Our results are in broad agreement with the results of Clark (1987), Cochrane
(1988), Dejong et al. (1989), and Rudebush (1990), and with the Bayesian analyses of
DeJong and Whiteman (1991) and Phillips (1991). It suggests that for many series the
existence of a unit root is in doubt, despite the failure of the Dickey-Fuller tests (and
other unit root tests) to reject the unit root hypothesis. Presumably other alternatives,
such as fiactional integration or stationarity around more general non-linear trend,

could be considered.

85

Table 3-0

Critical Values for Unit Root Tests

nu(0) n.(0)
0.010 0.0053 0.0021
0.025 0.0074 0.0027
0.050 0.0099 0.0033
0.100 0.0141 0.0043
0.200 0.0213 0.0058
0.300 0.0300 0.0072
0.400 0.0405 0.0086
0.500 0.0514 0.0100
0.600 0.0615 0.0116
0.700 0.0708 0.0135
0.800 0.0793 0.0156
0.900 0.0872 0.0183
0.950 0.0915 0.0199
0.975 0.0940 0.0211
0.990 0.0959 0.0221

A

.0001

.01

1.0

10000

T

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
200
500

30
50
80
100
120
200
500

86

Table 3-1

Size with iid Errors

£0

.854
.958
.992
.996
.998
1.00
1.00

.715
.734
.676
.647
.612
.509
.327

.097
.083
.072
.067
.057
.050

.045
.045
.045
.046
.049
.046
.046

"u

£4

.528
.610
.811
.787
.821
.921
.904

.406
.344
.374
.282
.274
.231
.141

.066
.049
.065
.052
.063
.062

.046
.041
.053
.049
.054
.060
.062

212

.000

.000

.054
.140

.222

.461

.558

.000
.000
.016
.030
.050
.059
.057

.000
.000
.004
.007
.021
.037

.000
.000
.003
.007
.013
.022
.037

£0

.814
.967
.997
.999
.999
1.00
1.00

.768
.901
.937
.930
.921
.868
.653

.131
.120
.104
.089
.068
.053

.034
.041
.042
.043
.044
.044
.044

’71

.126
.289
.754
.707
.789
.963
.994

.114
.209
.495
.372
.412
.416
.238

.016
.019
.045
.034
.059
.063

.008
.013
.037
.029
.036
.053
.061

£12

.000
.000
.000
.000
.000
.161
.684

.000
.000
.000
.000
.000
.023
.053

.000
.000
.000
.000
.004
.022

.000
.000
.000
.000
.000
.004
.022

87

Table 3—2 Size with AR(l) Errors (A - 10,000)

17,. m
P T £0 £4 £12 £0 £0 £12
0.8 30 .045 .046 .000 .034 .008 .000
50 .046 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .048 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .004
500 .046 .060 .036 .042 .061 .022
0.5 30 .045 .046 .000 .034 .008 .000
50 .045 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .049 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .005
500 .046 .060 .036 .043 .061 .021
0.2 30 .045 .046 .000 .034 .008 .000
50 .045 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .049 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .005
500 .046 .060 .036 .043 .061 .021
-0.2 30 .045 .046 .000 .034 .008 .000
50 .045 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .049 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .005
500 .046 .060 .037 .043 .061 .021
-0.5 30 .045 .046 .000 .034 .008 .000
50 .045 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .049 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .005
500 .046 .060 .036 .043 .061 .021
-0.8 30 .045 .046 .000 .034 .008 .000
50 .045 .041 .000 .041 .012 .000
80 .049 .060 .003 .043 .037 .000
100 .046 .049 .007 .043 .029 .000
120 .049 .054 .013 .044 .036 .000
200 .047 .062 .022 .044 .053 .005

500 .046 .060 .036 .043 .061 .021

0.8

0.5

0.2

—0.2

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

Table 3-2

10

.060
.059
.060
.058
.059
.053
.048

.078
.073
.067
.064
.065
.057
.049

.092
.081
.072
.065
.065
.059
.049

.103
.086
.074
.068
.067
.059
.049

.107
.088
.075
.069
.067
.059
.049

.106
.087
.074
.069
.067
.059
.049

(Continued) (A - 1)

52”..
2.

.060
.049
.069
.056
.062
.067
.063

.068
.054
.070
.056
.060
.066
.062

.067
.052
.067
.054
.060
.065
.061

.061
.048
.064
.051
.058
.063
.060

.057
.045
.062
.049
.056
.062
.060

.056
.043
.060
.049
.055
.062
.060

£12

.000
.000
.003
.007
.014
.022
.037

.000
.000
.004
.008
.013
.021
.036

.000
.000
.004
.007
.013
.021
.037

.000
.000
.004
.007
.013
.021
.036

.000
.000
.003
.007
.013
.021
.036

.000
.000
.004
.007
.013
.021
.036

£0

.042
.054
.058
.058
.061
.058
.050

.075
.086
.079
.077
.081
.065
.052

.112
.111
.092
.086
.081
.068
.053

.146
.127
.101
.091
.085
.070
.053

.163
.135
.103
.093
.087
.071
.053

.170
.139
.103
.092
.087
.071
.053

’71
£1.

.107
.016
.049
.037
.047
.065
.066

.015
.022
.055
.039
.046
.063
.064

.016
.021
.051
.035
.045
.061
.063

.014
.017
.045
.033
.040
.059
.062

.012
.015
.042
.030
.039
.057
.061

.014
.012
.039
.031
.038
.055
.061

£12

.000
.000
.000
.000
.000
.005
.024

.000
.000
.000
.000
.000
.005
.023

.000
.000
.000
.000
.000
.004
.022

.000
.000
.000
.000
.000
.005
.022

.000
.000
.000
.000
.000
.005
.022

.000
.000
.000
.000
.000
.005
.022

0.8

0.2

—0.2

Table 3-2
T £0
30 .162
50 .252
80 .342

100 .371
120 .387
200 .385
500 .290
30 .413
50 .540
80 .568
100 .563
120 .546
200 .472
500 .312
30 .616
50 .684
80 .651
100 .625
120 .594
200 .496
500 .317

30 .776
50 .772
80 .702
100 .662
120 .622
200 .512
500 .319
30 .841
50 .802
80 .717
100 .677
120 .632
200 .515
500 .321
30 .859
50 .807
80 .719
100 .676
120 .632
200 .513
500 .321

89

(Continued) (A - 0.01)

"M
£7.
.150
. 198
.323
.304
.333

.355
.264

.307
.341
.432
.363
.367
.336
.209

.386
.360
.406
.321
.316
.270
.163

.406
.321
.335
.241
.234
.196
.117

.404
.269
.267
.188
.178
.152
.094

.415
.170
.170
.148
.137
.121
.075

£12

.000
.000
.027
.052
.095
.137
.139

.000
.000
.028
.053
.086
.097
.089

.000
.000

.020
.038

.062

.071
.065

.000
.000
.012
.023
.040
.049
.049

.000
.000
.007
.016
.026
.035
.043

.000
.000
.004
.011
.020
.027
.039

£0

.082
.153
.282
.352
.416
.537
.541

.297
.540
.729
.765
.785
.784
.627

.603
.810
.891
.893
.891
.845
.647

.882
.947
.957
.950
.937
.876
.659

.964
.979
.975
.964
.952
.886
.664

.986
.986
.978
.966
.953
.889
.663

.7,
1.

.018
.045
.211
.202
.290
.464
.475

.060
.143
.452
.390
.468
.547
.389

.103
.199
.506
.400
.450
.472
.289

.115
.208
.466
.334
.362
.351
.189

.139
.182
.393
.263
.282
.259
.135

.308
.061
.229
.217
.220
.187
.093

£12

.000
.000
.000
.000
.000
.049
.175

.000
.000
.000
.000
.000
.044
.101

.000
.000
.000
.000
.000
.028
.064

.000
.000
.000
.000
.000
.015
.042

.000
.000
.000
.000
.000
.010
.033

.000
.000
.000
.000
.000
.007
.026

0.8

0.5

0.2

—0.2

Table 3-2

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

£0

.178
.300
.458
.544
.612
.790
.952

.471
.695
.846
.898
.929
.982
.997

.733
.895
.966
.984
.992
.998
1.00

.930
.987
.999
1.00
1.00
1.00
1.00

.986
.999
1.00
1.00
1.00
1.00
1.00

.999
.00
.OO
.00
.00
.00
.00

Hl-‘r-‘r-‘t-‘H

"u
£6

.165
.238
.436
.455
.535
.743
.919

.347
.464
.693
.690
.763
.896
.953

.470
.568
.778
.758
.820
.922
.932

.577
.645
.832
.797
.844
.917
.856

.674
.697
.862
.820
.853
.898
.755

.845
.709
.854
.869
.877
.869
.578

(Continued) (A

212

.000
.000
.038
.089
.177
.382
.710

.000
.000
.056
.129
.250
.463
.687

.000
.000
.056
.138
.265
.473
.608

.000
.000
.051
.143
.269
.440
.489

.000
.000
.048
.141
.266
.390
.376

.000
.000
.036
.141
.253
.303
.235

- 0.0001)
n.

20 2,
.083 .018
.161 .049
.317 .234
.410 .242
.504 .358
.764 .683
.982 .958
.313 .064
.601 .165
.837 .563
.907 .548
.944 .682
.992 .909
1.00 .993
.638 .109
.888 .252
.981 .702
.994 .663
.996 .781
1.00 .951
1.00 .994
.925 .135
.993 .320
1.00 .795
1.00 .740
1.00 .838
1.00 .968
1.00 .991
.990 .186
1.00 .356
1.00 .851
1.00 .800
1.00 .881
1.00 .974
1.00 .979
1.00 .503
1.00 .203
1.00 .864
1.00 .923
1.00 .956
1.00 .988
1.00 .917

£12

.000
.000
.000
.000
.000
.096
.656

.000
.000
.000
.000
.000
.148
.730

.000
.000
.000
.000
.000
.158
.708

.000
.000
.000
.000
.000
.158
.643

.000
.000
.000
.000
.000
.155
.546

.000
.000
.000
.000
.000
.149
.372

91

Table 3—3 Size with MA(l) Errors (A - 10,000)

0.8

0.2

-0.8

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

30
50
80
100
120
200
500

36

.045
.045
.049
.046
.049
.047
.046

.045
.045
.049
.046
.049
.047
.046

.045
.045
.049
.046
.049
.047
.046

.045
.045
.049
.046
.049
.047
.046

.045
.045
.049
.046
.049
.047
.046

.045
.045
.049
.046
.049
.047
.046

'7.
2.

.046
.041
.059
.049
.054
.062
.060

.046
.041
.059
.049
.054
.062
.060

.046
.041
.059
.049
.054
.062
.060

.046
.041
.060
.049
.054
.062
.060

.046
.041
.060
.049
.054
.062
.060

.046
.041
.060
.049
.054
.062
.060

£12

.000
.000
.003
.007
.013
.022
.036

.000
.000
.003
.007
.013
.022
.036

.000
.000
.003
.007
.013
.022
.036

.000
.000
.003
.007
.013
.022
.036

.000
.000
.003
.007
.013
.022
.036

.000
.000
.003
.007
.013
.022
.036

£0

.034
.041
.043
.043
.044
.044
.043

.034
.041
.043
.043
.044
.044
.043

.034
.041
.043
.043
.044
.044
.043

.034
.041
.042
.043
.044
.044
.043

.034
.041
.042
.043
.044
.044
.043

.035
.041
.042
.043
.044
.044
.043

~

'71
£2.

.008
.012
.037
.029
.036
.053
.061

.008
.012
.037
.029
.036
.053
.061

.008
.012
.037
.029
.036
.053
.061

.008
.012
.037
.029
.036
.053
.061

.008
.012
.037
.029
.036
.053
.061

.008
.012
.037
.029
.036
.053
.061

£12

.000
.000
.000
.000
.000
.005
.021

.000
.000
.000
.000
.000
.005
.021

.000
.000
.000
.000
.000
.005
.021

.000
.000
.000
.000
.000
.005
.021

.000
.000
.000
.000
.000
.005
.021

.000
.000
.000
.000
.000
.005
.021

0.8

0.2

Table 3-3
T 20
30 .085
50 .078
80 .070
100 .065
120 .065
200 .057
500 .049
30 .087
50 .079
80 .070
100 .065
120 .065
200 .058
500 .049
30 .081
50 .083
80 .072
100 .065
120 .065
200 .059
500 .049
30 .103
50 .086
80 .074
100 .068
120 .067
200 .059
500 .049
30 .108
50 .089
80 .075
100 .070
120 .067
200 .059
500 .050
30 .109
50 .091
80 .075
100 .069
120 .067
200 .059
500 .050

92

(Continued) (A = 1)

5,
2.

.072
.055
.069
.055
.061
.066
.062

.070
.054
.068
.055
.061
.066
.062

.051
.052
.066
.054
.059
.065
.060

.061
.047
.063
.051
.057
.063
.060

.057
.046
.062
.050
.056
.062
.060

.057
.044
.061
.050
.055
.062
.060

£12

.000
.000
.004
.008
.013
.022
.036

.000
.000
.004
.008
.013
.022
.036

.000
.000
.004
.007
.013
.021
.037

.000
.000
.004
.007
.013
.021
.036

.000
.000
.004
.007
.013
.021
.036

.000
.000
.003
.007
.013
.021
.036

£0

.088
.097
.086
.082
.079
.067
.053

.097
.102
.087
.084
.080
.067
.053

.115
.112
.093
.086
.082
.068
.053

.147
.128
.102
.092
.085
.070
.053

.166
.138
.104
.093
.088
.070
.053

.174
.143
.105
.094
.088
.070
.054

.000
.000
.000
.000
.000
.004
.023

.000
.000
.000
.000
.000
.004
.023

.000
.000
.000
.000
.000
.004
.022

.000
.000
.000
.000
.000
.004
.022

.000
.000
.000
.000
.000
.004
.022

.000
.000
.000
.000
.000
.004
.022

93

Table 3-3 (Continued) (A - 0.01)

m. n.
a T £0 £6 £12 £0 £2 £12
0.8 30 .520 .391 .000 .416 .109 .000
50 .625 .385 .000 .694 .207 .000
80 .619 .442 .024 .836 .530 .000
100 .603 .359 .046 .850 .432 .000
120 .577 .353 .074 .855 .493 .000
200 .487 .309 .083 .823 .529 .035
500 .316 .185 .073 .641 .342 .077
0.5 30 .553 .393 .000 .477 .109 .000
50 .646 .379 .000 .736 .207 .000
80 .630 .431 .022 .856 .526 .000
100 .610 .348 .043 .867 .424 .000
120 .583 .341 .080 .870 .481 .000
200 .491 .296 .079 .831 .514 .032
500 .317 .178 .070 .643 .324 .072
0.2 30 .633 .398 .000 .630 .111 .000
50 .694 .364 .000 .829 .207 .000
80 .658 .405 .019 .901 .513 .000
100 .630 .316 .037 .902 .400 .000
120 .598 .310 .060 .898 .448 .000
200 .498 .264 .069 .849 .464 .026
500 .318 .159 .063 .648 .281 .062
-0.2 30 .784 .414 .000 .893 .120 .000
50 .777 .323 .000 .953 .214 .000
80 .705 .333 .012 .960 .470 .000
100 .664 .237 .022 .953 .333 .000
120 .624 .230 .038 .940 .358 .000
200 .513 .192 .048 .878 .343 .015
500 .319 .115 .048 .660 .183 .041
-0.5 30 .867 .433 .000 .983 .142 .000
50 .820 .288 .000 .988 .227 .000
80 .727 .270 .006 .981 .426 .000
100 .682 .181 .013 .971 .265 .000
120 .638 .169 .023 .958 .276 .000
200 .519 .142 .031 .890 .244 .009
500 .322 .088 .041 .668 .125 .030
-0.8 30 .902 .445 .000 .997 .175 .000
50 .836 .263 .000 .995 .242 .000
80 .738 .235 .004 .987 .402 .000
100 .688 .150 .010 .976 .225 .000
120 .643 .138 .019 .963 .226 .000
200 .522 .120 .026 .896 .189 .006

500 .324 .076 .038 .671 .100 .025

0.8

0.5

0.2

-O.2

94

Table 3-3 (Continued) (A - 0.0001)

n, n.
T 2, 2, 2,2 2, 2,
30 .606 .458 .000 .439 .113
50 .821 .560 .000 .773 .250
80 .928 .773 .057 .942 .691
100 .962 .756 .139 .974 .655
120 .975 .820 .267 .987 .774
200 .995 .927 .479 .999 .950
500 .999 .949 .647 1.00 .995
30 .651 .465 .000 .505 .115
50 .848 .564 .000 .817 .253
80 .944 .777 .057 .959 .698
100 .972 .758 .140 .984 .660
120 .983 .821 .267 .991 .779
200 .996 .926 .478 1.00 .951
500 .999 .946 .635 1.00 .995
30 .757 .489 .000 .668 .118
50 .909 .583 .000 .907 .265
80 .973 .789 .056 .986 .719
100 .988 .768 .140 .996 .679
120 .994 .827 .267 .998 .792
200 .999 .926 .474 1.00 .956
500 1.00 .928 .596 1.00 .995
30 .940 .597 .000 .935 .143
50 .990 .661 .000 .995 .335
80 .999 .843 .051 1.00 .812

100 1.00 .807 .144 1.00 .755
120 1.00 .850 .271 1.00 .845
200 1.00 .919 .438 1.00 .971
500 1.00 .850 .479 1.00 .991

30 .997 .777 .000 .998 .209
50 1.00 .803 .000 1.00 .505
80 1.00 .922 .045 1.00 .941
100 1.00 .876 .152 1.00 .893
120 1.00 .894 .282 1.00 .944
200 1.00 .911 .377 1.00 .989
500 1.00 .726 .320 1.00 .979
30 1.00 .955 .000 1.00 .349
50 1.00 .963 .000 1.00 .817
80 1.00 .988 .036 1.00 .999
100 1.00 .945 .183 1.00 .994
120 1.00 .941 .316 1.00 .998
200 1.00 .898 .309 1.00 .999
500 1.00 .612 .186 1.00 .957

.000
.000
.000
.000
.000
.160
.727

.000
.000
.000

000

.000
.160
.723

.000
.000
.000
.000
.000
.159
.706

.000
.000
.000
.000
.000
.159
.637

.000
.000
.000
.000
.000
.167
.501

.000
.000
.000
.000
.000
.193
.322

95

Table 3-4

Power with iid Errors (A - O)

"p "1
T £0 24 £12 20 24 £12
30 .854 .528 .000 .814 .126 .000
50 .960 .615 .000 .969 .291 .000
80 .994 .823 .057 .997 .757 .000
90 .996 .860 .140 .999 .825 .000
100 .998 .802 .150 1.00 .712 .000
120 .999 .862 .286 1.00 .822 .000

200 1.00 .961 .535 1.00 .971 .172
500 1.00 .998 .854 1.00 .999 .811

 

96

Table 3-5 Power with AR(l) Errors (A — 0)

0,. m
P T 30 £4 £12 £0 24 £12
0.8 30 .177 .165 .000 .083 .018 .000
50 .301 .239 .000 .161 .049 .000
80 .461 .436 .038 .317 .234 .000
100 .546 .455 .089 .410 .244 .000
120 .614 .538 .179 .507 .358 .000
200 .799 .756 .391 .767 .687 .097
500 .976 .951 .771 .985 .964 .682
0.5 30 .472 .349 .000 .312 .063 .000
50 .697 .465 .000 .602 .166 .000
80 .851 .698 .057 .837 .561 .000
100 .904 .700 .134 .908 .551 .000
120 .935 .773 .256 .946 .684 .000
200 .988 .916 .494 .994 .919 .153
500 1.00 .993 .835 1.00 .997 .779
0.2 30 .733 .473 .000 .639 .109 .000
50 .898 .572 .000 .889 .254 .000
80 .969 .789 .059 .981 .704 .000
100 .987 .774 .146 .994 .668 .000
120 .994 .837 .276 .997 .785 .000
200 .999 .949 .521 1.00 .960 .168
500 1.00 .997 .849 1.00 .999 .802
-0.2 30 .933 .580 .000 .924 .135 .000
50 .989 .652 .000 .994 .322 .000
80 .999 .851 .056 1.00 .801 .000

100 1.00 .828 .154 1.00 .749 .000
120 1.00 .883 .296 1.00 .852 .000
200 1.00 .971 .546 1.00 .978 .178
500 1.00 .999 .862 1.00 1.00 .822

—0.5 30 .987 .679 .000 .990 .187 .000
50 .999 .713 .000 1.00 .361 .000
80 1.00 .891 .054 1.00 .857 .000
100 1.00 .870 .161 1.00 .819 .000
120 1.00 .918 .318 1.00 .900 .000
200 1.00 .984 .572 1.00 .990 .190
500 1.00 1.00 .874 1.00 1.00 .843

-0.8 30 .999 .856 .000 1.00 .504 .000
50 1.00 .753 .000 1.00 .704 .000

80 1.00 .927 .043 1.00 .888 .000

100 1.00 .956 .203 1.00 .952 .000

120 1.00 .978 .391 1.00 .978 .000

200 1.00 .998 .642 1.00 1.00 .240

500 1.00 1.00 .908 1.00 1.00 .891

 

97

Table 3-6 Power with MA(l) Errors (A - 0)

’7“ ”1

a T 20 £4 £12 £0 £4 £12
0.8 30 .606 .460 .000 .440 .114 .000
50 .823 .564 .000 .775 .250 .000

80 .933 .781 .059 .943 .693 .000

100 .966 .768 .146 .976 .660 .000

120 .981 .832 .275 .987 .778 .000

200 .998 .946 .519 1.00 .956 .166

500 1.00 .997 .848 1.00 .999 .799

0.5 30 .652 .468 .000 .505 .115 .000
50 .852 .569 .000 .819 .253 .000

80 .949 .786 .059 .960 .700 .000

100 .975 .771 .147 .985 .665 .000

120 .987 .835 .276 .992 .782 .000

200 .999 .948 .520 1.00 .958 .167

500 1.00 .997 .849 1.00 .999 .800

0.2 30 .757 .490 .000 .669 .119 .000
50 .912 .587 .000 .908 .265 .000

80 .976 .802 .058 .987 .723 .000

100 .990 .783 .148 .996 .684 .000

120 .996 .845 .279 .998 .798 .000

200 1.00 .954 .526 1.00 .963 .168
500 1.00 .998 .851 1.00 .999 .804

—0.2 30 .942 .599 .000 .936 .143 .000
50 .991 .667 .000 .995 .338 .000

80 1.00 .863 .055 1.00 .816 .000

100 1.00 .838 .156 1.00 .765 .000

120 1.00 .891 .301 1.00 .863 .000

200 1.00 .976 .552 1.00 .982 .179

500 1.00 .999 .865 1.00 1.00 .827

-0.5 30 .998 .783 .000 .998 .208 .000
50 1.00 .825 .000 1.00 .513 .000

80 1.00 .954 .052 1.00 .949 .000

100 1.00 .935 .188 1.00 .914 .000

120 1.00 .962 .371 1.00 .962 .000

200 1.00 .996 .627 1.00 .998 .227

500 1.00 1.00 .903 1.00 1.00 .888

-0.8 30 1.00 .964 .000 1.00 .351 .000
50 1.00 .987 .000 1.00 .836 .000

80 1.00 1.00 .063 1.00 1.00 .000

100 1.00 1.00 .385 1.00 1.00 .000

120 1.00 1.00 .688 1.00 1.00 .000

200 1.00 1.00 .903 1.00 1.00 .528

500 1.00 1.00 .996 1.00 1.00 .997

98
Table 3-7

Size Comparison of the KPSS Unit Root Tests to
The Dickey-Fuller Tests Under MA(l) Errors

(a) Tests That Allow Level But Not Trend

"u 7n
T 9 £0 £4 312 20 £4 312
25 -0.8 .463 .197 .000 .923 .522 .036
-0.5 .151 .070 .000 .418 .143 .038
0.0 .044 .036 .000 .050 .052 .039
50 -0.8 .418 .161 .000 .989 .471 .046
-0.5 .114 .059 .000 .523 .082 .035
0.0 .045 .041 .000 .051 .047 .036
100 -0.8 .311 .118 .012 .997 .434 .055
-0.5 .088 .057 .007 .573 .069 .039
0.0 .046 .048 .007 .053 .049 .043
250 —0.8 .193 .090 .036 .999 .371 .054
-0.5 .067 .063 .028 .604 .058 .045

0.0 .050 .060 .028 .049 .047 .044

500 -0.

00

.122 .077 .039 .999 .403 .058
.053 .062 .036 .610 .057 .046
0.0 .046 .060 .036 .053 .052 .046

I
O
u:

(b) Tests That Allow Trend
T o 20 2. 2.2 20 2. £12

25 -0.8 .549 .203 .000 .900 .466 .033
—0.5 .195 .011 .000 .514 .166 .034
0.0 .032 .001 .000 .050 .052 .041

50 -0.8 .673 .185 .000 1.00 .518 .045
-0.5 .186 .025 .000 .709 .099 .032
0.0 .041 .012 .000 .052 .045 .034

100 —0.8 .600 .141 .000 1.00 .568 .055
-0.5 .134 .040 .000 .794 .079 .039
0.0 .043 .029 .000 .054 .044 .040

250 —0.

00

.381 .111 .011 1.00 .551 .055
.082 .053 .006 .841 .064 .039
0.0 .043 .048 .006 .051 .050 .040

I
O
U1

500 -0.8 .237 .095 .028 1.00 .613 .057
-0.5 .063 .064 .023 .853 .065 .046
0.0 .043 .061 .021 .052 .049 .048

99

Table 3—8

Power Comparison of the KPSS Unit Root Tests To The Dickey-Fuller Tests

(a) Power Comparison With Level But No Trend

T 5 30 34 £12 £0 £4 312

25 .00 .798 .414 .000 .990 .794 .064 .989
.20 .667 .364 .000 .937 .656 .068 .923
.50 .409 .257 .000 .564 .380 .068 .510
.80 .151 .115 .000 .145 .135 .075 .111
.90 .093 .076 .000 .088 .091 .073 .054
.95 .067 .055 .000 .075 .078 .073 .037
.99 .048 .041 .000 .067 .069 .070 .026

50 .00 .960 .615 .000 1.00 .955 .135 1.00
.20 .898 .572 .000 1.00 .906 .127 1.00
.50 .697 .465 .000 .983 .687 .109 .991
.80 .301 .239 .000 .348 .224 .077 .404
.90 .162 .138 .000 .127 .105 .057 .147
.95 .101 .089 .000 .079 .067. .051 .081
.99 .056 .052 .000 .063 .060 .049 .044

100 .00 .998 .802 .150 1.00 1.00 .448 1.00
.20 .987 .774 .146 1.00 .999 .425 1.00
.50 .904 .700 .134 1.00 .979 .351 1.00
.80 .546 .456 .089 .875 .605 .199 .940
.90 .289 .265 .048 .320 .243 .113 .430
.95 .151 .150 .015 .123 .108 .075 .170
.99 .066 .068 .009 .064 .060 .050 .063

250 .00 1.00 .961 .617 1.00 1.00 .980 1.00
.20 1.00 .951 .607 1.00 1.00 .972 1.00
.50 .995 .924 .581 1.00 1.00 .949 1.00
.80 .861 .791 .490 1.00 .999 .778 1.00
.90 .609 .589 .352 .972 .852 .501 .994
.95 .350 .366 .197 .452 .357 .215 .606
.99 .095 .113 .054 .066 .061 .053 .096

500 .00 1.00 .998 .855 1.00 1.00 1.00 1.00
.20 1.00 .997 .849 1.00 1.00 1.00 1.00
.50 1.00 .993 .779 1.00 1.00 1.00 1.00
.80 .976 .951 .771 1.00 1.00 .999 1.00
.90 .849 .842 .656 1.00 1.00 .958 1.00
.95 .604 .636 .473 .967 .911 .690 .993
.99 .156 .194 .126 .109 .104 .087 .176

100

Table 3—8 (Continued)

(b) Power Comparison With Trend

~

"1 71 pr
T 3 30 £4 312 £0 £4 £12

25 .00 .713 .004 .000 .954 .600 .057 .956
.20 .511 .005 .000 .810 .459 .063 .797
.50 .221 .003 .000 .373 .248 .069 .343
.80 .061 .001 .000 .113 .112 .077 .080
.90 .038 .001 .000 .081 .085 .073 .055
.95 .031 .001 .000 .073 .079 .075 .042
.99 .030 .001 .000 .073 .075 .076 .040

50 .00 .969 .291 .000 1.00 .831 .072 1.00
.20 .889 .254 .000 1.00 .721 .068 1.00
.50 .602 .166 .000 .898 .447 .061 .924
.80 .161 .049 .000 .215 .136 .052 .221
.90 .075 .023 .000 .095 .075 .045 .092
.95 .052 .015 .000 .069 .060 .046 .055
.99 .042 .012 .000 .059 .052 .043 .045

100 .00 1.00 .712 .000 1.00 .996 .237 1.00
.20 .994 .668 .000 1.00 .985 .221 1.00
.50 .908 .551 .000 1.00 .896 .181 1.00
.80 .410 .244 .000 .646 .377 .106 .727
.90 .163 .103 .000 .192 .141 .068 .228
.95 .081 .052 .000 .087 .079 .051 .092
.99 .044 .030 .000 .051 .050 .042 .051

250 .00 1.00 .972 .338 1.00 1.00 .873 1.00
.20 1.00 .962 .326 1.00 1.00 .855 1.00
.50 .998 .931 .299 1.00 1.00 .790 1.00
.80 .855 .739 .206 1.00 .980 .516 1.00
.90 .497 .445 .101 .826 .607 .267 .900
.95 .208 .205 .083 .257 .206 .121 .341
.99 .052 .055 .037 .057 .053 .043 .058

500 .00 1.00 .999 .812 1.00 1.00 1.00 1.00
.20 1.00 .999 .802 1.00 1.00 1.00 1.00
.50 1.00 .997 .779 1.00 1.00 .999 1.00
.80 .985 .964 .682 1.00 1.00 .980 1.00
.90 .838 .825 .498 1.00 .997 .810 1.00
.95 .490 .528 .264 .813 .701 .424 .885
.99 .080 .110 .039 .077 .070 .058 .091

101

Table 3-9

The KPSS Unit Root Tests Applied To the Nelson—Plosser Data
The 91"“ Test for a Unit Root without Trend
(5% Critical Value is .0099)

Lag Truncation Parameter (2)

Series 0 l 2 3 4 5 6 7 8
Real GNP .0961 .0493 .0671 .0771 .0839 .0892 .0936 .0975 .1011
Nominal GNP .0937 .0481 .0657 .0755 .0823 .0876 .0921 .0961 .0998
PCR GNP .0893 .0459 .0627 .0723 .0791 .0844 .0888 .0928 .0965
IP .0972 .0493 .0666 .0759 .0819 .0863 .0898 .0927 .0953
Employment .0935 .0478 .0649 .0744 .0807 .0856 .0897 .0933 .0965
Unemployment .0039 .0022 .0034 .0042 .0050 .0058 .0067 .0076 .0085
GNP deflator .0916 .0466 .0631 .0720 .0779 .0824 .0860 .0892 .0921
CPI .0712 .0363 .0492 .0562 .0609 .0644 .0672 .0696 .0717
Nominal Wage .0086 .0045 .0062 .0073 .0082 .0090 .0098 .0105 .0114
Real Wage .0980 .0500 .0677 .0773 .0837 .0885 .0924 .0958 .0989
Money Stock .0976 .0497 .0673 .0768 .0831 .0878 .0917 .0949 .0979
Velocity .0824 .0421 .0570 .0651 .0705 .0744 .0775 .0801 .0824
Bond .0110 .0060 .0085 .0101 .0113 .0123 .0133 .0141 .0149
SP500 .0801 .0410 .0558 .0640 .0697 .0740 .0775 .0806 .0835
The 5; Test for a Unit Root with Trend

(5% Critical Value is .0033)

Lag Truncation Parameter (2)

Series 0 1 2 3 4 5 6 7 8
Real GNP .0102 .0054 .0078 .0096 .0112 .0127 .0143 .0159 .0177
Nominal GNP .0122 .0063 .0088 .0104 .0117 .0128 .0139 .0149 .0160
PCR GNP .0085 .0046 .0066 .0081 .0095 .0108 .0122 .0137 .0152
IP .0074 .0040 .0058 .0069 .0079 .0088 .0097 .0104 .0112
Employment .0065 .0034 .0049 .0059 .0067 .0075 .0083 .0091 .0100
Ilnemployment .0027 .0015 .0023 .0029 .0035 .0041 .0047 .0053 .0060
(SNP Deflator .0060 .0031 .0043 .0051 .0057 .0063 .0068 .0073 .0079
(SP1 .0167 .0085 .0115 .0133 .0145 .0154 .0162 .0170 .0177
Iqominal Wage .0946 .0484 .0657 .0752 .0816 .0864 .0904 .0940 .0972
lieal Wage .0135 .0072 .0103 .0124 .0142 .0159 .0176 .0192 .0208
lioney Stock .0054 .0028 .0038 .0045 .0051 .0056 .0061 .0067 .0073
‘Jelocity .0174 .0091 .0127 .0148 .0164 .0177 .0187 .0197 .0206
Bond .0119 .0064 .0091 .0108 .0120 .0131 .0140 .0149 .0157
SP600 .0123 .0065 .0091 .0108 .0121 .0132 .0142 .0151 .0159

Augmented
Series 0
Real GNP —2.03
Nominal GNP -1.35
PCR GNP -2.12
IP -3.08
Employment —2.17
Unemployment -3.36
GNP Deflator -1.83
CPI -0.65
Nominal Wage —l.46
Real wage -2.33
Money Stock -l.44
Velocity —l.66
Bond 1.86
SP500 -l.94

102

Table 3—10

Dickey-Fuller Unit Root (6,) Tests Applied to
the Nelson—Plosser Data
(5% critical value is -3.45)
Number of the AR Order (P)

1 2 3 4 5 6 7
—2.99 —2.94 —2.69 —2.43 -2.12 —2.38 -2.68 -2
-2.32 —2.04 —l.83 —1.53 -1.79 —2.20 —2.17 —2
-3.05 -3.00 —2.80 —2.56 —2.22 -2.48 —2.84 -2
—3.13 -2.66 -3.23 -3.23 -2.57 —3.36 —3.63 -3
—3.92 —3.14 —3.55 —3.09 —2.84 —2.98 -3.33 -2
-2.52 —2.57 -2.45 -2.39 -2.47 -2.52 -2.44 -2
-1.86 -l.44 -l.97 -2.75 -2.37 -2.28 -2.38 -2

—3.05 —2.97 -2.80 -2.54 -2.56 -2.26 -2.33 —l
—3.08 -2.79 -2.93 -2.91 -3.00 -3.40 -3.68 -3

1.46 0.69 0.49 0.66 0.60 0.55 0.85 0
-2.65 -2.12 -2.12 -l.60 -l.06 -0.97 -l.06 -l

.23
.26
.39
—3.36 —3.19 —3.27 —3.08 —2.53 -2.49 —2.67 —2.

68

.60
.97
.65
.41
—2.52 -2.24 -2.24 —2.07 -2.12 -2.62 -2.92 -2.

64

.93
.46
-1.75 —l.47 -l.40 —l.08 -0.74 -0.79 -0.90 -l.

08

.76
.02

CHAPTER 4

CHAPTER 4
ASYMPTOTIC DISTRIBUTION OF UNIT ROOT TESTS

WHEN THE PROCESS IS NEARLY STATIONARY

4.1 Introduction

The unit root hypothesis has recently attracted a lot of attention in time series
econometrics. Dickey and Fuller (1979, 1981, DP) develop several tests of the unit
root hypothesis. They use a Monte Carlo simulation to tabulate the sampling
distributions of the coefficient and t statistics, assuming iid errors. These distributions
are nonstandard; they are skewed to the left and have too many large negative values
relative to a normal distribution. Furthermore, it is well-known that many economic
time series are mixed processes, in the sense that they contain not only autoregressive
but also moving average components. In this case, the DF tests are not expected to be
robust because the distribution of test statistics is derived under the assumption of the
errors being white noise.

There have been some attempts to derive alternative testing procedures to
correct for the presence of autocorrelated errors. Said and Dickey (1984, 1985) have
suggested the use of augmented Dickey-Fuller tests (ADF), based on the DF regression
augmented with lagged differences of the dependent variable, which should have the

correct size asymptotically even in the presence of autocorrelated errors, if the number

103

104

of augmentations increases with the sample size at an appropriate rate. Phillips (1987)
and Phillips and Perron (1988, PP) allow for a wide class of weakly dependent and
heterogeneous errors and use semiparametric corrections to derive transformed
statistics which have the same limiting distributions that the DF statistics have under
iid errors.

However, the finite sample performance of most corrected unit root test
statistics is not robust when the data follow an ARIMA process. Schwert (1987,
1989) shows by extensive Monte Carlo simulations that most unit root tests have a
problem of size distortions when the errors are MA(l) with negative parameter. For
example, when the MA(l) parameter is -0.8, the size distortion of standard unit root
tests such as the DF and the PP tests is almost one.

This problem can be examined theoretically by using the "nearly stationary

model":

y.=By..1+8..B=1.£.=U.+9u..1

6=-1+C/I"

where u, is iid, and C > 0 and 8 > 0 are fixed numbers. Note that, even under the
maintained hypothesis of a unit root ([3 = 1), yt is white noise when 0 = -1, and so yt
approaches white noise as T -—> 00. By using this model we will investigate the
asymptotic behavior of uncorrected unit root tests when the process is nearly
stationary. It will be shown that the order in probability and the asymptotic

distributions of standard unit root tests in this case may depend on the value of 5 as

well :
Wci(i
apprc

of alt

but 5:

andt'

probe

study

3301p

SCCfiO
”Clio.

Elven

105
well as C. This approach is similar to the approach of Phillips(1988) and Chan and

Wei(l988) who consider the "nearly integrated process" in which the AR parameter [3
approaches unity as T —9 co, and who therefore investigate power against a sequence
of alternatives getting close to the null of a unit root.

Our model is an extension of Pantula (1991), who considers the same model,
but sets C = 1. He considers corrected versions of the unit root tests such as the ADF
and the PP tests, and he is primarily interested in finding and comparing the order in
probability of those statistics. However, our concern is different in the sense that we
study uncorrected unit root tests, and we are primarily interested in the finite sample
adequacy of the asymptotic results. The asymptotic results must be accurate in finite
samples if they are to explain the reason for the size distortions of unit root tests when
the process is nearly stationary.

The main purposes of this chapter are to derive the asymptotic distribution of
the Dickey-Fuller and the Schmidt-Phillips (1991, SP) unit root test statistics using the
local approximation of the MA(l) parameter to minus one, to tabulate the distributions
of the unit root test statistics predicted by our asymptotic theory, and then to compare
the predicted distributions with the actual sampling distributions. Based on our
findings we suggest directions for the further research.

In section 2 we discuss the model and derive the main asymptotic results. In
section 3 a Monte Carlo simulation is conducted to present the main results. In
section 4 we give some concluding remarks and discussion. The proofs and tables are

given in the appendix.

106
4.2 Model and the Asymptotic Results

To derive the asymptotic distribution of the standard unit root test statistics

when the process is ’nearly stationary’ we consider the following model:

(1) y. = By.-. + Ev 6t = n. + But-1

(2) 9=-1+C/I"

whereu,isiid(0,0u2),8=1,8>O,andC>0. AsT—)oo,0 —)-1. Infact,yt
becomes stationary when 6 = -1.

We consider two types of unit root tests. First, we use the Dickey-Fuller tests

based on the regressions:

(3) Y: = BYt-l + 8,
(4) y. = u + BY... + 8.
(5) yt=n+5t+Bytt+et

The 6, 6,, and 6, tests ate based on the coefficient statistic T03 - 1), where {3, 6,, and
8, are the OLS estimators of B in (3), (4), and (5) respectively, while the f, f", andfi
tests are based on the t statistics for the hypothesis 8 = 1 in the same three
regressions.

Next we consider the Schmidt-Phillips test which is based on the

parameterization:

tot}

lien

nude
and 1
is cm

them

in Cor

Csfimz

107
(6) yt=W+§t+XeXt=l3Xtt+€t

The SP test of the unit root hypothesis can be derived from the results of OLS applied
to the following regression:
(7) Ay, = intercept + 08,, + error
Here 8,, = y,, - Ty, - at - 1) is a residual, with E = (yT - y1)/(T - 1) and it, = y1 - E.
which are the restricted maximum likelihood estimators of i and ‘Vx = \y + X0. The
SP test statistics are defined as E = T6 and '1' = t statistic for the hypothesis 6 = O.
For later use, we define 8 = 6 + 1.
Remark 1

The reason for including the SP test is that its parameterization can avoid the
awkward interpretation of nuisance parameters that the DF tests have. While the
meaning of intercept and coefficient of time trend under the null is different from that
under the alternative in the DF regressions (3) - (5), \l’ and 5, always represent level
and linear trend in the SP model (6). In addition, the detrending method of the SP test
is different from that of the DF test. It is well-known from the general regression

theory that the inference on B in (5) is the same when we replace (5) by
A
(5)’ Ay, = intercept + (1)8H + error

A
where SM is the residual from the OLS regression of y,l on intercept and trend, and E
= B - 1. While the DF test uses the OLS estimators of coefficients on level and trend
A
in constructing the residual 8,,1 in (5)’, the SP test uses restricted maximum likelihood

estimators under the null of a unit root in constructing the residual S1 in (7).

108

We now show main our asymptotic results. Theorems 1 through 5 show the
limiting distribution of the estimates of the coefficients of the lagged dependent
variable 6,13,, 6., and E) under the maintained hypothesis that y, is generated by (1)
and (2). We use the mixing and moment conditions of Phillips and Perron (1988) to
derive the asymptotic results. Let us define functionals of the Brownian motion which
are used in deriving the following results: standard Brownian motion, W(r); demeaned
Brownian motion,W = W - 1W; demeaned and detrended Brownian motion,V=V = W +
(6IrW - 41W) + r(6IW - 12IrW); and demeaned Brownian Bridge,V = V - IV, where V
= W(r) - rW(1) is a Brownian Bridge. All integrals are understood to be taken over
the interval [0,1] and with respect to Lebesgue measure.

Theorem 1
For 0 < 5 < 1/2, the estimates of the coefficient [3 of the lagged dependent

variable have the following asymptotic distributions as T —> 00:

(8-1) T” (I? -1)_. -1/(czlw2>
A -
(8-2) T“215 (13,, - 1) -) -1/(C2!w2)
A =
(8-3) rm «3, - 1) -+ -1/(C21w2)
(8-4) T“” (13' - 1) —) -1/(C?l\72)
Theorem 2

For 8 = 1/2, the estimates of the coefficient [3 of the lagged dependent variable
have the following asymptotic distributions as T —> oo:
A
(9-1) (13 - 1) —> -1/(1 + CZIW2)

(9-2) (13‘, - 1) —> -1/(1 + czlwz)

109

A

(9-3) (13, - 1) —) —1/(1 + 0152)
(9-4) ('3' - 1) —> -1/(1 + czlx'n)
Theorem 3

For 1/2 < 5 < 3/4, the estimates of the coefficient [3 of the lagged dependent

variable have the following asymptotic distributions as T —> ca:

(101) Tm}? —> CZIW2
(102) T1“ 6,, -+ C2le
(103) T2'” 6 —) 011712
(104) T2er B —) €2le
Theorem 4

For 5 = 3/4, the estimates of the coefficient 8 of the lagged dependent variable

have the following asymptotic distributions as T —> oo:

(ll-1) Tmfi -) W(1)+C21w2
(11-2) Tmfi, —) W(1)+C2Nv2
(11-3) Tmfi, —+ W(1)+c21 v=v2
(ll-4) WE —) W(1)+ czh'n
Theorem 5

For 5 > 3/4, the estimates of the coefficient 8 of the lagged dependent variable
have the following asymptotic distributions as T —-) oo:
(12-1) Tmfi -> W(l)
(12-2) Tml’iu —> W(l)

(12-3) Tm 6 —) W(l)

110

(12-4) Tmfi —> W(l)

Basically the same results are obtained for each test, with the difference being
only the functional form of Brownian motion used.
Remark 2

For 0 < 5 < 1/2, (13, - l) —9 0, but T185431- - 1) converges to a random limit,
wherellij = [3, 8,, [3,, and 8. The speed of convergence is slower than in the case of
standard asymptotics (for fixed 0), where Talij - 1) —-> the limit. For 5 = 1/2, (131 - 1)
converges to a random limit and the speed of convergence is even slower.

When 5 > 1/2, 03, - 1) —> -1 so thatllij -) 0. Therefore, we get limiting
distributions for 6, instead of 6,. - 1). For 172 < 5 < 3/4, W113, has a limiting
distribution and the speed of convergence is between 0 and 1/2; i.e., 0 < 25 - 1 < 1/2.
For 5 = 3/4, T"2 8, has a limiting distribution which is a mixture of a functional of
Brownian motion and a standard normal process.

Finally, for 5 > 3/4, T"2 8, always converges to the standard normal process
W(l), as in the usual stationary case. Fuller (1976) also shows that the process can be
approximated by the standard normal process when 5 > 3/4.

Next we use the results in Theorems 1 through 5 to find what happens to
standard unit root tests when the process is nearly stationary. Corollaries 1 through 5
show that the standard unit root tests have different orders in probability, depending
mainly on the value of 5 and that their asymptotic distributions depend on the

functionals of Brownian motion and C.

1 1 1
Corollary 1

For 0 < 5 < 1/2, the coefficient and the t statistics of the DF tests and of the

SP test have the following asymptotic distributions as T -) co:

(13-1) '1‘”)? —> -1/(c21w2) and T‘f‘ —+ -1/(2C21w2)"2
(132) T” 6,, --) -1/(czlwz) and T‘fi —> .1/(2czlv'vzr'2
(13-3) T215 6, --) -1/(C21 1712) and T‘f, —+ 442021152)"2
(134) T25 5 —> -1/(c2!v2) and '1“ ‘i’ —> -1/(2@IV2)‘”

Coroll_a_ry 2

For 5 = 1/2, the coefficient and the t statistics of the DF tests and of the SP

test have the following asymptotic distributions as T —-> 00:

(14-1) T16 —) -1/(1 + czlwz) and TW‘ -—> -1/(1 + 2c21w2)“2
(14.2) T16, —> -1/(1 + czlwz) and T"2 i, —-> -1/(1 + 2c2hiv2)”2
(14-3) '1“ 6. —-) -1/(1 + 0111712) and Tmt‘, ——> -1/(1 + 2019112)“?
(144) T1 '5 —> -1/(1 + 01172) and T"2 't‘ —> -1/(1 + 291172)"2
Corolla_ry 3

For 1/2 < 5 < 3/4, the coefficient and the t statistics of the DF tests and of the

SP test have the following asymptotic distributions as T —> co:

05-0 Tm’é‘ + “D —9 CZIW2 and flame + T”) -) czlw2
(15-2) Tut-66” + T) _, Czfivz and Tamara + Tm) _) 01W;
(153) T"”’(6‘. + T) —> 01 \71/2 and Tmmd‘, + T”) —) c2! \7/2
054) T"“"(§ + T) —9 C2192 and remain“ + T”) —> czlvz

Corolla 4

1 12
For 5 = 3/4, the coefficient and the t statistics of the DF tests and of the SP

test have the following asymptotic distributions as T -> oo:

(16—1) T1949 + T) -) W(1) + czlw2 and (t‘ + T‘”) —> W(1) + CZIw2
(152) Tmcb‘. + T) -> W(1) + CW)!2 and (t; + T‘”) —) W(1) + (3216112
06-3) r146, + T) —> W(1) + czlv=v= and 0‘. + T“) -+ W(1) + 01%
(164) T‘”(6 + T) —> W(1) + czlvz and (i + Tm) ... W(1) + c2192
Corona 5

For 5 > 3/4, the coefficient and the t statistics of the DF tests and of the SP

test have the following asymptotic distributions as T —> 00:

(17-1) Tm(6 + T) —> W(1) and (P + T”) -) W(1)

(17-2) Tints, + T) —> W(1) and (6}, + T‘”) —> W(1)
(17-3) Twat, + T) —-> W(1) and (t, + T1”) —) W(1)
(17-4) “1“”(5 + T) —> W(1) and (i + T“) -> W(1)
32mg;

For 0 .< 5 < 1/2 the coefficient and t statistics have order in probability 0,0”)
and 0,,(1‘), respectively. When 5 2 1/2, the coefficient statistic is of order 0,.(T) and
the t statistic is of order 0,,(Tm). (Note that the 5 = 1/2 and 3/4 are the discontinuity
points dividing the limiting distributions which have different behaviors.) This implies
that all the standard unit root test statistics diverge to negative infinity as T —> oo,
when the process is nearly stationary.

Our results can be directly compared to Pantula (1991), in which he studies the

performance of the various unit root test statistics when the process is nearly

113

stationary. He uses the same model as (l) and (2) but restricts the value of C to 1.
However, there are some differences between Pantula’s analysis and ours.

First, Pantula analyzes only the case of random walk without drift, while we
extend it into the more general cases of random walk with drift and random walk with
drift and time trend. We also consider the SP unit root test.

Second, he divides 5 into three regions: 0 < 5 < 1/4 (nonstationary region), 1/4
< 5 < 1/2 (grey region), 1/2 < 5 (stationary region), but we unify first two regions and
consider additional cases of 5 = 1/2 and 5 = 3/4. As will be shown, the choice of 5 =
1/2 is important because the predicted distribution of standard unit root tests in this
case approximates the actual sampling distribution relatively well unless 0 is very
close to minus one.

Third, Pantula fixes C to one, but C is unrestricted inour model. In both our
model and Pantula’s, 0 —-) -1 as T —-) 00. From the point of view of using asymptotics
to approximate finite sample distributions, our model is obviously more flexible. For
example, if we pick 0 = 0.8 and T = 100, we pick 5 = 0.35 in Pantula’s model,
whereas in our model we can have 0 = —0.8 for 5 = 1/4 and C = 0.632, or 5 = 1/2 and
C=2,or5=5/8 andC=3.557,or5=3/4andC=6.325,or5=7/8andC=
11.247. In our calculations we typically set 5 (e.g., 5 = 1/2) and let C be the value
that is chosen to yield (0,T) pair.

Finally, Pantula compares the performance of various unit root tests such as the
DF tests, ADF tests, PP tests and Hall’s (1989) IV test, based on the criteria that good

unit root tests should accept the null of unit root when the process is in the

114

’nonstationary region’ (0 < 5 < 1/4) and reject the null when the process is in the
’stationary region’ (5 > 1/2). Based on limited simulations he suggest the use of the
ADF unit root test as best when the process is nearly stationary. We are rather
interested in examining the behavior of uncorrected DF and SP tests when the process
is nearly stationary in more detail to see how accurate asymptotic approximations of

this type are in finite samples.

4.3 Simulation Results

In this section we use extensive Monte Carlo simulations to study the finite
sample performance of standard unit root tests when the process is nearly stationary.

The basic data generating process we use is the ARIMA(0,1,1) process:
(18) Yr = Yt-l + en E! = “t + 9“H

where u,’s are serially uncorrelated standard normal random

variables. The data are generated by choosing the initial value of u, after discarding
the first 20 observations. The standard normal random numbers are selected using the
Fortran subroutine GASDEV/RAN3 of Press et_al. (1986). The actual sampling
distributions are tabulated by applying the standard DF and SP tests directly to data
generated according to the basic DGP (18), using 50,000 replications. Values of 6 = -
0.5, -0.8, -0.9, -0.95, -0.99, -1.0 and values of T = 25, 50, 100, 250, 500, 1000 are

used. The results for the 5 % and 95 % fractiles are given in Table 4-1.

115

Since the unit root test is a lower tail test, we are mainly interested in the
behavior of the lower 5 % fractiles. The 5 % critical values of the coefficient
statistics are diverging around -T, as 0 —> -1. This is expected, because if 0 = -1 then
y, reduces to the iid process and therefore 8 -) 0. In this case T43 - 1) almost acts
like -T; t statistics almost act like -T‘”. The speed of divergence depends on the
sample size. These critical values are far less than those of the original DF statistics.
This implies that standard unit root (coefficient) tests are expected to reject almost
always the null hypothesis of the unit root when 0 S -O.8 and T is large.

It is well-known that the more regressors such as constant and time trend we
include, the more negative critical values we get. When we use the 6, test, the speed
of divergence of the 5 % critical value to -T is very fast in finite samples. For
example, when 0 = -0.8 and T = 100, the 5 % critical value or6, is -101.9. This
indicates that there is a very strong bias of 8 toward 0 when the process is nearly
stationary with trend. Even though we increase the sample size to 1,000, this bias is
still quite large. Therefore, the direct application of uncorrected DF and SP tests to a
nearly stationary process is dangerous way as is well known.

We obtain basically the same results for the t statistics. The 5% critical values
of the t statistic become more negative as 0 —> -1, given sample size, and these critical
values are still far less than those of the DF and SP t statistics.

To tabulate the predicted distributions of the unit root test statistics 6, 6, 6,,

fl, 6,, 9,, 13, T) when the process is nearly stationary, a sample size of T = 2,000 is

used to find the limiting fractiles of standard Brownian motion, demeaned Brownian

116

motion, demeaned and detrended Brownian motion, and demeaned Brownian bridge.
Each experiment is replicated 50,000 times. We then use the formulas given in
Corollaries 1 - 5 to convert these fractiles into predicted fractiles for the unit root test
statistics. We use values of 5 = 1/4, 1/2, 5/8, 3/4, and 7/8, and the same values of T

and 0 as above. For given values of 0 and T, we use the following formula for C:
(19) C = (1 + an“

The results for the 5 % and 95 % critical values are given in Tables 4-2 through 4-6.

We now compare the predicted distributions with the actual sampling
distribution to see how closely they are. The results are based on the comparison of
Table 4-1 to Tables 4-2 through 4-6.
1) 0<5<1/2(5=1/4)

Table 4-2 shows that the predicted critical values of the unit root tests for 5 =
1/4 are independent of the sample size, given a value of 0. This is so, because (19)
implies that C is proportional to T1”, in which case T cancels from the expression in
Corollary 1 above. These critical values are generally far more negative than the
actual sampling values. Furthermore, the discrepancy is wider as 0 is closer to -1.
Even though we increase the sample size to 1,000, the actual sampling fractiles never
catch up with the predicted critical values. Therefore, the predicted distributions of
the unit root test statistics in this case are not a good approximation of the actual ones.
2) 5 = 1/2

The results for 5 = 1/2 are given in Table 4-3. The predicted distributions of

117

the test statistics in this case approximate the actual ones relatively well, especially
when 0 is in the range of -0.8 to -0.9. For example, for 0 = -0.8 and T = 100, the
actual 5 % critical values oi6 is -81.7 and the predicted value is -81.1. The
discrepancies between actual and predicted critical values is larger but not very great
for the other test statistics and for other values of T and 0. The actual critical values
change more rapidly than the predicted ones as 0 —> -1; that is, the speed of
divergence of the actual distribution is relatively faster than that expected from the
predicted distribution for 5 = 1/2 as 6 —-> -1.

3) 1/2 < 5 < 3/4 (5 = 5/8)

These results are given in Table 4-4. The predicted distributions for 5 = 5/8
show quite different behavior from the previous ones. The right tail critical values are
explosively positive except when 0 = -0.99, while the left tail critical values are
dependent upon the sample size and the value of 0 chosen. When 0 is -0.5 or -0.8,
the left tail predicted critical values are getting positive as the sample size increases.
This behavior is not present in the actual sampling distribution. More generally, the
quality of the approximations is fairly good in the left tail for 0 = -0.9 and -0.95, but
not usually as good as it is with 5 = 1/2.

4) 5 = 3/4

The predicted critical values for 5 = 3/4 given in Table 4-5 show the
intermediate behavior between those for 5 = 5/8 and for 5 = 7/8. When 0 2 -O.9, the
predicted critical values are close to those for 5 = 5/8, and therefore they have the

same problem of explosive positive right tail fractiles as before. However, when 0 =

1 18
-0.95 or -0.99, the predicted distributions are close to those for 5 = 7/8, whose

behavior will be explained below.
5) 5 > 3/4 (5 =7/8)

For 5 > 3/4, Corollary 5 indicates that all of the statistics (appropriately
normalized) converge to a standard normal distribution. Thus the predicted
distribution is the same for all test statistics and all values of 0.

We conclude that, as a general statement, 5 = 1/2 is the choice that leads to the
best match between predicted and actual sampling distributions. Pantula does not
consider this value, and thus this is an important and novel result. We proceed to
compare the predicted distribution for 5 = 1/2 with the actual sampling distribution in
more detail than above. For simplicity we do so only for the 6 and 6 statistics; similar
results are obtained for the other statistics.

For convenience we use AD to represent the actual distribution and PD to
represent the predicted distribution. We also denote AC as the actual critical value and
PC as the predicted critical value. We consider critical values of .01, .025, .05, .95,
.975, and .99. These results are given in Table 4-7. Note that Table 4—7 contains
results also for 5 = 1/4, 5/8, 3/4, and 7/8, but our discussion applies only to the results
for 5 = 1/2. Also, while Table 47 contains results for thefi andf statistics, we will
only discuss the results for the 6 test.

1) = -0.5
When T S 50, AD and PD have similar dispersion with PD a little more

concentrated, and for lower tail critical values AC is more negative than PC. However,

119

when T 2 100, PD becomes more negatively skewed than AD. Therefore, Pc is more
negative than AC and the difference becomes larger as T increases.
2) 6 = -0.8

When T S 100, PD is more concentrated than AD. AC is more negative than PC
in the left tail, while AC is less negative than PC in the right tail. When T = 250, PD is
similar in dispersion to AD, but PD begins to skew negatively. When T 2 500, PC
becomes more negative than AC.
3) 0 = -0.9

When T S 100, PD is more concentrated than AD. The smaller T is, the more
concentrated PD will be. AC is more negative than PC in the left tail, while PC is more
negative than AC in the right tail. When T = 250, P1) is similar in dispersion to AD,
but Ac is still more negative than PC in the left tail. When T = 500, PD becomes less
concentrated and negatively skewed, but still similar in dispersion to AD. When T =
1,000, PD becomes more negatively skewed than AD, and PC is more negative than AC.
4) = -0.95

When T S 500, PD is more concentrated than AD. AC is more negative than PC
in the left tail, while PC is more negative than AC in the right tail. However, when T
is small, the predicted distribution is too narrowly clustering around -T. For example,
when T = 25, PD ranges only from -24.95 to -21.33, while AD ranges from -34.12 to -
10.69. When T = 1,000, PD is finally similar in dispersion to AD, but AC is more
negative than PC in the left tail.

5) = -O.99

at
El

ex

97

4.4

Dii
stat
Par

Stu:

Slat

dep,

gent

Cho;

 

120

For all sample sizes, PD is more concentrated than AD. Therefore, A,3 is more
negative than PC in the left tail, and PC is more negative than AC in the right tail.
Except when T = 1,000, the problem of converging around the negative sample size
exists. Especially when T s 100, all the critical values of the predicted distribution
are almost the same as -T. For example, when T = 100, PD ranges from -99.97 to -

97.32, but AD ranges from -120.76 to -73.68.

4.4 Discussions and Concluding Remarks

We have examined the asymptotic and finite sample behavior of the standard
Dickey-Fuller and Schmidt-Phillips unit root tests when the process is nearly
stationary, using a local approximation of the MA(l) parameter around minus one.
Pantula (1991) has studied the same problem. However, some implications of these
studies are different, mainly due to the different treatment of the parameter C in the
local approximation of the MA(l) parameter around minus one and our inclusion of 5
= 1/2. The main findings we have obtained are as follow:

First, when 6 —> -1 as T —9 co, the coefficient statistics 6, 6,, 6,, 5 and the t
statistics 6, 6,, 6,, 7f diverge to negative infinity, but have different orders in probability
depending on the value of 5.

Second, we find by Monte Carlo simulations that the choice of 5 = 1/2 is
generally best, in the sense that its predicted distributions are closer than for other

choices of 5 to the actual sampling distribution, at least unless 0 is very close to -l.

121

When 0 < 5 < 1/2, the predicted distributions of the unit root test statistics are the
same for all the statistics we consider and for all T, which is clearly unsatisfactory.
When 1/2 < 5 S 3/4, the predicted distributions have a problem of critical values being
explosively positive, which again is not consistent with the actual sampling
distribution.

To sum up, the tendency of most unit root tests such as the Dickey-Fuller test
and the Phillips-Perron test and their modifications, to have considerable size
distortions in finite samples when the process is nearly stationary is not surprising, and
is well predicted by our asymptotics. It reflects the fact that a nearly stationary
process can be expected to behave approximately as a stationary process in finite
samples. This point has been made before by others, including Wichem (1973) and
Blough (1989). For example, Wichem shows that information about the sample
autocorrelations does not differentiate appreciably between stationary and
nonstationary ARIMA(0,1,1) processes, and argues that, in practice, it is not possible
to prove that the series is stationary or nonstationary in finite samples. Furthermore,
he argues that the use of either model will lead to very similar results in testing and
other applications for reasonable sample sizes.

Schwert (1989) concludes from his Monte Carlo evidence that the augmented
Dickey-Fuller statistic provides the most accurate unit root test in the presence of
strongly autocorrelated errors. Pantula (1991) also recommends the augmented
Dickey-Fuller tests, based on his asymptotics. We do not consider asymptotics for the

augmented Dickey-Fuller test in this chapter. However, we note that in our view we

122

cannot take granted for the superiority of the augmented Dickey-Fuller statistic over
any other unit root test statistic, as Pantula and Schwert have suggested. The
augmented Dickey-Fuller test requires a very large number of augmentations to have
correct size in finite samples when the process is nearly stationary, because otherwise
the OLS estimator of 6 is seriously biased toward zero when 0 is close to -1. It is
well-known that there should be a finite sample trade-off between size and power of
the test when the null is close to the alternative. That is, the cost of the good size
performance of the augmented Dickey-Fuller test should be very poor power in finite
samples. This has also been argued by Blough (1989), and the Monte Carlo evidence
to this effect is given by a number of authors, including Lee and Schmidt (1991).

One of the possible ways to overcome the above problems is the use of the IV
unit root test suggested by Hall (1989). He uses an instrumental variable approach to
handle the size distortion problem caused by moving average errors. The instrument
for y,,1 is y,,, , where k > q for MA(q). Hall applies the IV method to the Dickey-
Fuller regression and shows a significant improvement. Lee and Schmidt (1991) also
derive the IV versions of the Schmidt-Phillips test and show much more improvement.
Although the IV unit root tests still have some size distortions when the process is
nearly stationary and it is difficult to know the exact order of q, more research in this
direction would probably give further insights.

Another possibility is the construction of a unit root test by using a more
efficient estimator of B, which can also possibly reduce the bias in the estimation of B

when the process is nearly stationary. Basically, this would involve estimation of the

123
Dickey-Fuller or Schmidt-Phillips regression by GLS, given an assumption on the

order of the ARMA process of the errors. Choi (1990) considers a partial GLS
estimator, in which the AR portion of the process is handled by data transformation
while the MA portion is handled by IV estimation, and he gives some optimistic
results on the finite sample properties of the resulting test. How this compares to a

full GLS treatment is not clear. Again, future research is needed.

124

Apmndix

In this appendix we prove the main asymptotic results given in section 4.2.
The data generating process is given in equations (1) and (2) in the main text. First,

set u,, = y,, = 0 without loss of generality. Then equation (1) is solved for y,:
(A1) y, = u, + (1 + EDS,1

t
where S, =21 u, is a partial sum process. Substituting for (1 + 0) from
J:

(2) into (A1) we get
(A2) y. = u. + (0158.-.

Note that this is the solution for the nearly stationary process using our local
approximation for the MA(l) parameter to minus one. That is, as T —-> oo, y, —> u,.
This transformation will be intensively used in the proof. Using (A2) we will derive

the asymptotic results for the statistics:

A
(A3) 6, = T( B, - 1)
(A4) 6 = (3, - 1)/(sz-XX,,")”2
A

B, is the estimate of the coefficient of the lagged dependent variable in (3), (4), (5),
A A A ... T
and (6) so that B, = [3, [3,, and [3 ; s2 = (1/1‘).2‘i 6,2, where 6, are the appropriate residuals;
- J=
_ A A _
and XX1i = 2y,,2, 2(y,, -y_,)", 26,, -S,,)2, and Z(S,, - 8,)”, respectively for regressions

A
(3), (4), (5)’ and (7). S,, and 8,, are defined in text. As shown in Theorems 1

throug

functit

Philli;

(A5)

(A6)

 

 

 

125

through 5 in the main text, the asymptotic results depend mainly on C, 5 and
functionals of the Brownian motion. Note that we use the basic preliminaries of

Phillips and Perron (1988) (given on p. 377) extensively in the proofs.

[1] 6 and‘i‘tests
It is useful to write
A 2 l A 2 1
(A5) B = (ZYt-l ) zyt-IYt 01' (B " 1) = (234.1 ) ZYt-lel
A
(A6) 82 = (1/1)X(yt - 534.02

Case 1: 0 < 5 <1/2

Lemma 1.1 (1m2°)£y,2 —+ e,,ZCZIW2
Lemma 1.2 (l/I')2’.y,,e, —) -o,2 for all 8 > 0.
(proof)
(ImZYt-let
= (Immarut + (C/I“)(1/T)23t.2ut + (9/1")(1/1‘)23t.rur + (ii/1324.12

—> 43,2 since first three terms —) 0 and 0 —> -1 as T —) oo. Cl

Theorem 1.3 qu’i — 1) —> -1/(c21w2)

Lemma 1.4 s2 —> 20,2
(PTOOD A A
82 = (lfl'PJE.2 + (l3 - 1)2(1/1")4‘3yt.r2 - 2(13 - 1)(1fl‘)2yt-r€r

—-) 26,2 Cl

Corolla 1.5 T5 6 —9 .1/(2C21w2)"2

126
Case 2: 5 = 1/2

Lemma 1.6 (1mzy,2 —) 6,2(1 + czlw2)
Theorem 1.7 6- 1 —> -1/(1 + calwz)
Lemma 1.8 s2 —+ 6,2{2 - Na + CZIWW
(proof) ,t A
s2 = (N'DXE.2 + (3 - 1)2(1/I)2yt.t2 - 2(13 - 1)(1/1)Zy..t8.
(1) <6 - 120/0223 —> 0.2/(1 + ciwa

<2) -2<6 - Hummus. _. ~2o.2/(1 + oth)

Using (1), (2), and (1/'I‘)2‘.l»:,2 -) 20,}, we get the result. El
Corollgy 1.9 T"2 t? —> .170 + 201an2
Case 3: 1/2 < 5 <3/4

lemma 1.10 (1fI')£y,2 -> 6,,2
(proof)
T‘zy,2= T122113 + (cznzhrlzs,,,2 + (2crl‘)les,,u,

—>o,2 D

A
Lemma 1.11 6 -1-—) -1.

A
Since the above lemma implies that B —9 0 when 5 > 1/2, we derive
A A
asymptotics for [3 instead of (B — 1).
Leme 1.12 (rm-”)zy,,y, —> o,2C2Iw2
(proof) 28
Off” )Zytnyt
= (Inq'zsfilumut + (01036214 + (Cm)ut-lSt-l + (Czrrzs)St-lSl-2}

—> 0,30le

sine

Lari

(pro

Con

-

m

H\
O
C

C

127

since T228,S,, —) T228,2 -) 6,,le and the remaining terms —) 0. D

A
Theorem 1.13 T2!” 13 —> crlw2
Lemma 1.14 s2 —> 0,2 for 5 > 1/2
(proof) it A
82 = T‘Eetz + (l3 - 1)2(1/I)>3y..r2 - 2(13 - 1)(1/F)Zy..t€r

—> 20,2 + 0,2 - 20,2 = 0,2 El
Corollag 1.15 6+ Tm —> czlw2
Case 4: 5 = 3/4

Lemma 1.16 TmZy,,y, —-) 0,2[W(1) + czlwz}

(proof) I
T ”Zyury.

= Tmtzu..,u. + (C/r3")S..ru. + (CIT”‘)S...u... + (C2/F’”)S..tS.2)
-) 0,2W(1) + e,,zcrlw2 III

A
Theorem 1.17 Tmfl —> W(1) + 01w

Coroll 1.18 e + T"2 -> W(1) + czlwz

Case 5: 5> 3/4

Lemma 1.19 TmEy,,y, -—> 0,2W(1)
(proof)
Tmzyt-lyt

= T‘”2{u,,u, + (CH6)S,,2U, + (m6)ul-lsl-l + (CZ/T28)St-lSt-2}
-—) 0,2W(1) [1

A
Theorem 1.20 Tmfi —-> W(1)

128

CoroLary 1.21 6+ T”2 —> W(1)

[2] 6, andfi, tests

In this case it is useful to write
A
(A7) 13,. =

A

B): ' 1 = 2"(Vt-1 ”37.1)51 I 204-1 ‘37-1)2

2641 -y'.t)yt / fly... - 9-1)2 or

(A8) s2 = ammo. -y) - 6,0... -y'..>12
Case 1:0 < 5 <1/2

Lemma 2.1 Tm“)? —> CO’JW for all 5 > 0.

Lemma 2.2 (1fI‘2'25)2(y,,2 y,,): —> e,,zczhiv2
(proof)
(1m'”)rry..t 21/11)” = (mt-”>2? - (PM 9.02

Using Lemma 1.1 and 2.1 we get the result. C]
Lemma 2.3 (1/T)2(y,, -y7,,)e, —> ~03 for all 5.

(proof)
(ImEYt-let = (lmzlut-l 7' (1 '1' e)St-2}(ut + eut-l)

= (lmzut-lut + (W)(1fl)$t-zm + (9m)(1/1')23t-rut + (WHY-Uta2

Since first 3 terms —) 0 and 0 —) -1 as T -> 00, we get the result. C]
A -
Theorem 2.4 was, - 1) —> -1/(:21w2

Lemma 2.5 $2 —) 20',2
(P1000 A _ A -
82 = T1263 + (I3 - 1)2T12(yt.r -)'.t)2 - 203 - 1)le(yt-l “y-t)8t

Since T1263 —9 202 and the last two terms —-> O, we get the result.

Cl

129

Corollag 2.6 T“ 6,, -> -1/(2czlw2)”2
Case 2: 5 = 1/2

Lemma 2.7 T‘2(y,, -)7.,)2 -> 0.3(1 + C2171”)
Theorem 2.8 6, - l —> -l/(1 + C219”)
Lemma 2.9 s2 —) o,2{2 - 1/(1 + c2197?»

Coro11a_ry 2.10 T"2 6, ... -1/(1 , wimpy/2
Case 3: 1/2 < 5 <3/4

W Tlmyt-l '74? '2 Caz

Theorem 2.12 6,, -1-) -1
my; (W‘z‘my... -y'..><y. ~91 —> 0.201912
(proof)
(1m”)2(y,., -y'..)(y. -y)

= (lffwfilytty. -y'.ty. - ytti +y'.ty)

(1) (1W”)Eyt-tyt -> otzczlw2

(2) 6171225122» —» -(CoJW)2

(3) «were... —> 40on

(4) (1m“)zyz,y -) (CoJW)2

Using (1), (2), (3), and (4) we get the result. E]

A -
Theorem 2.14 T616 —) crlwr.

Lemma 2.15 32 —) 0,2 for 5 > 1/2

(pro

 

/?

 

130

Corollary 2.16 6, + T"2 -> criw2
Case 4: 5 = 3/4

___Lcmma 2.17 1“” 20- -y‘..><y. -y) -> 0.2mm + cilia}
(proof) _ _ _
Tm 26st -17.t)(yt-y) = T‘”2{yt.tyt -y.ty. -yt.ry +y-ty)
<1) 1“” Easy. —> dawn) + 01W}
(2) -Tmy'..>:y. —> «:0sz
(3) 4“” 29y... —> «€0sz
(4) T‘” 2M —r (COJWY

Using (1), (2), (3), and 4), we get the result. [:1

A -
Theorem 2.18 T166, —) W(1) + crlw2

Corolla_ry 2.19 6, + T"2 —) W(1) + c2191!2

Case 5: 5 > 3/4

___Lomma 2-20 T” 26.4 ->7.r)(yt -y) —> 0.2W(1)
(proof)
Tmflyta -y.t)(y. - y) = T‘”2{y.yt.t - yy..t - ytya +yy-t}

Since T‘”2y,y,, -> 6,2W(1) and the remaining terms —) 0 we get the result.

A
Theorem 2.21 TVZB,I —-) W(1)

Corollag 2.22 6, + T"2 —> W(1)

equ

dcfi

II is

OLS

 

131
[3] 6, andfi tests
The derivation is simpler when we use the transformation of regression

equation (5):
(A9) Ay. = a + 5t + (B - 1)y..1 + 8.

A
First, consider the regression of yH on X = [1,t] to get the residuals SH and
define the matrix D and Q as follow:

(A10) D=|'1“’2 0]
L0 TmJ

(All) D’X’XD—) r 1 1/2]=Q
L1/21/3]

(A12) Q" =r 4 -61
|_-6 12)

It is straightforward to obtain the asymptotic results for the

OLS estimator of the coefficients of level and trend as follow:

(A13) |' 3] FUN) ZyH‘I
I I =(1NT)<D’X'XD)D'X’Y —> Q"|
L161 L(1fI’)2ty1-Ll

—+ Q“ I'CouIW'I —> Con r4Jw - 6er 1
LCoJrWJ L-6IW + 12IrWJ

Using (A13) we get the asymptotics for the residual from the regression of yt on [1, t]:

A A A

A =
= um] + (CNT)S[T,] - a - r(Tb) —+ um] + CouW

C0

Ac

(A

Ca

132

where W(r) = W(r) - 4IW + 6IrW + 6rIW - 12rIrW is a detrended Brownian motion as
defined in the main text.

A
Next consider the regression of AyH on [1, SH] to draw an inference on the

coefficient 4) = B - 1.
(A15) Ay. = intercept + 1s... + error

which is the same as (5)’ in the text. We use (A15) to derive the asymptotics for the
6, andfi statistics. Note that we use the same symbol is‘ as the error in (A15) without

A
loss of generality. Therefore, the 6, statistic is defined as To," where

A K A K
we <6. = 2(8... -s..>Ay. / 2(8... -s..>2
A A A K
= 2(St-l 'S-1)£t I 2(81-1 'S-1)2

Accordingly, s2 is defined as

2 , A A x 2
(A17) 5 = (ImZKAY; ' Ay) '¢r(St-l '84)}
Case 1: O < 8 <1/2

Lemma 3.1 TWM] —+ Co“ |’ 41w - elrw 1
L19] L- 61 W + 121 rWJ
This is true for all 8 > 0.
Using Lemma 3.1 we get the following result:
(mm) s...) = (1/rm")um + (chinsrm - (mm-5m - (r/rlfl-5)'I€

—) C0}, V=V(r)

Us:

E

L

‘

C

C

C2

1...:

1H

Tl;

11::

Pk

133

Lemma 3.2 (1/I"*2‘)r.’s‘.2 —) 003117112
Lemma 3.3 (1/I‘”2'°)§ —> 0
Lemma 3.4 (1/I'2'”)2(’S\._1 -§,,)2 —+ ouzczl i172
Lemma 3.5 T1 >:(’s‘._l -§_,)e, —) of for all 5.
(proof)

(1) TIZYHQ ‘9 ‘Guz

(2) T! 52% = (T‘”*‘fi)(1‘"’“)28. —’ 0

(3) mm 32m. = (T‘fl+°)(t6)(1/r3fl+°)2te, —+ 0
Using (1), (2), and (3) we get (1m2§._,e, -+ of.

A A
Since '1‘1 8421151 = (T I’M) 8,,(1/1‘ 1”“)28. -> O, the result comes. D

A =
Theorem 3.6 T"”(B, - 1) —> -1/c2Iw2
Lemma 3.7 s2 -> 26.,2

Corollary 3.8 T‘fi —-) -1/(2CZI\7=V2)”2
Case 2: 8 = 1/2

Lemma 3.9 §,, —9 CoudW - 41w + olrw + 3Iw - oIrW) = 0
Lemma 3.10 Tl 2:(’s‘.,l -§,,)2 —) 03(1 + c2fi7v2)

Theorem 3.11 6; = a? - 1) —> -1/(1+ C2Iv=v2)

Lemma 3.12 s2 —> 0.3{2 - 1/(1 + ONT/2)}

Corolla_ry 3.13 Tme, —) -1/(1 + 2C2! 17:11:)“2

134
Case 3: 1/2 < 5 <3/4

A

Lemma 3.14 Sm] —> “m1
(WOOD A A

SIT!) = 11”,] + (CH6)S[T1.] " a " I'Tb

= “rm + (Gnomflqnsrm _ Terri/206105) _ Ton/2(161rerQ)

—-) “rm [:1

A
Lemma 3.15 '1“L(§.,l -s_,)2 —> of

A A
Lemma 3.16 41, = (131 - 1) —-9 -1

A
Since the above lemma implies that B -) 0 when 8 > 1/2, we use (A18) to

A
derive the asymptotics for [3,.

A

A A A K
(A18) B: = 2(St-1 'S-1)Yt/ 2x814 'S-1)2

Lemma 3.17 (rm-”ms... -§,,)y. —> ou2c21v=v2
(”000 A A A
zst-IYt = z{Yr-1 ' a ' b(t ' 1)}Yt = zyt-IYt ' 523’: ' b2(t'1)Yt

(1) (1m 2{Sp-"82.11% ‘9 OeZCZIW’

(2) (~1/r2'2")tir;yH -> -Go.2(4lw - 61rW)Iw

(3) (- 1n“) €2(t-1)y. -) -CZGu2(-61W + 121rW) Irw
Using (1), (2), and (3) we show that (1/rm)>:§.,,y. —> c203] v=v2

A
Since (1fI"‘Z’2‘5)S.,12yt —) O, we get the result. [:1

A =
Theorem 3.18 Ti“ 13, —) c21w2

Lemma 3.19 s2 —> of for 8 > 1/2

13S

Corolla 3.20 f; + T”2 —-) C21 {111”
Case 4: 8 = 3/4

W T‘”2(§... -§-.)y. —+ c.2{W(1>+ 02W]
(proof)
(1) T1’223y.,1y.= 'I‘mZyHy. —+ ou2{W(1) + c21w2)
(2) TwaZy. -) -C20u2(4IW - 6IrW)Iw
(3) -Tm€z(t - 1)yt —> -C20u2(-61W + 121rW)Irw
Using (1), (2), and (3) we can show that Tngmy. —) ou2{W(1) + 0161/2}. Since

A
-'I“’2S,,}.‘.yt —> O the results comes. E]

A =
Theorem 3.22 T"2 [3, —) W(1) + c2! W2

Corollary 3.23 t‘, + T"2 —-) W(1) + C21 31/2
Case 5: 8 > 3/4

A A

Lemma 3.24 Tm2(S._, -s_.)y, —> ou2W(1)
A

Theorem 3.25 T"2 [5, —> W(1)

Corollgy 3.26 r. + T"2 —-) W(1)

[4] E and 75 tests

Regression equation (7) can be transformed into:
(A19) Ay. = (B - DYt-l + W(1 - B) + £0 + B - tB) + 8.

= intercept + o3“ + error

136

where S”: y,_, - fix - at - 1), t 2 2, with wx = w + X0. As before, we use the same

symbol a, as the error in (A19) without loss of generality. Using the DGP of (1) and

(2) we get
(A20) x, = x0 + u. + (1 + 9)S.-r
(A21) yt = Wx + gt + (1 + 6)S,-1.

The restricted MLE’s of fit and E are derived under the null as follow:

(A22) 3 = mean Ay = (yr - y.)/CT - 1)

(A23) TVx = yr - E
Substituting yt from (A21) into (A22) and (A23) we get

(A22)’ E =§+(1+e)ti+op(1)

(A23), {itx = Wx ’ (1 + 0)fi + 111
By using the above results we show that

(A24) 5.. = y... - it. - £0 - 1)
= (1 + 63:21-12 (uj - fr) + (u,,l - ul)
Sm, = (1 + (9)8”, - [Tr](1 + 0)(1/r)sT + um] - u1
= (C/I“)Sm] - “WET + “(m ' u1
We use (A19) to derive the asymptotics for the 5 and 76 statistics. Note that 5 statistic

is defined as T$, where

137

(A25) E = B ‘ 1 = 2X3” '§-1)AYt I z(St-1 '§-1)2
= 261-1 'é-1)€t I 2“(St-1 '§-1)2

Now, 82 is defined as

(A26) s2 = (1m2{(Ay.- Ay) - $6... -§-.))2
Case 1: O < 8 <1/2

LeanaAl amazes... -§,,)2 —) 6.201112
(proof) (1/'I"’*"‘)2:(SH -§,,)2 = (1/1‘2'”)ZS,,,2 - (Tl/M §_,}2
Using (A22)’ we can show that

Tim‘s”, = (CNT)S - r(C/«/T)sT + o,(1)

-2 0..W(r) - r6..W(1) = 0.V(r)
é. = y". - )1". - €011)
=u'.1 - u. + (Cfl‘)(1/I')ZS..2 + (C/I“)Ii - (C/I“)Sr(1/I)(t31)

Therefore,

TW‘S —-) (CfI°’2)Xs,_2 - (Chins, (1/2) + op(1)

-) Con - (1/2)Co,W(1)

Combining the above results we get

T‘”*‘(Sm 43) —> Co..{W(r) + (1/2 - r)W(1) - 1W} = Cod/(r)

Therefore, the result follows. Cl

Lemma 4.2 'I‘1 }:.(S,,1 - Spa, —> ~03 for all 8.

(PFC

[ST I571 I57

IQ

the

 

138

(proof)

T1 z(gt-1 ‘§-1)€1
= Tl£{(C/Ié)su - r(C/16)S'r + “(.1 ‘ u1 " (CmyFlZSt-z
+ (1,2) (W)ST} ‘(ut + 9“1.1)

—> (l/’I')9).‘.u,_,2 ——) 45,} Cl

Theorem 4.3 T"”(B - 1) —> -1/C21\-/2
Lemma 4.4 $2 —) 26,,2

Coronary 4.5 ’1“? —> -1/(2c21v2r'2
Case 2: 8 = 1/2

W 'I‘1 FISH -S,,)2 -9 030 + C2! {12)
Theorem 4.7 B - 1 —-) -1/(1 + c2192)
Lemm_L“ 82 -+ 0.212 - 1/(1 + Cain);

Corollag 4.9 T"2 't‘ —) -1/(1 + 201%)"2
Case 3: 1/2 < 8 <3/4

14cmma 4.10 SUV] —) 11”,] ' 111
Lemma 4.11 T1 26,, -s,,)2 —) 0,2

(proof)
Using Lemma 4.10 and the following result

3 = ti. - u. +(CfI‘)(1/I')ES..2 + (cn“)u' - (1/2)(cn“)sr —+ -u..
then we can show that Sm] - S —) um]. Therefore,

T236.-. -§-.)2 =(1rr)2u...2 —> 6.2 D

139

Since $ _) -1 (B _) 0) when 8 > 1/2, we use (A27) to derive the relevant

asymptotics.

(A27) B = 26... -§-.)S. / 2(3... —§-.)2
= 2G“ '§-1)(§t 'é) I 26:4 '§-1)2

where we use the facts that S, = y, - E and ZXSH -S,1)§ = 0.

Lemma 4.12 (1rr””)z(s,_,-§-,)(S,-§) -—) 6,2C21vz
(proof) - -
(Sm 3.06. 3)
= (1 + t))2(s,,2 is) + (1 + 0)2 020 - t) - 2(1 + (9)2n'(s,,2 -§)(t - t‘)
+ (1 + 9)(S,,2 -S)(u, -u) - (1 + 6)1'1(t - 001. - 11) + (1 + 9)(S,_2 -S)(u,., -u' )
' (1 + 9) 1.10 ' 001m ' 1.1) + (“t-l ' fiXUr ' 1.1) + (“at -u)(1 + 9)u..r
+ (1 + 0)2(S,,2 -§)u,_, - (1 + (1)2 fi(t - t' )u,_l
Therefore,
(were... -§-.)(S. -§)
—-) c2m2(s,_, - s)2 + (1m8.2(1/r’)2(t - t‘ )2
- (2C'z/slr)s—.(1rr‘”)>:(8..2 - §)(t - f )

-> (220,21 92 1:)

Theorem 4.13 TWB —) C21 in
Lemma 4.14 s2 —-) of for 8 > 1/2.

Corollgy 4.15 'r' + T"2 -+ C21)?2

140
Case 4: 8 = 3/4

Lemma 4.16 T‘” 2(3... -§-.)(S. -§) —> 6,21wm + (22192}
Theorem 4.17 TmB -) W(1) + C2192

Coroll_a_rv 4.18 'i' + T"2 —-) W(1) + C2192
Case 5: 3/4 < 8 <1

Lemmg 4.19 T“2 26,, -§_,)(S, -§) —> e,,zwu)
Theorem 4.20 TmB -—> W(1)

Corollary 4.2; ‘i’ + T"2 —9 W(1)

25
50
100
250
500
1000

25
50
100
250
500
1000

25
50
100
250
500
1000

~0.95

25
50
100
250
500
1000

~0.99

25
50
100
250
500
1000

141

Table 4—1 (a)

5 % Fractiles of the Actual Sampling Distribution

~18.
~25.
~31.
.10
~40.
~41.

~38

~27.
~47.
~81.
~150.
~210.
~264.

~29

~54.
~100.
~220.
~382.
~606.

~30.
~57.
.43

~108

~250.
~47l.
.40

~872

~31.
~59.
~114.
~27l.
~525.

37
50
78

05
91

36
83
10
44
61
55

.81

7O
40
12
37
31

94
68

96
70

67
81
12
30
61

~1025.2

~23.

~33

~45.

~57

~62.
~65.

~30
~55

~92.

~180

~270.

~362

~32.

~58
~107
~238

~425.

~718

~32.

~59

~ll3.
~262.

~494

~925.

~32.
~60.

~116

~275.
~533.

87
.72
10
.30
20
30

.76

.07
8O
.71
20
.10

14

.47

.60
.50
00
.50

60
.82
50
11
.00
00

76
88
.04
20
60

~lO40.0

~25.
.87
~59.
~80.
~92.
.10

~42

~98

~32.
~58.
~101.
~206.
~328.
.40

~473

~33.
~61.
.43
.06
.80

~ll2
~252
~457

~806.

~33.
~61.
.85
~269.
~510.
.40

~115

~965

~33.
~62.
~117.
~276.
~536.

87

73
6O
20

76
00
9O
04
80

48
04

10

64
04

05
10

68
16
21
70
30

~1048.0

bl

~25.
.79
~53.
~70.
~79.
.90

~38

44

~30.
~54.
~95.
~19l.
~301.
~425.

~31.
~57.
~106.
~239.
~435.
~755.

~31
~58

~31.
~58.
.46
.64
.70

~111
~268
~519

85

05
60
30

53
37
6O
13
20
70

31
58
65
63
10
00

.49
.44
~110.
~258.
~491.
-928.

19
91
10
50

52
71

~1020.0

I
u:

LLLL

~5.
~6.
~8.
~10.
~11.
~11.

~6.
.88
~10.
~14.
~17.
~21.

~6.
~8.
.96
~15.
~21.
~27.

~10

~6.
.78
~11.
~17.
.70
~32.

~23

“'>

.86
.12
.32
.50
.60
.70

67
84
28
37
60
60

31

10
04
6O
10

63
39

91

20
90

86

62
30

50

~6

~9.
~11.
~13.
~14.

~6

~10.
~15.
~19.
~23.

~8.

~11

~16.
~22.

~29

-7

~11.
~17.
~24.

~33

‘1)

.83
.14
.40
.70
.80
.90

.44
.68
31
90
6O
90

.87
.52
80
10
3O
70

.01
83
.47
61
10
.40

.05
.94
80
58
00
.00

‘1)

...6.

~10.
~13.
~15.
~17.

.64
.07
.49
.00
.20
.20

86

.26

10
21
70
60

.10
.82
.30
.91
.60
.00

.15
.98
.71
.04
.80
.60

.17
.04
.86
.64
.10
.20

~41

.07
.54
.97
.39
.55
.69

.27
.65
.51
.45
.66
.36

.38
.16
.65
.15
.59
.62

.45
.35
.00
.33
.95
.47

.46
.40
.17
.79
.21
.23

142

Table 4—1 (b)

95 % Fractiles of the Actual Sampling Distribution

r a. p. z . ., .
~0.5
25 ~0.08 ~3.7l ~9.03 ~7.28 ~0.05 ~1.28 ~2.15
50 ~0.20 ~4.36 ~12.07 ~9.57 ~0.12 ~1.43 ~2.50
lOO ~0.29 ~4.77 ~14.10 ~11.3O ~0.17 ~1.48 ~2.69
250 ~0.33 ~5.00 ~15.60 ~12.50 ~0.30 ~1.50 ~2.80
500 ~0.37 ~5.2O ~16.10 ~l3.00 ~0.30 ~1.60 ~2.90
1000 ~0.40 ~5.2O ~l6.40 ~l3.10 ~0.30 ~1.60 ~2.90
~0.8
25 ~4.90 ~12.6l ~16.34 ~12.27 ~l.54 ~2.82 -3.30
50 ~7.10 ~21.19 ~31.89 ~23.50 ~l.85 ~3.62 ~4.73
100 ~8.83 ~29.34 ~54.6l ~40.30 ~2.05 ~4.12 ~6.07
250 ~lO.33 ~37.44 ~87.75 ~66.70 ~2.20 ~4.49 ~7.27
500 ~ll.09 ~40.40 ~107.60 ~83.9O —2.30 ~4.60 -7.80
1000 ~ll.20 ~42.40 ~121.80 ~95.20 ~2.30 ~4.7O ~8.10
~0.9
25 ~10.43 ~16.04 ~17.76 ~l3.19 ~2.53 ~3.35 ~3.53
50 ~19.13 ~33.56 ~37.73 ~27.40 ~3.39 ~4.98 ~5.40
100 ~29.05 ~61.62 ~76.58 ~53.66 ~4.08 ~6.65 ~7.82
250 ~40.79 ~108.13 ~l7l.20 ~120.80 ~4.67 ~8.30 ~11.38
500 ~45.96 ~l39.50 ~27l.70 ~l98.20 ~4.9O ~9.00 ~13.54
1000 ~49.56 ~163.00 ~378.80 ~285.70 ~5.10 ~9.50 ~15.3O
~0.95
25 ~13.92 —l6.93 ~18.10 ~l3.39 ~3.09 ~3.50 ~3.59
50 ~30.24 ~37.51 ~39.24 ~28.46 ~4.65 ~5.44 ~5.57
100 ~57.27 ~78.39 ~82.82 ~57.46 ~6.32 ~8.01 ~8.34
250 ~107.86 ~185.25 ~212.07 ~143.41 ~8.28 ~12.12 ~l3.64
500 ~145.26 ~306.70 ~406.50 ~272.30 ~9.20 ~l4.90 ~l8.50
1000 ~179.04 ~449.40 ~711.90 ~494.00 ~9.95 ~17.10 ~23.60
~0.99
25 ~15.94 ~l7.20 ~l8.23 ~l3.45 ~3.42 ~3.55 ~3.6O
50 ~37.00 ~38.70 ~39.68 ~28.75 ~5.44 ~5.58 ~5.62
100 ~80.77 ~83.53 ~84.64 ~58.68 ~8.26 ~8.46 ~8.51
250 ~215.3O ~222.90 ~224.70 ~l49.60 ~13.80 ~l4.28 ~14.30
500 ~433.05 ~458.8O ~462.70 ~301.00 ~19.60 ~20.60 -20.80
1000 ~821.89 ~923.8O ~940.80 ~606.10 ~26.50 ~29.30 ~29.80

‘31

~2.

-5.
-3.
~11.
~12

~2

-9.
~13
~18.

-2 .
—4

~10.
~14.
~20.

.98
.25
.43
.52
.57
.59

.76
.84
.00
.19
.75
.08

9O

.25

04
89
13

.88

.91
.35
.33

96

.71

09

91

.40
.40

31
64
95

Fractiles of

‘1) *>
t:

*1 *>
«4

b1

fl) ‘0)
t

W! ‘0)
q

~71.
~111.
~173.
~l48.

43
ll
91
15

.98
.45
.33
.61

.42
.73
.21
.51

.10
.09
.69
.28

143

Table 4-2

5 % fractile

~0.8

~446.43
~694.44
~1087.00
~925.93

~14.94
~18.63
~23.3l
~21.52

95

~15.14

~170.07
~l34.4l

~2.75
~5.22
~9.22
~8.20

~0.9

~1785.0
~2777.8
~4347.8
~3703.7

~29.88
~37.27
~46.63
~43.03

fractile

~0.9

—60.57
~218.34
~680.27
~537.63

~5.50
~10.45
~18.44
—16.4O

~0.95

~7143.0
~11111.0
~1739l.0
~l4815.0

~59.76
~74.54
~93.25
~86.07

~0.95

~242.28
~873.36
~2721.10
~2150.50

~ll.Ol
~20.90
~36.89
~32.79

the Predicted Distribution for 6 - 1/4

~0.99

~178571
~277778
~434783
~37037l

-298.81
~372.68
~466.25
~430.33

~0.99

~6056.9
~21834.0
~68027.0
~53764.0

~55.03
~92.69
~184.43
~163.96

T
~0.5
25 ~18.
50 ~29
100 ~41.
250 ~55.
500 ~62.
1000 ~66.
~0.8
25 ~23.
50 ~44.
100 ~81.
250 ~160.
500 ~235.
1000 ~308.
~0.9
25 ~24.
50 ~48.
100 ~94
250 ~219.
500 ~390.
1000 ~641
~0.95
25 ~24.
50 ~49
100 ~98.
250 ~241.
500 ~467.
1000 ~877.
~0.99
25 ~25
50 ~49.
100 ~99.
250 ~249.
500 ~498.
1000 ~994

144

Table 4~3 (a)

5 % Fractiles of the Predicted Distribution for 6 - 1/2

p

52
.41
67
56
50
67

67
96
69
26
85
64

65
64
.70
30
6O
.00

91
.65
62
55
29
19

.00
99
94
65
60
.40

~20
~34

~52.
~76.
~90.
~100.

~24.
~46.

~87

~l83.
~290.
~409.

~24.
~49.

~96

~229.
~423.
~735.

~24.
~49.

~99

~244.

~478
~9l7

~25.
~49.
~99.
~249.

~499
~996

A

Pu

.41
.48
63
92
91
00

13
64
.41
82
70
84

78
12
.53
36
73
30

94
78
.11
50
.47
.43

00
99
96
78
.10
.41

~21.
~38.
.49
~102.
.03
~l48.

~63

~129

~24.
~47.
~91.
~203.
.47
~520.

~342

~24.
.43
~97.

~49

~236
~448
~8l3

~24.
~49.
.43
~246.
.03
~945.

~99

~486

~25

~49.
~99.
~249.
.43
.71

~499
~997

86
83

56

15

44
80
58
25

83

86

75

.41
.43
.01

96
86

46

63

.00

99
98
86

b1

~21.
~37.
~59.

~93

~24.
.44

~90.
~l96.
~324.
~480.

~47

~24.
~49.
~97.
~234.
~440.

~787

~24.
~49.
~99.
~245.
~483.
~936.

~25

~49.
~99.
~249.
~499.
.31

~997

39
38
7O

.02
~114.
~129.

29
03

34

25
85
68
77

83
33
37
19
53

.40

96
83
33
85
68
77

.00

99
97
83
33

A
1'

.83
.56
.13
.59
.77
.87

-4.
~6.
.31
~10.
.42
~13.

~12

~4.
~6.
.48
~13.
~17.
~21.

-9

-4.
~7.
~9.

~15.

~20.
~27.

74
39

86

51

93
88

98
90
72

98
02
86
26
80
20

.00
.07
.99
.79
.30
.45

~4.
~6.
.81
.06

~12

~14.
~16.

-4.
-5.
~9.

~14.

~19.

—24

~7

~15
~21
~29

~5.
.07
~10.
~15.
~22.
~31.

~7

.15
.13
.98
.74
.07
.26

83
61

32
05

96
95
67
56
17

.11

.99
.04
~9.

91

.47
.42
.11

00

00
80
32
51

~13

-4.
.05

~9.
~15.
~21.
~29.

~7

~5

~10

.41
.63
.82
.03
.61
.94

.89
~6.
.19
.09
~16.
~18.

77

14
77

.97
.99

~9.
~14.
~20.
~26.

78
97
16
17

99

94
59
74
95

.00
~7.
.00
~15.
~22.
~31.

07

80
34
55

*1

~6.
~7.

~8.

~6
~9.
~12.
~15.
~17.

~9.
~14.
~19.
~25

~5.
~7.
~10.
~15.
~22.
~31.

.32
.46

52
56

.03

31

.87
.72

O7
74
50
79

.97
.98

74
84
84

.48

.99
.05
.93
.55
.64
.68

00
07
00
80
33
54

95 % Fractiles of

T
~0.5
25 ~2.
50 -2
100 ~2.
250 ~2
500 ~2
1000 ~2
~0.8
25 ~9
50 ~11.
100 ~13.
250 ~14.
500 ~14.
1000 ~14.
~0.9
25 ~17.
50 ~27.
100 ~37.
250 ~48.
500 ~54.
1000 ~57.
~0.95
25 ~22.
50 ~41.
100 ~70.
250 ~123.
500 ~163.
1000 ~195
~0.99
25 ~24.
50 ~49.
100 ~98.
250 ~240.
500 ~46l.
1000 ~858.

21

.31

37

.40
.41
.42

.43

62
15
28
70
92

70
39
72
76
03
11

66
45
78
O4
20

.03

9O
59
38
09
87
3O

~6

~8.
~8.

~8

~17.
~26.

~35

~44.
~49.
~51.

~22

~40.
~68.
~116.
~151.
~179.

~24.
~47.
~89.
~l94.

~317

~466.

-24

~49.
~99.

~247

~488.
~956.

pu

.47
.43
O3
44
.58
.66

15
10
.31
80
21
76

.43
68
59
55
98
21

30
29
73
36
.97
20

.97
89
54
.17
81
21

145

Table 4~3 (b)

the Predicted Distribution for 6 - 1/2

3. 3 4‘ r“. r“. ;'
~13.03 ~ll.56 ~l.07 ~l.93 ~2.97 ~2
~17.62 ~15.04 ~l.09 ~2.00 ~3.27 ~2
~2l.39 ~17.70 ~1.09 ~2.05 ~3.46 ~3
~24.54 ~19.80 ~1.10 ~2.07 ~3.59 ~3
~25.81 ~20.62 ~1.10 ~2.08 ~3.64 ~3
~26.49 ~21.05 ~1.10 ~2.09 ~3.66 ~3
~21.80 ~21.08 ~2.41 ~3.61 ~4.40 ~4
~38.64 ~36.44 ~2.56 ~4.20 ~5.61 ~5
~62.97 ~57.34 ~2.65 ~4.63 ~6.78 ~6

~101.22 ~87.41 ~2.7l ~4.96 ~7.97 ~7
~126.90 ~105.93 ~2.73 ~5.09 ~8.53 ~7
~145.35 ~118.48 ~2.74 ~5.15 ~8.85 ~7
~24.11 ~23.89 ~3.7O ~4.51 ~4.83 ~4
~46.58 ~45.75 ~4.34 ~5.86 ~6.60 ~6
~87.18 ~84.32 ~4.82 ~7.22 ~8.79 ~8
~182.82 ~170.65 ~5.20 ~8.72 ~12.00 ~11
~288.18 ~259.07 ~5.34 ~9.47 ~l4.23 ~13
~404.86 ~349.65 ~5.42 ~9.92 ~15.93 ~14
~24.77 ~24.71 ~4.55 ~4.86 ~4.95 ~4
~49.10 ~48.86 ~5.95 ~6.7O ~6.94 ~6
~96.46 ~95.56 ~7.4O ~9.02 ~9.65 ~9.
~228.96 ~223.96 ~9.03 ~12.61 ~14.53 ~14.
~422.39 ~405.68 ~9.88 ~15.27 ~19.12 ~18
~731.26 ~682.59 ~10.40 ~l7.43 ~24.0l ~22.
~24.99 ~24.99 ~4.98 ~4.99 ~5.00 ~5
~49.96 ~49.95 ~7.01 ~7.05 ~7.07 ~7
~99.85 ~99.81 ~9.84 ~9.95 ~9.99 ~9
~249.09 ~248.84 ~15.2O ~15.63 ~15.75 ~15
~496.35 ~495.39 ~20.72 ~21.85 ~22.20 ~22
~985.51 ~981.74 ~27.42 ~30.17 ~31.17 ~31

.74
.97
.12
.21
.24
.26

.27
.35
.34
.28
.70
.94

.78
.49
.54
.38
.22
.56

.94
.91

57
24

.47

76

.00
.07
.98
.74
.15
.03

~160.
~140.

5 % Fractiles of

Pu

~l9.38
~27.50
~10.00

312.5
1750.0
8000.0

~24.10
~46.4O
~85.60
00
00

440.00

~24.78
~49.10
~96.4O

~192.
~270.

146

Table 4~4 (a)

the Predicted Distribution for 6 - 5/8

~21.41
~35.63
~42.50
109.4
937.5
4750.0

~24.
~47.
~90.

43
70
80
50
00

~80.00

~24.86
—49.43
~97.70

~182.
~230.

~20.78
~33.13
~32.50

171.9
1187.5
5750.0

~24.
~47.
~89.

33
30
20
50
00

80.00

~24.
~49.
~97.

83
33
30

T p
~0.5
25 ~16.25
50 ~15.00
100 40
250 625
500 3000
1000 13000
~0.8
25 —23.60
50 ~44.40
100 ~77.60
250 ~110.00
500 60
1000 1240
~0.9
25 ~24.65
50 ~48.60
100 ~94.40
250 ~215.00
500 ~360.00
1000 ~440.00
~0.95
25 ~24.9l
50 -49.65
100 ~98.60
250 ~24l.25
500 ~465.00
1000 ~860.00
~0.99
25 ~25.00
50 ~49.99
100 ~99.94
250 ~249.65
500 ~498.60
1000 ~994.40

~227.
~410.
~640.

~24.
~49.
~99.
~244.
~477.
~9lO.

~25

~49.
~99.
~249.
~499.

~996

50
00
00

94
78
10
38
50
00

.00

99
96
78
10

.40

~235

~24.
~49.
.43
.41
~485.
~942.

~99
~246

~25

~499

.63
~442.
~770.

50
00

96
86

63
50

.00
~49.
~99.

~249.

.43

~997.

99
98
86

70

~233.
~432.
~730.

~24.
~49.
~99.

~245

~25
~49

~249

13
50
00

96
83
33

.78
~483.
~932.

13
50

.00
.99
~99.
.83
~499.
~997.

97

33
30

3>

~3.
.12
.00
.53
134.
411.

39

-4

~9

-4.
.02

~9.
~15.
~20.
.20

~27

~5

~22

25

16
10

.72
.28
~7.
~6.
.68
39.

76
96

21

.93
.87
.44
~13.
~16.
~13.

60
10
91

98

86
26
80

.00
.07

-9_
~15.

99
79

.30
~31.

45

*>

~3

~1.
19.
78.
252.

-4.
.04
~9.

~15

~28

—5
~7
~10

.88
.89

00
76
26
98

.82
.56
~8.
~10.
~6.
13.

56
12
26
91

.96
.94
.64
.39
.34
.24

99

91

.46
~21.

35

.77

.00
.07
.00
~15.
~22.
~31.

80
32
51

~5

~9

~12

*>

.28
.04
-4.
.92

41.
150.

25

93
21

.89
.75
.08
~12.
.08
~2.

18

53

.97
.99
.77
.90
.79
.35

.99
.05
.94
.58
.72
.80

.00
.07
.00
.80
.34
.55

*1

~21
~29

.16
.68
~3.
10.
53.
181.

25
87
11
83

.86
~6.
-3.

~11.

~10.

.53

69
92
54
29

.97
~6.
.73
~14.
~19.
~23.

98

74
34
09

.99
~7.
.93
~15.
.61
.49

05

55

.00
.07
.00
.80
.33
.54

147

Table 4-4 (b)

95 % Fractiles of the Predicted Distribution for 6 - 5/8

T S
~O.5
25 233
50 982
100 4028
250 25547
500 102687
1000 411750
~0.8
25 16.3
50 115.1
100 560.4
250 3877.5
500 16010.0
1000 65034.0
~0.9
25 ~14.7
50 ~8.7
100 65.1
250 781.9
500 3627.5
1000 15510.0
~0.95
25 ~22.42
50 ~39.68
100 ~58.73
250 8.0
500 531.9
1000 3127.5
~0.99
25 ~24.90
50 ~49.59
100 ~98.35
250 ~239.68
500 ~458.73
1000 ~834.90

A

p”

47
236

~13.

895.

1045
6906

28125
113500

—4.
83.

~21.
~35.
~41.
117.

3
26
111
455

~20.
~31.
~25.
215.

4080.
17320.

~2
~3

~54.

3
64

3580.

~24.
~47.
~88.

~178

~213.
145.

~24.
~49.
~99.
~247.
~488.
~954.

OOONNO‘

2.
8.

6.
5.

COWNO‘H

28
14
55
.44
75
00

97
89
54
14
55
20

970.
4880.

~24.
~46.
~85.
~158.
~l32.
470.

~24.
~49.
~96.
~227.
~408.
~632.

~24.
~49.
~99.
~249.
~496.
~985.

COUNUOLAJ

08
33
30
13
50
00

77
08
33
03
13
50

99
96
85
08
33
30

1360.
6440.

~23.
~45.

~81

~l33.
~35.
860.

~24.
~48.
~95.
~220.

~387

~535.

~24.
~49.
~99.
~248.
~495.

~981

*>

33

105
437
1258
3590

-2.

56

;

7 47
66 139
65 403
56 1616
25 4592
00 13021

4 3.3

4 16.3

6 56.0

0 245.2

0 716.0

0 2056.8
84 ~2.94
35 -1.23
.42 6.5
75 49.5
00 162.2
00 490.5
71 —4.48
84 —5.61
30 ~5.87
94 0.50
.75 23.79
00 98.90
99 ~4.98
95 -7.01
81 —9.84
84 —15.16
35 —2o.52
.40 —26.40

182
547

113.

-4.
~6.
~8.

~11.

\IU'IO‘UOO‘N

.43
.45
.42
.29
.85

21

86
67
86
29

.56
.59

.99
.05
.95
.63
.85
.17

9)
q

~0.

26.
129.
388.

1130.

.4.
~6.
_9_

~14.

~18.
~20

LflLflU‘Q‘Op

.27
.99
.12
.43
.38
.48

.82
.55
.53
.00
.93
.86

95
94
63
36
25

.00

.00
.07
.99
.75
.20
.16

‘H

36.
168.
497.

1438.

.07
.44
~2.56
13.60

60.
203.

~6.
~9.
~13
~17.
~16.

~5.
~7.

~15.
~22.
~31.

82
65

.77
.41
.14
.46
.57
.20

.94

91
54

.97

16
92

00
O6

.98

74
15
O3

50
64
00
48
48
96

OPVOUDU)

.00

29
00
22
13
95

.40

O3
00

.09
.46

86

50
O7
70
23
28

T
~0.5
25 ~16.
50 ~14.
100 42.
250 ~25.
500 ~182.
1000 ~550.
~0.8
25 ~23.
50 ~44.
100 ~78.
250 ~107.
500 72.
1000 1280.
~0.9
25 ~27
50 ~49.
100 ~95.
250 ~215.
500 ~359.
1000 ~433.
~0.95
25 ~29
50 ~54.
100 ~100.
250 ~242
500 ~466
1000 ~860.
~0.99
25 ~29.
50 ~57.
100 ~109.
250 ~264.
500 ~511.
1000 ~996.

84—1015.

~19.
~27.
~9.
305.
1555.
13205.

~26.
.17
~86.
~158.
~128.
483.

~47

~29.
~53.
~97.
~227.
~408.
~630.

~30.
~56.
.00
~246.
~477.
~908.

~107

~30.
.07
~109.
~265.
.01

~57

~519

OOOOJ-‘UI

00

00
29
81
11

50
54
00
86
32
01

00
29

84

64
29

00

90
02

~24.
~37.
~42.
116.
971.
4888.

148

Table 4~5 (a)

NMQCWU‘I

‘bi

~23.0
~34.4
~32.0

1259.8 136

~3.
~2.

4.
180.1 40.
.44
6045.6 417.

~29.
~53.
.00
.08

~95
~193

~265.
~57.

~30

~242

~446.
~765.

~29.
~56.
~110.
.07
~500.
.40

~261

~949

~30.
~57.
~110.
~265.
~521.
81-1028.

50
54

21
64

.00

~56.
~107.
.09

36
00

33
99

95
86
00

00

00
07
00
65
69

~29.
~52.
~93.
~183.
~224.
106.

~30

~56.

~106

~238.
~435.
~728.

~30.
~56.

~109
~259

~493.
~939.

~30.
~57.
~110.
~265.
~522.
46—1025.

00
83
00
59
96
80

.00
36
.00
93
15
04

00
86
.00
.49
29
92

00
07
00
81
36
30

-4.
~6.
~7.
~6.
.14
.47

40

~5

~27

~5.
-3.
~10.
~16.
~22.
.49

~31

‘\>

30
O7
20
29

58

7O
27
80
81

.40
~6.
~9.

~13.

~16.
~13.

97
50
61
O6
72

.88

~7.
~10.
~15.
~20.
.22

64
00
31
86

90
07
97
66
82

~3.
~3.
~0.
19.
69.
417.

~5.
~6.
~8.
~10.
~5.
15.

~5

~6.
~8.
~10.
~16.
~23.
~32.

‘1)

9O
87
9O
29
54
58

20
67
60
01
76
28

.90
~7.
~9.

~14.

~18.
~19.

57
70
41
26
92

.00

~7.
~10.
~15.
~21.
~28.

96
70
61
36
72

00
07
99
76
21
12

43

~5.
~7.
~9.
~12.
~11.
.82

~6.
~7.
~10.
~15.
~19.
~24.

~8.
~11.

~16

~22.
~30.

~6.
~8.
~11.
~16.
~23.
~32.

9)
§

.90
_5_
.20
.39
.44
154.

27

58

90
57
50
21
86

00
97
70
31
96
22

.99
04
00
.51
36
02

00
07
00
8O
33
52

5 % Fractiles of the Predicted Distribution for 6 - 3/4

~4.
.87
.20
11.
56.
191.

~5.
.47

~7

-9.

~11

~6.
~7.
~10.
~15.
.46
~23.

~19

~6.
~8.
~11.
~16.
~23.
~32.

~11

6O

39
34
18

80

30

.61
~10.
.38

06

00
97
60
11

02

.99
.04
.90
.41
.06
.72

00
07
00
81
36
52

149

Table 4—5 (b)

95 % Fractiles of the Predicted Distribution for 6 - 3/4

T 3 3p 91 3 7 :6 71 ;
-0.5
25 233 47 -1 5 47 9 -0.2 0.9
50 982 236 42 67 139 33 6.0 9.4
100 3998 1043 269 366 400 104 26.9 36.6
250 23388 6984 2055 2662 1479 410 130.0 168.4
500 107635 22307 8723 11047 4636 997 390.0 494.0
1000 414130 112370 35907 45195 13063 3324 1135.5 1429.2
-0.8
25 16.5 -11.0 -18.0 -17.0 3.3 -2.2 -3.60 -3.40
50 113.3 -2.6 -32.3 -28.8 16.0 —0.4 -4.57 “4.07
100 555.0 84.0 -40.0 -25.0 55.8 8.4 -4.00 -2.50
250 3840.4 897.9 120.0 216.4 243.4 56.8 7.59 13.69
500 15885.9 4090.7 973.6 1358.2 711.3 182.9 43.54 60.74
1000 64020.0 17385.5 4885.0 6431.4 2111.3 549.8 154.48 203.38
-0.9
25 -12.5 -18.0 -19.50 -19.50 -2.5 -3.60 —3.90 -3.90
50 -7.6 -34.4 -40.81 -40.10 -1.1 -4.87 -5.77 -5.67
100 65.0 —51.0 -80.00 -77.00 6.5 -5.10 -8.00 ~7.70
250 780.9 37.8 -155.13 -133.00 49.4 2.39 -9.81 -8.41
500 3625.6 647.1 -128.81 -32.66 162.1 28.94 -5.76 -1.46
1000 15506.0 3588.5 473.62 865.74 490.3 113.48 14.98 27.38
-0.95
25 -18.2 -19.58 -19.90 -19.85 -3.63 -3.89 -3.98 -3.97
50 -35.4 -40.74 -42.43 -42.22 -5.00 -5.76 ~6.00 -5.97
100 -57.0 -82.00 -88.00 —88.00 -5.70 -8.20 -8.80 -8.80
250 7.7 -174.11 -218.38 -213.63 0.49 -11.01 -13.81 -13.51
500 533.1 ~211.55 -403.85 -381.49 23.84 -9.46 -18.06 -17.06
1000 3126.8 151.07 -630.01 -531.98 98.88 4.78 -19.92 -16.82
-0.99
25 -19.50 -20.00 -20.00 -20.00 -3.90 -4.00 -4.00 -4.00
50 -42.93 -42.93 -42.93 -42.93 -6.07 -6.07 -6.07 -6.07
100 -89.00 -89.70 -90.00 -89.90 -8.88 -8.97 -9.00 -8.99
250 -227.86 '231.98 -233.71 -234.19 -14.33 -14.67 -14.78 -14.81
500 -448.57 -468.92 -475.40 -475.40 -19.98 -20.97 -21.26 —21.26
1000 -829.24 -932.96 -959.52 -958.89 -26.21 -29.50 -30.34 -30.32

Fractiles of the Predicted Distribution

T

p tests

1’ tests

5%
95%

5%
95%

25

~33.15
~l6.80

~6.63
~3.36

50

~61.53
~38.40

~8.70
~5.43

150

Table 4~6

100

~116.30
~83.60

~11.63
~8.36

250

~275.77
~224.07

~17.44
~14.17

for 6 - 7/8

500

~536.45
~463.33

~23.99
~20.72

1000

~1051.55
~948.l4

~33.25
~29.98

25

50

100

250

500

1000

151

Table 4~7

(a)

Comparison of the Predicted Distribution with

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

The Actual Sampling Distribution (6 Test)

Actual

~23.
~20.
~18.
~0.
0.
0.

~34.
~29.
~25.
~0.
O.
0.

~46.
~38.
~31.
.29
0.
O.

~0

~58.
~46.
~37.
~O.
0.
.40

0

~65.
~50.
~40.
~0.
0.
0.

~67.
~53.
~41.
.40
0.
0.

~0

96
84
37
08
29
68

90
85
50
20
18
55

17
O4
78

10
47

13
60
51
33
O7

16
62
05
37
02
37

39
70
91

00
32

6-1/4

~117

~1

~117

~90.
.43
.42
.88
.45

~71
~2

~l

~117

~117.
.91
.43
.42
.88
.45

.65
~90.
~71.
.42
~1.
.45

91
43

88

.65

91

.65
.91
.43
.42
.88
.45

.65
.91
.43
.42
.88
.45

65

.65
.91
.43
.42
.88
.45

6 - ~0.5

6-1/2

~20.
~19.

~18

~35

~32.

~29

~54.
~47.
~41.
~2.
~1.
~1.

~80.
~66.
~55.
.40
-1.
-1.

~2

~95.
~76.
~62.
.41

~2

~1.
~1.

~105.

~83.
.67
.42
.88
.45

~66

62
61

.52
~2.
~1.
~1.

21
75
37

.09

26

.41
~2.
~1.
~1.

31
82
41

05
62
67
37
85
43

00
67
56

87
44

24
92
50

88
45

26
33

6=S/8 6—3
-19 69 -20.
-18.12 ~18.
-16.25 ~16.
232.97 233.
306.72 308.
405.16 409.
—28.75 ~28.
-22.50 -22.
-15.00 -14.
981.88 982.

1276.88 1227.
1670.63 1623.
—15.00 -13.
10.00 11.
40.00 42.
4027.50 3998.
5207.50 5106.
6782.50 6555.
281.25 293.
437.50 452.
625.00 637.
25546.87 23388.
32921.87 32368.
42765.62 44123.
1625.0 1680.
2250.0 2283.
3000.0 3050.
102687.5 107635.
132187.5 139912.
171562.5 183725.

7500 7721.
10000 10137.
13000 13204.

411750 414130.
529750 536124.
687250 696323.

/4

6-7/8

~36.
~34.
~33.

~16

~66.
~63.
~61.
~38.
~36.
~33.

~123.
~119.
~116.
.60
~80.
~76.

~83

~286.
~280.
~275.
~224.
~219.
.16

~213

~551.
~543.
.45
~463.
.40
.90

~536

~456
~447

~1073.
~106l.
~1051.
~948.
~938.
~926.

55
80
15

.80
~15.
~13.

25
35

33
86
53
40
21
52

10
60
30

50
70

52
99
77
07
17

65
83

33

05
98
55
14
34
32

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

~31.
~29.
~27.
~4.
~3.
~2.

~54.
~50.
~47.

~7.
.45

~5

~3.

~93.
.42

~87

~81.
~8.
~6.

—4

~183.

~l66

~368.

~312

37
19
36
90
69
61

66
96
83
10

88
88
12

83
71

.95

01

.15
~150.

~10.
.00
.88

44
33

.29
.72
.68
.09
.39
.16

93

.54
~264.
~11.
~8.
~6.

55
20
36
24

6~1/4

~735.
~568.
.43
~15.
~11.
.08

~446

~9

~735.
~568.
.43
~15.
~11.
.08

~446

~735.
~568.

~446

~9

~735.
~568.
.43

~446

~15.
~11.
~9.

~735.
~568.
.43
~15.
~11.
.08

~446

~735.
~568.
.43
~15.
~11.
.08

~446

29
18

14
78
29
18
14
78

29
18

.43
~15.
~11.

14
78

.08

29
18

14
78
08

29
18

14
78
29
18

14
78

152

o - -0.8

6-1/2 6-5
—24.18 —24.
-23.95 -23.
-23 67 -23.

-9.43 16.

-8.01 28.

-6.66 43.
—46.82 -46.
—45.96 —45.
-44.96 -44.
~11.62 115.

-9.53 162.

~7.69 225.
—88.03 —86.
—85.03 -82.
-81.70 -77.
-13 15 560.
-10.54 749.
- 8.33 1001.
—186.57 —165.
—173.61 -140.
-160.26 -110.
—14.28 3877.
—11.25 5057.

~8.76 6632.
—297 62 -160.
-265.96 ~60.
-235.85 60.
—14 70 16010.
-11.51 20730.

~8.92 27030.
-423.73 360.
-362.32 760.
~308.64 1240.
—14.92 65040.
—11.64 83920.

-9.00 109120.

Table 4~7 (a) (Continued)

/8

6-3/4

~24.
~24.
~23.
16.
29.
44.

~47.
~45.
~44.
113.
162.
225.

~87.
~82.
~77.
560.
753.
1009.

~163.
~l39.
~107.
3840.
5007.
6574.

~151.
~55.
72.
15885.
20523.
26795.

372.
777.
1280.
64020.
84110.
110821.

6-7/8

~36.
~34.

~33

~66.

~63

~38

~123.
~119.
~116.

~83.

~80

~286.
~280.
~275.
~224.

~219
~213

~551.
~543.

~536
~463
~456

~1073.
~1061.
~1051.
~948.
~938.

~926

55
80

.15
~16.
~15.
~13.

80
25
35

33

.86
~61.

53

.40
~36.
~33.

21
52

10
60
30
60

.50
~76.

70

52
99
77
07

.17
.16

65
83

.45
.33
.40
~447.

90

05
98
55
14
34

.32

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

~33.
~31.
~29.
.43
~8.
~7.

~10

~60.
~57.
.70
~19.
~16.
~12.

~54

~109
~104
~100

~240.
~230.
~220.

~40.
.47

~32

~25.

~431.
.41
~382.
~45.
~36.
~26.

~407

~727

~39

~30.

28
44
81

71
04

17
28

13
06
74

.41
.42
.43
~29.
~23.
~18.

05
73
97

37
28
12
79

8O
02
37
96

07
90

.44
~667.
~606.

~49.

21
31
56

.42

86

6-1/4

~294l.
~2272.
~1785.
~60.
~47.
~36.

~2941.
~2272.
~1785.
~60.
~47.
~36.

~2941.
~2272.
~1785.
~60.
~47.
~36.

~2941.
~2272.
~1785.
~60.
~47.
~36.

~2941.
~2272.
~1785.
~60.
~47.
~36.

~2941.
~2272.
~1785.
~60.
~47.
~36.

ODD-'O‘VVN WHO‘VNN WHO\\J\JN wHO‘VNN WHO‘VVN

WHChVVN

153

0 - ~0.9

6~1/2

~24.
~24.
~24.
.70
~16.
~14.

~17

~49.
~48.
~48.
~27.
~24.
~21.

~96.
~95.
~94.

~37
~32

~230

~225.
~219.
~48.
~39.
~31.

~427.
~409.
~390.
.03
.05

~54
~43

~33.

~746

~64l

79
73
65

33
81

16
92
64
39
25
04

71
79
70

.72
.02
~26.

65

.42

23
30
76
64
72

35

84
63

86

.27
~694.
.03
~57.
~44.
~35.

44

ll
98
05

6~5

~24.
~24.
~24.
~14.
~11.

-7.

~49.
~48.
~48.

18.

~96.
~95.
~94.

65.
112.
175.

~228.
~222.
~215.

781.
1076.
1470.

~415.
~390.
~360.
3627.
4807.
6382.

~660.
~560.
~440.

15510
20230
26530

Table 4~7 (a) (Continued)

/8 6=3/4
79 -29.00
73 -28 00
65 —27 00
68 —12.50
73 —8.00
80 -2.00
15 -51.41
90 —49.29
60 —49.29
.72 —7.57
.08 5.86
80 22.83
60 -97.00
60 -96 00
40. —95.00
10 65.00
30 114.00
30 179.00
75 -229.45
50 —223.12
00 -215.22
88 780.90
88 1081.32
63 1481.35

0 —412.79

0 -390.43

0 —359.13

5 3625.55

5 4812.90

5 6407.21

0 -652 15

0 -557.28

0 —433.95
.0 15506.00
.0 20360.72
.0 26642.80

6-7/8

~36.
~34.
~33.
~16.
~15.
~13.

~66

~38

~36.
~33.

~123.
~119.
~116.
~83.
~80.
~76.

~286.
~280.
~275.
~224.
~219.
~213.

~551.
~543.
.45
.33
.40
~447.

~536
~463
~456

~1073.
~106l.
~1051.
~948.
~938.
~926.

55
80
15
80
25
35

.33
~63.
~61.
.40

86
53

21
52

10
6O
30
60
50
70

52
99
77
07
17
16

65
83

90

05
98
55
14
34
32

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

~34.

~32

~62
~60

~57.
~30.
.26
~23.

~27

~116.
~111.
.43
~57.
.99
~43.

~108

~50

~265.

~257

~250.
.86
.43

~107
~ 92

~ 77.

~497.
.06
~47l.
~145.
~121.
~ 98.

~484

~932.
~903.
.40
.04
~l45.
~110.

~872
~179

12

.42
~30.
~13.
~12.
~10.

94
92
35
69

.53
.06

68
24

64

12
93

27

91

13

.74

96

91

73

70
26
22
83

60
10

15
93

154

Table 4~7 (a) (Continued)

0 - -o 95

6-1/4 6-1/2
—11764.0 -24.95
—9090.9 —24.93
-7142.8 —24.91
-242.2 -22.66
-188.4 -22.07
-145.3 —21.33
—11764.0 -49.79
-9090.9 -49.73
-7142.8 —49.65
-242.2 —41.45
-188.4 -39.51
-145.3 —37.20
~11764.0 -99.13
-9090.9 -98.91
-7142 9 ~98.62
—242.2 ~70.78
-188.4 -65.33
-145.3 -59.23
-11764.0 —244.80
-9090.9 -243.31
-7142.9 -241.55
—242.2 —123.04
-188.4 -107 44
-145.3 - 91.89
-11764 0 -479.62
—9090 9 -473.94
-7142.9 —467.29
-242.3 —163.20
—188.4 -136 85
-145.3 —112.58
-11764.0 —921.66
—9090.9 -900.90
-7142.9 -877.19
-242.3 —195.03
-188.4 -158.54
-145.3 -126 86

6-5/8

~24.
~24.
~24.

-22
—20
-49
—49
—49

~36
~32

~99.
~98.
~98.
~58.
~46.

~31

~244.
~243.

~241

~478.
~472.
~465.
531.
826.
1220.

~915

95
93
91

.42
~21.

68

.70

.79
.73
.65
~39.
.73
.80

68

15
90
60
73
93

.18

69
13

.25
.97

81.
180.

72
16

75
50
00
88
88
63

.00
~890.
~860.
3127.
4307.
5882.

00
00
50
50
50

6~3/4

~32

~29

~55.
~55.
~54.
~35.
~30.
~22.

~104

~101.
~100.
.00
.00

-57
—42

~24.

~245.
.68
~242.
.73

83.
186.

~243

~479.
~473.
~466.

533.
.46
.42

830
1237

~9l4.
~889.
~860.
3126.
4213.
5912.

.45
~29.

75

.40
~18.
~16.
~11.

15
50
75

87
16
03
36
06
71

.40

00
00

00

26

09

62
39

88
17
46
06

62
32
86
77
63
74

6~7/8

~36.
~34.
~33.
~16.
~15.
~13.

~66.
~63.
~61.
.40
~36.
~33.

~38

~123

~116

~286.
~280.
~275.
~224.

~219

~551.
.83
.45

~543
~536

~463.

~456
~447

~1073
~1061.
~1051.
~948.

~938

55
80
15
80
25
35

33
86
53

21
52

.10
~119.
.30
~83.
~80.
~76.

60

60
50
70

52
99
77
07

.17
~213.

16

65

33

.40
.90

.05

98
55
14

.34
~926.

32

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

~34.
~33.
~31
~15
~14.
~13

~64.
~61.
~59.
~37
~34.
~32

~120.
~117.
~114.

~80.
~77.
~73.

~281.
~276.
~271.
~215.
~209.
~201.

~542.
~534.
~526.
~432.
~419.
~401.

—1049.
~1036.
~1025.

~821
~785
~733

75
17

.67
.94

57

.04

37
83
81

.00

88

.46

76
24
12

3
1
2
.9
5
9

155

Table 4~7 (a) (Continued)

6-1/4

~294118
~227273
~17857l
~6057
~4710
~3632

~294118
~227273
~178571
~6057
~4710
~3632

~294118
~227273
~178571
~6057
~4710
~3632

~294118
~227273
~178571
~6057
~4710
~3632

~294118
~227273
~l78571
~6057
~4710
~3632

~294118
~227273
~178571
~6057
~4710
~3632

0 - ~0.99

6-1/2

~25.
.00
.00
~24.
~24.
~24.

~25
~25

~49.
~49.
~49.
~49.

~49

~99

~249.

~249

~499.
~498.
~498.
~461.
.02
~439.

~452

~996

00

90
87
83

99
99
99
59

.47
~49.

32

.97
~99.
~99.
~98.
~97.
~97.

96
94
38
92
32

79

.73
~249.
~240.
~237.
~233.

65
O9
40
9O

15
90
60
87

50

.61
~995.
~994.
~858.
~824.
~784.

62
43
30
88
13

6-5/8

~25
~25
~25

~24.
~24.
~24.

~49.
~49.
~49.
~49.

~49

~49.

~99.
~99.

~99

~98.
~97.
~97.

~249.
~249.
~249.

~239

~236.
~232.

~499.

~498

~498.

~458

~446.
~431.

~996
~995

~994.
~834.
~787.
~724.

.00
.00
.00
90
87
80

99
99
99
59
.47
30

97
96
.94
35
88
25

79
73
65
.68
73
79

15
.90
60
.73
93
18

.60
.60
4O
90
70
70

6-3/4

~35.
~30.
~29.
.50
.00
.00

~19
~19
~15

~64.
.07
.07
~42.
~42.
~35.

~57
~57

~119

~110.
.70
.00
.00
.00

~109
~89
~88
~78

~272.
~265.
.23
~227.
~221.
~205.

~264

~517.
~515.
~511.
~448.
~428.
~403.

~1009
~1000.
~996.
~829.
~775.
~705.

00
00
50

14

93
93
86

.00

00

14
81

86
54
73

89
65
18
57
45
85

.49

00
84
24
48
91

6-7/8

~36.
~34.
~33.
~16.
~15.
~13.

~66.
.86
~61.
.40
~36.
~33.

~63

~38

~123.
~119.
~116.
~83.
~80.
~76.

~286.
~280.
~275.

~224

~551.
~543.

~536

~456

~447.

~1073
~1061.
~1051.

~948

55
80
15
80
25
35

33

53

21
52

10
60
30
60
50
70

52
99
77

.07
~219.
~213.

17
16

65
83

.45
~463.
.40

33

90

.05

98
55

.14
~938.
~926.

34
32

156

Table 4~7 (b)

Comparison of the Predicted Distribution with

The Actual Sampling Distribution (6 Test)

a - -0.5

T Actual 6-1/4 6-1/2 6-5/8 6~3/4 6-7/8
25 .010 ~4.81 -7.67 —4 19 -3.94 -4.00 -7.31
.025 -4.23 -6 74 —4.02 -3 63 —3.70 -6 96
.050 -3.78 -5.98 -3 83 -3.25 -3 30 -6.63
.950 -0.03 -1.10 —1 07 46.59 46.60 ~3.36
.975 0.21 -0.97 -0.95 61.34 61.60 -3 05
.990 0.50 ~0.85 -0.84 81.03 81.80 -2.67

50 .010 —5.19 ~7.67 —5.20 —4.07 —4.07 -9.38
.025 -4.61 —6.74 -4.88 —3.18 —3.17 -9.03
.050 —4.12 ~5.98 —4.56 -2.12 -2 07 -8 70
.950 —0.12 -1.10 -1.09 138.86 138.93 -5.43
.975 0.11 -0.97 —0.96 180.58 180.73 -5.12
.990 0.36 ~0.85 ~0.85 236.26 229.53 -4 74
100 .010 -5 47 -7 67 ~6.09 -1.50 —1.30 -12.31
.025 -4 82 -6.74 -5.59 1.00 1.10 -11.96
.050 ~4.32 -5.98 —5.13 4.00 4.20 —11.63
.950 -0.17 —1.10 -1 09 402.75 399.80 ~8.36
.975 0.03 —0.97 -0.97 520.75 510.60 -8.05
.990 0.29 -0 85 -o.85 678.25 655.50 -7 67

250 .010 ~5.68 ~7.67 ~6.90 17.79 18.59 ~18.12
.025 ~5.02 ~6.74 ~6.20 27.67 28.59 ~17.77
.050 ~4.46 ~5.98 ~5.59 39.53 40.29 ~17.44
.950 ~0.19 ~1.10 ~1.10 1615.73 1479.19 ~14.17
.975 0.04 ~0.97 ~0.97 2082.16 2106.29 ~13.86
.990 0.28 ~0.85 ~0.85 2704.74 2784.59 ~l3.48

500 .010 ~5.88 ~7.67 ~7.25 72.67 75.14 ~24.67
.025 ~5.11 ~6.74 ~6.45 100.62 102.14 ~24.32
.050 ~4.52 ~5.98 ~5.77 134.16 136.44 ~23.99
.950 ~0.21 ~1.10 ~1.10 4592.33 4636.14 ~20.72
.975 0.01 ~0.97 ~0.97 5911.60 5981.84 ~20.41
.990 0.25 ~0.85 ~0.85 7672.51 7767.94 ~20.03

1000 .010 ~5.88 ~7.67 ~7.45 237.17 244.18 ~33.93
.025 ~5.21 ~6.74 ~6.59 316.23 320.58 ~33.58
.050 ~4.58 ~5.98 ~5.87 411.10 417.58 ~33.25
.950 ~0.23 ~1.10 ~1.10 13020.68 13662.98 ~29.98
.975 0.00 ~0.97 ~0.97 16752.17 16883.98 ~29.67
.990 0.22 ~0.85 ~0.85 21732.75 22741.18 ~29.29

157

Table 4~7 (b) (Continued)

0 - ~0.8

T Actual 6=1/4 6-1/2 6-5/8 6~3/4 6-7/8
25 .010 -6 77 -19 17 -4.84 ~4.83 -4.90 -7 31
.025 -6.15 ~16 85 -4.79 ~4.78 —4 80 -6.96
.050 -5.67 -14.94 -4.74 -4.72 —4.70 ~6.63
.950 -1.54 -2.75 —2.41 3.25 3.30 -3.36
.975 ~1.28 -2.43 ~2.18 5.61 5.80 -3 05
.990 -1.00 -2.13 -1.96 8.76 8.80 —2.67

50 .010 -8 62 -19.17 -6 63 —6.59 —6.67 —9.38
.025 —8.12 —16.85 —6 52 —6.45 —6.47 —9.03
.050 —7.68 —14.94 ~6.39 -6.28 -6.27 —8.70
.950 ~3.62 -2 75 —2.56 16.28 16.03 —5.43
.975 -3 29 -2.43 -2.30 22.95 22.93 -5.12
.990 —2 93 —2.13 -2.04 31.86 31.83 —4.74
100 .010 —9.45 -19.17 —8.87 -8.64 -8 70 -12.31
.025 —8.85 —16.85 ~8.60 -8.24 -8.30 -11.96
.050 ~8.28 —14.94 ~8.31 ~7.76 -7.80 ~11.63
.950 —2.05 -2.75 —2.65 56.04 55.80 ~8.36
.975 -1.75 -2.43 -2.36 74.92 72.90 ~8.05
.990 —1.45 —2.13 —2.08 100.12 96.80 -7 67
250 .010 -12.04 -19 17 -12.20 -10.44 —1o.41 -18.12
.025 —11.16 -16.85 -11.53 —8.85 -8.81 -17.77
.050 -10.37 -14.94 —10 86 —6.96 —6.81 -17.44
.950 —2.20 —2 75 -2 71 245.24 243.39 -14.17
.975 -1 88 -2.43 -2 40 319.86 313.79 -13 86
.990 -1.59 -2.13 —2.11 419.48 369.19 -13.48
500 .010 -13.77 -19.17 —14.56 -7.16 ~7.61 ~24.67
.025 —12.62 ~16.85 —13.46 —2.68 ~2.56 -24.32
.050 -11 53 -14.94 —12.42 2.68 3.14 -23.99
.950 -2 27 -2 75 -2.73 715.99 711.34 —20.72
.975 -1 94 -2.43 -2.41 927.07 862.64 —20.41
.990 -1.63 —2.13 —2.12 1208.82 1210.14 -20.03

1000 .010 ~15.03 ~19.17 ~16.40 11.38 11.78 ~33.93
.025 ~13.58 ~16.85 ~l4.87 24.03 24.58 ~33.58
.050 ~12.33 ~14.94 ~13.51 39.21 40.47 ~33.25
.950 ~2.26 ~2.75 ~2.74 2056.75 2111.28 ~29.98
.975 ~l.93 ~2.43 ~2.42 2653.78 2667.48 ~29.67
.990 ~1.64 ~2.13 —2.13 3450.68 3542.58 ~29.29

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

—7
~6

.43
.79
~6.
-2.
-2.
~1.

31
53
24
93

.84
.32
.88
.39
.03
.63

.02
.49
.03
.08
.63
.18

.25
.64
.04
.67
.11
.63

.46
.54
.63
.86
.27
.66

.91
.37
.85
.00
.43
.89

158

Table 4~7 (b) (Continued)

6-1/4

~38.
~33.
~29.
~5.
~4.
-4.

~38
~33

~29.
~5.
~4.

—4

~38.
~33.
~29.
~5.
~4.
-4.

~38.
~33.
~29.
-5.
-4.
-4.

~38.
~33.
~29.
~5.
~4.
-4.

~38.

~33

~29.

-5
-4
—4

35
71
88
50
85
26

.35
.71
88
50
85
.26

35
71
88
50
85
26

35
71
88
50
85
26

35
71
88
50
85
26

35
.71
88
.50
.85
.26

0 - ~0.9

6—1/2

~14.
~14.
~13.

_5_

-4.

~19.
~18.

~17

_5_
-4.

-24

~23.

-21
-5
-4
-4

.96

.95

.93
.70
.48
.24

.95
.92
.88
.34
.00
.65

.68
.59
.48
.82
.37
.92

62
32
98
20
.64
12

32
64
.90
34
74
.19

.40
06
.72
.42
.80
.22

6-5/8

-14

49

~18.
.44

~17

~16.
.23
.00
.43

162
215
285

~20.

~17

490

.96
.95
.93
.94
.35
.56

.95
.92
.87
.23
.43
.66

.66
.56
.44
.51
.23
.53

.47
~14.
~13.
.45
68.
93.

07
60

11
01

56

10

87

.71
~13.

91

.47
639.
838.

73
95

6-3/4

-9.
.60
-9.
.50
.40
.90

~9

11
17

~14.
~14.
.61
49.
68.
.69

~13

93
~18

~17
~16

286

~20.
.62
.72

~17
~13

490.
650.
.72

877

.80
.60
.40
.50
.60
.40

.27
.97
.97
.07
.83
.23

70

50

51
11

39
39

.46
.46
.06
162.
215.
.54

14
24

62

30
72

6-7/8

~12.
~11.
~11.

-3.

~8
~7

~18.
~17.
~17.
.17
~13.
.48

~14

~13

~24.
~24.
~23.
~20.

~20

~20.

~33.
~33.
~33.
~29.
.67
.29

~29
~29

.31
.96
.63
.36
.05
.67

.38
.03
.70
.43
.12
.74

31
96
63
36

.05
.67

12
77
44

86

67
32
99
72

.41

03

93
58
25
98

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actual

_7_
~7.
~6.
~3.
~2.
~2.

-9.
-3.
~8.
-4_
-4.
_3_

~11.
~11.
~10.
~6.
~5.
~5.

~16.
~16.
~15.
~8.
~7.
-5.

~22.
~21.
~21.

-9.
.27
~7.

~8

~29.
~28.
~27.
-9.
_3_
~7.

71
10
63
09
84
55

32
83
39
65
32
90

85
35
96
32
84
28

84
35
91
28
51
77

27
66
13
19

37

56
70
81
90
75
63

159

Table 4~7 (b) (Continued)

6-1/4

~76.

~67

~59.
~11.
-9.

~8

~76.

~67

~59.
~11.
~9.
~8.

~76.

~67

~59.

~11

_9.
-8.

~76.

~67

~59.
~11.
-9.
-3.

~76.

~67

~59.
~11.

~9

~8.

~76.

~67

~59.

~11
~9
~8

70
.42
76
01
71
.52

7O
.42
76
01
71
52

70
.42
76
.01
71
52

70
.42
76
01
71
52

70
.42
76
01
.71
52

70
.42
76
.01
.71
.52

0 - ~0.95

6-1/2

~29

~28.
~27.

~10
~9
~8

.99
.99
.98
.55
.44
.31

.04
.03
.02
.95
.72
.44

.92
.89
.86
.40
.97
.49

.49
.39
.29
.03
.27
.50

.47
.22
.94
.88
.90
.96

.24
63
95
.40
.28
.23

6=5/8

£11411

~7.
~7.
~7.
-5.
.19
.64

-9.
.89
-9.
-5.
.69
~3.

~15

~15.
.26
.50
.17
11.

~15

~21

~28.
~28.
~27.

98.
136.
.02

186

.99
.99
.98
.48
.34
.14

04
03
02
61

92

86
87

12

.48

38

39

.41
~21.
~20.
23.
36.
54.

13
80
79
98
59

94
14
20
90
22

6-3/4

~6
~5
~5
~3

~7.
.80
.64
-5.
-4.
~3.

-7

~10

~10.
~10.

-5.
.20
.40

~15.
.41

~15

~15.
.49
.29
11.

~21

~21.
~20.
23.
37.
.34

55

~28.
~28.
~27.

98.
136.
186.

.49
.95
.88
.63
-3.
-2.

30
35

90

00

25
21

.40

10
00
70

51

31

79

.46

16
86
84
14

92
12
22
88
38
98

6-7/8

~7.
.96
-5.
-3.
-3.
-2_

-9.
.03
~8.
.43
-5.
.74

~9

~12.
~11.
~11.
~8.
-3.
.67

~18.
~17.
.44
~14.
~13.
.48

~17

~13

~24.
~24.
~23.
~20.
~20.
~20.

~33.
~33.
~33.
~29.
~29.
~29.

31

63
36
05
67

38

70

12

31
96
63
36
05

12
77

17
86

67
32
99
72
41
03

93
58
25
98
67
29

25

50

100

250

500

1000

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

.010
.025
.050
.950
.975
.990

Actua

~17.
~17.
~17.
~13.

~13

~12.

~24.
~24.
~23.
~19.

~19

~18.

~33.
~32.

~32
~26
~25

~24.

1

.92
.35
.86
.42
.18
.92

.69
.18
.78
.44
.18
.90

.43
.93
.56
.23
.96
.68

98
58
21
74
.40
95

33
90
55
54
.03
38

24
80
.43
.40
.43
06

160

Table 4~7 (b) (Continued)

6-1/4

~383

~337.
~298.
~55.
~48.
.62

~42

~383.
~337.
~298.
~55.
~48.
~42.

~383

~337.
~298.
~55.
~48.
~42.

~383
~337

~298.
~55.
~48.
~42.

~383

~337.
~298.
.03
~48.
~42.

~55

~383

.48

10
81
03
53

48
10
81
03
53
62

.48

10
81
03
53
62

.48
.10

81
03
53
62

.48

10
81

53
62

.48
~337.
~298.
~55.
~48.
~42.

10
81
03
53
62

~0.

99

6-1/2

~15.
~15.
~15.
~15.
~15.
~14.

~22.
~22.
~22.
.72
~20.
~19.

~20

~31.

~31
~31
~27
~26
~25

.00
.00
.00
.98
.97
.97

.07
.07
.07
.01
.00
.98

.00
.00
.99
.84
.79
.74

80
79
79
20
03
82

32
31
30

31
80

52

.49
.45
.42
.49
.40

6-5/8

~10

~15.
~15.
~15.
~15.
~14.

—14

~22.
~22.
~22.
~20.
~19.
~19.

~31.
.48
.45
.40
~24.
~22.

~31
~31
~26

.00
.00
.00
.98
.97
.97

.07
.07
.07
.01
.00
.97

.00
~10.
~9.
~9.

00
99

84
.79
~9.

73

80
79
79
16
97

.72

32
31
29
52
99
28

52

91
92

5-3/4

~17.
~16.
~16.
~14.
~13.
~12.

~23.
.01
.82

~23
~22

~19.
~19.
.92

~17

~31.
~31.
.49
.21
.58
.47

~31
~26
~24
~22

.00
.00
.90
.90
.80
.00

.07
.07
.07
.07
.07
.07

.85
.99
.97
.88
.96
.80

20
74
66
33
55
99

14

98
04

91
55

6-7/8

~12.

~11

~18.

~17
~17

~14.
~13.
.48

~13

~24.
.32
~23.
~20.

~24

~20

~33.
~33.
~33.
~29.

~29

.31
.96
.63
.36
.05
.67

.38
.03
.70
.43
.12
.74

31

.96
~11.
~8.
-3.
.67

63
36
05

17

.77
.44

17

86

67

99
72

.41
~20.

03

93
58
25
98

.67
~29.

29

 

CHAPTER 5

CHAPTER 5

CONCLUSION

This dissertation has developed two main topics. The first topic is the question
of testing the univariate properties of economic time series; that is, testing whether
they have a unit root or are trend stationary. Economically, the distinction between a
trend stationary process and a unit root process is important, because the latter implies
long run persistence (current shocks have permanent effects), while the former implies
trend reversion (current shocks have temporary effects). This distinction is also
important statistically, due to the ’spurious regression phenomena’ described by
Granger and Newbold (1974) and Phillips (1986). In other words, if there are
variables that are 1(1) processes and no cointegrating relationship exists among the
variables, standard regression inferences, based on the assumption that all variables are
(trend) stationary, are not valid any longer. For a further discussion see Park and
Phillips (1988, 1989).

There have been a lot of attempts to test the unit root hypothesis, including the
Dickey-Fuller (1979, 1981), the Phillips-Perron (1988), and the Schmidt-Phillips
(1991) unit root tests. Furthermore, the null hypothesis of a unit root has not been
rejected for many economic time series. Since the seminal paper by Nelson and

Plosser (1982), in which the null of a unit root is rejected for only two out of fourteen

161

162

series, many applied economists have tended to take views that most economic time
series are 1(1) processes.

However, it is well-known from a large body of Monte Carlo studies that
standard unit root tests are not very powerful against plausible trend stationarity
alternatives. Therefore, it is sensible to check (from the different direction) whether or
not the data are trend stationary in order to have a complete view of the properties of
economic time series. In Chapter 2 we derive an LM test for the null of trend
stationarity under very general assumptions about the stationary enors; that is, the
mixing and moment conditions of Phillips and Perron (1988). We use a components
model in which the series of interest is decomposed into the sum of the deterministic
trend, the random walk, and the stationary error. This test has a nonstandard limiting
distribution, which depends on a functional of a Brownian bridge. By a Monte Carlo
simulation we derive the critical values for the LM statistic for both level stationarity
and trend stationarity, and we consider finite sample (size and power) performance of
the test statistic in the presence of autocorrelated errors. We consider both AR(l) and
MA(l) errors. Generally, we find that there is a finite sample trade-off between size
and power of the stationarity test statistic, and that the choice of lags used in
calculating the long run variance has a major impact on the outcome of the test for all
reasonable sample sizes. We find that the use of shorter lags (e.g., l = 4 when T
=100) is suggested unless it appears that the stationary errors follow an AR(l) process
with a large positive parameter. Empirically, however, p 2 0.8 is very plausible since,

if we take most series to be trend stationary (which is the null), their first order

163

autocorrelations will often be in this range. According to our simulation results, the
use of longer lags (e.g., l = 12 when T =100) is needed for the test to avoid severe
size distortions in this case. This substantially reduces the power of the test. In our
empirical work, we attempt to compromise between the possible size distortions from
using 2 = 4 and the power loss from using 2 = 12 by picking 1 = 8. We find that the
null of trend stationarity is rejected only for five series (industrial production,
consumer prices, real wages, velocity, and stock prices) of the 14 series considered by
Nelson and Plosser.

Next we consider the consistency of the stationarity test under the unit root
alternative. In doing so, we derive the different limiting distribution of the test
statistic under the alternative of a unit root (the random walk component has positive
variance), which depends on detrended (or demeaned) Brownian motion. Therefore, in
Chapter 3 we consider the use of the KPSS statistic as a unit root test statistic.
Simulation results show that the main determinant of the finite sample size
performance of our unit root tests is the relative variance ratio A (variance of the
stationary error divided by variance of the random walk innovation) rather than the
autocorrelation of the stationary errors. Since the null is simply A > O, the exact
location of the null is important for the quality of inference. Generally, the use of
longer lags is preferred in terms of correct size when the process is nearly stationary,
but the use of no lags is preferred in terms of good power when the process is nearly
integrated, which again confirms the finite sample trade-off of the unit root test. We

have compared the finite sample performance of the KPSS unit root test with that of

164
the Dickey-Fuller test and found that the Dickey-Fuller test is more powerful, but also

has more size distortion. Since our main interest lies in the search for a more
powerful unit root test, we conclude that dual-use statistics are not very promising.
Finally, since the values of the long run variance under the null of a unit root
(normalized by W) are almost constant at all lags, we use the results for 2 = 0 (or E
= 1) for the unit root test in empirical applications. The main finding is that a unit
root is strongly rejected only for the unemployment series, which is almost the same
as Nelson and Plosser’s results.

Combining the empirical results for the Nelson-Plosser data from both
stationarity and unit root tests, we reach the following conclusions. Three series
(unemployment rate, GNP deflator, and money) appear to be trend stationary. Four
series (consumer prices, real wages, velocity, and stock prices) appear to have a unit
root. Two series (nominal GNP and bond yield) probably have unit roots, while two
more series (employment and real per capita GNP) are probably trend stationary. For
the nominal wage we cannot reject either the unit root or the trend stationarity
hypothesis. There are two interesting cases: industrial production and real GNP series.
For industrial production we can reject the trend stationarity hypothesi at the 5 % level
and the unit root hypothesis at the 10 % level. For real GNP we can reject the trend
stationarity hypothesis at the 10 % level and the unit root hypothesis at the 20 %
level. It seems that for many economic time series it is not very clear whether they
have a unit root or are trend stationary. Probably other alternatives such as fractional

integration or a nonlinear trend model are needed in further research.

 

165

The above empirical results may indicate that many economic time series could
be in the region of ’near stationarity’, in which the series have combinations of a
random walk component and a stationary component. Schwert (1989) also gives some
empirical evidence of near stationarity. The second t0pic of my thesis (Chapter 4) is
the study of the asymptotic and finite sample behavior of standard unit root tests such
as the Dickey-Fuller and the Schmidt-Phillips tests when the process is nearly
stationary. A lot of Monte Carlo studies including Schwert (1987, 1989) show that the [
size distortion of the Dickey-Fuller and the Phillips-Perron tests is almost one when
the process is nearly stationary. By using a local approximation of the MA(l)
parameter to minus one, we derive the asymptotic distribution of the Dickey-Fuller and
the Schmidt-Phillips unit root tests. We then examine their finite sample performance
when the process is nearly stationary by a Monte Carlo simulation. We find that
standard unit root tests have considerable size distortions when the process is nearly
stationary, because OLS estimation strongly biases the coefficient of the dependent
variable towards zero when the MA(l) parameter is near minus one. Our simulation
results show that this bias could be more severe in small samples, but also
considerable even for large sample sizes. Furthermore, our asymptotic results predict
the extent of the finite sample size distortions quite accurately.

Recently, various attempts have been made to reconsider the important problem
of distinguishing trend stationary and unit root processes. In particular, Perron (1989)
has suggested that a time series structure with very infrequent changes in slope or

intercept can be a useful approximation in empirical applications and argued that the

166

conclusion of Nelson and Plosser can be reversed if one time structural break is
allowed within the Dickey-Fuller test framework. On the other hand, Amsler and Lee
(1991) apply the same logic to the Schmidt-Phillips test and find that a suitably
modified Schmidt-Phillips test allowing for a structural change reverses the results of
Perron (1989). However, the implicit assumption in these studies is that there is only
one such "big shock", which is given exogenously. Banerjee M (1990) and Zivot
and Andrews (1990) criticize this assumption and develop a framework that
endogenizes the structural change. Even more general models can be entertained if we
adopt nonlinear structural models suggested by Hamilton (1989) and Lam (1990).
However, if we want to test for a more general type of trend specification, the testing
procedure for trend stationarity (proposed in this dissertation) is also relevant, and the
extension of the analyses of Banerjee ml. and Zivot and Andrews to the stationarity
test is an interesting path for future research.

Bayesian methods can also be used to distinguish trend stationary and unit root
processes. Dejong and Whiteman (1991) use a Monte Carlo based Bayesian
methodology and study the posterior distributions of dominant roots corresponding to
an AR(3) representation of the time series using flat (uninformative) priors. They find
that eleven of fourteen Nelson-Plosser series support trend stationarity over integration.
Similar results are obtained by Sims and Uhlig (1991). Phillips (1991) criticizes this
inference, claiming that the results are sensitive to the model and prior distribution.
Generally, he challenges the assertions (made by Bayesian econometricians such as

Sims (1988) and Sims and Uhlig (1991)) about the impropriety of classical methods

167

and the superiority of flat prior Bayesian methods. He employs ignorance (objective)
priors rather than flat priors, because under flat priors the bias towards the trend
stationary model is shown to be substantial, and especially in models with a fitted
deterministic trend. He finds under ignorance priors that Bayesian inference accords
more closely with the results of classical methods and that seven of the 14 Nelson-
Plosser series show evidence of stochastic trends.

Generally, the conclusion that many economic time series are not very
informative about stationarity is quite consistent with the above (inconclusive)
findings. Therefore, this leaves a lot of rooms for the further research.

First, it is interesting to see what happens to the empirical results of the
stationarity test if allowance is made for the presence of structural change. A simple
extension is straightforward, but adjusting for endogeneity of the structural break,
considering Hamilton’s nonlinear trend function, or combining our analysis with
Bayesian methods is more complicated.

Another second promising extension of this thesis is to a test of cointegration.
This would involve extending the test of stationarity to the error in a cointegrating
regression, instead of an observed series. This would be one of the first attempts of a
direc__t test of cointegration rather than a test of the hypothesis of no cointegration
(which is a direct extension of the unit root test). See Engle and Granger ( 1987),
Phillips and Ouliaris (1990), Stock and Watson (1988), and Johansen (1991). We
expect such a test of cointegration to give further light on the true relationships among

important economic relationship such as the consumption function, purchasing power

168

parity, the term structure of interest rates, the quantity theory of money, and so forth.
The asymptotic theory for the case that the generalized cointegrating vector is

efficiently estimated must be derived and we are currently working on this topic.

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169

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