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This is to certify that the
dissertation entitled

Unit Root Tests In the Presence of
Autocorrelated Errors and Structural Change

presented by

Junsoo Lee

has been accepted towards fulfillment
of the requirements for

Ph.D. . Economics
degree in

 

 

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MSU Is An Affirmative Action/Equal Opportunity institution '
ciieireMprna-DJ

 

 

BY

Junsoo Lee

A DISSERTATION

Submitted to

Michigan State University
in partial fulfillment of

the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1991

 

ABSTRACT

UNIT ROOT TESTS IN THE PRESENCE OF
AUTOCORRELATED ERRORS AND STRUCTURAL CHANGE

BY

Junsoo Lee

Whether observed series contains a unit root or not has

profound implications in macroeconomics. While the existing

unit root tests have mostly been based on one variant or
another of the Dickey-Fuller (1979, hereafter DF) tests,
they do not survive in the presence of strongly
autocorrelated errors in terms of both size and power of the

tests. Thus, a recent survey paper by Campbell and Perron

(1991) suggests the necessity of finding new tests which

alleviate the size distortion problem, while retaining good

power properties.
This dissertation suggests new IV tests which have

surprisingly small size distortions in the presence of.

strongly autocorrelated errors, and are more powerful than

Therefore, this remarkable

other tests of similar size.
performance of the new IV tests may well provide the

solution to the necessity of finding robust tests.
This dissertation shows that their improved performance
is driven from adapting an appropriate testing procedure.

The new IV tests are based on the Schmidt-Phillips (1990,

hereafter SP) tests, not variants of the DF tests. Thus

this suggests that there are perceivable advantages to

 

operating in the SP framework rather than in the DF

framework.
The most distinguished difference between the DF and SP

framework lies in the parameterization. It has been argued

by SP that the DF parameterization is "clumsy" in the sense

that it handles exogenous variables in a potentially

confusing ways. This argument has been made more visible

from in-depth discussion (in Chapter 5) on the

parameterization issue. The problem of the clumsy DF

parameterization gets more serious especially in the

presence of exogenous structural break. Specifically, a

suitably modified SP tests allowing for a structural change
reverse the results of Perron (1989) who finds that most of

the Nelson-Plosser series are trend stationary if allowance

is made for a structural break. We show that the contrary

results of Perron arise not from the presence of an

exogenous structural break in the DGP, but from the way in

which the DF tests allow for such a break. In this

dissertation, some new theoretical findings on the
asymptotic behavior of the SP and DF tests in the presence

of structural break are also provided to clarify this issue.

 

Dedicated to my parents

iv

 

ACKNOWLEDGEMENTS

First of all, I would like to thank my dissertation

chairman, Peter Schmidt for his excellent and dedicated
supervision throughout the preparation of this dissertation.
I am especially grateful for his kindness and intimate
guidance.

I wish to thank the other dissertation committee

members, Richard T. Baillie, Robert H. Rasche and Christine

Amsler for helpful comments. I also want to thank Peter

Schmidt, Daniel Hamermesh and Richard T. Baillie for
supporting me as a research assistant during summers. Peter
Schmidt kindly allows me to use the joint works as Chapters
3 and 4 of this dissertation, and has suggested many ideas

developed in Chapter 5 which is a joint work with Christine

Amsler.
My best thanks must go to my parents for their

invaluable support. I also wish to express my another
greatest thanks to my wife, Eunjoo and my beloved two sons,

Hosung and Justin for their encouragement and patience.

Finally I am grateful to my brothers and sister for their

help.

 

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES ... x

CHAPTER 1 : INTRODUCTION ...
......OOOOOOO. 11

LIST OF REFERENCES ......OOOOOOOO

CE OF SCHMIDT-PHILLIPS

CHAPTER 2 : FINITE SAMPLE PERFORMAN
UNIT ROOT TESTS IN THE PRESENCE OF

AUTOCORRELATION
13

1. INTRODUCTION ........................
2. TEST STATISTICS ........................... 13
................. 16

17

3. EXPERIMENT DESIGN ........
4. SIMULATION RESULTS ......OOOOO
......OOOOOOOOOOOOO 22

5. SUMMARY AND CONCLUSIONS
33

A MODIFICATION OF THE SCHMIDT-PHILLIPS UNIT

ROOT TEST
35

36

CHAPTER 3 :

1. INTRODUCTION ...... .......
2. THE NEW TESTS AND THEIR DISTRIBUTIONS .....
39

3. POWER OF THE TESTS eooeeeeooeeeeeeeeeoeeooo
4- CONCLUDING REMARKS Deco-eeeoeooeeeeeeeeo 42
47

50

APPENDIX ...
LIST OF REFERENCES

UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLES

CHAPTER 4 :
ESTIMATION

10 INTRODUCTION oeooeeeeoeeeeeeeeeeoooeoeo 51
eeeeeeeeeoeoeoooeeooeeeoee 54
62

2. THE NEW IV TESTS

3. PERFORMANCE OF IV TESTS 0.0000000000000000.

4. CONCLUSION ............ 71
78

81

APPENDIX ..............
LIST OF REFERENCES eoa.ooeooooeooooeooeoooeoo...

vi

CHAPTER 5 : STRUCTURAL CHANGE AND PARAMETERIZATION
PROBLEMS IN UNIT ROOT TESTS

INTRODUCTION

ocoo0.0000000000000000000000o.

1.
THEORETICAL RESULTS

2.
3. EMPIRICAL RESULTS
COMMON FACTOR RESTRICTIONS

4.
CONCLUDING REMARKS ........................

5.

APPENDIX ........
LIST OF REFERENCES

CHAPTER 6 :
LIST OF REFERENCES

......OOOOOOOO.

0.0.0..........OOOOOOOOOOOOOO

CONCLUSION O.......OOOOOOIOOCOOOOOOOI....

00............OOOOOOOOOOOOOOO

vii

83
85
93
95
104

110
126
128
132

 

CHAPTER 2

Table 1

Table 2

Table 3
Table 4
Table 5
Table 6
Table 7

Table 8

Table 9

CHAPTER 3

Table 1

Table 2

LIST OF TABLES

5% Sizes of Schm
and Dickey-Fulle

Errors ........

5% Sizes of Schm
and Dickey—Fulle
Errors 00......

5% Sizes of Corr
t-tests Under MA

5% Sizes of Corr
Coefficient Test

5% Sizes of Corr
Augmented Tests

5% Powers of Tes
(T=100, p=.9) ..

5% Powers of Tes
(T=100, p=.9) ..
5% Sizes of Test
Variances and Bi
Variances Under
5% Sizes of Test

Variances and Bi
Variances Under

CRITICAL VALUES

SIZE AND POWER,

idt-Phillips Tests
r Tests Under MA

00......

idt—Phillips Tests
r Tests Under

ected and Augmented
Errors

ected and Augmented
5 Under MA Errors

ected Z Tests and
under AR Errors

ts Under AR Errors

s Using True Error

as of Est
MA Errors (T=100)

5 Using True Error
as of Estima
AR Errors

000000....

imates of Error

tes of Error
(T=100) 000......

FOR ?’ zand§000000000000

5% LOWER TAIL TESTS

viii

24

25

26

27

28

29

30

31

32

44

45

CHAPTER 4
Table 1

Table 2

Table 3

Table 4

CHAPTER 5

Table 1

Table 2

Table 3

Table 4

Critical

5% Sizes
Presence
Critical

5% Sizes
Presence
Critical

values 0000000000000000 00000000000

of the IV tests In the
of MA errors (Asymptotic

Values Used)

of the IV tests In the
of MA errors (Finite Sample

Values Used)

5% Powers of the IV tests (B=0.9)
In the Presence of MA errors .............

Critical Values of the Schmidt-Phillips
Tests With Structural Change ..............

Schmidt-Phillips Tests for a Unit Root
With the Structural Change ...............

Critical Values of the Wald test ......,..

Results of Wald Test for a Unit Root
With the CFR 0.000000000000000000000..0...

ix

74

‘75

76

77

106

107

108

109

LIST OF FIGURES

CHAPTER 3

FIGURE 1 Power of Various Tests, T=100, B=0.9 ..... 46

CHAPTER 1

CHAPTER 1

INTRODUCTION

Since the pioneering work of Nelson and Plosser (1982),
which found that most U.S. macroeconomic time series are
nonstationary, there has been a vast amount of work done on
the macroeconomic and statistical implications of unit
roots, and on the question of how to test whether observed
series contains a unit root. From a macroeconomic point of
View, unit roots imply long-run persistence, in the sense
that the effects of random shocks on macroeconomic variables
are not just temporary. Many economists have argued that
theories designed to explain the properties of economic time
series with unit roots should necessarily be different from
theories designed to explain the properties of stationary
time series. From a statistical (or econometric) point of
View, correct treatment of the stationary or nonstationary
nature of the data is necessary for meaningful statistical
inference, owing to the spurious regression phenomenon
pointed out by Granger and Newbold (1974) and Phillips
(1986). Thus, for both economic and statistical reasons, it
is important to be able to distinguish a series with a unit
root from a stationary series. A considerable elaboration
of this point, along with a general survey of the unit root
literature, is given by Campbell and Perron (1991). See

1

2
also Diebold and Nerlove (1990).

Efforts to distinguish a unit root from stationarity
have most often taken the form of the test of the null
hypothesis of a unit root against the alternative of
stationarity. While there is an enormous literature on the
problem of testing for a unit root, most of the existing
unit roots tests are variants of the Dickey and Fuller
(hereafter, DF) tests provided by Fuller (1976) and Dickey
and Fuller (1979). The DF unit root tests are based on the

following regressions:

(1) y} = Byv1-+ at
(2) Yt
(3) Yt

a + ByH + et

a + fiyt“1 + 6t + 6t

for t = 1,...,T. In each case, the unit root hypothesis is
B = 1. Two types of test statistics are defined from each
regression equation. One is the normalized coefficient test
statistic T(§—1), where 5 is the OLS estimate of 3: this
yields the DF statistics 2, 21 and.3, from regressions (l),
(2) and (3), respectively. The other type of statistic is
the usual t-statistic for testing the hypothesis B = 1,
which yields the DF statistics ?, "F“ and ’f, corresponding to
the same three regressions.

The BF regressions differ in what they assume (or
allow) with respect to level and deterministic trend. The

regression (1) does not allow for non-zero level or trend

3

under the alternative, though the initial value (Yb) is
effectively the level of the series under the null. The
regression (2) allows for linear deterministic trend (drift)
under the null, since the solution for'y} contains the
deterministic trend term at, but it allows only for non-zero
level under the alternative, since then y is stationary
around the value a/(l-B). As a result, tests derived from
regression (2) are inconsistent against trend-stationary
alternatives; see West (1987). Similarly, regression (3)
allows for non—zero level and trend under the null, even
when 6 = 0, but implies nonlinear trend under the null when
6 f 0; under the alternative 6 f 0 simply allows for linear
trend.

This thesis will largely be concerned with extensions
of an alternative set of unit root tests suggested by
Schmidt and Phillips (1990, hereafter, SP). Their tests are

based on the parameterization
(4) yt = u. + at + xt, xt = fix” + at,

which was also used by Bhargava (1986). The unit root
corresponds to 8 = 1. SP criticize the DF parameterization
as "clumsy", because the meaning of the parameters a and 6
in (2) and (3) is different under the null and the
alternative. This type of difficulty does not arise in the
SP parameterization. The parameterization allows for non-

zero level and trend under both the null hypothesis and the

4
alternative hypothesis; u and 5 in (4) always represent
level and deterministic trend respectively, independently of
whether 6 = 1 or not. Since the distributions of the
Schmidt-Phillips tests do not depend on the nuisance
parameters P and 5, the test statistics require just one
tabulation for each statistic independently of level or time
trend. This is in contrast to the DF tests, since the
distributions of 3“ and.?“ under the null depend on the
‘values of a in (2), while the distributions of 3, and ?1
under the null depend on the values of 6 in (3).

The Schmidt-Phillips test statistics are derived from
the LM test of the hypothesis 6 = 1 in equation (4). The
restricted MLE's of 5 and ¢x(= 11’ + X0) are given by: E =
(yT - y1) / (T-l), and ix = y1 - 2. Then the test statistics

are given from the following regression:

(5) Ay} = constant + cpétfl1 + error

where Sp1 = yt“1 - ix - E(t-1). The test statistics are

defined by:

(6) 5 Ti.
(7) t-statistic for the hypothesis ¢=0.

*2
II

Since ASt== Ayt'- E, the 5 and ? tests are not affected

if we make the dependent variable in (5) ASt instead of Ayt;

that is, we can also derive 5 and i from the regression

(3) ASt== constant' + ¢Sb1 + error.

In Chapter 3, we show that the population intercept
(constant') in (8) equals zero, and we consider the modified
SP tests that set the intercept to zero in running
regression (8). We derive asymptotic distributions for
these modified tests, and we compare their powers in finite
samples to the powers of the DF and SP tests.

Chapters 2 and 4 deal with the problem of
autocorrelated errors in the DF or SP regressions. Since
the tabulated distributions for the test statistics assume
the errors in the model are i.i.d., they are affected by the
presence of autocorrelated errors even asymptotically. This
problem is commonly dealt with in two ways.

First, the so-called augmented Dickey-Fuller (hereafter
ADF) tests accommodate error autocorrelation by adding
lagged differences in the variable to the regression. Said
and Dickey (1984, 1985) showed that, if the number of lagged
differences is suitably chosen, the ADF statistics have the
same asymptotic distribution as the original statistics
would have under iid errors, so the usual tabulations apply,
at least asymptotically. We consider similarly augmented
versions of the SP tests, in which lagged values of AS (or
Ay) are added to the regression (5). We presume, though we
do not present the proof, that the result equivalent to the
Said-Dickey result holds, so that asymptotically the usual

SP tabulations are correct.

Second, Phillips (1987) and Phillips and Perron (1988)

6

derive the asymptotic distributions of the DF statistics
under assumptions that allow for a wide class of weakly
dependent and heterogeneous errors, and they provide
transformed (corrected) versions of the statistics that have
the same asymptotic distribution as the original DF
statistics would have under iid errors. Thus, for these so-
called Phillips-Perron (hereafter PP) statistics, the usual
DF tabulations apply, at least asymptotically. SP provide
the corresponding transformed versions of the SP tests.

While the ADF tests and the PP tests are valid
asymptotically, they do not necessarily perform very well in
finite samples of reasonable size. Simulation evidence
presented by Godfrey and Tremayne (1988), Phillips and
Perron (1988), Schwert (1989) and Kim and Schmidt (1990) has
shown that the uncorrected DF tests reject the null
hypothesis (when it is true) too often in the presence of
negative autocorrelation and too seldom in the presence of
positive autocorrelation. These size distortions can be
quite considerable. The PP corrected tests perform somewhat
better than the uncorrected tests, but still suffer from
considerable size distortions even for surprisingly large
sample sizes. The ADF tests also perform somewhat better
than the uncorrected tests, with the extent of the
improvement depending on the number of augmentations. With
a small number of augmentations, the behavior of the ADF

tests is similar to the behavior of the PP tests noted

7
above. 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 simulation results in
Chapter 2 indicate that the augmented tests with many
augmentations have almost no power.

In Chapter 2 of this thesis, we present simulation
results for the SP tests that are similar to the simulation
results that others have presented for the DP tests. The
performance of the augmented and corrected SP tests is not
very different from the performance of the augmented and
corrected DF tests, as summarized above. This is a
discouraging result.

Therefore it seems that no tests survive in the
presence of strongly autocorrelated errors in terms of bggh
size and power of the tests. The tests with correct size
have poor power and the tests with high power have serious
size distortions. This problem is also discussed in the
recent survey paper on the unit root literature by Campbell
and Perron (1991). They assert that none of the variants of
the DF tests seems to solve the problem, and suggest the
necessity of finding new tests which modify the PP procedure
in such a way as to alleviate the size problem, while

retaining good power properties.

In Chapter 4, we consider unit root tests based on

8

instrumental variables (IV) estimation. Hall (1989) and
Pantula and Hall (1991) have considered IV tests that are
based on IV estimation of the DF regressions (1) - (3).
They assume moving average (MA) errors, and the instrument
for yM is ybk, where k is chosen to be at least as large as
one plus the order of the MA process. These IV tests are
valid asymptotically in the presence of autocorrelation of
MA form. We propose similar IV versions of the SP
statistics, where the instrument for St, in (5) is Sbk, and
where again k is chosen to be at least as large as one plus
the order of the MA process for the errors. Hall's
simulation results in finite samples indicate size
distortions that are less than those of the ADF or PP tests,
though they are still substantial. Our simulation results
are more optimistic. IV versions of the SP tests have
surprisingly smaller size distortions, and they are more
powerful than other tests of similar size. The new IV tests
therefore show that it is still possible to improve the
performance of the unit root tests when the appropriate
testing procedure is used, and this improvement seems to be
implicitly related to the parameterization issue discussed
above.

Despite the difference in parameterization between the
DF and SP tests, however, in practice their implications
have not been very different. Specifically, both the DF and

SP tests have given the similar conclusions when they are

9
applied to the Nelson-Plosser data. Both tests, like a
large number of other tests, have failed to reject the null
hypothesis of a unit root in many macroeconomic time series.
A contrary conclusion is asserted by Perron (1989), who
finds most of the US macroeconomic series to be trend-
stationary if allowance is made for a structural break at
the time of the Great Depression. He shows that the DF
tests are biased toward accepting the null when a structural
break occurs in the data, and his modified tests which allow
for such a change reject the null hypothesis of a unit root
for eleven out of the fourteen Nelson-Plosser time series.

In Chapter 5, we consider Perron's analysis in the SP
setting. We argue that the clumsy way in which the DF
parameterization handles explanatory variables, like
intercept and trend, becomes more serious as we add
additional explanatory variables, like a structural break
dummy. Using modified versions of the SP tests which allow
for the same structural break reverses Perron's results.

Thus, the distinction between the two parameterizations
becomes more important when additional exogenous variables
are considered. It is shown that though both SP and DF
tests are affected by the structural break if the
alternative hypothesis of stationarity is true, the
asymptotic distributions of the SP tests are not affected if
the null of a unit root is true. Chapter 5 deals with this

parameterization issue in depth, and provides theoretical

10

and empirical results. These results support the argument

of SP that the DF parameterization is "clumsy" in the sense

that it handles exogenous variables in a potentially

confusing way.
The structure of the dissertation is as follows.

Chapter 2 investigates the finite performance of the SP

tests in the presence of autocorrelation. Chapter 3

considers modified SP tests based on the fact that the

constant term in regression (5) equals zero in the

population. Chapter 4 provides new IV tests based on the SP

framework, and shows their remarkable performance in the

presence of autocorrelated errors. Chapter 5 examines the

parameterization issue closely, with new theoretical

findings on the asymptotic behavior of the SP and DF tests

in the structural break model and with an empirical

application to the Nelson-Plosser data. Chapter 6 gives our

concluding remarks.

11

LIST 0? REFERENCES

Bhargava, A. (1986), "On the Theory of Testing for Unit
Roots in Observed Time Series," Rexigg_gf_§ggngmig

figggigg, 52, 369-384.

Campbell, J.Y. and P. Perron (1991), "P1tfalls and
Opportunities: What Macroeconomists should know about

Unit Roots", unpublished manuscript, prepared for NBER

Macroeconomic Conference.
"Distribution of the

Dickey, D.A. and W.A. Fuller (1979),
Estimators for Autoregressive Time Series with a Un1t
Root," Jounnal of the Anerican Statistical Association,

74, 427-431.
Diebold, F.X. and M. Nerlove (1990), "Unit Roots in Economic
Time Series: ASelective Survey," in Adynnge§_in

Esenemetriss. A8. 3- 69-
t ' t'c

Fuller, W.A. (1976), t o 0
Series, New York: Wiley.

Godfrey, L G. and A.R. Tremayne (1988), "On the Finite

Sample Performance of Tests for Un1t Roots,"
unpublished manuscript, University of York, England

Granger, C.W.J. and P. Newbold (1974), "Spurious Regressions
in Econometrics", Journal of Eggngnetnigs, 2, 111-120.
Hall, A. (1989), "Testing for a Unit Root in the Presence of
Moving Average Errors," fiiomgtnika, 76, 49-56.
"Some Evidence on the

Kim, K. and P. Schmidt (1990),
Accuracy of Phillips-Perron Tests Using Alternative
Estimates of Nuisance Parameters," Eggngnig§_ngngg;§,
34, 345-350.

Nelson, C R. and C.I. Plosser (1982), "Trends versus Random

' Some Evidence and

walks in Macroeconomic T1me Series:
Implications," i2urnel_2f_neneterx_Egengmis§. 10. 139-

162.
Hall (1991), "Testing for Unit Roots in

Pantula, S.G. and A.
Autoregressive Moving Average Models: An Instrumental
Variable Approach." Journsl_2f_Es2nemetris§. 48. 325-

353.
"The Great Crash, The Oil Price

Perron, Pierre (1989),

12
Shock, and The Unit Root Hypothesis," Eggngngnzign, 57,

1361-1401.
Phillips, P.C.B. (1986), "Understanding Spurious
Regression." leurnal_2f_322n2metriss. 33. 311-340-

Phillips, P.C.B. (1987), "Time Series Regression with Unit
Roots," Esenemstrisa. 55. 277-301-

Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit
Root in Time Series Regression," Bigngnnikg, 75, 335-
346.

Said, S.E. and D.A. Dickey (1984), "Testing for Unit Roots
in Autoregressive - Moving Average Models of Unknown

Order," Bignetrikg, 599-608.

Said, S.E. and D.A. Dickey (1985), "Hypothesis Testing in
ARIMA(p,1,q) MOdels," gounnal of the Amenignn
Stgtistigai Associatign, 80, 369-374.

Schmidt, P. and P.C.B. Phillips (1990), "Testing for a Unit
Root in the Presence of Deterministic Trends,"
Econometrics and Economic Theory Workshop Paper No.
8904, Department of Economics, Michigan State

University.
Schwert, G.W. (1989), "Tests for Unit Roots: A Monte Carlo
Investigation," gournni gt Business and Egongnig

§£§£i§£12§1 7, 147’159-

West, K.D. (1987), "A Note on the Power of Least Squares
Tests for a Unit Root," Egononigs Lentens, 24, 249-252.

CHAPTER 2

CHAPTER 2

PINITE SAMPLE PERFORMANCE OP
SCHMIDT-PHILLIPS UNIT ROOT TESTS
IN THE PRESENCE OP AUTOCORRELATION

1. INTRODUCTION

This chapter considers a new unit root test proposed
recently by Schmidt and Phillips (1990) (hereafter, SP).
The tabulated distributions for their test statistics assume
that the errors in their model are iid. The SP tests can be
made asymptotically robust to autocorrelation by a Phillips-
Perron (hereafter, PP) type correction of Phillips and
Perron (1988), or by augmenting the regression that yields
the test statistics like the augmented Dickey-Fuller tests
(hereafter, ADF) of Said and Dickey (1984, 1985). This
chapter investigates the finite sample performance of these
extended versions of the SP tests. It is basically found
that the corrected and augmented SP tests behave very much
like the corresponding PP-corrected and ADF tests, in terms

of size distortions and power under autocorrelated errors.

2. TEST STATISTICS

The SP tests employ the parameterization:

(1) yt = ¢ + gt + Xv Xt = fixt,1 + 5::

The unit root corresponds to B = 1. As discussed in Chapter

13

l4

1, this parameterization is different from the
parameterization of Dickey and Fuller (1979, hereafter DF)
tests. It consistently allows for non-zero level and trend
under both the null hypothesis and the alternative
hypothesis: 5 and 5 always represent level and deterministic
trend respectively, independently of whether 6 = 1 or not.

The SP test statistics are derived from the LM test of
the hypothesis 6 = l in the equation (1). The restricted
MLE's of e and ¢x(== (5 + X0) are given by E = (y, - y1)/(T-l),
and $X:= y,-- E. Then the test statistics are derived from

the following regression:
(2) Ay} = constant + ¢St4 + error

where St.1 = yr”1 - 5x - §(t-l). The test statistics are

defined by:

(3)
(4)

Té
t-statistic for the hypothesis o s 0.

‘bt
ll

'4:
I!

The corrected test statistics 2(5) and Z(f) are
designed to allow for a wide class of weakly dependent and
heterogeneous errors by employing correction term which is
the ratio of the estimates for the innovation variance of
and the long-run variance 02, respectively. Specifically,

define w2 = a 2/a2 and

(5) 2(a) = 2/82

(6) Z(?)

ll
~41
\
8 >

15

where 82 --= 33/32. A choice of the truncation lag parameter
a is required for 32; see Phillips (1987) or Schmidt and
Phillips (1990) for details.

The augmented versions of the test statistics are
derived from the following regression, which includes lagged
differences of g: in the regressors:

P
2

(7) Ay} = constant + ¢SM+j

61.68:. . + error.

1 J

The coefficient test statistic Aug(5) is defined as T5, and
the statistic Aug(i) is the t-statistic for ¢ = 0, where
both are based on regression (7). Note that including Aybj
instead of Aérq will lead to the same statistics, since

~

astj = AYS) + E and E is absorbed into the intercept. To
capture the autocorrelation structure of the errors, it is
required to choose the number of augmentations properly. If
the error (at) is AR(p), then p augmentations are necessary
in (7) for the t-test to be asymptotically valid. If the
error is MA or ARMA, validity of the test requires that the
number of augmentations rise with sample size, though more
slowly than sample size; for example, p can be taken to be
proportional to TV‘.

For comparison, our simulations also consider the DF

test statistics 5, and ?,, the PP test statistics 2(5,) and

Z(?,) and the ADF test statistics Aug(5,) and Aug(?,).

16
3. EXPERIMENT DESIGN
To examine the sensitivity of the test statistics to
autocorrelated errors, the data generating process will
allow AR(l) disturbances as well as MA(1) disturbances.

Thus we consider the ARMA(1,1) error process.

(8) e = 56M + ut + out,

where ut are iid with mean zero and variance of and [5] <
1, Isl < 1. The parameters 15, 5 and initial value x0 are
assumed to be zero, whereas we assume qf = l. Schmidt and
Phillips (1990) and DeJong §£_§l- (1989) showed that the
distributions of the test statistics under the null
hypothesis are independent of 6. :5, x0 and of. Under the
alternative, X0" = XO/Ou is relevant but the other parameters
are still irrelevant. When the autoregressive (AR)
parameter p is zero, the data will follow an ARIMA(0,1,1)
process. Then the construction of the data is exactly the
same as in Schwert (1987, 1989), Phillips and Perron (1988),
and Hall (1989). Schwert (1987) discusses the reasons to
believe that an economic time series contains MA
disturbances, rather than being a pure AR process.

Nicholls, Pagan and Terrell (1975) have provided a survey on
the use of moving average errors in modelling of economic
time series. On the other hand, when the moving average
(MA) parameter 0 is zero, the data will follow an AR(l)

process with autoregressive errors, as considered also by

17
Kim and Schmidt (1990). In the simulations, process (8) is
initialized by drawing an initial value of u, setting the
initial value of 6 equal to the initial value of u, and then
drawing and discarding 20 values of u and e (to remove
possible effects of these initial values) before drawing the
values that are actually used.

All the simulation results are calculated using 20,000
replications, except that 10,000 replications are used for
the simulations with sample sizes equal to or greater than
500. A 95% confidence interval around a rejection rate of
.05 is calculated as approximately [.047,.053] with 20,000
replications and [.046,.054] with 10,000 replications. The
random numbers were generated using the subroutine
GASDEV/RAN3 of Press, Flannery, Teukolsky and Vetterling

(1986).

4. SIMULATION RESULTS

The first simulation is performed on the basic SP test
statistics 5 and f and the DF tests statistics}?f and ?,,
none of which correct for the autocorrelation. Table 1
shows the 5% sizes of these tests (i.e., the proportions of
rejections using 5% critical values) under MA errors. Of
course, no size distortion is found when the MA parameter 0
is zero. But when 0 is negative, the sizes are close to
one, which implies too many rejections of the (true) null

hypothesis of a unit root. It is a curiosity that the

18
problem becomes more severe as the sample size grows.
Conversely, when 0 is positive, the sizes are less than the
nominal size 5%, resulting in too few rejections of the null
hypothesis. The same patterns are found in Table 2 for the
case of AR errors. These results for the DF tests 5,.and ?,
are already known, from the results of Schwert (1989),
Phillips and Perron (1988) and Kim and Schmidt (1990). The
results in Tables 1 and 2 indicate that the severity of the
problem is about the same for the SP test statistics 5 and f
as for the DF test statistics 5, and ?,.

The next question is whether there is any significant
improvement from using the corrected and the augmented
versions of the tests. These simulations are executed for
values of -.8, -.5, .0, .5 and .8 for each of the parameters
0 and p, and for sample sizes 50, 100, 500 and 1000. The
number of truncation lags (a) used for estimating the long-
run error variances was chosen in the same manner as Schwert
(1989). That is: 24 = int[4(T/100)""] and :12 2-
int[12(T/100)”%], where 'int' stands for integer. The
values of 24 are 3, 4, 6 and 7, and the values of £12 are
10, 12, 18 and 21 for T = 50, 100, 500 and 1000
respectively. Tables 3 and 4 present the simulation results
along with those for the PP tests, quoted from Schwert for
comparison. It is interesting how similar the results for
the SP tests are to the results for the corresponding PP

tests. The sizes of the corrected SP tests Z(?) and 2(5)

19

are very close to the sizes of the PP tests Z(?,) and 2(5,).
This similarity is consistent for each set of values of 9, T
and 2. Comparing the sizes of 2(?) and 2(5) with the sizes
of f and 5, which do not use correction terms, we can see a
small improvement. For example, when T = 100 and 0 = -.5,
the sizes of the statistics 2(?) and 2(5) are .713 and .708,
which are slightly less than the sizes of i and 5, .796 and
.800 respectively. The improvement is larger for larger
sample size. For example, when T = 1000 and 0 = -.5, the
sizes of 2(5) and 2(5) are .433 and .428, which are much
smaller than the sizes of f and 5, .860 and .861
respectively. Such an improvement can be found in the PP
tests as well. As a whole, we can conclude that the
corrected versions of the tests are not very accurate in the
presence of significant autocorrelation. They still show
most of the problems that the uncorrected tests showed.
Simulation results on the augmented test statistics
Aug(f) and Aug(5) are also presented in Tables 3 and 4 along
with simulation results of the ADF tests for comparison.
The performance of the augmented tests depends strongly on
the number of augmentations. The simulation results
indicate that the performance of the augmented SP t-test
statistic Aug(i) is more or less same as that of the ADF t-
test statistic Aug(?,), and these augmented t-tests with
enough augmentations are generally more accurate than the

corrected tests. But, this result is less optimistic under

20

as we see later in this section.

the alternative hypothesis,
the augmented coefficie
The augmented coeffic

nt test is not as good as

Meanwhile,
ient test shows

the augmented t-test.

a similar pattern as 2(i) and 2(5).
Simulation results under AR errors, as presented in
ors. The test

Table 5, show similar patterns as under MA err

istics 2(7) and 2(5) result in too many rejections of

ypothesis with negative
Aug(5) under AR errors

stat
AR errors and too few

the true null h
rejections with positiv

imilar problems as under
since the size 0

e AR errors.
MA errors. But Aug(i) seems

shows s

to be relatively accurate, f Aug(i) is close

to nominal size with enough augmentations even when the AR

is close to -1.
k at the powers of the test

100 and fi = 0.9 are given in

parameter
s when 5 f 1.

Now, let us loo

Simulation results for T =

For positively autocorrelated errors (a > 0

Tables 6 and 7.

most tests, exce

pt Aug(i) and 2(5), tend to

or p > O),
t root when it is not

accept the null hypothesis of a uni
e tests does

true. This tendency of decreased power of th
not disappear in large samples. For negatively

rrors (a < 0 or p < 0), the power of Aug(f),

autocorrelated e
ull

the only test which is acceptably accurate under the n

Note that the power of Aug(?)

hypothesis, is not very good.
y with more augmentations.

ower of Aug(i) is 0.551 for 2

decreases considerabl For

the p
- 12. That is,

example, when 0 = —0.8,
more augmentations

4, but it is 0.106 for t

21

make the test better under the null hypothesis, but worse

under the alternative. Thus adding all these lags has

diminished the ability of the test to reject anything. This

fact is also true for the ADF test statistic Aug(?,).
A final question of some interest, though not of

immediate practical importance, is why the corrected tests

2(5) and 2(f) are not more accurate. The corrections depend
2 - 03/02, and one

on an estimate of the nuisance parameter w

may suspect that the denominator of this fraction, the long
run variance 02, is difficult to estimate accurately. It is

therefore possible that the poor performance of the
corrected tests just reflects difficulty in estimating wz.

However, it is also possible that the problem is simply slow

convergence of the statistics to their asymptotic

distributions. In order to distinguish between these two

explanations of the poor performance of the corrected tests,

we follow Kim and Schmidt (1990) in considering the
corrected tests that are based on the true value of wz.
Because we is unknown, these are not feasible tests in

actual applications, but we can consider them in our
simulations. For the MA(l) process with parameter a, we
have w2== (1+flz)/(1+0)Z, and for the AR(l) process with
parameter p, we have :02 = (l-p)2/(1-p2) . We will denote the
corrected versions of 5 and i, using the true value of oz,

by 2*(5) and 2*(i).
Tables 8 and 9 give the 5% sizes of the corrected tests

22

that use the true value of wz, for T = 100 and for the MA

and AR errors, respectively. They also give the mean values
of the estimates of w, :02 and 0‘2 and the corresponding
biases (mean values minus true values). The bias results

indicate that we do indeed estimate we poorly when there is

substantial autocorrelation, and especially when there is
For example, when 0

substantial negative autocorrelation.

-0.8 in Table 8, the true value of m2 is 41, while the mean
value of estimated w2 is less than one. These substantial
differences between true and estimated w2.lead to

substantial differences between 2(5) and 2*(5) and between

Z(i) and 2*(f). However, it is still true that the

corrected tests based on the true values of «:2 do not

In the MA case, 2*(5) and 2*(f) are

perform very well.
In the AR

reasonably accurate for a 2 0 but not for a < 0.
< 0 but not for p >

case, 2*(7) is reasonably accurate for p

0, while 2*(5) is never very satisfactory. This must

reflect a very slow convergence of these statistics to their

asymptotic distributions. The poor performance of the

corrected tests even when the nuisance parameter w2 is

assumed known suggests the need for new tests, not just new

estimates of the nuisance parameters.

5. SUMMARY AND CONCLUSION
This chapter has investigated the sensitivity to error

autocorrelation of the new unit root tests proposed by

23

Schmidt and Phillips (1990). Unfortunately, the performance

of the tests is found to be similar to that of the DF tests.
Corrected versions of the tests, similar to the PP versions
of the DF tests, reject too often under negative

autocorrelation and too seldom under positive

autocorrelation. Augmented versions of the tests, similar

to the Said-Dickey versions of the DF tests, have similar

problems. The augmented version of the t-test, irrespective

of whether it is the augmented SP test or the ADF test, has
much smaller size distortions than the other tests in the
presence of negative autocorrelation, and is probably the
best test overall, but it turns out that including the

augmentation terms diminishes the power of the test

considerably.
An alternative approach to the problem of testing for a

unit root in the presence of autocorrelation has recently

been advocated by Hall (1989), based on IV estimation of the

DF regressions. IV versions of the SP tests are also

possible, and results to be reported in Chapter 4 are more

optimistic for these IV tests than for either the original

SP tests or Hall's IV tests.

50

100

500

1000

5% Sizes 0
Dickey-Eu

§

. g . . .
0°mcaumo

0....
mU'IOUlm

000100100

24

TABLE 1

S-P Tests
? 5
.993 .993
.708 .711
.049 .050
.002 .002
.001 .001
1.000 1.000
.796 .800
.051 .052
.003 .003
.001 .001
1.000 1.000
.849 .854
.049 .051
.002 .002
.001 .001
1.000 1.000
.860 .861
.050 .051
.002 .002
.001 .001

f Schmidt-Phillips Tests and
ller Tests Under MA Errors

D-F Tests
A A
’1 P,
.998 .999
.702 .751
.048 .048
.010 .002
.009 .001
1.000 1.000
.801 .832
.050 .051
.010 .003
.009 .001
1.000 1.000
.845 .827
.048 .077
.008 .004
.008 .002
1.000 1.000
.851 .916
.046 .076
.008 .004
.007 .002

25

TABLE 2

5% Sizes of Schmidt-Phillips Tests and
Dickey-Fuller Tests Under AR Errors

S-P Tests D-F Tests
'1' p i 25 ’1‘, 3,
100 -.8 .957 .958 .956 .967
-.5 .570 .574 .570 .611
.0 .051 .052 .050 .051
.5 .000 .000 .009 .000
.8 .000 .000 .003 .000
500 -.8 .977 .977 .972 .980
-.5 .589 .593 .585 .549
.0 .049 .051 .048 .029
.5 .000 .000 .008 .000
.8 .000 .000 .000 .000

50

100

500

1000

-.8
-.5
.0
.5
.8

-.8
-.5
.0
.5
.8

-.8

500 and 1000
the Phillips

from Schwert

z<7)

:4 e12
.991 .963
.679 .718
.057 .025
.006 .000
.005 .000
1.000 1.000
.713 .840
.064 .045
.016 .001
.013 .000
1.000 1.000
.533 .709
.058 .065
.032 .024
.030 .022
1.000 1.000
.433 .600
.055 .061
.036 .035
.035 .033

The numbers of au
for £4 and 10, 12, 18 a
respectivel
-Perron test
(1989).

TABLE 3

5% Sizes of Correct
t-tests Under

ed and Augmented

MA Errors
Aug(i) W.)
e4 :12 24 £12
.292 .073 .999 1.000
.082 .076 .669 .753
.058 .086 .056 .038
.071 .092 .013 .010
.110 .085 .010 .009
.330 .064 1.000 1.000
.066 .054 .704 .831
.052 .059 .060 .050
.047 .060 .020 .011
.028 .057 .016 .009
.306 .054 1.000 1.000
.054 .049 .545 .704
.055 .066 .057 .067
.050 .050 .030 .028
.036 .048 .027 .026
.258 .049 .999 1.000
.052 .049 .453 .600
.049 .048 .056 .063
.050 .048 .036 .037
.049 .065 .039 .038

gmentations are 3, 4
nd 21 for £12 when T
y. The simulation results for
s and the ADF tests are quoted

I

Auqfi.)

:4 e12
.518 .045
.099 .032
.045 .034
.033 .039
.020 .044
.568 .055
.079 .039
.044 .040
.061 .040
.096 .043
.613 .057
.065 .046
.049 .048
.042 .046
.029 .048
.350 .051
.053 .047
.051 .046
.048 .049
.041 .051
6 and 7
50, 100,

50

100

500

1000

5% Sizes of C
Coefficient

from Schwert (1989).

TABLE 4

orrected and Augmented

Tests Under MA Errors

0 2(5) Aug(5) 20?.) Aug(fi.)
£4 £12 84 £12 4 £12 84 £12
-.8 .990 .921 .813 .666 .999 .998 .858 .292
-.5 .667 .707 .354 .414 .673 .776 .320 .283
.0 .063 .024 .090 .153 .056 .024 .171 .273
.5 .010 .001 .018 .063 .010 .001 .121 .259
.8 .007 .000 .011 .035 .008 .000 .077 .251
-.8 1.000 1.000 .893 .747 1.000 1.000 .821 .408
-.5 .708 .844 .363 .387 .707 .852 .220 .343
.0 .067 .047 .064 .104 .061 .045 .143 .355
.5 .020 .001 .012 .029 .016 .002 .171 .353
.8 .017 .000 .003 .015 .015 .001 .238 .363
-.8 1.000 1.000 .939 .839 1.000 1.000 .711 .186
-.5 .529 .711 .352 .356 .551 .719 .095 .152
.0 .060 .067 .055 .066 .059 .068 .072 .151
.5 .034 .025 .009 .012 .030 .025 .062 .147
.8 .033 .023 .002 .004 .027 .023 .037 .149
-.8 1.000 1.000 .937 .853 .999 1.000 .431 .114
-.5 .428 .598 .347 .348 .455 .611 .070 .099
.0 .055 .062 .054 .057 .055 .063 .065 .107
.5 .036 .036 .008 .010 .035 .035 .062 .109
.8 .035 .034 .003 .003 .035 .034 .046 .103
Note: The numbers of augmentations are 3, 4 6 and 7
for £4 and 10, 12, 18 and 21 for 212 when T 50, 100,
500 and 1000 respectively. The simulation results for
the Phillips-Perron tests and the ADF tests are quoted

100 -.8
-.5
.0
.5
.8
500 -.8
-.5
.0
.5

08

Note: The numbers 0
12, 18 for 212 when

5% Sizes of Corre
Augmented Tests u

TABLE 5

t-tests
Z(f) Aug(?)
24 £12 £4 £12
.926 .973 .050 .057
.479 .632 .051 .058
.064 .045 .052 .059
.004 .000 .053 .061
.000 .000 .055 .064
.804 .926 .502 .049
.277 .439 .050 .050
.058 .065 .055 .066
.016 .014 .051 .049
.001 .004 .051 .050

cted Tests and
nder AR Errors

Coefficients Tests

84

.924
.475
.067
.006
.000

.801
.275
.060
.018
.001

2(5)

£12

.975
.638
.047
.000
.000

.926
.441
.067
.016
.005

AUQ(5)
24 £12
.288 .338
.203 .255
.064 .104
.003 .011
.000 .000
.287 .297
.190 .204
.055 .066
.001 .002
.000 .000

and

f augmentations are 4, 6 for £4
T = 100, 500 respectively.

29

TABLE 6

5% Powers of Tests Under MA Errors

(T = 100, p = .9)

t-tests

7 2(7)
1.000 1.000
1.000
.991 .985
.996
.264 .315
.232
.019 .095
.005
.010 .077
.001

A(I)

.551
.106

.226
.113

.191
.125

.170
.125

.113
.116

5

1.000

.991

.270

.019

.010

2(5)

1.000
1.000

.985
.997

.326
.234

.114
.005

.096
.001

Coefficients Tests

A(5)

.953
.820

.778
.733

.294
.350

.071
.131

.025
.071

30

TABLE 7

5% Powers of Tests Under AR Errors
(T = 100, fi = .9)

t-tests Coefficient Tests

I 2(7) M?) i5 2(5) ME)
1.000 1.000 .162 1.000 1.000 .678
1.000 .109 1.000 .665

.947 .922 .181 .949 .921 .593
.972 .117 .973 .603

.264 .315 .191 .270 .326 .294
.232 .125 .234 .350

.001 .028 .175 .011 .039 .024
.000 .122 .000 .059

.000 .001 .137 .000 .002 .000
.000 .107 .000 .003

31

TABLE 8

5% Sizes of Tests Using True Error Variances

5% Sizes
Z"‘(5‘) 2*(5)
.000 .000
.000 .000
.012 .000
.012 .000
.051 .052
.051 .052
.051 .060
.051 .060
.050 .061
.050 .061

A
w

.94
.83

1.05
.90

.98
.98

081
.86

.78
.83

bias

-5.46
-5.57

-1.18
-1.33

-.02
-.02

006
.11

.07
.12

.88
.71

1.11
.81

.96
.98

066
.76

.61
.71

and Bias of Estimates of Error Variances
Under MA Errors (T = 100)

"2

.97
.97

1.03
1.03

.93
.93

1.17
1.17

1.53
1.53

bias

-.67
-.67

'.20
-.20

-007
“.07

-.08
-.08

-.11
-.11

32

TABLE 9

5% Sizes of Tests Using True Error Variances
and Bias of Estimates of Error Variances
Under AR Errors (T = 100)

5% Sizes

2*(7) 2*(5)

.049

.047

.051

.065

.115

.000

.020

.052

0081

.147

A
w

.89
.70

.08
.94

.98
.98

.71
073

056
.51

bias

~2.11
-2.30

-.65
-.79

-.02
-.02

.13
.15

.23
.18

.81
.50

.18
.09

.96
.98

.51
.55

.32
.27

Bias
bias

-8.19
'8.50

-1.82
-2.11

-.04
-004

.18
.22

021
.16

3‘2 bias
1.89 -.89
1.89 -.89
1.15 -.18
1.15 -.13
.93 ”.07
.93 -.07
1.22 -.11
1.22 ‘.11
2.24 -.54
2.24 -054

33
LIST OF REFERENCES

Bhargava, A. (1986), "On the Theory of Testing for Unit

Roots in Observed Time Series," Egyi§w_gf_zggngni§
Studies, 52, 369-384.

DeJong, D.N., J.C. Nankervis, N.E. Savin and C.H. Whiteman
(1989)," Integration Versus Trend Stationarity in
Macroeconomic Time Series," Working Paper 89-99,
Department of Economics, University of Iowa.

Dickey, D.A. and W.A. Fuller (1979), "Distribution of the
Estimators for Autoregressive Time Series with a Unit

Root," J urna e e c n t 's ' ss c' ' ,

74, 427-431.

Hall, A. (1989), "Testing for a Unit Root in the Presence of
Moving Average Errors," fiionettika, 76, 49-56.

Kim, K. and P. Schmidt (1990), "Some Evidence on the
Accuracy of Phillips-Perron Tests Using Alternative

Estimates of Nuisance Parameters," Eggngnig§_tgttg;§,
34, 345-350.

Nicholls, D.F., A.R. Pagan and R.D. Terrell (1975), "The
Estimation and Use of Models With Moving Average

D1sturbance Terms: A Survey," intetnntionai Ecgnonig
Review, 16, 113-134.

Phillips, P.C.B. (1987), "Time Series Regression with Unit
Roots,“ Econometr'ca, 55, 277-301.

Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit

Root in Time Series Regression," Binngtnikn, 75, 335-
346.

Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T.

Vetterling (1986), Numetical Recipes: Eng Ant g:
Scientifig Qompnting, Cambridge: Cambridge University
Press.

Said, S.E. and D.A. Dickey (1984), "Testing for Unit Roots
in Autoregressive — Moving Average Models of Unknown
Order,“ Aigngttikn, 599-608.

Said, S. E. and D. A. Dickey (1985), "Hypothesis Testing in
ARIMA(p,1,q) Models,“

Statistical As sgtigtion, 80, 369- 374.

scumidt, P. and P.C.B. Phillips (1990), “Testing for a Unit

34

Root in the Presence of Deterministic Trends,"
Econometrics and Economic Theory Workshop Paper, No.
8904, Department of Economics, Michigan State

University.

Schwert, G.W. (1987), "Effects of MOdel Specification on
Tests for Unit Roots in Macroeconomic Data," J9n:na1_gf

Monetnty Econonigs, 20, 73-103.
A Monte Carlo

Schwert, G.W. (1989), "Tests for Unit Roots:
Investigation." l95rn9l_25.5551n55§_snd_fisenemis
5555155155. 7. 147-159-

CHAPTER 3

1 . INTRODUCTION
Schmidt and Phillips (1990, hereafter, SP) have

proposed a unit root test based on the parameterization

(1) yt=¢+5t+xt, X=BXH+et.

according to the LM (score) principle. Specifically, define
7, = 4 + X,. E = ‘17 = (y, - y,)/('r-1). 13, = y, - E. and 5. = y.
- 5x - 5t; 5x1and 5 are the restricted MLE's of the
corresponding parameters, and St is the residual from ( 1)

with these estimates. SP consider the regression

(2) Ayt = c1 + (58“ + error, t = 2,...,T,

= Té, i = usual t statistic for testing the hypothesis ¢=o.

Under the null hypothesis that B = 1, it is obvious
from (1) that ayt = 5 + 6:, so that in (2) ¢ = 0, c1 = 5 and
the error equals 6t. In particular, the intercept c1 is
nonzero unless 5 = 0. It is also clear from the definition
Of St that Aét = Ayt‘- 5, so that the SP test can equally

Well be performed using the regression

35

36

(3) AS = c2 + (58“ + error , t = 2,...,T.

The estimate of ¢ and its t-ratio will be unaffected by the
change in dependent variable, while the estimates of c1 and
62 will differ by E.

The motivation for this chapter is the observation that
the intercept c2 in (3) equals zero (in the population).

Thus we will consider new tests based on the regression

~ ~

(4) ASt== cpSt.1 + error , t = 2,...,T.

In addition, we also consider the F—test of the null
hypothesis that both.c§ and ¢ in (3) equal zero. The F-test
statistic, to be denoted by E, is defined in the usual
manner. Intuitively, the parameter c5 is redundant in (3),
and setting it to zero could be expected to lead to a more
powerful test. However, the comparison of the power of
these tests to the power of the original SP tests or the DF
tests will be seen to depend on the value of the

(unobserved) initial conditions parameter x0* = Xo/a‘.

2. THE NEW TBBTS AND THEIR DISTRIBUTIONS

Let 3 be the least squares estimate of ¢ in (4). We
define the new unit root test statistics 5 = T5 and 7 =
usual t-statistic for testing the hypothesis ¢ = 0 in (4).

Also, as above, we define E to be the usual F statistic for

testing the hypothesis c2 = 6 = o in (3) -

37
The basic parameters of the model are 5, 5, B, a“.)%*
= XNHH and T. It is known from SP and from DeJong,

Nankervis, Savin and Whiteman (1989) that the distributions

independent of 25, 5, a‘ and xoir, and therefore depend only
on T. The same is true of the tests proposed in this
chapter. Under the null hypothesis that H = 1, it is easy
to demonstrate that St = E}=2(ej-7) . Clearly this does not
depend on the parameters )6, 5 or Xo*, and it has the same
scale as a}. The statistics 5, 7 and E are functions of 8,,
t = 2,...,T, so their distributions do not depend on w, 5 or
Xo*° Furthermore, the scale factor a‘ cancels in all
expressions for 5, 7 and E. Thus their distributions under
the null hypothesis depend only on T.

The finite sample distributions of the statistics
depend on the assumption that the (Et are iid normal, but
their asymptotic distributions can be derived under weaker
assumptions. We follow SP in assuming that the error
sequence {6:} satisfies the regularity conditions of
Phillips and Perron (1988, p. 336), and we define the
nuisance parameters 0‘2 and 02 as in Phillips and Perron (p.
337) . (Our 0"- is their auz. Under iid errors, 02 = 0‘2 and
q} is the variance of e as above.) Define the Wiener
process W(r) and the Brownian bridge V(r) = W(r) - rW(l),

for r 6 [0,1]. It then follows (SP, Appendix 3) that

-12~
(5) T ’8‘”) -> aV(r)

38

where [rT] is the integer part of rT, and where "4"
indicates weak convergence. Using (5), we can establish the

asymptotic distributions for the statistics 5, 7 and R:

(6a) 5 » -(1/2) (of/oz) [f3 V(rfier"
(6b) 7 -> -(1/2> (er/a) If; V(rfi’clrr"2
(6c) F -> (1/8) (of/62mg wrfidrr‘

Here, y(r) = V(r) - jg‘V(s)ds is a demeaned Brownian
bridge. The proof is given in the Appendix. The asymptotic
distributions of 5 and 7 are the same as the asymptotic
distributions given for 5 and f in SP, except that in SP the
Brownian bridge V(r) must be replaced by the demeaned
Brownian bridge 2(r). In SP, the Brownian bridge is
demeaned by the inclusion of intercept in the regression (2)
or (3). Here, as in SP, the simple manner in which the
ratio of nuisance parameters (a‘/a) enters the asymptotic
distribution leads to very simple corrections for error
autocorrelation. One needs simply to multiply 5 or E by a
consistent estimate of (of/02)", or 7 by a consistent
estimate of (a/a)". Estimation of the nuisance parameters

Q} and 02 is accomplished in exactly the same way as in SP

or Phillips and Perron.

We obtain 0‘2 = 02 under iid errors so that we can
tabulate the critical values. Table 1 gives critical values

for the new test statistics 5, 7 and E, as a function of T.

39
These are calculated via simulation, as discussed in SP,
section 3. The number of replications is 50,000. Note that
5 and 7 are lower tail tests if the alternative hypothesis

is stationarity, while E is an upper tail test.

3. POWER OF THE TESTS

Under the alternative that B 7 1, the powers of the SP
tests 5 and f and of the DF tests 5, and ’7‘, are known to be
independent of the parameters 5, 5 and a},.but they depend
on B, T and the initial conditions parameter Xfi*. If the
errors are symmetrically distributed, they depend on xpk
only through lx5*l. The same is true of the powers of the
tests proposed in this chapter. This is most easily seen

from the SP (Appendix 2) representation of 8,:

t j-l
- —_ = 1'1 . - i . ..
t 733- (wj w), where WI. )6 3X0 + 61 + (B ”12:08 6“

(n1

(7)

This depends on 8, T and Xh*, and given these parameters,
has the same scale as a‘; (a then cancels from all
expressions for 5, 7 and R.

We now compare the power of the new tests 5, 7 and E
With the power of the SP tests and DF tests. This is done
Via a Monte Carlo simulation, using essentially the same
experimental design as in SP. The results of the experiment
are given in Table 2. The results in Table 2 for the SP
tests and for the DF tests are as in SP, while the results

for this chapter's new tests (5, 7 and E) are new.

40

The power of the F-test E is quite similar to the power
of the SP test 5 for all of our experiments. Therefore, we
do not need any separate explanation on the power of the F-
test. In the remainder of this section, the "new tests"
will mean the new 5 and 7 tests.

Experiment 1 takes B = 1, T = 100, 200 and 500, and
simply confirms that, apart from randomness, the sizes of
the test are correct.

Experiment 2 considers the power of the tests, for X55
= 0, T a 100, and B = 1, .95, .90 and .80. This variation
in B induces a substantial variation in the power of the
tests. From SP it is known that the SP tests are more
powerful than the DF tests for these parameter values. The
comparison of the new tests to the SP tests is more
ambiguous. Basically, the new tests are more powerful than
the SP tests when power is low (B = .95 and .90) and less
powerful when power is high (B = .80). Experiments 3 and 4
provide similar results and essentially the same
conclusions, but for T = 200 and T = 500 respectively.

We now turn to the effect of the initial conditions
parameter X5* on power. Because we use normal errors, which
are symmetrically distributed, only IX0*I matters.

Experiment 5 considers T = 100, B = .90, and X0* = 0, -1, -
2, -5 and -10, while Experiment 6 considers B = .95 instead
of .90 (so that powers are lower). From SP it is known that

the SP tests and the DF tests have powers that are monotonic

41
in 'Xo*‘ , though in opposite directions. Over the range we
consider, the power of ’1‘, increases with lXo*l , while the
powers of the other tests decrease as lxo*l increases. In
both Experiment 5 and Experiment 6, we can see that the
power of the 5 and 7 tests also decreases as 'Xo*l
increases, but their power decreases faster than the power
of the original SP tests. The new tests 5 and 7 are about
equally powerful, and they are marginally more powerful than
the SP tests or the DF tests when lxo*l == 0, but less
powerful when IXOH is large. For example, the new 7 test
is more powerful than the SP 9‘ test for IXO*I S 1.7
(approximately), and it is more powerful than the DF ’7‘, test

for |X0*| < 2.8 (approximately). The above discussion is

summarized in Figure 1.

If X0* is treated as a fixed parameter, there is not
much more to say. However, following SP, when B 7! 1 we can
also think of X0* as random and drawn from the stationary
distribution of xt/o“ which is N[0, l/(l-B2)]. When X0* is
treated as random, we can calculate the probability that the
new tests are more powerful than the others; for example,
the probability that the new 7 test is more powerful than
the SP 7 test is about .54 and the probability that the new
test is more powerful than the DF test 3“, is about .78.

Experiments 7 and 8 treat Xoir as random and drawn from
the stationary distribution of X,/a,, and therefore

calculate the powers of the tests n9_t conditional on Xoir.

42
For these experiments the number of replications used is
50,000. Experiment 7 has T = 100 while Experiment 8 has T =
500, and in each case a variety of values of B is considered
so that power ranges from nominal size to nearly unity.

The basic result of these experiments is that the power
of the tests proposed in this chapter is greater than the
power of the SP or DF tests when power is low, but it is
less than the power of the competing tests when power is
high. However, comparing the new tests to the SP tests, the
gain in power when power is low is small compared to the

loss in power when power is higher.

4. CONCLUDING REMARKS
The calculation of the SP test statistics involves

estimation of a redundant intercept, and the new tests 5 and
7 proposed in this chapter successfully eliminate the need
to estimate it. This might be expected to lead to a gain in
power. However, our Monte Carlo power calculations do not
support this expectation. Treating the initial conditions
term as fixed, the new tests are better than the SP tests
when the initial condition term is small and worse when it
is large, and neither the magnitude of the power differences
nor the range over which the tests are better provides a
strong basis for preferring the new tests to the SP tests or
vice-versa. Treating the initial conditions term as random,

the new tests are slightly more powerful than the SP tests

43
when power is low, and moderately less powerful when power
is high. These results would appear to support the general

use of the original SP tests rather than the new tests.

44

TABLE 1

CRITICAL VALUES FOR 7

T .01 .025 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .975 .99
25 -3.40 —3.01 -2.71 -2.37 -2.01 -1.78 -1.59 -1.43 ~1.29 -1.15 -1.00 '0.84 -0.74 -0.66 -0.59
50 -3.29 -2.94 -2.66 -2.36 -2.02 71.79 -1.61 -1.45 -1.30 '1.16 -1.02 -0.85 -0.74 -0.66 -0.59
100 -3.24 -2.89 -2.64 -2.35 -2.01 '1.79 -1.61 -1.46 -1.31 -1.17 -1.02 '0.85 -0.74 -0.66 -0.58
200 -3.19 -2.89 -2.63 -2.34 -2.01 '1.79 -1.61 -1.45 -1.31 -1.17 -1.02 -0.85 -0.74 -0.66 -0.59
500 -3.21 -2.90 -2.63 -2.34 '2.01 -1.79 -1.61 -1.45 -1.31 -1.17 -1.02 -0.85 -0.74 -0.66 -O.59
1000 -3.20 -2.89 —Z.62 -2.34 -2.00 71.78 '1.60 '1.45 -1.31 ~1.17 -1.02 -0.85 -0.74 -0.66 -0.59

CRITICAL VALUES FOR 25

T .01 .025 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .975 .99

25 ~16.7 -14.1 ~12.0 -9.82 -7.46 -6.02 -4.92 -4.07 -3.33 -2.68 -2.08 -1.47 -1.14 -0.93 -0.74
50 -18.3 -15.2 -12.8 '10.4 -7.79 -6.22 -5.08 -4.17 ’3.39 -2.71 -2.09 '1.47 -1.12 -0.90 :0.71
100 -19.3 -15.7 -13.2 -10.6 -7.92 -6.30 -S.14 -4.21 -3.41 -2.72 -2.07 -1.45 -1.09 -0.88 -0.69
200 -19.5 -16.1 -13.5 -10.7 -8.00 -6.35 -5.14 -4.20 -3.40 '2.72 -2.07 -1.44 -1.08 -0.86 -O.69
500 -20.3 -16.5 -13.6 -10.8 -7.99 -6.33 —5.14 -4.21 -3.42 -2.71 -2.07 -1.43 -1.09 -0.87 -0.69
1000 -20.2 -16.5 -13.6 -10.8 '7.97 -6.31 '5.11 -4.17 '3.40 -2.71 -2.07 -1.45 -1.09 -0.88 -0.68

CRITICAL VALUES FOR E

T .01 .025 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .975 .99
25 0.48 0.57 0.66 0.80 1.04 1.27 1.51 1.78 2.10 2.50 3.05 4.00 4.96 6.01 7.45
50 0.47 0.56 0.66 0.82 1.06 1.29 1.53 1.80 2.11 2.49 3.02 3.93 4.84 5.73 6.85
100 0.46 0.56 0.67 0.82 1.07 1.30 1.54 1.80 2.11 2.48 2.98 3.81 4.65 5.49 6.56
200 0.47 0.56 0.67 0.82 1.07 1.30 1.54 1.80 2.10 2.47 2.97 3.80 4.60 5.38 6.48
500 0.47 0.56 0.67 0.82 1.06 1.29 1.53 1.78 2.08 2.45 2.94 3.78 4.59 5.34 6.38
1000 0.46 0.56 0.67 0.82 1.07 1.29 1.53 1.79 2.09 2.45 2.94 3.75 4.54 5.29 6.28

 

No.

T

100
200
500

100

200

500

100

100

100

500

45

TABLE 2

SIZE AND POWER, 5% LOWER TAIL TESTS

.6 x;
1 0
1 0
.95
.90
.80
1 O
.95
.90
.80
1 O
.95
.90
.90 0
-1
-2
-5
-10
.95 O
-1
-2
-5
.95 rd
.90
.85
.80
.70
.99 rd
.98
.97
.95

.90

‘9)

.048
.048
.051

.048
.082
.186
.644

.048
.178
.617
.999

.051
.819
1.00

.186
.188
.191
.211
.304

.082
.083
.082
.083

.081
.187
.390
.647
.961

.081
.184
.374
.828
1.00

A

p,

.050
.048
.051

.050
.098
.239
.734

.048
.233
.724
1.00

.051
.897
1.00

.239
.239
.234
.198
.120

.098
.097
.095
.085

.090
.225
.464
.732
.982

.094
.225
.452
.892
1.00

.051
.048
.049

.051
.105
.264
.759

.048
.254
.751
.997

.049
.910
1.00

.264
.260
.248
.161
.032

.105
.104
.101
.086

.096
.238
.476
.722
.952

.095
.230
.455
.863
.998

.052
.050
.051

.052
.108
.270
.765

.050
.266
.763
.997

.051
.914
1.00

.270
.267
.252
.165
.033

.108
.106
.104
.088

.098
.242
.483
.728
.954

.098
.237
.464
.869
.999

.050
.050
.050

.050
.113
.288
.719

.050
.290
.742
.972

.050
.874
.993

.288
.276
.238
.095
.003

.113
.113
.108
.082

.098
.236
.441
.634
.854

.100
.234
.433
.762
.973

.052
.051
.050

.052
.116
.294
.723

.051
.292
.744
.973

.050
.874
.993

.294
.280
.242
.097
.003

.116
.115
.111
.084

.100
.241
.446
.638
.857

.099
.234
.433
.762
.973

.051
.051
.049

.051
.108
.273
.767

.051
.269
.765
.998

.050
.913
1.00

.273
.268
.251
.165
.032

.108
.105
.102
.089

.098
.244
.483
.725
.954

.096
.233
.460
.865
.998

.30

.25

.20

.15

.10

.05

46

FIGURE 1

POWER OF VARIOUS TESTS, T = 100, fi - 0.9

 

 

 

 

 

\

‘1)

b)
‘4

fl

‘1

47

APPENDIX

We employ the same functional limit theory and assume
the same regularity conditions as Phillips and Perron (1988,

p.336). First, we want to show the following results:

(A.1) T'2 SM?- -* 02 f; V(r)2dr

-1 §

N1.")0-1 NMr-l

(A.2) I- Met -> -(1/2)a‘2

Since SM = St_1 - (t-2) - E - 61 and A§t = Et - E from (17)

of Schmidt and Phillips (1990), we have:

I ~ ,, _ '1‘ - _
(A.3) T" 2 St_1ASt = T‘ 5: {st,1-(t-2)e-61}(6t-6)

N

T
—_ -1 - -1 - .1- -
T a: 3:46: T e '22 SM 'r e gm: 2)et

_ '1‘ 1 T _1_ 'r
+ T’162.2‘3(t-2) - T' )23616t + T e '2861.

The following results follow (where summations are over t

from 2 to T):
(i) T" 33.46. .. %[02W(1)2-o‘2]
" '1' - "’2’ T3”, 25 -+ oW(1)-af1W(r)dr
(11) T 6 25p4 — T 6 t4 0
_ _ _ - _ 2-
(iii) '1‘"? Eur-2).:t = 'r'VZe T3” zztet - 2T1l2€T1TV e
1
-v - aW(1)a[LW(r)dr - W(1)]

(iv) T"? 2(t-2) = ('1"’22E)2'1"2 2(t-2) -v (1/2)[¢IW(1)]2

48
and the last two terms cancel each other. Then, it is

straightforward from above to see that

T - -
(11.4) T" 2 St,1ASt -» 3.0291(1)? - ‘10} - aZW(1)_I:W(r)dr - 02W(1)2

N

+ 02W(1)L1W(r)dr + wwuv

- _1 2
_ 10"

which proves (A.2). The result in (A.1) is similarly
obtained as in Schmidt and Phillips (1990, p.23), while a
standard Brownian bridge V(r) replaces a demeaned Brownian
bridge y(r) used for the original SP test.

From the results in (A.1) and (A.2), it is

straightforward to see that the results in (6a) and (6b)

hold:

(A's) 7’ = T5 = IT" Sgt-1A§.1/[T'2 2?»..121
-* “1(0.2/02)[]:V(r)2dr]"

(A.6) 3? = .11-1;. [o‘Z/zét.12]‘/2

" "1 (0.2/02) [1/L1V(r)2dr]"[02L1V(r)2dr]‘/Za"1
1 ‘ 2 -v2
= “1(0,/0) [LV(r) dr] .

The F-statistic for testing the joint hypothesis c2 = (P = o
in (3) is given by:

RSS/Z

 

ESS/(T-B)

49

where

2'
RSS - 2W5..1-T9)]2
2
2'
355 ' 2:] (”c-A?) -$<$’.-1-§) ]’

Using the result in (A.1), we have

—RSS - éjz (gt- 1

T
- -21- (fiWT-zzz: (gm-3)

2 2 1

" i --1--?-'-; 02f1(r)2dr

 

fjflflzdr °
0
0: 1
- 802 1
fidflzdr
0
1'
E33 - 21
—T— - ;(A§c-A§)3%, + (735) T22(.S7_-)2-1—,
T ~ — ~ — 1
- 21$ T422: (SH-s) (ASt-AS) 7,
Therefore

since the last two terms vanish asymptotically.
the asymptotic distribution of the F-statistic in (Go) can

be obtained from the ratio of the last two expressions

above. D

50

LIST 0? REFERENCES

DeJong, D.N., J.C. Nankervis, N.E. Savin and C.H. Whiteman
(1989), "Integration Versus Trend Stationarity in
Macroeconomic Time Series," Working Paper 89-99,
Department of Economics, University of Iowa.

Phillips, P. C. B. and P. Perron (1988), "Testing for a Unit

Root in Time Series Regression,"fligmg_113§,75, 335-
346.

Schmidt, P. and P.C.B. Phillips (1990), "Testing for a Unit
Root in the Presence of Deterministic Trends,"
Econometrics and Economic Theory Workshop Paper No.

8904, Department of Economics, Michigan State
University.

CHAPTER 4

CHAPTER 4

UNIT ROOT TESTS BASED ON INSTRUMENTAL
VARIABLES ESTIMATION

1. INTRODUCTION

In this chapter, new tests for a unit root, based on
instrumental variables (IV) estimation are developed. The
tests are similar in spirit to the IV tests of Hall (1989)
and Pantula and Hall (1990), but they are derived from the
unit root tests of Schmidt and Phillips (1990) rather than
from the Dickey-Fuller tests of Fuller (1976) and Dickey and
Fuller (1979). The new tests perform remarkably well. In
the presence of moving average errors, the tests are quite
accurate under the null hypothesis of a unit root.
Furthermore, their power against stationary alternatives is
better than the power of other tests that are accurate under
the null.

It is well known that the tabulations of the Dickey-
Fuller (hereafter, DF) tests assume iid errors. When the
errors are autocorrelated, the distributions of the test
statistics are affected, even asymptotically. Subsequent
research has suggested modifications of the DF tests that
have the same asymptotic distributions under autocorrelated
errors as the DF tests have in the presence of iid errors,

so that the usual DF tabulations can be used. As discussed

51

52
in Chapter 1, these modifications can be put roughly into
three categories.

First, Said and Dickey (1984, 1985) have suggested
augmented DF (hereafter, ADF) tests based on the Dickey-
Fuller regression augmented with lagged differences of the
dependent variable. The ADF tests were intended to
accommodate autoregressive errors, and their size should
also be correct asymptotically in the presence of moving
average errors, if the number of augmentations increases
with sample size at an appropriate rate. Second, Phillips
(1987) and Phillips and Perron (1988) developed transformed
tests (hereafter, PP tests) that are valid asymptotically in
the presence of a wide class of weakly dependent and
heterogeneous errors. Third, Hall (1989) and Pantula and
Hall (1991) proposed IV tests that easily allow for moving
average errors. They are based on the IV estimation of the
Dickey-Fuller regression, where the instrument is the
dependent variable with a lag greater than the order of the
MA process of the errors. If there are ARMA errors, they
can be handled by a combination of augmentations and IV
estimation.

A considerable body of simulation evidence, including
Phillips and Perron (1988), Schwert (1989) and Kim and
Schmidt (1990), has documented serious problems with the
finite sample properties of the PP and ADF tests. The PP

tests reject the null too often in the presence of

53
negatively autocorrelated errors and too seldom in the
presence of positively autocorrelated errors. These size
distortions are considerable even for rather large sample
sizes, such as T = 1000. The size distortions of the ADF
tests can be more or less removed by using a large enough
number of augmentations, but then the tests have very little
power against reasonable alternatives. Hall (1989) presents
simulation evidence showing that the size distortions of the
IV tests are less than those of the PP tests, though they
are still considerable at reasonable sample sizes if the
errors are strongly autocorrelated.

Schmidt and Phillips (1990, hereafter, SP) have
proposed unit root tests and discussed the advantages of
their tests relative to the DF tests. The SP tabulations
also assume iid errors, and error autocorrelation can be
handled in essentially the same ways as for the DF tests.
Chapter 2 of this dissertation provides simulation evidence
which shows that the augmented SP tests and the SP tests
with PP-type corrections suffer from the same problems as
the corresponding DF tests did. Specifically, the SP tests
with PP-type corrections and the augmented SP tests with a
small number of augmentations both suffer from very
substantial size distortions. Furthermore, while the
augmented SP tests with enough augmentations do not have
large size distortions, they also have very little power.

This chapter provides IV tests based on the SP

54
framework. Our simulations show that these tests have
desirable operating characteristics: they have surprisingly
small size distortions and are more powerful than other
tests of similar size. This chapter provides an alternative
interpretation of the IV tests, as PP-type tests but with
unusual estimates of the relevant error variances. This
interpretation may explain why IV versions of the SP tests

outperform IV versions of the DF tests.

2. NEW IV TESTS
The SP tests are based on the following

parameterization, as discussed in Chapter 1:
(1) Yt=¢+€°t+xtr xt=flxt-1+etl t=1l”lT‘

The unit root hypothesis is, Eh : p = 1. The basic
parameters are 16, g, 0‘ and x; = Xo/a‘, where X0 is the
initial value of X. The only observable is yt, t=1,..,T.

The SP tests are tests of B = 1 derived according to
the LM (score) principle. Specifically, define ¢x== 6 + X0,
E= 23; = (y, - Y1)/(T-1). ~13, = 171-5. and 1:3. = Y. - 13x - 3t:
'7, and E are the restricted MLE's of the corresponding

parameters, andét is the residual from (1) with these

estimates. SP consider the regression
(2) ayt== c + ¢étq + error , t = 2,..,T ,

which is estimated by least squares, yielding an estimate 6.

55
Then the SP statistics 5 and 7 are defined as follows: 5 =
T5, T = usual t statistic for testing the hypothesis ¢ = 0.
In this chapter we actually employ a slightly modified
version of the SP tests, discussed in Chapter 3. Since ASt
= ayt- E, the dependent variable in (2) can equivalently be
taken to be ASt instead of Ay}. However, if this is done,

the population intercept is zero, so that we can consider

the alternative regression:

(3) ASt== cpr1 + error , t = 2,..,T.

If this is estimated by least squares, yielding the estimate

5, the modified SP statistics are
(4) '5 = T-5

usual t-statistic for the hypothesis ¢ = 0.

*1
H

(5)

Define the "innovation variance" of and the "long-run

variance" 0'2 as in PP:

(53) 0‘2 = lim T’1 26,2
Tww t

(610) o’- = lim Wager)? .
Tam t

Then, assuming the regularity conditions of PP, p. 336, the

asymptotic distributions for z and 7 (as given in Chapter 3)

are:

(Va) 3 -> -<1/2) (of/oz) [f3 V(rVer"

56

(7b) 'r' -» -(1/2) wow; V(r)2dr]"’2 .

Here V(r) is a Brownian bridge: V(r) = W(r) - rW(1), where
W(r) is a Wiener process. The asymptotic distributions in
(7) are the same as those given for 5 and f by SP, except
that in SP the Brownian bridge V(r) must be replaced by the
demeaned Brownian bridge V(r) = V(r) - fg‘V(s)ds.

In this chapter we assume that the errors at follow an

MA(q) process:

‘1

(8) e = u

t t b]

with u.t iid. We assume that the errors satisfy conditions 1
and 2 of Hall (1989, p. 51). We will discuss in most detail
the special case q = 1, so that et==tx + ”“64 with Iol < 1.
For the MA(1) case, we note that a‘z/a2 = (1+02)/(1+0)2.

We derive the IV versions of the (modified) SP tests by
estimating (3) by instrumental variables estimation, where
the instrumental variable is Sb,, k 2 q+1- Let éw,be the IV
estimate of o, and define the consistent estimator 6} of

the error variance of as follows:

1'

(9) ai-i 1157—6 -2.
T.;1( c 1.5.1)

57

Then the test statistics are given by

T
T - +
T

X 51-3.-.

t-k+1

(105) 51v " Téiv "'

$11!

T
afilg: £32
t- +1

(>5 Mar

-k*1

 

(10b) 71v -

 

Theorem 1: Assume that the data are generated by (l) with
B = 1 and (8), and that the errors in (8) satisfy conditions
1 and 2 of Hall (1989, p. 51). Suppose that 5,, is the IV

>

estimate of o from (3), based on the instrument Sbk, k _

 

 

q+1. Then,
(11) p19 .. -. 1 1
2 Tflzflzdr
J
(12) {IV 7. “—0— 1 1 I

2 V(r)2dr
[

58

where "e" indicates weak convergence as T 4 w.

2:99;: Write 5w as
1

51V - T 6.1V T

T" 321 $-13.-.

 

The denominator converges weakly to 02 f 3V(r)2dr for k 2 0;

that is, regardless of whether or not k 2 q+1. However, the

probability limit of the numerator depends on whether k 2
q+1. We show in Appendix 1 that

-02/2 , k'z q+1

“'1‘“ -af/2 , k s q .

For k 2 q+1, the asymptotic distribution given by (11)

follows by joint convergence of the numerator and

denominator. The proof for 7w follows exactly the same

lines. B

Our Theorem 1 is very similar to Theorem 1 of Hall

(1989, p. 51). But, the asymptotics are expressed in much

simpler form than those of Hall's statistics.
Theorem 1 shows that the IV estimator successfully

handles moving average errors. The intuition behind it is

appealing. For k .>. q+1, SM is uncorrelated with 5: and so

a bias correction is asymptotically unnecessary. In other

words, in estimating by IV we have taken advantage of the

59

maintained assumption that Cov(e,ebj):= 0 for j 2 q+1. This

restriction is not so easily exploited in the PP framework.

The proper choice of the order of moving average

errors, q, is obviously important. As asserted in Hall

(1989), the choice of q should ideally rely on the economic

theory on which the model is based. If theory does not

provide any information on q, one may empirically determine

q; for instance, the Wald tests developed by Hall (1990) can

be applied by using the IV residuals in (3). It would be

safe to select q conservatively (i.e., higher q rather than
lower), since choosing q too low causes a serious problem,
but choosing q too high does not, at least asymptotically.

Theorem 1 indicates that the asymptotic distribution of
the test statistic 5w does not depend on the nuisance

parameters 02 and of, so that the in, test is valid

asymptotically in the presence of MA errors. This is

similar to Hall's result for IV versions of the DF
Also similarly to Hall's t-test, the new
If

coefficient tests.
IV t-test iw still depends on the nuisance parameters.

cn/a, the asymptotic distribution of i“
It is very interesting that

we define A =

involves the scale factor A”.

the asymptotic distribution of the OLS-based sp t-test ?
(given in (7b)) involves the scale factor A. These factors
can be eliminated easily by multiplication, and so we define

the statistic:

-2_-..
(13) r -— T-TW .

60

The asymptotic distribution of 72 is therefore free of

nuisance parameters, and is the same as the distribution

(under iid errors) of the square of the SP statistic 7.

Thus, unlike Hall's case, there is a simple way to obtain a

nuisance-parameter-free t-statistic, without having to

estimate MA parameters.

Interestingly, the fa statistic can be interpreted as a

PP-corrected version of the SP statistic E, but with an

unusual estimate of the nuisance parameters. Recall that

62 is defined above in (9) as the error variance estimate

for IV estimator of equation (3). Let 5f be the

corresponding estimate from OLS (that is, 5 replaces 6W,in

(9)). Then we have

2 3",,113", )3 g 4,157,
,2 _ 2&1. , Eats.-.
_ 1 E g7
a: "—7—— 5. ~ it"
23 3‘11 J (z: St-kSt-l)2

r12 .315; . T )3 52-1w".
6,0, 76: 37—1”: 5:24)

(14)

where all summations are over t from k+1 to T. Notice the

second equation. The second part of this expression is

essentially 3. Furthermore, since T"? SMASt converges in
probability to -a%/2, the first part of the expression is a
consistent estimate of -0.5(05Ng?), and cancels the factor

A2 = age/02 that appears in the asymptotic distribution of 72,

61

as given by equation (7a) above. Thus the 1‘2 test is

equivalent to a PP-corrected version of 3, where the

estimate of 12 implicitly used is

T
1'2 — a ’ -1 ~ ~
(15) A —- «1‘0, / 2T 2: SWASt .
t=k+1

Interestingly, the 72 test performs well, as we shall

see in the next section. However, the estimate 52 in (15)

is intuitively unappealing because it estimates 02 in an

unconventional way, but it estimates 0‘2 in a conventional

way (or, actually in two conventional ways, as 6‘3‘) . We

can easily consider estimating 0‘2 analogously to the way

that 02 is estimated, in which case we have the consistent

estimate of 12 given by

(16) 12 =

T _ ~ T __ ~

2 3 AS / z 3 AS .
-1 -k
t=k+1t ‘ t-k+ ‘ ‘
Using this estimate, we can correct either the SP

coefficient test 3 or the SP t-test 7. This gives

(17a) 5,, = m2

(17b) 7an = 72/12 .

It should be noted that while 12 > o, i?- < o is

possible. In fact, 22 < 0 occurs with non-negligible

probability when the errors are strongly negatively

autocorrelated. This point will be discussed more in the

next section. For now, we will simply observe that we

62

consider inf in (17b), rather than in" to avoid taking the

square root of 32. Since the distribution of 7 is

essentially completely negative, there is in any case no

loss in information in squaring it.

3. PERFORMANCE OF THE IV TESTS

In this section we consider the finite sample

performance of the new IV tests. We are interested both in

their size under the null hypothesis, in the presence of

moving average errors, and also in their power against

stationary alternatives.
The model to be considered is the SP model (1), with

the errors following the MA(1) process 5t==‘% + auP4. The

Parameters of the problem are therefore ¢, 5, 0‘, X0* =

)%/a‘, 0 and T. From SP and from DeJong gt_al, (1989) it is

known that the distributions of the SP and DF statistics

under the null do not depend on #5. 5, 0‘ or xo*. The same

is easily seen to be true of Hall's and our IV statistics,
and so the only relevant parameters under the null are a and

The distributions of the test statistics under the
If the errors

T.
alternative (8 < 1) also depend on a and X *.

are symmetrically distributed, they depend on Xo* only

through I x0* I .
We will investigate the size and power properties of

the tests in a Monte Carlo simulation, with the data

generating process being the model (1), as just described.

In

(2

63

We will use random normal errors generated using the FORTRAN

routines GASDEV/RAN3 of Press, Flannery, Tuekolsky and

Vetterling (1986). For each sample, MA(1) errors are

generated by initializing u and e at zero, and then

generating and discarding 20 observations. Our simulations

use 20,000 replications, except that only 10,000

replications are used for sample size T = 1000. We consider

sample sizes of T = 25, 50, 100, 200, 500 and 1000, and

moving average parameters a = 0, 10.5 and 10.8.

The statistics that we consider are the SP statistics 3

and 7, Hall's IV statistics 3w and 7”, and the new IV

statistics is", 7“, 7‘2, ,3”, and fppz, as defined in this

chapter in equations (10a), (10b), (13), (17a) and (17b),

respectively. All IV tests use as their instruments the

relevant dependent variable lagged two periods (She for our

tests and Yhz for Hall's). We will also make some reference

to previously-reported simulation results for the ADF and PP

tests.
In Table 1 we present the critical values of the test

These are generated by a simulation

(The

statistics just listed.
with iid errors (a = 0), using 50,000 replications.

critical values for the E and 7 statistics agree with those

presented in Chapter 3.) Asymptotically there should be

simple relationships between the critical values of the

various statistics. For example, the critical values for

the iZTand in} statistics should be the square of the

64

critical values for the 7 and 7" statistics, and this is

more or less so for T = 1000. (For example, at the 5%

level, 2.61? and 2.622 are nearly equal to 6.80.)

Furthermore, the critical values for the 72 and fmz
statistics should be equal to minus one half of the critical

values for the 3, 5w and Z",statistics, and again this is

more or less so for T a 1000. (For example, at the 5%

level, 6.80 is nearly equal to minus one half of -13.6, -

13.7 and -13.5.) For smaller sample sizes these

relationships hold less well, as would be expected.
Although the justification for the IV tests under
autocorrelated errors is only asymptotic, the use of

critical values simulated at the appropriate sample size may

still increase the accuracy of the tests, as we will see

shortly.
Table 2 gives the size (proportion of rejections under

the null) of the tests, at the 5% level, in the presence of

MA(1) errors, when the "asymptotic" critical values (i.e.,

those of the modified SP statistics 7 and 7, or of the DF

statistics 2, and 7,) are used. Similarly, Table 3 gives

the size at the 5% level, in the presence of MA(1) errors,

when the "finite sample" critical values (i.e., those from

Table 1 for the appropriate value of T) are used. As noted

above, the only two relevant parameters are 0 and T. The

asymptotic distributions of the 3, 7, f“ and 7” test

statistics depend on nuisance parameters and so we do not

65

expect these tests to have correct size, even

asymptotically, in the presence of autocorrelated errors.
In fact, it is true in Tables 2 and 3 that all of these

tests suffer from large size distortions, even for large

values of T, except when a = 0. Accordingly, we will focus
7 2 and

our attention on the performance of the 5w 72, 5”, pp

3w tests, which should have correct size asymptotically in

the presence of MA(1) errors.
Comparing the results in Tables 2 and 3, it is apparent

that the sizes of the tests tend to be more nearly correct
when the "finite sample" critical values are used (Table 3)
than when the "asymptotic" critical values are used (Table

Using the "asymptotic" critical values in small samples

2).
2
and i to

causes 5w and 3w to reject too often, and ,2, 5m

reject too seldom: see particularly the results for T S 100

and a = 0. We will therefore focus on the results using the

"finite sample" critical values, as given in Table 3.
The main conclusion from the simulation results in

Table 3 is that the new IV tests perform better than Hall's

IV test. The 599 test appears to be the best overall. For

-0.8 and T = 100, the 5% sizes of the new

example, when 0 -

statistics 5“, 72, is” and *an are 0.235, 0.125, 0.077 and

0.137, respectively, while the size of Hall's 3w test is

0.446. A similar pattern is found for larger sample sizes,

though, as Schwert (1989) and others have found, the

improvement of the various tests is neither rapid nor

66

monotonic in sample size. When 0 = -0.5, the size

distortions are minimal for the 1‘2, 599 and 9‘an tests, so

that serious size distortions occur only in the presence of

a very large negative MA parameter. The 5”,test tends to

over-reject a bit when 0 > 0 and T s 100, but this problem

is not very serious. If a > 0 is a reasonable possibility,

the 5w and?2 tests appear to be best, but their size

distortions are clearly larger than those of the A» test

when a < 0.
To put the optimistic results for the new IV tests in

perspective, it is instructive to compare them to the

corresponding results for the PP and ADF tests. For

example, for T = 100 and 0 = -0.8, Schwert (1989) reports

size of 1.00 for the PP—corrected 3, and ’1‘, tests, with

either four or twelve covariances used to estimate the long

run variance. (Note that 0 = -0.8 in our notation is a

0.8 in Schwert's notation.) For the ADF 7, test, he reports

size of 0.568 with four augmentations and 0.055 with twelve

augmentations. Chapter 2 reports similar results for the

PP-corrected and augmented SP tests; for example, for T =

100 and 0 = -0.8, the size of the SP test is 0.330 for the

SP 7 test with four augmentations and 0.064 with twelve

augmentations. Similar results obtain for other sample

sizes. Our new IV tests clearly have better size properties

than the corresponding PP tests or the ADF tests with only a

few augmentations.

h«

67
We noted at the end of Section 2 the possibility that
the nuisance parameter estimator 12 could be negative. This
outcome is related to the signs of all of the IV-type tests,
and is worth a brief discussion. Let St be as defined in

Section 2, and let St be the residual in an OLS regression

of yt on [1,t]: the various SP statistics depend on the St,
while the various DF statistics depend on the St.

With a probability of essentially one, we have )gSt_1/.\St
< 0 and ¥§MA§t < 0, which implies that the uncorrected
statistics 3, 7, ’p‘, and ’1‘, are all negative with a
probability of essentially one. It is also usual to have
{3St_2ASt < o and géwém > 0, which implies a" < o, iiv < o,

A A
72 > 0, 12 > 0, < 0 and fmz > 0: and to have 7362453: < 0

5w
and ¥§t-2§t-1 > 0, which implies 3w < 0. We will call these
the "usual signs" for these statistics. For example, with T
= 100, and o = 0, all of these signs obtain in our

simulations with probabilities in excess of 0.99, and many

obtain in every replication of the simulation.

When the errors are strongly negatively autocorrelated,

however, unusual signs sometimes appear. For example, with

T = 100 and o .-.-. -o.3, we find 8.592118, > 0 with frequency

- 2
0.385, which therefore yields 32 < 0. Pm > 0 and 7m < o

- 2
with this same frequency. We have piv > 0, 7w > 0 and 7‘ < 0

With slightly lower frequency, 0.379, because occasionally

(frequency = 0.007) we also have )gsmspz < 0, and these

cases tend to overlap with cases in which 7362453: > O, 80

68

that the unusual signs cancel. Similarly, we have SSPZASgO
with frequency 0.418, but 3w,> 0 with frequency only 0.360,
because a few cases with ¥§t-2A§t > 0 also have {SSWSM < 0.

Because unit root tests are one-tail tests, an unusual
sign for a statistic implies an acceptance of the null
hypothesis. The occurrence of unusual signs when the errors
are strongly negatively autocorrelated is therefore helpful,
as it counteracts the tendency of unit root tests to reject
the null too often. This is true for our new IV tests and
for Hall's IV tests as well.

We now turn to the power of the tests against
stationary alternatives. In addition to T and 0, the
parameters 3 and X5* are now relevant. We pick 8 = 0.9 and
15* = 0 as reasonable values to consider. Some limited
experimentation (not reported here) with other values of p
and X5* indicates that the power comparisons are not changed
markedly by other plausible choices for these parameters.

Our results for power against the alternative B = 0.9
are given in Table 4. These are somewhat difficult to
interpret because the tests with upward size distortions
will tend to appear more powerful, and the tests with
downward size distortions will tend to appear less powerful,
than the tests with no size distortions. For example, for T
= 25 and o = -0.8, Hall's 3w test appears to be more
powerful than our zfi,test, but this difference in power is

in fact just the same as the difference in size: if

69

properly size-corrected, neither test has appreciable power
against 6 = 0.9 for T = 25. Thus, power comparisons are
meaningless when there are size distortions. The simplest
way to handle this problem appears to be to consider power
when 0 = 0, in which case none of the tests we are
considering suffers from appreciable size distortions. If
this is done, we find in Table 4 that the powers of all of
the tests are rather similar. Among the set of tests that
are asymptotically robust to HA errors (i.e., 5“, 1‘2, 5",,
7“} and 5“), there is no clear basis for preferring one to
the other in terms of power when a = 0. There is some
indication that the new IV tests are more powerful than
Hall's 5“ test for T s 200, but this difference is not
large. Furthermore, it is known from Schmidt and Phillips
(1990) that the parameter value x;'= 0 favors SP tests over
DF tests. However, it is noteworthy that these tests are
just as powerful when 0 = O as the OLS-based SP 5 and 7
tests (or the SP 5 and 7 tests, or the DF 5f and ’1‘, tests ,-
see SP or DeJong g;_al& (1989) for the relevant power
tabulations), which do not allow for MA errors. In other
words, robustness in the presence of MA errors is purchased
at no loss in power in the presence of iid errors. This is
a strong argument in favor of the new IV tests.

A fairly clear-cut comparison exists between the powers

of the new IV tests and the augmented DF and SP tests.

Following Schwert (1989), define the numbers of

70
augmentations £4 = integer[4(T/100)”‘] and £12 =
integer[12(T/100)L“], which yields £4 = 4, 5, 6 and £12 =
12, 14, 18 for T = 100, 200 and 500, respectively. When a =
0, the power against 3 = 0.9 of the ADF t-test with £4
augmentations is 0.118 for T = 100, 0.385 for T = 200 and
0.991 for T = 500. The 5“,test is more powerful at all
three sample sizes (with powers of 0.259, 0.691 and 0.992),
as are our other asymptotically valid tests. This is so
despite the fact that the ADF t-test with £4 augmentations
suffers from greater size distortions than the 5“,test.
The ADF t-test with £12 augmentations is even less powerful:
its power against 6 = 0.9 is only 0.058, 0.166 and 0.778,
for T = 100, 200 and 500, respectively. Chapter 2 reports
similar results for the augmented SP t—test. For example,
for T = 100, B = 0.9 and 9 = 0, the power of the augmented
SP t-test is 0.191 for with £4 augmentations and 0.125 with
£12 augmentations. Thus, if enough augmentations are used
to make the size distortions of the augmented DF and SP
tests small, the tests have very little power. Unlike the
case for the IV tests, the robustness of the augmented tests
in the presence of MA errors is purchased at the cost of a
considerable loss in power. This is not surprising, since
the augmented tests were really developed to be suitable in
the presence of autoregressive errors, while the IV tests
were specifically designed to be useful in the presence of

MA errors: in particular, they take advantage of the fact

71
that, with MA errors, error covariances at lags greater than

the order of the MA process equal zero.

4. CONCLUSION

In this chapter we have provided new IV tests for a
unit root. We extend to the Schmidt-Phillips (SP) testing
framework the IV techniques that Hall (1989) and Pantula and
Hall (1990) had previously applied in the Dickey-Fuller (DF)
framework. Like Hall's IV tests, ours are asymptotically
robust to MA errors. However, there are several advantages
to operating in the SP framework rather than in the DF
framework. Primary among these, in the present context, is
that in the SP framework the nuisance parameters that
reflect error autocorrelation enter only through scalar
multiplication of the asymptotic distribution that would
otherwise apply. As SP note, this makes Phillips-Perron
type corrections much more straightforward for the SP tests
than for the DF tests. As this chapter has shown, it also
makes IV versions of the SP tests much more straightforward
than IV versions of the DF tests. As a result we have been
able to provide asymptotically robust IV statistics based
22th on the estimated coefficient and on the t-statistic.
Furthermore, we have been able to interpret our IV
statistics as Phillips-Perron corrected SP statistics, using
unconventional but consistent (given MA errors) estimates of

the nuisance parameters. Neither the t-statistic versions

72
of the IV tests nor the Phillips-Perron correction
interpretation appear to be easy in the DF framework.

Our tests have generally smaller finite sample size
distortions than Hall's IV test. They are comparable in
their accuracy to the augmented DF (or SP) statistics with a
moderate to large number of augmentations. However,
compared to the augmented tests, they are much more powerful
against stationary alternatives. This constitutes the main
argument in favor of using our new IV tests.

Many authors since Wichern (1973) have pointed out
that, as the MA(1) parameter a approaches minus one, the
variable approaches stationarity (indeed, white noise) and
so it is natural that unit root tests should have
considerable size distortions with large negative MA
parameters. Indeed, it may therefore be natural to think in
terms of a trade-off between accuracy (lack of size
distortion) and power. However, our results show that this
viewpoint should not be taken too far. Some tests do
dominate others, in the sense that they can have smaller
size distortions with very highly autocorrelated errors and
higher power with iid errors: greater accuracy under the
null is not necessarily purchased with lower power under the
alternative.

An important extension of our work is to allow for more
general ARMA errors. Pantula and Hall (1991) have taken a

step in this direction, in the DF framework, by using

73
augmentations to handle the AR component of the errors and
IV estimation to handle the MA component. However, this is
awkward in the DF framework, because augmentations make the
t;§;gti§t1§ test robust to AR errors, while IV estimation
makes the gggffiigien; test robust to MA errors. Making the
coefficient test robust to the AR component of the errors
requires multiplication by a function of the sum of the
coefficients of the lagged difference regressors. However,
in this chapter we have shown how to construct the IV t-
statistic tests in the SP framework. Thus it is potentially
straightforward to construct IV-based t-statistic tests that
should be robust to ARMA errors. This will be the subject

of future research.

100 1
10
200 1
10
500 1
10
1000 1

10

-1607
“12.0
“9.82

“18.3
“12.8
“10.4

“19.3
“13.2
“10.6

“19.5
“13.5
“10.7

“20.3
“13.6
“10.8

“20.2
-1306
“10.8

Critical Values

‘11

“3.40
“2.71
“2.37

“3.29
-2066
-2036

“3.24
“2.64
“2.35

“3.19
“2.63
“2.34

-3.21
“2.63
“2.34

“3.20
-2062
“2.34

74

TABLE 1

Pw

-25.9
“15.3
“11.6

“22.1
“14.3
“11.2

-20.9
“14.0
“11.0

“20.2
“13.7
.71-009

“20.5
“13.7
“10.9

“20.4
-1307
-1008

TN

“2.59
“2.25
-2004

“2.97
-2047
“2.20

“3.10
“2.55
“2.27

“3.14
“2.58
“2.30

“3.16
“2.61
“2.32

“3.17
“2.61
“2.32

6.36
4.99
4.12

8.39
5.90
4.76

9.23
6.36
5.10

9.50
6.56
5.25

9.97
6.74
5.35

10.0
6.80
5.37

-1108
“9.00
-7o49

“15.0
“10.9
“8.96

-1702
“12.2
’9088

“18.3
“12.8
“10.4

“19.6
“13.4
“10.6

-1909
“13.5
“10.7

-3008
“22.4
“18.8

“30.2
“22.1
“18.5

“29.8
“21.7
“18.2

25 -

50 -

100 -

200 -

500 -

1000-

.8

.0
.5
.8

.8
.5
.0
.5
.8

.8
.5
.0
.5
.8

.8
.5
.0
.5
.8

.8
.5
.0
.5
.8

.8
.5
.0
.5
.8

75

TABLE 2

5% Sizes of the IV tests

In the Presence of MA errors

(Asymptotic Critical Values Used)

5

.730
.417
.049
.004
.002

.931
.553
.048
.004
.002

.987
.632
.051
.005
.002

.997
.658
.051
.003
.002

1.00
.681
.052
.004
.002

1.00
.686
.049
.003
.002

7

.726
.413
.048
.004
.002

.931
.552
.047
.004
.002

.987
.628
.050
.005
.002

.997
.656
.050
.003
.002

1.00
.678
.051
.004
.002

1.00
.684
.048
.003
.002

p“

.288
.164
.093
.085
.085

.266
.114
.067
.065
.066

.246
.092
.062
.060
.060

.226
.075
.054
.052
.052

.189
.061
.052
.053
.054

.160
.055
.050
.050
.050

‘11

W

.004
.001
.004
.044
.068

.003
.003
.029
.141
.179

.003
.002
.039
.185
.223

.002
.001
.043
.201
.239

.000
.000
.047
.212
.249

.000
.000
.049
.216
.254

.023
.004
.000
.001
.001

.071
.033
.023
.025
.024

.107
.040
.036
.038
.038

.124
.044
.042
.041
.041

.128
.045
.047
.047
.047

.119
.046
.047
.047
.048

.007
.005
.009
.022
.027

.029
.016
.024
.037
.039

.062
.026
.037
.046
.048

.093
.036
.042
.045
.045

.111
.042
.048
.050
.050

.109
.044
.048
.049
.049

.020
.011
.009
.014
.015

.068
.032
.024
.029
.030

.116
.042
.036
.040
.041

.141
.048
.042
.043
.042

.141
.048
.047
.048
.048

.131
.048
.047
.047
.048

.012
.002
.004
.024
.034

.003
.000
.015
.145
.199

.000
.001
.036
.242
.301

.001
.000
.041
.273
.331

.000
.000
.046
.290
.349

.000
.000
.049
.297
.360

50

100

200

500

76

TABLE 3

5% Sizes of the IV tests

In the Presence of MA errors

(Finite Sample Critical Values Used)

.730
.417
.049
.004
.002

.931
.553
.048
.004
.002

.987
.632
.051
.005
.002

.997
.658
.051
.003
.002

1.00
.681
.052
.004
.002

1.00
.686
.049
.003
.002

.726
.413
.048
.004
.002

.931
.552
.047
.004
.002

.987
.628
.050
.005
.002

.997
.656
.050
.003
.002

1.00
.678
.051
.004
.002

1.00
.684
.048
.003
.002

Pw

.241
.121
.051
.042
.042

.245
.095
.049
.047
.046

.235
.081
.051
.051
.051

.223
.071
.051
.048
.048

.187
.059
.051
.051
.051

.158
.054
.048
.049
.049

.009
.007
.049
.199
.248

.006
.006
.049
.202
.244

.004
.003
.050
.212
.253

.002
.001
.049
.216
.256

.000
.000
.050
.218
.257

.000
.000
.049
.213
.250

.079
.048
.052
.057
.057

.108
.055
.050
.052
.052

.125
.052
.051
.054
.054

.137
.051
.051
.050
.050

.131
.049
.051
.052
.052

.121
.048
.048
.049
.049

.034
.028
.050
.089
.097

.050
.033
.050
.069
.072

.077
.038
.050
.062
.063

.102
.042
.051
.055
.055

.115
.045
.051
.053
.054

.109
.045
.048
.050
.050

.070
.048
.052
.067
.070

.102
.053
.050
.057
.059

.137
.054
.051
.056
.057

.154
.055
.051
.052
.052

.147
.051
.051
.052
.053

.132
.049
.048
.049
.049

‘1)

W

.030
.007
.049
.267
.351

.005
.001
.050
.282
.348

.001
.001
.051
.295
.357

.002
.000
.050
.299
.355

.000
.000
.050
.302
.361

.000
.000
.050
.301
.364

50

100

200

500

77

TABLE 4

5% Powers of the IV tests (p=0.9)

bl

.741
.474
.066
.006
.003

.948
.745
.113
.011
.007

.997
.949
.293
.036
.021

1.00
.998
.744
.228
.156

1.00
1.00
.994
.914
.879

1.00
1.00
1.00
.997
.995

In the Presence of MA errors

‘11

.738
.470
.065
.006
.003

.947
.744
.113
.011
.007

.997
.948
.288
.035
.020

1.00
.998
.742
.226
.154

1.00
1.00
.993
.912
.877

1.00
1.00
1.00
.997
.995

Pw

.255
.139
.066
.058
.057

.279
.158
.108
.107
.107

.300
.253
.268
.279
.279

.353
.477
.698
.731
.733

.445
.874
.989
.992
.992

.525
.989
1.00
1.00
1.00

TW

.011
.007
.060
.245
.303

.007
.008
.093
.358
.428

.005
.009
.205
.656
.725

.007
.013
.574
.942
.962

.006
.045
.976
1.00
1.00

.005
.199
1.00
1.00
1.00

;2
.084
.057
.066
.078
.077

.121
.091
.106
.120
.117

.165
.164
.264
.292
.292

.227
.372
.695
.738
.740

.343
.827
1.00
.992
.992

.445
.985
1.00
1.00
1.00

O»
.035
.032
.065

.118
.129

.054
.052
.101
.147
.155

.100
.119
.259
.319
.327

.169
.319
.691
.750
.755

.298
.804
.992
.992
.992

.415
.983
1.00
1.00
1.00

2
H’

.073
.056
.066
.092
.096

T

.112
.089
.105
.128
.130

.173
.174
.264
.299
.303

.250
.397
.696
.743
.746

.370
.842
.992
.992
.992

.468
.986
1.00
1.00
1.00

.032
.007
.044
.277
.374

.007
.001
.060
.369
.458

.002
.003
.131
.619
.710

.003
.005
.419
.956
.977

.005
.016
.996
1.00
1.00

.007
.100
1.00
1.00
1.00

78
APPENDIX

In this Appendix, we prove that, under the assumptions

of Theorem 1,

1. -02/2 , k 2 (1+1
(A.l) pll'm T'1 2 S'HrAfi -

““1 “OZ/2 , k s q .

Here q is the order of MA error process, 02 is the long-run

variance, and 0‘2 is the innovation variance.
Since SM = Sb1 - (t-2)7 - 61 from (17) in Schmidt and

- 7, we obtain

0 0 t ~
Phllllps (1990), where St =j216j, and AS: = e:

T T
'1 ~ ~ _ '1 - - - - - - -
(Ii-2) T k2‘11st13t - T 1315... (t k 1)£ ¢-:,][¢st e]
‘ T 1 T t k ‘ T1; 3 "
=T’ZSe-T‘Z --lce-‘ _e+
k+l "" ' k+l( ) ‘ k+l ‘ "

1 T "
T‘ 2 (t-k-1)e2 .
k+1

First of all, we can show that, for k 2 (1+1:

T k-l
(IX-3) T" 2 2 art,
k+l j-l ’

= 0.5 {T'1[osk,1-i~--'i-6T]2 - T"[€,.12+- +6.21}

'1‘ 'I‘
- T" 2 z: Et_.£t
k+1j-k ’

.. 0.5 [a2 - of] .

Here, we have applied the AM restriction, Mamet) = 0

79

2 k 2 q+1, to show that

TT
T": 26

e -» o .
k+lj-k "1 '

This reflects the fact that error covariances of order

higher than the order of the MA process should be zero.

But, this g_prigri restriction cannot be imposed, when k 5

Now, each term in (A.2) can be calculated as follows:

(i) The first term:

4 T 2 2 2
For k S q, T k24:48“;t ~+ 0.5 a [W(1) - 0‘]

from (A3.1) of SP (1990).

4 T 4 T 4 ksl
For k .>. q+1, T kHHS“;t = T 1318,46, - T J‘zt'ilewet

.. 0.5 [0291(1)2 - of] - 0.5 [a2 - of]
= 0.5 O‘Z[W(1)2 - 1].
(ii) 'I'"§ (t-k—1)Ee = (TVZE) TM; (t-k-l)e
k+l ' k+l '
2 2 2 1
-» aW(1) - oW(1)LW(r)dr

T T
' ' ° -1 ‘ _ 1/2“ 812
(111) T k243b,; — (T e) T 2; St...

k 1
.. 02W(l)]:W(r)dr

(iV) T" 7: (t-k-l)?Z = (TVZE)z T’2 T(T-1)/2 + o (l)
k+1 p

-> 0.5 02W(l)2

80

Therefore, for k 2 q+1,

T .. ..
T ‘13".flst_k2\st -» 0.502[W(1)2-1] - 02W(1)2 +

02W(l)j:W(r)dr - 02W(1)L1W(r)dr + 0.5::2m1)2

= - 0.502

I

and for k S q,
4 T “ " 2 2 2 2 2
T kfilst'kAS‘ -v 0.50 [W(1) -a‘ ] - aW(1) +

02W(1)]:W(r)dr - 02W(1)]:W(r)dr + 0.5::2wu)2

= - 0.5qf. D

81

LIST 0? REFERENCES

Bhargava, A. (1986), "On the Theory of Testing for Unit

Roots in Observed Time Series," Bgyigw_gfi_figgngmig
Studies. 52. 369-384-

DeJong, D.N., J.C. Nankervis, N.E. Savin and C.H. Whiteman
(1989), "Integration versus Trend Stationarity in
Macroeconomic Time Series,” Working Paper 89-99,
Department of Economics, University of Iowa.

Dickey, D.A. and W.A. Fuller (1979), "Distribution of the
Estimators for Autoregressive Time Series with a Unit

Root," Journal of the American Statistical Association,
74, 427-431.

Fuller. W-A- (1976). Intr2du2ti2n_t2_§teti§tigal_lime
Series, New York: Wiley.

Hall, A. (1989), "Testing for a Unit Root in the Presence of
Moving Average Errors," fiigmgtzikg, 76, 49-56.

Hall, A. (1990), "Model Selection and Unit Root Tests Based
on Instrumental Variable Estimators," unpublished
manuscript, Department of Economics, North Carolina
State University.

Kim, K. and P. Schmidt (1990), "Some Evidence on the .
Accuracy of Phillips-Perron Tests Using Alternative
Estimates of Nuisance Parameters," Economig§_Lg§§§z§,
34, 345-350.

Lee, J. (1990), "Finite Sample Performance of Schmidt-
Phillips Unit Root Tests in the Presence of
Autocorrelation," Econometrics and Economic Theory
Paper No. 8817, Department of Economics, Michigan State

University.

Pantula, S.G. and A. Hall (1991), "Testing for Unit Rootslin
Autoregressive Moving Average Models: An Instrumegzg-
Variable Approach," on a co om c p 48:

353.

Phillips, P.C.B. (1987), "Time Series Regression with Unit
Roots," Econometrics. 55. 277-301-

"Testing for a Unit

Phillips, P.C.B. and P. Perron (1933)! 75' 335-

Root in Time Series Regression," nigmetrika.
346.

82

Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T.
Vetterling (1986), l e °
Scicntific_ccmcutinc

, Cambridge: Cambridge University
Press.

Said, S.E. and D.A. Dickey (1984), "Testing for Unit Roots
in Autoregressive - Moving Average Models of Unknown

Order," Bigmgtzikg, 599-608.

Said, S.E. and D.A. Dickey (1985), "Hypothesis Testing in
ARIMA(p,1,q) Models," 0
Stcticticcl_8ccccicticn. 80. 369-374.

Schmidt, P. and J. Lee (1991), "A Modification of the
Schmidt-Phillips Unit Root Test," forthcoming,

Eccncmicc_Lcttct§-

Schmidt, P. and P.C.B. Phillips (1990), "Testing for a Unit
Root in the Presence of Deterministic Trends,"
Econometrics and Economic Theory Paper No. 8904,
Department of Economics, Michigan State University.

Schwert, G.W. (1989), "Tests for Unit Roots: A Monte Carlo
Investigation," a 0 us' ess co 0

Stcticticc. 7. 147-159-

Wichern, D.W. (1973), "The Behavior of the Sample
Autocorrelation Function for an Integrated Moving
Average Process," Biometrika, 60, 235-239.

Illi’.i"llfr‘i I Ill, Iii;

.Il-l'l Elli-L. .l‘

CHAPTER 5

CHAPTER 5

STRUCTURAL CHANGE AND PARAMETERIZATION PROBLEMS
IN UNIT ROOT TESTS

1. INTRODUCTION

Following the pioneering work of Nelson and Plosser
(1982), a large number of studies have failed to reject the
hypothesis of a unit root in many macroeconomic time series.
These studies have mostly been based on one variant or
another of the Dickey-Fuller (1979, hereafter, DF) tests.
However, other tests generally give more or less the same
results. Specifically, the unit root tests recently
proposed by Schmidt and Phillips (1990, hereafter SP) lead
to the same conclusions as the DF tests when they are
applied to the Nelson-Plosser data.

A contrary conclusion is reached by Perron (1989), who
finds that most of the Nelson-Plosser series are trend
stationary if allowance is made for a structural break in
1929, at the beginning of the Great Depression. He shows
that the DF tests are biased toward accepting the null when
there is a structural change, and when he allows for such a
change, he finds eleven of the fourteen Nelson-Plosser time
series to be trend stationary. From this perspective, the
persistent effects of the great crash make the series appear

to have a unit root, when in fact they do not. He also

83

84
reaches similar conclusions for more recent quarterly
series, treating the oil price shock of 1973 as an exogenous
structural break.

Banerjee, Lumsdaine and Stock (1990) and Zivot and
Andrews (1990), among others, have criticized Perron for
treating the structural break as exogenous and its time of
occurrence as known a_pzigzi (independently of the data).
They show that the asymptotic critical values for the unit
root test must be modified if the structural change occurs
at an unknown time, so that its placement is decided from
the data. When this is done, the null hypothesis of a unit
root is no longer rejected.

In this chapter Perron's result is also reversed, but
in a different way. It is shown that his results disappear
if we use a suitably modified version of the SP unit root
tests instead of the DF tests. This chapter shows this
result empirically for the Nelson-Plosser series, and also
provides theoretical results that explain why these
empirical results are not surprising. Those results support
the argument advanced by SP that the DF parameterization is
"clumsy", in the sense that it handles level, trend and
other relevant variables in a needlessly complicated and
potentially confusing way. Allowing for a one-time
structural break affects the distributions of the DF
statistics under the null, even asymptotically, and allowing

for a break at an unknown time affects the asymptotic

Wu

85

distribution even more. These difficulties simply do not
arise in the SP framework. A one-time structural break does
not affect the asymptotic distributions of the SP tests, and
this is true irrespectively of whether the break is allowed
for in the model or not. This constitutes a powerful reason
for preferring SP tests to DF tests in the presence of
possible structural change.

The plan of the chapter is as follows. In Section 2
the DF and SP parameterizations are presented, and the
relevant asymptotic results for the SP tests in the presence
of a possible structural break are provided. In Section 3
the SP tests are applied to the Nelson-Plosser data and
Perron's conclusions are reversed. In Section 4 we consider
and test the common factor restrictions that are implied by
the SP model; they are not rejected. Section 5 gives the
concluding remarks. Throughout this chapter, "a" indicates

weak convergence as T 4 w.

2. THEORETICAL RESULTS

To allow for structural change, Perron (1989) considers
three kinds of models. Here, we will examine only his first
crash model, which allows for a one-time change in the level
of the variable's time path. We consider Perron's crash

model using the following data generating process (DGP):

(1) yt = 51 + 62t + 539“: + xt , xt = pXt_1 + at.

86
where DUt = 1 for t 2 TB+1 and zero otherwise, and D(TB)t = 1
for t = T3+1 and zero otherwise, while.Tg stands for the
time period when the structural change breaks out. Note
that D(TB)t = ADUt = DUt - DUN. The unit root hypothesis is

p = 1, and this is the null hypothesis to be tested. Note

that this DGP is consistent with Perron's crash model:

(2a) Null yt p + d-mTB)t + y“ + vt

(213) Alternative yt p, + fi-t + (1.42-u1)-DUt + vt .

This is so, since equation (1) implies equation (2a) under
the null (p = 1) with p -- 62, d = 63, yo = 61+ x0, and vt =
a}; and it implies equation (2b) under the alternative (p <
1) with u, = 61, fl = 62, (p2 - #1) == 63, and vt = Xt. More

generally, the DGP (1) implies:

(3) Y: = 71 + 72yt4 + 73t + 1,.DUt + 75001.8): + 6t
where

(4a) 71 = 51(1-p) +629

(4b) '72 = P

(40) 73 = 52(1‘9)

(4d) "n, = 53(1‘9)

(43) '75 = P63 0

Also, note that equation (3) is Perron's nested equation
(12) which includes the null and the alternative models. It

is important to notice that a unit root imposes three

re

tw

87

restrictions on (3): 12 = l, 13 = 0, and 1,. = 0. The first
two restrictions are the usual unit root restrictions, and
the last restriction is the common factor restriction (CFR)
to be discussed in Section 4. Perron tests only the first
unit root restriction (p = 1), ignoring the two additional
restrictions. The missing restrictions can potentially lead
to quite different inference, as discussed in Section 4. It
could well be that the unit root hypothesis is true, but
that the other restrictions do not hold, or vice-versa.

It is typical of the DF regressions that they include
variables which are relevant under the alternative but not
under the null. Here, the dummy variable (DUt) representing
changed level and the time trend variable are variables that
belong in (2b) but not (2a), and thus are relevant in (3)
under the alternative, but not under the null. The
inclusion of these variables affects the asymptotic
distributions of the DF-type unit root statistics, and so
Perron's Theorem 2 shows that the asymptotic distributions
of his test statistics depend on A ==1%/T even under the
null.

This problem does not arise in the SP framework. We

may consider the SP-type test in the structural change model

by writing (1) as:
(5) yt=¢+zt6+xt, Xt=pxm+et

where y, = 51, 2t = [(3, out], and 6 = [62, 631'. The unit root

 

 

88
restriction is p = 1. According to the EM (score)

principle, the restricted MLE's of 6 and 45(1- (6 + X0) are

given by:
(6a) 3 = coefficients of the regression of 13yt on AZt
(6b) wbx = y1 - Z16 .

Then the SP-type test statistics are obtained from the

following regression:

 

(7) AY} = AZty + ¢§p4 + error

- - _ t ,
where St a yt - 2px - Zt6 = j€410in - AZj6), t = 2,..,T. The

test statistics are accordingly defined by:

T-d
t-statistic for the hypothesis ¢ = 0.

‘b(
ll

(8a)

A:
II

(813)

See SP for more details.

To establish the asymptotic distributions of the SP-

type tests that allow for structural break, we need the

following assumption.

Aim The data are generated according to (5), and

the innovations 6t satisfy the regularity conditions of

Phillips and Perron (1988, p.336).

Tim; Suppose that Assumption 1 holds, and p = l.

89

Let b and i be defined as in (8). Then,

 

1 03 1
(9a) 8’” “757;;1
fflfl’dr
0
1 0' 1
(9b) -' ’37 1 1/2
fiflrfidr]
0

 

where y(r) = V(r) - L1V(r)dr is a demeaned Brownian bridge,

and V(r) = W(r) - rW(1) is a Brownian bridge: where W(r) is

the Wiener process on [0,1]; and where the nuisance

parameters a2 and 0‘2 are defined as in Phillips and Perron

(1988).

Proof: See Appendix.

Note that the above asymptotic distributions are the
same as those of the usual SP tests (hereafter the "usual"
SP or DF tests refer to those which do not allow for the

break), as given in equations (20) and (21) of Schmidt and

Phillips (1990). So, unlike the case in Perron's model,

allowing for structural break at a single known time does

not affect the asymptotic distributions of the test

statistics under the null. The intuitive reason is that the

SP-type tests are based on a regression in differences, and

ADUt = D(TB)t equals one at only one point, so that its

inclusion has no effect asymptotically. Perron's DF-type

tests are based on a regression that includes DUt in levels,

 

 

90
and DUt equals one for a constant fraction (1-1) of the
sample even asymptotically.

Next, we turn to a question that was not asked in
Perron's paper. What will be the effects on the asymptotic
distributions of the usual DF and SP tests under the null
hypothesis, if there occurs a one-time structural break, but
it is ignored? The answer, perhaps surprisingly, is that

there will be no effect on either the DF or SP-type tests.

Thooxom_z Suppose that Assumption 1 holds, and p = 1 so
that the null hypothesis is true. Suppose that there occurs
a structural break at time To! and that Ta/T -> A as T -v on.
Then the asymptotic distributions of the usual DF and SP

tests are unaffected by the structural break.
Proof: See Appendix.

Theorem 2 indicates that the asymptotic distributions
under the null of both the usual DF and SP tests are not
affected by the presence of a one-time structural break.

Under the null hypothesis, it is unnecessary

(asymptotically) to worry about the presence of a one-time

structural break, and this is so for both the DF and the SP-

tYPe tests. Note that this result does not conflict with

our Theorem 1 and Perron's Theorem 2, which deal with the

question of whether asymptotics under the null are affected

if one does allow for a structural break. The changes

demonstrated by Perron in the null distribution of the DF

91
tests arise not from the presence of a structural break in
the DGP, but from the way in which the DF tests allow for
such a break. The SP tests allow for the break differently,
and their asymptotics under the null are unaffected by
either the presence of a break or by allowance for such a
break.

This raises an obvious question: why should we allow
for a break in performing the SP tests? The answer is that
we do so to increase the power of the tests. Perron shows,
in his Theorem 1, that when the alternative of stationarity
is true, the probability limit of the coefficient estimate
of the lagged dependent variable in the usual DF regression
depends on the magnitude of the structural change, and the
coefficient estimate gets closer to one, as this magnitude
increases. Thus, the usual DF tests will be biased toward
accepting the null. This fact is also true for the SP
tests. When the null is true, the usual SP tests are not
affected by the presence of the break, but when the
alternative is true, they are affected. The following
theorem formally shows that the equivalent of Perron's

Theorem 1 also holds for the usual SP tests.

Itcctcm_§ Suppose that Assumption 1 holds with p < 1.

Suppose that a structural break occurs at time T3! and that

Ta/T '* A as T .. co. Define

T
a 2 = lim T‘1t21Var(Xt)

T+w

92

‘1'
-1
7, 1TB: T tz_:11«:(xtxt,1)

_ 2
P1-71/Ox -

Then, if 5 is the coefficient of St,1 in the regression of

ay. on [1. em].

01( - 1)
(10) 6 I 2 1 2 2 p11 1
ax+—3— (X.+X1X.+X1) + (AZ-1+3)6§+(x_-X1) (AZ-.3.) 53+x153

Proof: See Appendix.

Theorem 3 shows that when the alternative is true (p <
1), the asymptotic distribution of the estimated coefficient
in the usual SP tests depends on the magnitude of the
structural change. In particular, a: -* 0 as 63 -v on, which
corresponds to a potential bias toward accepting the null.
Thus, ignoring the break reduces power. Therefore it is
better to allow for the structural break if one exists.
However, note that whether or not we allow for the break is
not a question of size of the SP tests. That is, if we put
the break in the wrong place or if we do not consider the
break at all, the size of the SP tests (under the null) is
not affected asymptotically: only power is affected. This
advantage does not hold for Perron's tests which are based

on the DF framework.

 

93
3. EMPIRICAL RESULTS

In this section, we now apply the SP tests with
structural change to the Nelson and Plosser time series
analyzed by Perron. Specifically, we analyze the same
eleven series to which Perron applied his crash model (A).

First, we calculate some critical values. Table 1
gives critical values calculated by simulation, using iid
normal errors, for various relevant values of T and T3“ The
simulation used 50,000 replications for each value of T and
I}. The FORTRAN subroutine program GASDEV/RAN3 of Press ot
oi; (1986) has been used for generating the iid N(0,1)
random numbers.

These critical values are almost same as those of the
usual SP test statistics. For example, the 5% critical
values of the usual SP statistics 5 and i are -17.5 and -
3.06 respectively, when T = 100. These numbers are quite
similar to -17.6 and -3.06, which are the critical values of
the statistics )3 and i in (8) , when T = 102 with T3 = 61.
This similarity is to be expected from Theorem 1, for large
sample sizes, and it is comforting to see in sample sizes
found in the data we consider, since it reinforces our faith
in the asymptotic theory.

To allow for autocorrelated errors, we consider the
augmented versions of the SP tests which include the
augmented terms ASP}, j = 1,..,k in regression (5). We will

consider the augmented t-test version only, not the

 

94
coefficient version. The augmented tests critically depend
on the choice of k. In this chapter, the values of k are
determined as the same numbers that Perron selects, so that
differences between our results and Perron's will not be due
to differences in the choice of k.

As it was mentioned earlier, only the first crash model
(A) of Perron is considered. The natural logarithm has been
taken for all of the series except for the "interest rate"
series. Table 2 presents the test results. They show that
at the 5% level, we can reject the null hypothesis of a unit
root only for two series: the "money stock" and "employment"
series. At the 1% level, we cannot reject the null
hypothesis of a unit root for any of the Nelson-Plosser time
series considered here. The striking results of Perron
disappear.

The overall results of the SP tests indicate that the
"unit root" model is more relevant for the Nelson-Plosser
macroeconomic time series than the "trend-stationary“ model,
contrary to Perron's results, even when we consider
structural change in the model.

As the discussion of the last section indicates, we
note, as a practical implication, that if the null is true,
allowing for the break should change the DF test statistics,
but should not change the SP test statistics much. This is
exactly what we find in the data. The usual DF test

statistics ignoring the break are quite different from the

95
DF test statistics allowing for the break. For example,
when the same number of augmentations is chosen, the usual
augmented DF t-statistics are -2.27 and -2.47 for the "real
GNP" and "GNP deflator" series, while the augmented DF t-
statistics allowing for the break are -5.03 and -4.04 for
the same time series. On the other hand, for the same time
series, the augmented SP statistics allowing for the break
are -1.96 and -2.29, while the SP statistics ignoring the
break are -1.91 and -2.23. Comparing the two sets of SP
statistics, we note that they are not quite different. Thus
the pattern in the DF and SP statistics with and without
allowance for structural break is exactly as the asymptotic

theory would predict, if the null were true.

4 . COMMON FACTOR RESTRICTIONS

It is well known that AR models imply common factor
restrictions (hereafter, CFR); for example, see the classic
paper of Hendry and Mizon (1978) . For instance, if yt = 2,6

+ ut, ut = pu + 6t, then

P1
(11) Yt = Pyt-1 + ZtS + Zt-1(-P6) + 6t'

Thus the coefficients of Zt,1 contain no new parameters. The
coefficients of y“, 2t and zt,1 are subject to nonlinear
restrictions: (p)(6) + (-p6) = 0. These are CFR's, and they

are testable. In general, if zt contains k variables, there

are k CFR's.

l.

 

96

The same principle applies, with some modification, to
our model. Our equation (1), or (5), is an AR(1) model, and
Perron's "nested“ equation, our equation (3), is the
unrestricted equation that corresponds to (11). In our
model, zt contains two variables, t and DUt. The time trend
variable is special in the sense that it does not generate a
CFR; given intercept and time trend, the lagged time trend
variable is redundant (perfectly collinear). The variable
[Mk does generate a CFR: the coefficient of DUb4tmust be
equal to the negative of the coefficient of yt,1 times the
coefficient of DU,. Recall that D(TB)t = ADUt, and note that
(3) actually includes DUt and D(TB)t. This is equivalent to
inclusion of DUt and DUN, but it changes the form of the

CFR slightly, so that it becomes:
(12) 121‘ - 75(1-12) = 0.

There are also two unit root restrictions:
(13) 72 = 1, 13 = 0.

These restrictions transform the CFR in (12) into'1‘== 0.
This means that although the CFR is nonlinear, it becomes
linear when the unit root restrictions are imposed. The
validity of the CFR in its general form (12) does not depend
on the unit root restrictions. However, we cannot test the
CFR in (12) independently of the unit root restrictions, at

least based on asymptotics, since 75 in equation (3) cannot

97
be consistently estimated. The reason is that the regressor
D(TB)t is asymptotically degenerate. Therefore, we settle

for testing the joint restrictions of the unit root and the

CFR. Thus we wish to test
(14) H0: 12=1, 73=1‘=0.

Note that the SP test amounts to an LM test of 12==Jq
imposing the restriction 13== 0 and the CFR. Perron's test
is a test of 1} = 1, not imposing 73== 0 or the CFR. In

this section, we develop a generalized Wald statistic which
considers the joint hypothesis (14) consisting of the unit

root restriction and the CFR. To do so, we define:
(15) r(‘/’) = [r1(¢)l 1.200): r3(¢)]'
where

r1(¢) = '72 " 1'0
rzhl’) = '13
r3(¢) = '74.

'l’ = [72! '73: 71.1 751'

Also define:

iooo
R.__‘Ldr()-01oo

my 0010

98

and Xt = [yt_1*, t', DUt", ADUJ], where the asterisk (*)

represents deviations from means, and let X be the Tx4

matrix with t“ row Xr'

estimates in the regression of yt on Xt,

Lat ; '3 (;2I ‘;31 T4! ;5) be the OLS
t=1,..,T, and let

r($) be r(¢) evaluated at 3. Let a} be the usual error

variance estimate from the regression of yt on Xt.

Then,

the generalized Wald test statistic is defined as:

(16) E, =

Ih£2££!_1

hypothesis of a unit root (p =

is satisfied.

at time T3! and that TD/T -v A as T -v on.

1).

rmta 3.2(x'xr' R'J“r(£s)'

Let {Y}} be generated by (1) under the null
Assume that Assumption 1
Suppose that there occurs a structural break

Then, the

asymptotic distribution of the Wald statistic in (16)

follows:
_. 02 / H
(17) 3., ——2 E (3) E
a!
where
“312+31-1 H2 H1
12 ”1+ 2 2
H: _
H - 14.2 3

 

H2

+_____..
121-12).2

5&32 H2

- 1-12 2

H

H2--—3
1 12
1—12

1

 

 

99

and

B = H12 + H1112 + fizz/(1214212) + H3(-312+3i-1)/12

L1rW(r)dr - 0.5]:W(r)dr

3:
ll

:1:
N
II

LW(r)dr -A]:W(r)dr
H; = fo'wm2 - (fo'mmdrfi

E = [12,, F2, F3]'

F1 0.5[W(l)2-a‘2/02] — W(1)]:W(r)dr

F2 0.5W(l) - L'mndr

F3 W(i) - mu)

and where W(r) is a Brownian motion defined on [0,1]. D

Efoof: See Appendix.

The theorem shows that the asymptotic distribution of
the Wald test statistic depends on A, 02, 0‘2 and T. We
obtain 02 =- 0‘2 under iid errors so that we can tabulate the
critical values, as presented in Table 3. These are the
values that correspond to k = 0 in the table.

To allow for general innovations, one may use an
augmented version of the Wald test by including the
augmentation terms Aym, j = 1,..,k in (3) . We may presume
(though no proof will be given) that the standard asymptotic

results on augmentations apply. Specifically, if the number

 

100

of augmentations is made a suitable function of sample size
(e.g., k proportional to TV“), the asymptotic distribution
of the augmented test in the presence of innovations
satisfying Assumption 1 is same as (16), the asymptotic
distribution of the non-augmented test in the presence of
iid innovations. This should be so because the relevant
asymptotic covariance matrix is block-diagonal with respect
to the main regressors and the augmented terms. In small
samples, however, this distributional equivalence may not be
true. The simulated critical values for different values of
k show significant discrepancies in small samples. For
example, when the sample size is T = 62 (with the break
point T; = 21, thus A a .338), the simulated critical values
under iid errors are 16.4, 31.2 and 35.2 at the 5% level for
k = 0, 5 and 8 respectively. But when the sample size is
large enough, such differences disappear, as our conjectured
asymptotics predict. When T = 1000 (with T8 = 338, so A =
.338), the 5% critical values are 1.48, 1.42 and 1.37 for k
= 0, 5 and 8, for example. Therefore, as long as the
critical values are not same for the different values of k
in small samples, we will use the critical values
corresponding to the actual choice of k.

The results of the augmented Wald tests applied to the
Nelson-Plosser data are given in Table 4. Again we use the
same number of augmentations as Perron did. They show that

at the 5% level the joint hypothesis is rejected only for

101
the "nominal wage" and "industrial production" series. At
the 1% level, we cannot reject the joint hypothesis for any
of the time series except the "industrial production"
series.

The slight difference between these results and the
results of the SP-type tests in the last section is that
rejections of the null hypothesis occur for different series
using different tests. For the augmented Wald test, at the
5% level, the joint null hypothesis of a unit root and the
CFR is rejected only for the "industrial production" and
"nominal wage" series; the SP test does not reject the unit
root null for these series. For the SP test, at the 5%
level, the null hypothesis is rejected only for the "money
stock" and "employment" series, for which the Wald test did
not reject the joint null of a unit root and the CFR.
Overall, the results of the augmented Wald test are similar
to the results of the SP tests: most of the Nelson-Plosser
time series are nonstationary. This is favorable evidence
that the restrictions that the SP tests impose are in fact
not rejected by the data.

The strong dependence of the critical values on k is a
considerable disadvantage of the augmented Wald test. It
presumably indicates that the relevant asymptotic theory
does not apply well at the sample sizes found in the Nelson-
Plosser data. From a more pragmatic vieWpoint, the choice

of critical values matters: using the "asymptotic" values

102

for k = 0 would cause the Wald test to reject for most
series. Therefore, we alternatively consider the
"corrected" Wald test which is based on a Phillips-Perron
type correction of Phillips and Perron (1988). Define S2
and 82(2) as in (27) and (28) of SP, using'gtias the
residuals from a unrestricted regression on (3): these are
consistent estimates of the innovation variance cf and the
long-run variance 02 that appear in ( 17) . Define the
corrected Wald test by:

$2

(18) E; 2 r‘h‘mn 33mm" R'J"r‘({3)'
s (a)

 

where

rm?) = r075) — i i,‘ D,*[R(X'X)R'1
1} = 0.5[sz(e) - 37-]

i3 = [1, o, 01'

n,‘ = diagonal[T, Tm, T1’2].

Then the asymptotic distribution of the corrected Wald test

is given in the next theorem.

W Assume that the conditions in Theorem 4 are

satisfied. Then, the asymptotic distribution of the

corrected Wald test in (18) follows:

103

(19) 6‘,» E"(-§) E"

where E‘ = [F1, F2, 1231' with F,‘ = 0.511(1)2 - 0.5 -
W(1)j:W(r)dr, and with F2, F3, H and B as defined in Theorem

4.
gfoof: See Appendix.

The theorem indicates that the asymptotic distribution
of the corrected Wald test does not depend on 02 or of.
Furthermore, it is the same as that of the uncorrected Wald
test under iid errors. Thus, the critical values for k = 0
in Table 3 can be used.

The results of the corrected Wald test applied to the
Nelson-Plosser data are also provided in Table 4. We supply
the results for Z = 4 and 12, which are popular choices for
data sets of this size. The results again indicate that we
cannot reject the joint hypothesis of a unit root and the
CFR at the 5% level for all of the Nelson-Plosser series
except two: the "industrial production" and "GNP deflator"
series. At the 1% level, the null is rejected only for the
"industrial production" series. These results are basically
similar to those of the augmented Wald tests with the minor
difference that once again the null is rejected for
different series using different tests. But, most of the
time series considered here remain nonstationary. Once

again the overall results agree with those provided by the

104
SP tests; namely, that most series appear to contain a unit

root.

5. CONCLUDING REMARKS

In this chapter we have adapted the SP unit root test
to allow for a one-time structural change. This is
analogous to Perron's adaptation of the DF unit root tests.
However, we find important advantages to operating in the SP
framework rather than in the DF framework. Most
importantly, allowing or not allowing for a one-time
structural change does not affect the asymptotic
distribution theory for SP statistics under the null
hypothesis. As a result, the asymptotic validity of the SP
tests is not affected by possibly incorrect placement of the
structural break, or by allowance for a break when there is
not a break (or vice versa). These findings reflect and
reinforce the SP argument that the DF parameterization is
clumsy, in the sense that explanatory variables (such as
level, trend, or in this case a structural change dummy) are
treated in a statistically inconvenient way.

When the modified SP tests are applied to the Nelson

and Plosser data, Perron's striking results favoring the

trend-stationary model disappear. Most U.S. time series now

appear to have a unit root even when structural change is

allowed for. In fact, the effect of allowing for structural

change in the SP framework is minimal, and this is as

105
predicted by the asymptotic theory if the null is true.
Finally, the common factor restrictions implied by the SP
model appear to hold in these data.

Phillips (1988) and others have argued that models
(such as Perron's) that allow for exogenous structural
change actually support the unit root hypothesis, in the
sense that the innovation that is labelled as structural
change is obviously persistent. That is, the exogeneity of
the structural change can be argued to be questionable.
Models that endogenize structural change have been
considered in the DF framework by Zivot and Andrews (1990)
and Banerjee of_o_. (1990). Similar extensions of the SP
model with structural change are presumably possible, and

this is a topic for further research.

Critical Values of the Schmidt-Phillips Tests

106

TABLE 1

With Structural Change

T To
62 21
71 3o
32 41

102 51

111 70

1.0%
2.5%
5.0%

1.0%
2.5%
5.0%

1.0%
2.5%
5.0%

1.0%
2.5%
5.0%

1.0%
2.5%
5.0%

“17.52

-23.82
-20.40
-17.60

-24.29
-20.68
~17.81

-23.53
-20.63
-17.73

-24.26
-20.54
-17.83

-3.69
-3.37
-3.09

-3.68
-3.36
-3.08

-3.69
-3.36
-3.09

'3.66
'3.32
'3.06

-3.63
-3.31
-3.06

**
*

107

TABLE 2

Schmidt-Phillips Tests
With Structural

Series

Real GNP

Nominal GNP

Real Per Capita GNP
Industrial Production
Employment

GNP Deflator

CPI

Nominal Wage

Money Stock

Velocity

Interest Rate

T

62
62
62
111
81
82
111
71
82
102

71

for a Unit
Change
T} k
21 8
21 8
21 7
70 8
40 7
41 5
70 2
30 7
41 6
61 1
3O 2

significant at the 2.5% level
significant at the 5% level

Root

~44

-1.90
-2.38
-2.26
-2.83
-3.24*
-2.23
-l.78
-3.04
-3.60**
-1.74

-1.12

 

 

108

TABLE 3

Critical Values of the Wald test

T TB k 1% 2.5% 5% 10%
62 21 0 22.1 18.8 16.4 13.9
5 42.9 36.4 31.4 26.6

8 46.4 39.3 33.9 28.4

71 30 o 18.6 16.0 14.0 11.9
5 34.5 29.9 26.2 22.3

8 38.0 32.6 28.3 24.0

82 41 o 15.7 13.5 11.8 10.0
5 28.7 24.8 21.5 18.3

8 30.5 26.1 22.8 19.3

102 61 o 12.3 10.6 9.3 8.0
5 21.3 18.7 16.5 14.1

8 22.0 19.2 16.8 14.4

111 70 0 11.3 9.8 8.6 7.3
5 19.6 17.0 15.0 12.8

8 20.0 17.3 15.3 13.0

Results of Wald Test for a Unit Root

109

TABLE 4

with the CFR

 

 

 

 

 

 

 

 

Augmented Corrected
Series T TB ~—— .
k (=4 (=12
Real GNP 62 21 8 26.90 9.99 8.61
g 1
Nominal GNP 62 21 8 30.13 13.25 15.08 1
Real Per Capita 62 21 7 16.86 7.97 6.64
GNP
Industrial 111 70 8 33.18"“ 18.97“" 18.40“"
Production

Employment 81 40 7 20.89 10.27 9.37
GNP Deflator 82 41 5 16.67 13.61“ 13.64“
CPI 111 70 2 4.28 2.74 2.76
Nominal Wage 71 30 7 29.70‘ 12.73 12.86
Money Stock 82 41 6 18.67 10.00 10.39
Velocity 102 61 1 5.39 6.52 7.79
Interest Rate 71 30 2 7.52 9.94 10-19
*** significant at the 1% level

** significant at the 2.5% level

* significant at the 5% level

 

110

APPENDIX

ASYMPTOTICS

x t , we have

ut = ASt = Ayt - AZt6 = 6t - AZt(6-6)
- t - t -
St =j§2[€j - AZj(6-6)] = 7:26] - (Zt-Z1) (5‘6) .

t
Letting St =j22¢aj and [rT] be the integer part of rT, for r

5 [0,1], we have

(A-l) T'Vzétm = T‘Wsm] - ([rT]-1)T".T'/2(62 - 62)

- T'1(DU[m - 001) -T'/2(33 - 63).

We consider the three terms on the right hand side of

(A.l) separately. First,

-1/2
(A.2) T s,m -» 0W(r),

a standard result. For the second term, note that AZt = [1,

D(TB),], and D(TB)t equals zero except when t = Tait-1. Thus
" - - 12 7 _

(52 - 62) is asymptotically equivalent to e, and T/ (62 62)

7 “W(l). Therefore,

(11.3) ([rT]-1)T-1.T1/2(32 - 62) -» arW(1) .

111

For the third term, we have

0 if r < A
DUtrTJ - DU1 -' { .
1 if r 2 A .

We also have
(33 - 63) = [D(TB)'M1D(TB)]'1D(TB)'M16

with D(TB) a Txl vector of zeros except for a one in
position T3+l, and with M1 = I -i(i'i)"i' = I - T"i'i, where

i is a vector of ones. Then,
D(TB) 'M1D(TB) = 1 - T"
D(TB) 'M16 = 51m - e
T1’2(33 _ 63) = [1-T'11'1T1’2(em+1 _ g)
e - aW(1) .
Thus,
(1L4) T"(DU[m - DU,)T‘/Z(6'3 - 63) .. 0
Combining (A.2), (A.3) and (A.4), we get
4 0[W(r) - rW(1)] = aV(r),

'1/2~
(A.5) T SIrTJ

where V(r) is a Brownian bridge. Then, similarly as in

SChmidt and Phillips (1990, p. 24), we obtain

- 2 2
(A.6) T2t§2(st"2 - s) .. a foyr) dr

112

T _
-1 " - ~ ~ - 2' - 2
T go‘s“ S) (ASt AS) -> 0.50‘

1 .
where _V_(r) = V(r) - foV(r)dr is referred to as a demeaned
Brownian bridge. Therefore, the result in (9a) can be
obtained from the ratio of the above expressions, and the

result in (9b) similarly can be obtained.

W
The DGP is, similarly as in (1):

 

(11.7) yt = 4 + 5t + 63DUt + xt, xt = pr + at,

where p = 1 under the null. When the usual SP test
statistics (ignoring the break) are calculated from data

following this DGP, we have

Ayt = f + 63I)('.'I'.'B)t + 6,:

~

5 = mean Ay = 6 + T463'+ 3

and
133. = (yt - Y1) — é(t-1)
= (xt - x1) + 531mt - (E - ()(t-i)
= €253 - (t-1)'e' + 530”: - (t-l)63/T
Then,
T'Wé = T"’2 [rifle] _ ([rT]-1)T'1°T1IZE + T‘1’263DUt

[TI] 3'2

113
- ([rT]-1)T'3’263 .
Noticing that the last two terms vanish as T —. no, we obtain
(A.8) T"/2§[m -+ a[W(r) - rW(1)] .

This is the same result as we obtain from the usual SP tests
when the DGP does not contain a break; see the equation just
before (A3.1) in SP (Appendix 3, p. 24). Therefore,
ignoring the break in the usual SP tests does not matter
asymptotically under the null.

Next, we want to show that this result also holds for
the DF tests, when the DGP is again (A.7) with p = 1.
Define Wt I [1, t]. Since the DF tests are based on the
regression yt = a + 7t + pyt.1 + ct, consider the projection

residual:

S (residual from regression of yt on Wt)

t

d- (residual from regression of DUt on W)

+ (residual from regression of Xt on Wt) .

(The properties of the DF tests will depend on the
Properties of these residuals, since the DF tests are

A A 'der
obtained by a regression of St on SM.) Then conSi ,

(1L9) T'VZSt = d-T’1’2(residual from regression of DUt on Wt)

+ T'1’2(residual from regression of Xt on Wt) .

Notice that the second term converges to a demeaned and

114

detrended Wiener process, shown in equation (24) in
Kiatkowski ot_o1f (1990, p.16) and in equation (16) in Park
and Phillips (1988, p. 474). This is exactly same
asymptotic result as we encounter in considering the usual
DF test statistics which do not allow for the break.
Therefore, to complete the proof, we only need to show that
the first term vanishes asymptotically. To do so, define

the convergence rate matrix,
R = diagonal[T“’2, T'3/2].
Then, the first term in (A.5) follows:
(11.10) d~T‘1/2DUt - d-WtR(RW'WR)'1RW'DU
= d-T'VZDUt - d.(T"/2, tT'3’2)(RW'WR)"(T'1§DUt, T'Zstnutw
Here, notice that the following limits exist:

1

1
3
(RW’WR) .. 1
3

1

2
T’1JEDUt .. A
Tag tDUt -> (l-A2)/2 .

But since T'Vzout e 0, and (T'VZ, tT'3/2) -» 0, the whole term

in (A.10) vanish asymptotically.

W

115
The DGP is (A.7) under the alternative hypothesis. The

usual SP tests statistics are calculated from:

CD!

, = (y. - Y1) - 2(t-1)

(xt - x1) + 63DUt - (E - §)(t-1)

(x - x1) - (t-1)fi + 63mm - (t-1)/T] .

t

Define 0x2, 11 and p1 as in the statements of the Theorem.
Note that H = Op(T"), since T H = x, - X1 = 0p(1). Thus,
Xi and JT Xi vanish asymptotically. Now, consider the

following expression for the denominator of <33
(A.11) T")§§t2 = T'1)E[(Xt-X1) - (t-1)K)? + 63(DU,-(t-1)/T)]Z
= T"); [(xt-x1)-(t-1)fi]2 + 632T'1}E[DUt-(t-l)/T]2
+ 263T"); [(xt-x1)-(t-1)fi][DUt-(t-1)/T].
Evaluating each term of the above expression, we get:
(i) T“§[(X,-x1) - (t-nfilz = T"§<X.-X1>2
+ T"'[fx'2 )t:(t-1)2 - 2T"fi (30‘.
+ 223 + 2X1A_X T"{;t - 2x133?
-> (0.2892) + (X.'X1)2/3 + o + 0 + X1X. - X12 + 0
= 0x2 + (X."’ +X1X, + x12)/3

2 = +1 60
where we have used )Et = T(T+1)/2 and ¥t T('I'+1) (2T )/

116
(ii) 632m"); 00,2 + T3); (t-i)2 - 2T'2)E DUt(t-l)]
.. 632[(l-A) + 1/3 - (142)]
= (A2 - A + 1/3).s32
(iii) 263T'1)E(Xt-X1)DUt - 263T"; (t-1)H DUt
- 263T"); (xt-x1) (t-1) + 263ET‘2); (t-—1)2
-> 0 - 63(x,-x,)(1-,\2) + X163 + (263/3)(X.-X1)
= 63(X.-X1) (-1/3+12) + X16.
Thus, the whole expression in (A.11) follows:
(11.12) T")t:§.t2 .. 0x2 + (x.2+x,x,+x,2)/3 + (AZ-A+1/3)632
+ (x,-x,) (-1/3+A2)63 + X163 .

Note that the last three terms of the above expression do
not exist for the denominator of 5 when there is no break.

For the numerator of 6, consider,
(A-13) T-1¥§t-1A§t = T'1¥[(AXt-Z—)-(-) + 63(D(TB)t-l/T)]-[(XM-X1)
- (t-2)7fi +63(DUt_1-(t-2)/T)]
= T'1}E(AXt-Z_i)[(xt_1-X1) - (t-2)Z')E]
+ 6;"); (Axt-fi) [DUt_1-(t-2)/T]

+ 63T"§ [D(TB)t-1/T][(Xt,1-X1) - (t-2)Z')E]

117

+ 632T"); [D(TB) t-l/T] [DUt_,-(t-2)/T] .
The first term is as follows:
T"); AXt(Xt_1-X1) - T'1¥E(Xt-X1) - T'1§AXt(t-2)H
+ T");fiz(t-2)
-» (71 - of) + 0 + 0 + o = Ox2(p1-1) .

The other terms in (A.13) vanish asymptotically. Thus the

whole expression in (A.13) follows:

(11.14) T"§J§MA§t -> aim-1)

Thus, the asymptotic distribution of 6 is given by the ratio

of two expressions in (A.12) and (A.14).

ai<p1-1)
(11.15) 6‘. 2 1 2 2 1 2 2 1
ox+75(AL+AQAL+X3)+(12-1+?;)03+(2L-X3)(A —?;)63+Xp

From above we see that 6 is not a consistent estimator of
(P1 - 1). Interestingly, this is so even when there is no
break (63 = 0). Since (AZ-A+l/3) > 0, 63 7e 0 will tend to
move the distribution of 5 toward zero, and this is

certainly so for large enough {63:-

rem

Consider the demeaned version of equation (3):

118

(A.16) yt' = azyH' + a3.t* + (2,130: + asADUt' + 6:.

t

where Xt = [yb1 , t', DUt', ADUt'], while the asterisk (*)

represents deviations from means. Also, X is the Tx4 matrix
with tth row xt. Then consider the convergence rate matrices
D1 = diagonal[T, T3”, T1”, 1] and D,’ = diagonal[T, Tm, TVZJ.

Notice that RDT'1 = DTMR. So, we obtain:
(A.17) 2" = rd.)[R0,"(D,"(x'X)0,")"D,"R'1"r($) 73,2
= r($)DT'[R{D,"(X'X)D,"}"R'14013013)73," .

Note that the dummy variable D(TB)t does not affect the
asymptotic distribution of the test statistic, since the
matrix D1'1(X'X)DT'1 is asymptotically block diagonal with
respect to the sub-matrices D,*"(x,'x,)0,*" and (xz'xz) , where
X1 and X2 are Tx3 and Txl matrix with t‘" rows being X1t =
[Y,-1*, t', DUt'] and X2: = [D(TB)t*], respectively. That is,
D,"(X1'X2) -+ 0 as T -' 00. Therefore, R[D,"(X'X)D,"]'1R' can be
replaced with the sub-matrix [0,*"(x,'x1)0,‘"]". (But it
does not necessarily mean that we can exclude D(TB)t in
equation (3).)

Also notice that under the null, r($)DT' is same as
E,'D,*, where E, = [)gyt-1*'e*, {:t*'e*, snuf'efi'. Then,

flag-1)

... Q- - 3/2a
1 _ 3/2 ' Tina [D 1(X’X )DT 1] T 3
(24.18) 5,, - —2-[T(82 1) , T 83 4] T i 1 Tl/za‘

 

119
= (1/6}) (I),""x1 'E1) ' [1),"'"(x1 'x1)0,"'"]"(1),“):1 '13,) .

Denote

E Vial 2:: t.y;-1 E yngUz.
c

 

 

f0 f.“1 f2
'2 e ‘
X1X1' f1d11d12 ' ' gt gtDUt
f d
2 12 d22 . . EDI]?
c J
and
X11 X12 X13
- 1
(Xixl) 1 ' 'TBT X12 X22 X23
X13 X23 X33

where :D: is the determinant of X1UQ. Then,

T«X11 T-7/2Xu T-9/2x13

(A. 19) [D;v-1 (X1x1)D;'-1] -1 - 34% T-7/2X12 T-3X22 T'4X23
I I
T.9/2X13 T4X23 T 5X33

Now, consider the asymptotic results.
(a) T’Zfo = T'2 )EyH'Z .. a"-H3
(b) T'S/2f1 = T'5/2 ¥t*yt,1. _. 0H1

(C) T'3/2f2 = T'3/2 gyt-‘liDUtt _. 0H2

(d) T'3d" (Ir-3 ¥t*2 -9 1/12

(e) T'Zd12 = T‘2 );.t"1)Ut .. -0.5(1-A)A

120

(f) T"d22 = T" 170th -» (l-A)A

where H1, H2 and H3 are defined in Theorem 4. Let E1 s [e1,

e2, e3] '. Then we also have the additional asymptotic

results:
(9) ‘1"‘e1 = T" {Silt-1.6: .. 622',
(h) T‘s/262 = T'3/2 §ttett ... OFZ

(i) T"/2e3 = T"/2 {:DUt'et‘ -> 0F3

where 1“,, F2 and F3 are defined in Theorem 4. Then, we

obtain for each term in (A.19):
(i) T":D: = -(T'2d12)2T'2fo + T'3d11-T’1dzon'2fo
_ T"d22 (TS/2131)? + 2-T'2d12-T'5/2f1-T'3/2f2
_ T'3d"(T'3’2f2)2
-» -(1-A)AaZ-B
(ii) T"X" = -(T'2d12)2 + T'3d11-T'1d22
-+ -(1-A)A02[(-3A2+3A-1)/(1202)]
(iii) T'mx12 = -T"d22-T'5/2f1 + T’2d12-T'3’2f2
.. -(1-A)AaZ[H,/a + 112/(26)]

' - - - - -3/2
(W) T9’2X13 = T 261,2.T5/2f.1 - T3d,,.T f2

121

.. -(1-A)AoZ[H,/(2a) + HZ/(12(l-A)Aa)]
(v) T'3X22 = T'1d22-T‘2fo - (T's/2f2)2

.. -(l-A)A02[H22/((l-A)A) - H3]
(vi) T"X23 = -T'2d12-T'2fo + T'5’2f1-T'3’2f2

.. -(1-A)AaZ[-H,/2 - H,HZ/((1-A)A)]
(vii) T'5X33 = T'3d11-T’2fo - (T'5’2f1)2

.. -(l-A)Aoz[(H12 - H3/12)/((1-A)A)]

Then, we obtain from the above,

 

 

H
(A.20) [DT*"(x1vx1)DT*-1]-1 _. _
B 0
where
[31“31-1 fi.& 5. ”2
1202 o 20 20 6(121-1212)
H22 -1; _H.H.__I{._
H‘ - 1-12 3 1-12 2
H
2-_3
H1 12
1-12 J

 

 

Also, we have:
(A.21) D,"’"1i:1 -> E’ = [02171, oFZ, OF3J' .

Now, it is useful to note that

 

 

122

*

H=I°H'Io

where I, = diagonal[l/a, 1, 1]. Also, note that 1,13" = 013,
where E = [F1, F2, F3] '. Then it is directly shown from

(A.20) and (A.21) that

 

 

A 1 . H’ 02 H
s, -» E*( )E" = —2 E'(—) E
()2 B a B

C C

which proves Theorem 4.

2£22£_2£_IL£2££E_§_

Define F: = F1 - I‘, where I‘ = 0.5(02 - afl/az. Also
define E‘r = [F1', F2, F3] ' such that E = E' + P-i, where 13 =
(1, 0, 0)’. Then it is easy to show that under iid errors

the Wald test in (16) follows:
(A.22) 2" .. E"(H/B)E* .

Here, H, B and E' do not depend on error variances, so that
the above expression is free of the nuisance parameters. We
want to show that the corrected Wald test follows the same
asymptotic distribution in (A.22) as the uncorrected Wald
First, we alternatively

test in (16) does under iid errors.

show that the asymptotic distribution in (A.22) of the

uncorrected Wald test is expressed by:

(A.23) 2H .. (oz/a‘z)oE'(H/B)E

123

= (OZ/0‘2) - (E*+ri3) ' (H/B) (saris)
= (oz/o,2)-E"(H/B)E* + 2(oZ/o.2)-[(oZ-a,2)/2a"’1
i3'(H/8)E* + (Oz/0‘2)-[(02-0‘2)/202]213'(H/B)13 .
Consider the corrected Wald test in (18) as:
(L24) 2* = r($)*[R-3.2(X'X)"oR'1"r({i>)*'
= (r0700; - iii,’[D("(xcanfh'U'[D,"‘(X.'X.)D,*“J
.{D,*-r($)' - 1?[I),""(x1'x1)1),""]"i3 } / 3‘2
= [D,*r($)'1'[D,*"(X.'x1)o,*"1[D(rd)'1 / 3.2
- 2iis'tofrdm / 3.2
+ {813'[1),""(x,'x1)1),""]"i3 / 3‘3 .

Now, notice that the first term of the last equation is same
as the expression for the "uncorrected" Wald test Eu in

(16) , and its asymptotic distribution is expressed in

(A.23) . Also notice that a? -r 1r = P-az. Then the

distribution of the second term is given by:
(A.25) -21}13'[0,*r($)'1/3‘2

.. -2sri3'(H"/13)E"’/a,Z

= - (2/03) [ (oz-0.2m] ~13' -I.<H/B> RE”

= *1/63) (aZ-af) .13- 01/3) E‘

 

124
- (1/0‘2) (oz-of) -i3' (H/B) [(az-a‘Z)/202] i3 .
And, the third term is as follows:
(A.26) {Rig[D,""(x1'x1)0,*"]"i3'/3‘2
a [(az-a,2)/2azlz(aZ/af) .13' - (H/B) is.

Finally, from (A.23), (A.25) and (A.26), the asymptotic

distribution of the whole expression in (A.24) is given by:

3* = r($)*[Ro3,2(x'X)"-R'firm“

-+ (oz/a,2)-E"(H/B)E*
+ (1/0‘2) - (oz-of) -i3' (H/8)1-:*
+ (Oz/0‘2) . [ (oz-a‘z)/20212- 13' 01/3) i3.
- (IL/a3) (oz-of) .13: (Ii/am"
- 2 (oz/0‘2) [(02-0‘2)/202]2- 13' (H/B) i3

+ (oz/of) [ (oz-0‘2)/202]2- i3' - (H/B) i3
= (oz/a‘Z)-E*'(H/B)E'.

Therefore, the asymptotic distribution of the corrected Wald
test statistic €J’in (18) is the same as the distribution
(which is expressed in (A.22)) of the uncorrected Wald test

under iid errors:

125

(A.24) 8‘ .. E" (g) E"

V

which is free of nuisance parameters.

126

LIST OF REFERENCES

Banerjee A., R.L. Lumsdaine, and J.H. Stock (1990),
"Recursive and Sequential Tests of The Unit Root and

Trend Break Hypothesis: Theory and International
Evidence", N353 Working Paper No. 3510.

Dickey, D.A. and W.A. Fuller (1979), "Distribution of the
Estimators for Autoregressive Time Series with a Unit

Root,"
74, 427-431.

Fuller. w.A- (1976). Intrcducticn_tc_§tsticticcl_Timc

Sofios, New York: Wiley.

Hendry, D.F. and G.E. Mizon (1978), "Serial Correlation as a
Convenient Simplification not a nuisance: A Comment on
A Study of the Demand for Money by the Bank of

England", Economic Joufooi, 88, 549-563.

Kiatkowski, D., P.C.B. Phillips and P. Schmidt (1990),
"Testing the Null Hypothesis of Stationarity Against
the Alternative of A Unit Root: How Sure Are We That
Economic Time Series Have A Unit Root?", Econometrics
and Economic Theory Paper No. 8905, Department of
Economics, Michigan State University.

I

Nelson, C.R. and C.I. Plosser (1982), "Trends versus Random
Some Evidence and

Walks in Macroeconomic Time Series:
Implications." Iccrncl_cf_ncnctcry_zccncmicc. 10. 139-
162

Park, J.Y. and P.C.B. Phillips (1988), ”Statistical
Inference in Regressions with Integrated Processes:

Part I," Econometfic ThoOLy, 4, 468-498.

Perron, Pierre (1989), "The Great Crash, The Oil Price
Shock, and The Unit Root Hypothesis,“ Eoooomotfioo, 57,

(6), 1361-1401.
Phillips, P.C.B. (1988), "Reflections on Econometric
Methodology." Thc_8ccncmic_Bcccrc. 64. (187). 344-359.
Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit
Root in Time Series Regression," Biomotfiko, 75, 335-
346.
Flannery, S.A. Teukolsky and.W.T.

Press, W.H., B.P.
Vetterling (1986). utmcriccl_3cciccct__Thc_Art_cf

127

aoionfifio_gomonfing, Cambridge: Cambridge University
Press.

Said, S.E. and D.A. Dickey (1984), "Testing for Unit Roots
in Autoregressive - Moving Average Models of Unknown

Order," Biomotfiko, 599-608.

Schmidt, P. and P.C.B. Phillips (1990), "Testing for a Unit
Root in the Presence of Deterministic Trends,"
Econometrics and Economic Theory Workshop Paper No.

8904, Department of Economics, Michigan State
University.

Zivot, E. and D.W.K. Andrews (1990), "Further Evidence on
the Great Crash, the Oil Price Shock and the Unit Root
Hypothesis," Working Paper, Cowles Foundation for
Research in Economics, Yale University.

CHAPTER 6

CHAPTER 6

CONCLUSION

Whether observed series contains a unit root or not has
profound implications in macroeconomics. Thus, finding
better unit root tests has been a hot issue in time series
econometrics. However, many authors, including
Schwert(1989), find that no unit root test perform
satisfactorily in the presence of strongly autocorrelated
errors, in terms of oofh size and power of the tests. The
tests with correct size under the null have poor power under
the alternative, and the tests with high power do not have
correct size under the null. Therefore, a recent survey
paper by Campbell and Perron (1991) stresses the necessity
of finding new tests which alleviate the size distortion
problem, while retaining good power properties.

While existing unit root tests have mostly been based
on one variant or another of the Dickey-Fuller (1979) tests,
other tests which are not based on the DF tests have been
proposed recently by Schmidt and Phillips (1990). They
employ a different parameterization. Unfortunately, Chapter
2 of this thesis finds that they also exhibit similar
problems of size distortion or lower power in the presence
of autocorrelated errors.

This dissertation, however, finds that significant

128

129
advantages can be drawn from the SP parameterization. The
new IV tests based on the SP framework have surprisingly
small size distortions in the presence of strongly
autocorrelated errors, and they are more powerful than other
tests of similar size. This result may well provide an
acceptable solution to the problem of finding new tests
which are adequate in terms of bofh size and power of the
tests, and it suggests that it is still possible to improve
the performance of unit root tests when the appropriate
testing procedure is employed.

The good performance of the new IV tests indicates
tangible advantages to operating in the SP framework rather
than in the DF framework. The SP tests have simpler
asymptotic distributions than the DF tests, and the simple
structure of the asymptotic distributions of the SP tests
may account for their remarkably improved performance, since
it may result in smaller finite sample bias terms.

The perceptible difference between the DF and SP
frameworks lies in their respective parameterizations. It
has been argued by SP that the DF parameterization is
"clumsy", in the sense that it handles exogenous variables
in a potentially confusing way. This argument has been
elaborated in Chapter 5, in the presence of possible
structural break. Specifically, a suitably modified SP test
allowing for a structural change reverses the results of

Perron (1989), who found most of the Nelson-Plosser series

130
to be trend-stationary if allowance is made for a structural
break. The simplicity of the SP parameterization compared
to the DF parameterization is reflected in simpler
properties of the SP tests under the null hypothesis. Both
DF and SP tests are unaffected by a structural break if the
null hypothesis of a unit root is true, while they are
affected if the alternative hypothesis is true. However, if
we allow for one time break under the null, the asymptotic
distributions of the SP tests are unaffected by the presence
of a break, although the asymptotic distributions of the DF
tests are affected. Thus, if the null is true, allowing or
not allowing for a break does not matter asymptotically in
the SP tests. This result does not hold for the DF tests.

Therefore, this dissertation has addressed the
importance of employing an appropriate parameterization in
unit root testing procedures, and explained why the SP
framework is superior to the DF framework.

The exogeneity of a structural change can be argued to
be questionable, so the models that endogenize structural
change can be extended from the SP framework, just as
Banerjee ot o1, (1990) and Zivot and Andrews (1990) did in
the DF framework. This remains as a topic for further
research. Another extension of our work is to modify the
new IV tests provided in Chapter 4 in order to allow for
more general ARMA errors. Pantula and Hall (1991) have

developed a coefficient test which allows for ARMA errors,

131
in the DF framework. However, we have shown in Chapter 4
how to construct IV t-statistic tests in the SP framework.
Thus it is potentially straightforward in the SP setting to
construct IV-based t-statistic tests that should be robust
to ARMA errors. This will also be the subject of future

research.

132

LIST OF REFERENCES

Banerjee A., R.L. Lumsdaine, and J.H. Stock (1990),
"Recursive and Sequential Tests of The Unit Root and
Trend Break Hypothesis: Theory and International
Evidence," NBER Working Paper No. 3510.

Campbell, J.Y. and P. Perron (1991), "Pitfalls and
Opportunities: What Macroeconomists should know about
Unit Roots", unpublished manuscript, prepared for NBER
Macroeconomic Conference.

Dickey, D. A. and W. A. Fuller (1979), "Distribution of the
Estimators for Autoregressive Time Series with a Unit

Root, " Journai of the Ameticao Statisticol Assooiotioo,
74, 427-431.

Fuller, W.A. (1976), o t t s
Ser'es, New York: Wiley.

Nelson, C.R. and C.I. Plosser (1982), "Trends versus Random
Walks in Macroeconomic Time Series: Some Evidence and

Implications," goutnol of Mooetaty Ecooomioo, 10, 139-
162

Pantula, S.G. and A. Hall (1991), "Testing for Unit Roots in
Autoregressive Moving Average Models: An Instrumental

Variable Approach." lccrnsl_cf_tccncmctrics. 48. 325-
353.

Perron, Pierre (1989), "The Great Crash, The Oil Price
Shock, and The Unit Root Hypothesis," Eoooomottioo, 57,
(6), 1361-1401.

Schmidt, P. and P.C.B. Phillips (1990), "Testing for a Unit
Root in the Presence of Deterministic Trends,“
Econometrics and Economic Theory Workshop Paper No.
8904, Department of Economics, Michigan State
University.

Schwert, G.W. (1989), "Tests for Unit Roots: A Monte Carlo
Investigation." 1csrnsl_cf_tusincss_snc_nccncmic
Ststistics. 7. 147-159-

Zivot, E. and D.W.K. Andrews (1990), "Further Evidence on
the Great Crash, the Oil Price Shock and the Unit Root
Hypothesis," Working Paper, Cowles Foundation for
Research in Economics, Yale University.

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