Post on 06-Apr-2022
Estimating Deterministic Trends with an Integrated orStationary Noise Component∗
Pierre Perron†
Boston University
Tomoyoshi Yabu‡
Bank of Japan
October 30, 2004; This version: March 27, 2007
Abstract
We propose a test for the slope of a trend function when it is a priori unknownwhether the series is trend-stationary or contains an autoregressive unit root. Let αbe the sum of the autoregressive coefficients in the autoregressive representation of theseries. The procedure is based on a Feasible Quasi Generalized Least Squares methodfrom an AR(1) specification with parameter α, the sum of the autoregressive coeffi-cients. The estimate of α is the OLS estimate obtained from an autoregression appliedto detrended data and is truncated to take a value 1 whenever the estimate is in a T−δ
neighborhood of 1. This makes the estimate “super-efficient” when α = 1 and impliesthat inference on the slope parameter can be performed using the standard Normaldistribution whether α = 1 or |α| < 1. Theoretical arguments and simulation evidenceshow that δ = 1/2 is the appropriate choice. Simulations show that our procedure hasgood size properties and greater power than the tests proposed by Vogelsang (1998)and Roy, Falk and Fuller (2004). Applications to inference about the growth rates ofGNP for many countries show the usefulness of the method.
JEL Classification Number: C22.
Keywords: linear trend, unit root, median-unbiased estimates, GLS procedure,super efficient estimates.
∗Pierre Perron acknowledges financial support from the National Science Foundation under Grant SES-0078492. We also wish to thank Eiji Kurozumi for useful comments.
†Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215 (perron@bu.edu).‡Institute for Monetary and Economic Studies, Bank of Japan, Chuo-Ku, Tokyo, 103-8660, Japan (To-
moyoshi.Yabu@boj.or.jp).
1 Introduction
Many time series are well captured by deterministic linear trend. With a logarithmic trans-
formation, the slope of the trend function represents the average growth rate of the time
series, a quantity of substantial interest. To be more precise, consider the following model
for the time series process {yt}:yt = μ+ βt+ ut (1)
where ut is the deviations of the series from the trend. The parameter β is then of primary
interest.
Hypothesis testing on the slope of the trend function is important for many reasons.
First, assessing whether a trend is present (i.e., whether β = 0 or β 6= 0) is of direct
interest. One application pertains to global warming. For example, Vogelsang and Fomby
(2002) found that global temperatures have increased roughly 0.5 degree Celsius per 100
years, and Vogelsang and Franses (2005) analyzing monthly trends in temperature series
for the Netherlands found the largest positive trends in the winter. Another application
relates to the Prebisch-Singer hypothesis, which states that over time the net barter terms
of trade should decline between countries that primarily export commodities and countries
that primarily export manufactured products. Bunzel and Vogelsang (2005) found strong
evidence to support the Prebisch-Singer hypothesis.
Second, Perron (1988) showed that the correct specification of the trend function is
important in the context of testing for a unit root in the noise component ut. He shows that
unit root tests that omit a trend when one is present in the data leads to an inconsistent
test. On the other hand, including a trend when one is not needed leads to a substantial
loss of power (e.g., DeJong, Nankervis, Whiteman and Savin, 1992)
Third, it is often of considerable interest to form confidence intervals on the rate of
growth of series such as real GNP or other index of real aggregate production. For example,
this allows cross-country comparisons or sub-period comparisons to determine if structural
changes are present. Canjels and Watson (1997) analyzed the annual growth rates for real
per-capita GDP in 128 countries. Also, Vogelsang (1998) provides estimates and confidence
intervals for postwar real GNP quarterly growth rates for the G7 countries.
There is a large literature on issues pertaining to inference about the slope of a linear
trend function, most of which is related to the case where the noise component is stationary,
i.e. integrated of order zero, I(0). A classic result due to Grenander and Rosenblatt (1957)
states that the estimate of β obtained from a simple least-squares regression of the form
1
(1) is asymptotically as efficient as that obtained from a Generalized Least Squares (GLS)
regression when the process for ut is correctly specified.
However, it is now recognized that many economic time series of interest are potentially
characterized as having a noise component ut with an autoregressive root that is unity, i.e.
integrated of order one, I(1) (e.g. Nelson and Plosser, 1982), or a root that is close to one. In
the former case, the least square estimate of β obtained from (1) is no longer asymptotically
efficient. Instead, the estimate of the mean of the first-differenced series is efficient in large
samples. In the case of a root close to one, the standard Grenander and Rosenblatt (1957)
result still holds but the limiting Normal distribution is a poor approximation in sample
sizes of interest.
It is important to realize that in all these cases the noise component can be either I(0) or
I(1) and that in general no a priori knowledge about which holds is available. The limiting
distribution of test statistics depends on the I(0) or I(1) dichotomy so that methods of infer-
ence that are robust to both possibilities are needed. Consider the standard t-statistic based
on the OLS regression. The t-statistic for testing β converges in distribution to a N(0, 1)
random variable when ut is I(0). On the other hand, as shown by Phillips and Durlauf
(1988), the t-statistic normalized by T−1/2, has a non-degenerate non-Normal limiting dis-
tribution when the errors are I(1). This dichotomy is one of the major source of problems
that makes inference about the slope of the trend function a difficult issue.
Several papers have tackled the issue of constructing tests and confidence intervals about
the parameter β when it is not known a priori that the noise component ut is I(1) or not.
Most closely related to our work are the following. Sun and Pantula (1999) proposed a
pre-test method which first applies a test of the unit root hypothesis and then chooses the
critical value to be used for the t-statistic according to the outcome of the test. Using this
method, however, the probability of using the critical values from the I(0) case does not
converges to zero when the errors are I(1), and the simulations reported accordingly show
substantial size distortions remaining. Canjels and Watson (1997) consider various Feasible
GLS methods. Their analysis is, however, restricted to the cases where ut is either I(1) or
the autoregressive root is local to one (i.e., ut = αTut−1+ et with αT = 1+ c/T and et being
I(0)). Hence, they do not allow I(0) processes for the noise. Also, even with I(1) or near
I(1) processes, their method yields confidence intervals that are substantially conservative
with common sample sizes. Roy et al. (2004) consider a test based on a one-step Gauss
Newton regression that uses a truncated weighted symmetric least-squares estimate of the
autoregressive parameter (as suggested by Roy and Fuller, 2001). The limit distribution of
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their test is not the same in the I(1) and I(0) cases, but their simulations show that the size
is basically the same in finite samples. Vogelsang (1998) is the only one, to our knowledge,
who proposes a test that is valid with either I(1) or I(0) errors (see section 2.5.2) but its
power is very low in the I(1) case. He then advocates the uses of two statistics which have
complementary properties with the usual drawback induced by the use of multiple tests.
We propose a new robust test statistic that is valid with either I(0) or I(1) noise in
the sense that the limit distribution is the same in both cases. It is based on applying a
Feasible GLS procedure (e.g., the Cochrane-Orcutt procedure) with an estimate of the sum
of the autoregressive coefficients that is truncated to take a value of one when the usual
estimate is in a neighborhood of one. Hence, our estimate of the sum of the autoregressive
coefficients is super-efficient when the true value is 1. As a result, the limiting distribution
of our statistic does not depend on whether the noise is I(0) or I(1) and is the usual standard
Normal. Theoretical arguments are provided to show that the size of the neighborhood where
truncation applies should be of order T 1/2. Also, to improve the finite sample performance, we
advocate the use of a median unbiased estimate of the sum of the autoregressive coefficients
(see Andrews, 1993, and Andrews and Chen, 1994). The resulting statistic is very easy to
implement, its size is close to the nominal level in finite samples and its power exceeds that
of previously suggested procedures.
The outline of the rest of the paper is as follows. In section 2, we analyze the simple
AR(1) case and lay out the basic framework, the test, its large properties and simulations
about its finite sample size and power. In section 3, we consider extensions for a general class
of processes for the noise ut. Section 4 considers the usefulness of our test in the context of
the proper specification of the trend component when conducting unit root tests. Section
5 contains applications of our procedure to the data analyzed by Vogelsang (1998) and
Canjels and Watson (1997). Section 6 contains brief concluding comment and an appendix
some technical derivations.
2 The AR(1) Case
We begin by considering the leading case with AR(1) errors so that the univariate time series
{yt; t = 1, ..., T}, is generated by
yt = μ+ βt+ ut (2)
ut = αut−1 + et
3
where et ∼ i.i.d.(0, σ2) and E(e4t ) < ∞. For simplicity we let u0 be some finite constant.Here −1 < α ≤ 1 so that both stationary and integrated errors are allowed.
Remark 1 The analysis extends easily to more general trend functions of the formnXi=0
βiti +
mXj=1
qXi=0
θi(t− TB,j)i1(t > TB,j)
where 1(·) is the indicator function. When m = 0, we have a polynomial trend function of
order n. If m 6= 0, m breaks are present occuring at some breaks dates TB,j (j = 1, ...,m).
For the analysis to carry over, we need these break dates to be known. For the case of
unknown break dates, the reader is refered to Perron and Yabu (2007). All of our main
theoretical results remain valid in this more general case, though some equations may have
a different form. For simpilcity of exposition, we consider throughout the simpler first-order
linear trend, which is the leading case of interest in practice.
The GLS estimator considered is the one that applies Ordinary Least Squares (OLS) to
the regression
yt − αyt−1 = (1− α)μ+ β[t− α(t− 1)] + et (3)
for t = 2, ..., T , together with
y1 = μ+ β + u1 (4)
Consider the infeasible Generalized Least Squares (GLS) estimate of β that assumes a
known value of α. It is well known that the t-statistic for testing the null hypothesis that
β = β0, tGβ , is then asymptotically distributed as a N(0, 1) random variable for any values
of α in the permissible range.
Canjels and Watson (1997) use a framework with α local to unity, i.e., α = 1+ c/T , and
an initial value u0 having a variance that can depend on the sample size. Two methods are
advocated. First, the estimator described above is the GLS estimator under the assumption
that u0 = 0 and, accordingly, works best when the variance of u0 is small. The second is the
Prais-Winsten (1954) estimator which also incorporates the first observation but with
(1− α2)1/2y1 = (1− α2)1/2μ+ (1− α2)1/2β + (1− α2)1/2u1
This estimator is superior when the variance of u0 is not small. We shall not be concerned
about the effect of the initial condition in this paper and, hence, we have assumed a fixed
value for simplicity and will use the specification (4). Nevertheless, all our results remain
valid using one or the other GLS estimate.
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2.1 The Feasible GLS estimate
We consider the Feasible GLS (FGLS) estimate of β that uses the following estimate of α:
α =TXt=2
utut−1/TXt=2
u2t−1 (5)
where {ut} are the OLS residuals from a regression of yt on {1, t}. When |α| < 1, T 1/2(α−α)→d N(0, 1 − α2) and from the Grenander-Rosenblatt (1957) result, the OLS and FGLS
estimates of β are asymptotically equivalent and the t-statistic obtained from the FGLS
procedure, denoted tFβ , still has a N(0, 1) limit distribution. Things are different when
α = 1. From standard results,
T (α− 1)⇒Z 1
0
W ∗(r)dW (r)/Z 1
0
W ∗(r)2dr ≡ κ (6)
with W ∗(r), 0 ≤ r ≤ 1, the residual process from a continuous time regression of a unit
Wiener process W (r) on {1, r}, and where ⇒ denotes weak convergence in distribution
under the Skorohod topology. It is shown in the appendix that, provided T (α− 1) = Op(1),
the t-statistic from the Feasible GLS procedure can be expressed as
tFβ =
"[T−1/2
TXt=2
et − T (α− 1)T−3/2TXt=2
ut−1] (7)
−T (α− 1)[T−3/2TXt=2
tet − T (α− 1)T−5/2TXt=2
tut−1]
#/£σ2(1− T (α− 1) + T 2(α− 1)2/3)¤1/2 + op(1)
Given (6), T−1/2PT
t=2 et ⇒ σW (1), T−3/2PT
t=2 ut−1 ⇒ σR 10W (r)dr, T−5/2
PTt=2 tut−1 ⇒
σR 10rW (r)dr and T−3/2
PTt=2 tet ⇒ σ
R 10rdW (r), the limit can be expressed as
tFβ ⇒W (1)− κ
R 10W (r)dr − κ[
R 10rdW (r)− κ
R 10rW (r)dr]
(1− κ+ κ2/3)1/2(8)
which is, indeed, different from a Normal distribution. Now suppose that κ, the limit of
T (α − 1), is zero. Then tFβ ⇒ W (1) =d N(0, 1), we would recover in the unit root case the
same limit distribution as in the stationary case and no discontinuity would be present. Our
main idea exploits this feature.
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2.2 A super-efficient estimate when α = 1
The estimate of α that we propose is a super-efficient estimate when α = 1. It is defined by
αS =
⎧⎨⎩ α if |α− 1| > dT−δ
1 if |α− 1| ≤ dT−δ(9)
for some δ ∈ (0, 1) and d > 0. Hence, when α is in a T−δ neighborhood of 1 it is assigned
a value of 1. The fact that we use a super-efficient estimate when α = 1 warrants some
comments. As is well known, such estimates have better properties only on a set of measure
zero, here α = 1, and have less precision in a neighborhood of this value. This is not a
problem here since α is a nuisance parameter in the context of our testing procedure which
pertains to the coefficients of the trend function, in particular β. Hence, the fact that αS is
less efficient than α for values of α close to one does not imply that tests related to β will
be badly bahaved, as we will show.
When the GLS procedure is applied with αS as an estimate of α, we denote the resulting
estimate by βFSand the t-statistic by tFSβ . The following Theorem, proved in the appendix,
shows that this indeed changes nothing in the stationary case and delivers a t-statistic with
a Normal limit distribution when α = 1.
Theorem 1 a) T 1/2(αS − α) →d N(0, 1− α2) when |α| < 1; and b) T (αS − 1) →p 0 when
α = 1. Hence tFSβ →d N(0, 1) for both |α| < 1 and α = 1.
Constructing the GLS regression to estimate β using the truncated estimate αS effectively
bridges the gap between the I(0) and I(1) cases, and the standard Normal asymptotic
distribution applies.
2.3 The case with α local to unity
The result obtained in Theorem is pointwise in α with −1 < α ≤ 1 and does not hold
uniformly, in particular in a local neighborhood of one 1. Hence, it is of interest to see what
happens when the true value of α is close to but not equal to one. Adopting the standard
local to unity approach, we have the following result proved in the Appendix.
1It is, however, possible to show, using the results of Mikusheva (2006), that our test has an asymptoticsize no higher to the stated nominal size uniformly over |α| < 1, though as we show below the test isconservative for values of α local to one.
6
Theorem 2 Suppose that the Data Generating Process is (2) with α = αT = 1 + c/T for
some c ≤ 0, then T (αS − 1)→p 0 and tFSβ →d N(0, (exp(2c)− 1)/2c).
The results are fairly intuitive. Since the true value of α is in a T−1 neighborhood of
1, and αS truncates the values of α in a T−δ neighborhood of 1 for some 0 < δ < 1 (i.e.,
a larger neighborhood), in large enough samples αS = 1. Then, the estimate of β is the
first-difference estimator, βFD
= (T − 1)−1PTt=2∆yt and the t-statistic has the following
properties:
tFD =βFD − β
std(βFD)=
(T − 1)−1PTt=2∆utq
((T − 1)−1PTt=2(∆ut)2)(T − 1)−1
=(T − 1)−1/2uT − (T − 1)−1/2u0q
(T − 1)−1PTt=2(∆ut)2
⇒ Jc(1)
since T−1/2uT ⇒ σJc(1), T−1/2u0 = op(1), and T−1PT
t=2(∆ut)2 →p σ2. Here Jc(1) ≡R 1
0exp(c(1− s))dW (s) ∼ N(0, (exp(2c)− 1)/2c).Note that when c = 0, we recover the result of Theorem 1 for the I(1) case. However,
when c < 0, the variance of the limiting distribution is different from 1. In fact, the variance
is lower than 1, so that, without modifications, a conservative test may be expected for
values of α close to 1, relative to the sample size.
Also, one can note that the limit of the variance as c → −∞ is 0, not 1, and we do not
recover the same result that applies to the I(0) case. As noted by Phillips and Lee (1996),
the local to unity asymptotic framework with c→ −∞ involves a doubly infinite triangular
array such that the limit of the statistic depends on the relative approach to infinity of c and
T . To address this issue and also provide some guidelines on the choice of the truncation
parameter δ, we shall analyze the properties of the OLS, First-Difference (FD) and GLS
estimate of β in a local asymptotic framework whereby c is allowed to increase with the
sample size. To that effect, we specify that c = bT 1−h for some 0 < h < 1, and accordingly
α = αT = 1+ b/T h. When h approaches 1, we get back to the local to unity case, and when
h approaches 0, we get closer to the fixed |α| < 1 case. We start with the following Theorem.
Theorem 3 Suppose that ut = αTut−1 + et with et as defined by (2), and αT = 1 + b/T h
for some b < 0 and some 0 < h < 1. a) Let βolsbe the OLS estimate of β in the regression
yt = μ + βt + ut, then T 3/2−h(βols − β) ⇒ N(0, 12σ2/b2) and the associated t-statistic is
such that tolsβ ⇒ N(0, 1); b) for the feasible GLS estimate, T 3/2−h(βF − β)⇒ N(0, 12σ2/b2)
7
if 0 < h < 1/2, and T 3/2−h(βF − β) ⇒ N(0, 3σ2/b2) if 1/2 < h < 1, while tFβ ⇒ N(0, 1)
for 0 < h < 1; c) for the first-difference estimate, T 1−h/2(βFD − β) ⇒ N(0, σ2/(−2b)) for
0 < h < 1, while tFDβ ⇒ 0.
This Theorem is helpful to analyze the effect of different truncation, i.e., different values
of δ, on the properties of the estimate βFSand the t-statistic tFSβ in a local framework where
c gets large. First note that βFS= β
FDwhen truncation applies, and β
FS= β
Fwhen it
does not.
Consider first, the issue of the relative efficiency of the estimates for different increasing
sequences for c, i.e., different values of h. When h < 1/2, βFis as efficient as β
ols 2 and both
are more efficient than βFD. On the other hand, β
Fis more efficient than β
olswhen h > 1/2.
The feasible GLS estimate βFis more efficient than β
FDfor all 0 < h < 1. Hence, if the goal
was to maximize power, we would never truncate. So we must rely on size considerations
since a well documented fact is that the size distortions of tFβ increases as α approaches 1
if one is using the standard Normal critical values. To minimize these size distortions, one
needs to truncate. So the issue is when to stop truncating. We recommend to do so when the
region for which no truncation applies is also the region for which we attain the asymptotic
equivalence between βFand β
ols. This occurs when h < 1/2, which suggests that one should
set δ = 1/2 so that truncation only applies when h > 1/2. The rational for not truncating
more is that when h < 1/2, βFis as efficient as β
olsand we are then in a region that can
be labeled stationary and one can expect the standard Normal distribution theory to be a
good approximation. The obvious price to pay is that the test will be conservative when
1/2 < h < 1 (in a neighborhood of 1), but as a c increases further and we get away from
the non-stationary region, we effectively ensure a test with a limiting N(0, 1) distribution
and estimates that are as efficient as the OLS, thereby bridging the gap between the non-
stationary and stationary regions. Our simulations will confirm that this is indeed the best
choice that ensures small liberal size distortions when α is close to one, while also restricting
the region where the test is conservative to a reasonably small neighborhood near α = 1.
2.4 Useful Modifications for Improved Finite Sample Properties
In finite samples, a further problem needs to be addressed. It is well known that the OLS
estimate is biased downward and that the bias is especially severe when the data are linearly2This is consistent with the result of Phillips and Lee (1996) who considered the case with α = 1 + c/T
known. They showed to that establish the equivalence between the GLS and OLS estimate as c increasesone must consider value of c such that c2/T is large.
8
detrended. A solution is to raplace the OLS estimate by one with less bias. We consider two
such modifications. The first one is the median unbiased estimate as described by Andrews
(1993). The second is a truncated version of the weighted symmetric least-squares estimate
as described by Roy and Fuller (2001) and used by Roy et al. (2004).
2.4.1 The median-unbiased estimate
When using the median unbiased estimate, we replace the OLS estimate α by a value that
would produce α as the median of the distribution assuming Normal errors. The specific cor-
rection depends on the nature of the deterministic components and Andrews (1993) provide
tabulated values for the linear trend case. More precisely the estimate is defined as follows.
Let m(α) be the median function of α, then:
αM =
⎧⎪⎪⎪⎨⎪⎪⎪⎩1 if α > m(1),
m−1(α) if m(−1) < α < m(1),
−1 if α < m(−1),where m(−1) = limα→−1m(α), and m−1 : (m(−1),m(1)]→ (−1, 1] is the inverse function ofm(·) that satisfies m−1(m(α)) = α for α ∈ (−1, 1].It is important to remark that the construction of the median unbiased estimate also
applies a truncation device given that the parameter space is restricted to |α| ≤ 1. Thisoccurs since for values of α above some threshold the assigned value is 1. Now this threshold
depends on the sample size T and shrinks as T increases. It is easy to verify that this
implies a truncation of the form specified by (9) with δ = 1, i.e., the neighborhood where
the truncation applies is of order T−1. Since our truncation imposes 0 < δ < 1, we can
substitute the OLS estimate by the median-unbiased estimate without changing any of the
stated large sample results.
2.4.2 The truncated weighted symmetric least-squares estimate
The weighted symmetric least-squares estimate of the autoregressive parameter α is defined
by
αW = [T−1Xt=2
u2t +1
T
TXt=1
u2t ]−1
TXt=2
utut−1.
An estimate of its variance is given by
σ2W = [T−1Xt=2
u2t +1
T
TXt=1
u2t ]−1(T − 3)−1
TXt=2
(ut − αW ut−1)2
9
and the associated t-ratio for testing that α = 1 is
τW =αW − 1σW
The modification proposed by Roy and Fuller (2001) is similar in spirit to our superefficient
estimate in that it also replaces estimates which are in a T 1/2 neighborhood of one. The
modification is however discontinuous and depends on the value of the t-ratio τW , making
the procedure explicitly dependent on some pre-test. The modified value is given by
αTW = αW + C(τW )σW ,
where
C(τW ) =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
−τW if τW > τ pct
T−1τW − 3[τW + k(τW + 5)]−1 if −5 < τW ≤ τ pct
T−1τW − 3[τW ]−1 if −(3T )1/2 < τW ≤ −50 if τW ≤ −(3T )1/2
with k = [3T − τ 2pct(Ip + T )][τ pct(5 + τ pct)(Ip + T )]−1, where in the case of a general noise
component with an AR(p) structure, Ip is the integer part of (p + 1)/2, and τ pct is the
percentile of the limiting distribution of τW when α = 1. The form of the WSLS estimator
for an AR(p) process is given in Fuller (1996, p. 572). If τ pct = −1.96, the median of thedistribution of τW when α = 1, αTW is approximately median unbiased, in the sense that it
is nearly unbiased when α < 1 and has a median of 1 when α = 1.
2.4.3 Recommended Procedure
The procedure we recommend is the following:
1. Detrend the data by OLS to obtain residuals, say ut;
2. Estimate an autoregression of order one for ut yielding the estimate α (or construct
the weighted symmetric least-squares estimate αW ) ;
3. Get the corresponding median-unbiased estimate, say αM (or the modified version of
αW , αTW );
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4. Apply the truncation
αMS =
⎧⎨⎩ αM if |αM − 1| > dT−δ
1 if |αM − 1| ≤ dT−δ
(with αM replaced by αTW for the second option).
5. Apply a GLS procedure with αMS to obtain an estimate of the trend parameter β and
construct the standard t-statistic which we shall denote by tFMSβ .
2.5 Simulation evidence
We conducted simple Monte Carlo experiments to illustrate the size and power of the t-
statistic constructed from a GLS regression with the truncated estimate of α. The Data
Generating Process is
yt = βt+ ut
where ut = αut−1 + et with et ∼ i.i.d. N(0, 1) and u0 = 0. In all cases, 50, 000 replications
are used, and the nominal size is 5%. Under the null hypothesis H0: β = 0 while under the
alternative hypothesis H1: β > 0. We first dicsuss the properties of the procedure when the
median unbiased estimate αMS is used and return later with a comparision with the method
that uses the truncated weighted symmetric least-squares estimate.
2.5.1 Size
Consider first the size properties of the t-statistic tFSβ without the median-unbiased adjust-
ment. The null rejection probabilities were simulated for values of α in the range of [0, 1] with
increments of 0.05 in the range [0.0, 0.90] and in increments of 0.01 in the range [0.90, 1.0].
The sample sizes used are T = 100, 250, and 500. We consider four cases for the value of δ,
namely δ = 0.3, 0.4, 0.5, and 0.6. d is set to 1.
The results are summarized in Figure 1. Consider first the cases with δ < 1/2. The
results show that the test is conservative when α is close to 1 even when the sample size is
large (T = 500). This is in accordance with the theoretical explanations given in Section 2.3.
As discussed, when δ < 1/2, the result of Theorem 2 applies even for moderate values of c so
that, in a local neighborhood of 1, a conservative test results. When δ > 1/2, the opposite
applies. The size distortions are liberal even at α = 1. This is due to the fact that large
value of δ imply that we barely truncate so that the usual size distortions that arise from
using the standard OLS estimate apply. The results for the case δ = 1/2 are encouraging,
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in the sense that when T = 500 only very small size distortions remain. However, when the
sample size is smaller, important distortions can be seen local to α = 1. This is due to the
fact that in small to moderate sample sizes, the OLS estimate of α is biased downward so
that no truncation applies when some would be desirable.
Figure 2 presents the size of the t-statistic tFMSβ when the OLS estimate of α is replaced
by the median unbiased estimate. As expected, when δ < 1/2, the test is conservative in a
neighborhood of 1. When δ > 1/2, the liberal size distortions are greatly reduced but remain
significant. What is more important is that the test constructed with δ = 1/2 now shows
basically no size distortion even with T = 100.
Note that the size simulations show that the choice of d = 1 is appropriate if the goal
is to reduce size distortions to a minimum. For example, when T = 100, setting d = 1/2
would yield results similar to the case δ = 0.6 and setting d = 2 would be almost the same
as using δ = 0.3. Hence, we shall continue to use d = 1.
Further unreported simulations showed the procedure to have similar properties when
using two-sided 10% tests. With two-sided 5% tests, size distortions are somewhat higher
when T = 100 and α = 1, but decrease rapidly as T increases.
2.5.2 Description of Vogelsang’s tests
An alternative procedure that offers a test with the same critical values under both the
I(0) and I(1) cases is that of Vogelsang (1998). His statistic, labelled t-PST here (and
corresponding to the t-PS1T version), is based on the partial sums (PS) regression:
zt = μt+ β[1
2(t2 + t)] + St
where zt =Pt
j=1 yj and St =Pt
j=1 uj. Let tz be the standard t-statistic for testing β = 0
in the above regression. Then, T−1/2tz has well defined asymptotic distributions for both
I(0) and I(1) errors, though they are different (and non Normal). To have identical asymp-
totic critical values in both cases for a given nominal size, Vogelsang suggests the following
adjustment. Let JT denote the standard Wald statistic normalized by T−1 for testing the
joint hypothesis γ2 = γ3 = ... = γm = 0 in the regression: yt = μ + βt +Pm
i=2 γiti + ut
(with m = 9, chosen based on simulations). JT is a unit root statistic proposed by Park
and Choi (1988) with the following properties: JT → 0 if the errors are I(0) while JT has a
nondegenerate asymptotic distribution if the errors are I(1). The t-PST is then defined by
t-PST = T−1/2tz exp(−b∗JT )
12
where b is some positive constant. When the errors are I(0), exp(−b∗JT ) →p 1. Therefore,
b∗ can be chosen to shrink the I(1) asymptotic distribution of t-PST without changing the
I(0) asymptotic distribution. Thus, given a choice of the nominal size of the test, b∗ can be
chosen so the I(0) and I(1) critical values are same.
The power of the t-PST test is, however, very low when the errors are I(1). The main
reason is that fact that while the statistic T−1/2tz has indeed good properties, that fact
that the correction factor exp(−b∗JT ) is random in the I(1) case makes the t-PST test a
noisy version of a good test, thereby reducing its ability to discriminate between the null
and the alternative hypotheses and consequently power is low. Hence, Vogelsang (1998) also
advocates the use of the t-statistic on β scaled by T−1/2 from the regression yt = μ+βt+ut
estimated by OLS, labelled T−1/2t-WT . This test has good power with I(1) errors but has
zero asymptotic size with I(0) errors. This leads to problems related to the use of multiple
testing procedures where the size of each test needs to be modified (e.g., use both at the
2.5% level to have an overall 5% test).
2.5.3 Power
Consider now the power of the tests. Since δ = 1/2 was shown to be adequate using theo-
retical arguments and simulations, we henceforth consider only this value and also consider
only the test constructed with a median unbiased adjustment, tFMSβ . The power of our test is
compared to three other tests: the t-test based on the infeasible GLS estimate which uses the
true value of α, as well as the T−1/2t-WT and the t-PST tests of Vogelsang (1998) for which
we use a 5% nominal size and, hence, the proper comparisons should be made assuming they
are applied independently. The power curves are plotted for α = 1.0, 0.95, 0.90 and 0.80 for
a range of values of β > 0. The number of replications is again 50, 000.
Figures 3.1-3.3 present the results for T = 100, 250 and 500, respectively. Consider first
the case α = 1. The test tFMSβ has basically the same power as the infeasible GLS test
(slightly higher when T = 100 because of a small liberal size distortion). The test is more
powerful than Vogelsang’s (1998) T−1/2t-WT test which is the preferred one for this value of
α. Consistent with the results reported in Vogelsang (1998), the t-PST test has substantially
lower power. When α = 0.95 or 0.90, the power of tFMSβ is not close to that of the infeasible
GLS test but it is higher than either of Vogelsang’s tests. As T increases, its power gets
closer to that of the infeasible GLS test. For instance, when T = 500 and α = 0.9, the power
of our test is as good as the infeasible test. But when α = 0.8 for any T , the tFMSβ has again
power close to the infeasible test and still higher than the test t-PST (the test T−1/2t-WT is
13
so conservative that power is non zero only for high values of β).
Remark 2 From Figures 3.1-3.3, the power functions of our test have some kinks and thustheir shapes are unusual. This is due to the fact that our procedure involves a mix of the
standard FGLS (using the non-truncated estimate of α) and the first difference estimator.
From unreported simulations for the case α = 0.9 and T = 100, the power function of the test
based on first differences is almost zero before β = 0.12 but increases rapidly thereafter. The
power function of the test based on FGLS procedure with α non-truncated increases rapidly
for β ≤ 0.06, but afterwards the increase is small. Hence, the overall procedure has a powerfunction with two kinks at 0.06 and 0.12.
2.5.4 Comparisons with the procedure of Roy et al. (2004)
Roy et al. (2004) consider inference about the slope of a linear trend function when the
noise component is an AR(p) processes that is either stationary or has one unit root. Their
procedure is based on a one step Gauss Newton regression using the truncated weighted
symmetric least-squares estimate. Consider a first-order expansion of the regression (3)
around initial estimates (eμ0, eβ0, eα0)eet = μ∆(1− eμ0) + β∆[t− eα0(t− 1)] + α∆eyt−1 + ωt (10)
where eyt = yt− eμ0− eβ0t, eet = eyt− eα0eyt−1, and {ωt} are the errors. Estimating (10) by OLSyields the estimates (eμ∆, eβ∆, eα∆) and the one-step Gauss-Newton estimates are
(μGN , βGN , αGN) = (eμ0, eβ0, eα0) + (eμ∆, eβ∆, eα∆)
The test statistic is
tGN =βGN − β0
se(βGN)
where se(βGN) is essentially the same as the standard error from the regression (10), see
Roy et al. (2004) for details.
Roy et al. (2004) suggest as the initial estimate the truncated weighted symmetric least-
squares estimate with τ pct = −2.85 (which is approximately the 85th percentile of the limitingdistribution of τW when α = 1, implying that the estimate takes value 1 about 85% of the
time when α = 1). The initial values of (μ, β) are obtained from the Feasible GLS regression
using the truncated weighted symmetric least-squares estimate with τ pct = −1.96.The limit distribution of tGN is standard Normal when |α| < 1 but not so when α = 1.
Nevertheless, the simulations of Roy et al. (2004) show that even in small samples, there are
14
only small departures from the nominal size in the I(1) case when two-sided 5% tests are
used. The goals in this sub-section are twofold: a) to evaluate whether there is any benefit
in using the truncated weighted symmetric least-squares estimate instead of the median
unbiased estimate in our procedure; b) to assess how our procedure performs relative to that
of Roy et al. (2004) in terms of size and power in finite samples.
We conducted simulation experiments for the size and power of the tests as specified at the
beginning of Section 2.5. Table 1 presents the size of one-sided 5% tests for the Gauss Newton
statistics tGN , the tFMSβ test using a median unbiased estimate, labelled tFMS
β (MU), and the
tFMSβ test using the truncated weighted symmetric least-squares estimate with τ pct = −2.85,labelled, tFMS
β (TW ). The exact sizes of the tests tGN and tFMSβ (MU) are comparable, except
when T = 100 and α is close to or equal to one. The test tFMSβ (MU) has slightly higher
liberal size distortions when α = 1. On the other hand, the test tGN is more conservative
for values of α between 0.8 and 1.0. When comparing the tests tFMSβ (MU) and tFMS
β (TW ),
one sees a similar trade-off.
Figures 4.1 through 4.3 present the power of the tests. It is seen that the test tFMSβ (MU)
dominates tFMSβ (TW ). Hence, in the sequel, we shall continue to use the version with the
median unbiased estimate. Comparing the tests tFMSβ (MU) and tGN , the results show that
power is higher (and close to that of the infeasible GLS procedure) with our test for all values
of α when T = 100. When T = 250 or 500 the power of our test is, in general, higher unless
α = .95. In particular, our test is noticeably better when α = 1.
3 Generalization of the model
We now consider an extension of the analysis to the case where the error term ut is allowed
to have a more general structure than the simple AR(1) process with i.i.d. shocks assumed
so far. The data generating process is now given by
yt = μ+ βt+ ut (11)
ut = αut−1 + vt
vt = d(L)et
with
d(L) =∞Xi=0
diLi,∞Xi=0
i|di| <∞, d(1) 6= 0,
and {et} a martingale difference sequence with respect to a filtration Ft to which it is
adapted. Also, E[e2t |Ft] = σ2 and suptE[e4t ] < ∞. Again, we assume for simplicity that u0
15
is some constant. These conditions imply that the following functional central limit theorem
hold for the partial sums of vt, T−1/2P[rT ]
t=1 vt ⇒ σd(1)W (r). Under the stated conditions, uthas an autoregressive representation, say A(L)ut = et where A(L) = 1−
P∞i=1 aiL
i. In the
representation (11), we wish to have the parameter α pertain to the sum of the autoregressive
coefficients. Accordingly we have the representation
ut = αut−1 +A∗(L)∆ut−1 + et
with A∗(L) =P∞
i=0 a∗iL
i where a∗i = −P∞
j=i+1 aj.
3.1 The estimation of α
Given that α represents the sum of the autoregressive coefficients, we cannot use the estimate
(5) based on an autoregression of order one since it is inconsistent for α when the errors utare a general I(0) process. Instead, we base our estimate on a truncated autoregression of
order k. Let ut be the residuals from a regression of yt on {1, t}, then the estimate of αconsidered is the OLS estimate eα obtained from the regression
ut = αut−1 +kXi=1
ζi∆ut−i + etk (12)
The estimate eα has the following properties provided k →∞ and k3/T → 0 as T →∞, seeSaid and Dickey (1984) and Berk (1974). When ut is I(0), T 1/2(eα − α) = Op(1). On the
other hand, if α = 1 + c/T , T (eα− 1)⇒ c+ d(1)R 10J∗c (r)dW (r)/
R 10J∗c (r)
2dr where J∗c (r) is
the residual function from the regression of Jc(r) ≡R r0exp(c(r − s))dW (s) on {1, r}. The
truncated estimate eαS defined by (9) with eα instead of α is therefore still superefficient undera local unit root, i.e. T (eαS − 1)→p 0 when α = 1 + c/T .
As in the AR(1) case, to improve the finite sample performance of the test, we consider
an approximately median unbiased estimate of α as described in Andrews and Chen (1994).
Since the distribution of eα depends on (α, ζ1,...,ζk), the method is based on an iterativeprocedure that jointly estimates α and the nuisance parameters (ζ1,...,ζk) based on the
regression (12) and then treats the estimate (eζ1,..., eζk) as though they were the true valuesof (ζ1,..., ζk) and computes the median unbiased estimate eαMS. One then treats eαMS as
given and re-computes the OLS estimates of (ζ1,...,ζk). The procedure is repeated until
convergence.
16
3.2 The estimation of β
The estimate of β considered is a quasi-FGLS estimate assuming AR(1) errors, i.e. the OLS
estimate in the transformed regression:
yt − eαMSyt−1 = (1− eαMS)μ+ β[t− eαMS(t− 1)] + (ut − eαMSut−1) (13)
for t = 2, ..., T together with
y1 = μ+ β + u1
Denote the resulting estimate of β by eβ. Since vt exhibits serial correlation in general, weneed to correct for this. Hence, the statistic considered is (where the subscript RQF stands
for Robust Quasi Feasible (GLS)):
tRQFβ =eβ − β0qhv(X 0X)−122
where (X 0X)−122 is the (2,2) element of the matrix (X0X)−1 with X = [x1, ..., xT ]
0, x0t =
[(1− eαMS), t− eαMS(t− 1)] for t = 2, ..., T and x01 = (1, 1). Also, hv is a consistent estimate
of (2π) times the spectral density function of vt at frequency 0. Hence, the usual estimate
of the variance of the residuals is replaced by the so-called long-run variance estimate hv.
We consider two classes of estimates hv. One is based on a weighted sum of the autoco-
variance function given by:
hv = T−1TXt=1
v2t + T−1T−1Xj=1
λ(j,m)TX
t=j+1
vtvt−j (14)
for some weight function λ(j,m) and bandwidth m, where vt are the OLS residuals from
the regression (13). The quadratic spectral window is used with the bandwidth m selected
according to the “plug-in method” advocated by Andrews (1991) using an AR(1) approxi-
mation.
The other estimate considered is an autoregressive spectral density estimate (at frequency
zero). Note that the residuals from the regression (13) are (1− eαMSL)ut and, hence, in large
samples, they are equivalent to vt = (1−αL)ut. Consider first the case where |α| < 1. A(L)is then invertible so that ut = A(L)−1et and thus vt = (1−αL)A(L)−1et. Since A(1) = 1−α,the spectral density at frequency zero of vt is simply σ2. To obtain a consistent estimate, we
use the following approximate regression
yt − eαMSyt−1 = μ∗ + β∗t+kXi=1
ρi∆yt−i + etk
17
with etk the corresponding OLS residuals. The estimate of σ2 is then σ2 = hv = (T −k)−1
PTt=k+1 e
2tk. This will be a consistent estimate provided k → ∞ and k3/T → 0 as
T → ∞. Consider now the case with α = 1. Then, in large samples, (1 − eαMSL)ut is
equivalent to vt = ∆ut and an autoregressive spectral density estimate at frequency zero can
be obtained from the regression
vt =kXi=1
φivt−i + ηtk
where vt are the OLS residuals from the regression (13). Denote the estimate by (φ1, ..., φk)
and σ2ηk = (T−k)−1PT
t=k+1 η2tk, hv = σ2ηk/(1−
Pki=1 φi)
2. The decision rule to select whether
to use the estimate for the case α = 1 or |α| < 1 is based on whether the truncated value eαMS
is 1 or not. Asymptotically, this results in a correct classification and a consistent estimate
for both cases. Hence, the procedure we recommend is the following:
1. Detrend the data by OLS to obtain residuals ut;
2. Consider the autoregression (12) with k selected using an information criterion (we
recommend using the BIC with k allowed to be in the range [0, 12(T/100)1/4], see
Section 3.3). The corresponding estimate is denoted eα. If the order selected is k = 0,the procedure of Section 2 applies, otherwise, the next steps are applied.
3. Use the method of Andrews and Chen (1994) to obtain the approximately median
unbiased estimate eαM and apply the truncation
eαMS =
⎧⎨⎩ eαM if |eαM − 1| > T−1/2
1 if |eαM − 1| ≤ T−1/2
4. Apply the quasi GLS procedure with eαMS to obtain the estimate of the trend parameter
β and construct the t-statistic tRQFβ using one of the estimate hv suggested to construct
the spectral density function at frequency zero of vt.
3.3 Finite sample simulations
This section presents results about the finite sample size and power of the test with an AR(2)
error component. The DGP is assumed to be
yt = βt+ ut
ut = αut−1 + ψ(ut−1 − ut−2) + et
18
where et ∼ i.i.d.N(0, 1) and u0 = u−1 = 0. Our hypotheses are H0: β = 0 and H1: β > 0.
Given the high computation costs, 1000 replications are used.
3.3.1 Size
We first consider the size properties of our test at the nominal 5% level. These are obtained
for α = 1, 0.95, 0.90, 0.80, ψ = 0.0, 0.3, 0.5, 0.7, and T = 100, 250. We consider positive AR
coefficients since this is the most relevant case in practice. We consider four variations of our
test pertaining to the choice of the estimate hv. In the first case, the estimate (14) is used
and is referred to as “NP” (for Non Parametric) and eα is obtained from (12) with k = 1 (i.e.,we impose the true value). The other three are autoregresive based estimates with k chosen
as follows: 1) fixed at k = 1 (to assess the effect of estimating k); 2) with the Akaike (1969)
information criterion, AIC; 3) with the Bayesian Information Criterion, BIC, see Schwarz
(1978).
The results, presented in Table 2, show some features of interest. When the test is based
on the weighted sum of autocovariances to construct the estimate hv, it is very conservative
for ψ > 0 and α < 1, and the distortions become more serious as ψ increases. When α = 1,
the test is liberal with the distortions reducing only slightly as T increases from 100 to
250. Using an autoregressive spectral density estimate, the size distortions are very small
even with T = 100 unless α = 1 and ψ is small (in which case the test is slightly liberal) or
α = 0.95 and ψ is large (in which case the test is conservative). However, these size distortions
diminish when T is increased to 250. Also, there is no noticeable difference between the cases
with the lag length assumed known or estimated. Hence, our recommendation is to use the
autoregressive spectral estimate with an information criterion to select the lag length.
3.3.2 Power
We now consider the power of our test and compare it with three other tests: the t-test
based on the infeasible GLS using the true value of α, as well as the T−1/2t-WT and t-PSTtests of Vogelsang (1998), again using a 5% nominal size so that their properties pertains
to the case when both tests are applied independently. The power curves are plotted for
α = 1.0, 0.95, 0.90, and 0.80 for T = 100 and a range of values of β > 0.
Figures 5.1-5.3 present the results for ψ = 0.3.0.5 and 0.7, respectively. Consider first
the case α = 1. Our test is as powerful as the infeasible GLS test for any ψ (again slightly
higher due to a small liberal size distortion). When α = 0.95 or 0.90, the power of our test
is not close to that of the infeasible GLS but is still competitive with the most powerful of
19
the T−1/2t-WT and t-PST tests. Note that all tests lose power as ψ increases. As before,
when α = 0.8, the tRQFβ has power close to the infeasible GLS test.
4 Specifying the trend function for unit root tests
As discussed in the introduction, the correct specification of a trend function is important
when testing for unit roots. Tests that omit a trend when one is present are inconsistent.
On the other hand, including a trend when one is not present leads to a substantial loss of
power. Therefore, it is important to know if a trend is present prior to assessing whether
the noise component is stationary or not.
Ayat and Burridge (2000) used Vogelsang’s tests to select the specification of the trend
function as part of sequential unit root testing procedure. If the null hypothesis of no trend
is not rejected unit root tests are performed with only a constant, whereas if the null is
rejected, a constant and a trend are included. They concluded that there is no advantage
in using Vogelsang’s tests due to their lack of power. It is of interest to see if there is any
advantage in using our trend test as part of a sequential unit root testing procedure. In this
section, we document the size and power properties of such a sequential procedure.
We use simulations and the Data Generating Process is
yt = βt+ ut
where ut = αut−1+et with et ∼ i.i.d. N(0, 1) and u0 = 0. We consider α = 1.0, 0.95, 0.90, 0.80
and a range of values of β ≥ 0. The sample size is T = 100 and 10, 000 replications are used.We consider unit root tests with only a constant, a constant and a time trend, as well as
sequential unit root tests performed as follows: 1) test the null hypothesis of no trend, β = 0,
using Vogelsang’s tests or our proposed test with a 5% nominal size (note that Vogelsang’s
procedure involves the combine use of the statistics of T−1/2t-WT and t-PST , hence 2.5%
critical values are used for each statistic to have an overall test with nominal size no larger
than 5%); 2) if the null hypothesis is not rejected, we conduct a unit root test with only a
constant, whereas if the null is rejected, we include a constant and a trend. The nominal
size of the unit root test is also 5%.
Figure 6.1 reports the size and power of the four variants when the Dickey-Fuller (1979)
unit root test is used. Consider first the size properties with α = 1. When β = 0, the unit root
test with only a constant has size close to 5%. However, as β increases, the misspecification
of the trend function becomes more serious so that the test has an increasing bias toward
not rejecting the unit root. On the contrary, the unit root test with a constant and a trend
20
always has exact size close to nominal size since there is no misspecification in the trend
function for any values of β. The sequential tests have small liberal size distortions when β
is near zero but otherwise maintain an exact size close to nominal level.
Consider now the power of the tests with α < 1. When β = 0, the unit root test with
only a constant is the most powerful test. However, as β increases, the misspecification of
the trend function becomes more serious and the power of the test decreases to zero. On
the contrary, the unit root test including a trend has non-trivial power for any values of β
since no misspecification occurs. However, when β is small the power is substantially lower
than with a constant only. The sequential unit root test with Vogelsang’s trend tests works
well when β is very small or very large. When β is very small, it is as good as the unit root
test with a constant only and when β is very large, it is as good as the test with a constant
and a time trend. However, when β is moderate size, the most relevant case in practice, the
procedure evidently lacks power. In contrast, the sequential unit root test procedure with
our trend test works very well for any values of β due to the higher power of the trend test
applied in the first step. Of special interest is the fact that the power function of the unit
root test with our trend test is higher than the best of the “constant only” or “constant
and trend” specifications for any values of β. This shows that our trend test adapts well to
the particular sample in selecting the appropriate specification of the trend.
Figure 6.2 reports a similar exercise with the GLS-detrended unit root tests of Elliott,
Rothenberg and Stock (1996). As expected, power is higher but otherwise the same quali-
tative features and conclusions apply.
5 Empirical applications
We consider two empirical applications related to the data sets analyzed in Vogelsang (1998)
and in Canjels andWatson (1997) 3. Vogelsang (1998) estimated postwar real GNP quarterly
growth rates for the G7 countries (U.S., Canada, France, Germany, Italy, Japan, and the
U.K.). The exact sample period for each country is indicated in Table 3. He considered the
possibility of a structural change in the slope of the trend function in 1973 (see, e.g., Perron,
1989). Hence, the interest is in comparing the estimates and confidence intervals for the pre-
1973 and post-1973. Table 3 reports the estimates of real GNP quarterly growth rates for
both periods and the full sample with the associated 95% confidence intervals (contrary to
Vogelsang (1998) who presents 90% confidence intervals). The columns with headings “βmin3In all cases, the estimate hv is the autoregressive spectral estimate using the BIC to select the lag length.
21
and βmax” refer to the lower and upper bound of the confidence interval and the column
with the heading “width” refers to its width. There are several features of interest to note.
First, the confidence intervals obtained using of test statistic are much tighter than those of
Vogelsang for 19 of 21 cases, as expected from our simulation results. Second, in all cases, the
point estimates of β are higher in the pre-1973 sample compared to the post-1973 sample.
For France, Germany, Italy, and Japan, the pre-1973 and post-1973 confidence intervals
using our test statistic do not overlap, indicating a statistical significant decline in the rate
of growth for these countries. This contrasts with the results of Vogeslang’s procedure for
which non-overlapping 95% confidence intervals occur only for France and Japan, and only
using the T−1/2t-WT statistic. With three exceptions, (pre-1973 for France and the U.K,
and post-1973 for Japan), the truncated median unbiased estimate of α is 1. The very poor
properties of the test t-PST when α is close to one is reflected in the wide confidence intervals
associated with this test (in many cases, the upper and lower limits of the confidence interval
exceed ±99). When no truncation applies, which implies a value of α far from 1, as expectedfrom the simulations, the test t-PST gives tighter confidence intervals than the test T−1/2t-
WT in 2 out of 3 cases. However, these are between 3 and 41 times as wide as the confidence
intervals provided by our test.
Consider now the data set analyzed by Canjels and Watson (1997) which consists the
annual real GDP per capita over the post-war period from the Penn World Table database
(version 5.5). The analysis is limited to the 128 countries for which at least 20 years of
data is available. Table 4 reports the results, which reproduces the estimates and confidence
intervals from Table 5 of Canjels and Watson (1997) and presents those using our method.
The first column is the country identification number. The third column shows the sample
period. Columns 4-7 show the results related to the Prais-Winsten estimator, recommended
by Canjels and Watson (1997). The fifth column shows the estimate of average trend growth
and the fourth and sixth show the lower and upper limits of the associated 95% confidence
interval, respectively. The seventh shows the width of confidence intervals. Columns 8-14
present the results based on our method (estimates eα and eαMS, order k selected, estimates
β with associated confidence intervals and their width). The main feature of interest is that
our method provides tighter confidence intervals for 117 countries out of the 128 considered.
Improvements are very large when eαMS is not close to one, although there are still some
improvements when the truncation is effective and eαMS is one. When no truncation is applied
(i.e., eαMS < 1, occuring for 24 countries), the ratio of Canjels and Watson’s confidence
interval relative to ours varies from 1.11 (Burundi) to 12.09 (Luxembourg) with the average
22
being 5.36. When the truncation is in effect (i.e., eαMS = 1, occuring for 104 countries), this
ratio varies from 0.35 (USSR) to 1.51 (Argentina) with the average being 1.09.
6 Conclusions
We proposed a new procedure to carry inference about the slope of a trend function that is
valid without prior knowledge about whether a series is I(0) or I(1). The test is based on a
Feasible quasi GLS method with a superefficient estimate of the sum of the autoregressive
coefficients α when α = 1. Simulations and empirical applications have shown its usefulness
and the fact that it provides a clear improvement over existing methods. The power of
our test is basically equivalent to that based on the infeasible GLS estimate when α is
assumed known, unless α is near but not equal to one. It may appear that there is room for
improvements in this case but as argued by Roy et al. (2004, p. 1088), this seems unlikely
if we require a test that controls size in both the I(1) and I(0) cases.
As mentioned in Section 2, our method extends with obvious modifications to the case
of a general polynomial trend function that can include multiple structural changes occuring
at known dates. An extension of practical importance is to consider the case where the
trend function can have structural changes at unkown dates. In this case, the analysis does
not extend in a strightforward fashion since the limit distribution of the t-statistic on the
slope coefficeint still depends on the I(0)-I(1) dichotomy. However, some modifications can
be applied to have useful procedures for that case as well, see Perron and Yabu (2007).
23
Appendix: Technical Derivations
Proof of equation (7): Under the hypothesis that the data are generated by (2), usingstraightforward algebra we can express the t-statistic as (see also, Canjels andWatson, 1997):
tFβ =q11T
−1/2r2 − T−1/2q12r1[σ2q11(q11T−1q22 − T−1q212)]1/2
(A.1)
where
q11 = 1 + (T − 1)(1− α)2 = 1 + op(1)
T−1/2q12 = T−1/2[1 + (T − 1)α(1− α) + (1− α)2TXt=2
t] = op(1)
T−1q22 = T−1[1 + (T − 1)α2 + 2α(1− α)TXt=2
t+ (1− α)2TXt=2
t2]
= 1 + T (1− α) + T 2(1− α)2/3 + op(1)
r1 = u1 + (1− α)TXt=2
u∗t = u1 + op(1)
T−1/2r2 = T−1/2[u1 + αTXt=2
u∗t + (1− α)TXt=2
tu∗t ]
= T−1/2TXt=2
u∗t + T (1− α)T−3/2TXt=2
tu∗t + op(1)
and where u∗t = ut− αut−1, σ2 = T−1PT
t=1 e2t with et the residuals from a regression of
y∗t = yt − αyt−1 on x∗t = [1 − α, t − α(t − 1)]0 for t = 2, ..., T and x∗1 = [1, 1]0, y∗1 = y1.It is straightforward to show that σ2 = σ2 + op(1). Equation (7) follows substituting theseexpansions in (A.1).
Proof of Theorem 1: Let A = {T δ|α− 1| > d} and let A = {T δ|α− 1| < d}. For part (a),it suffices to show that T 1/2(αS − α)− T 1/2(α− α)→p 0. We have
limT→∞
Pr(|T 1/2(αS − α)| > ε) =
limT→∞
Pr(|T 1/2(αS − α)| > ε|A) Pr(A) + limT→∞
Pr(|T 1/2(αS − α)| > ε|A) Pr(A)
24
The first term is zero given that, if A is true, we have αS = α so that Pr(|T 1/2(αS − α)| >ε|A) = 0. The second term is zero since α→p α 6= 1 implies limT→∞ Pr(A)→ 0 as T →∞.Therefore, Pr(|T 1/2(αS − α)| > ε)→ 0 as T →∞. For part (b), we have
limT→∞
Pr(|T (αS − 1)| > ε) =
limT→∞
Pr(|T (αS − 1)| > ε|A) Pr(A) + limT→∞
Pr(|T (αS − 1)| > ε|A) Pr(A)
Now the fact that T (α−1) = Op(1) and δ ∈ (0, 1) imply that limT→∞ Pr(A) = Pr(T δ−1|T (α−1)| > d) → 0 as T → ∞, so that the first term is zero. For the second term, if A is true,αS = 1 so that Pr(|T (αS − 1)| > ε|A) = 0. Thus, Pr(|T (αS − 1)| > ε)→ 0 as T →∞.Proof of Theorem 2: To prove part a) first note that with αT = 1 + c/T ,
T (α− 1)⇒ c+
Z 1
0
J∗c (r)dW (r)/Z 1
0
J∗c (r)2dr
with J∗c (r), 0 ≤ r ≤ 1, the residual process from a continuous time regression of an Ornstein-Uhlenbeck Jc(r) on {1, r}, where Jc(r) =
R r0exp(c(r − s))dW (s). Since the true value of α
is in a T−1 neighborhood of 1, and αS truncates values of α in a T−δ neighborhood of 1 forsome 0 < δ < 1 (i.e., a larger neighborhood), the results of Theorem 1 continue to apply inthe local to unity case and T (αS − 1)→p 0. Now the t-statistic can be written a
tFβ =
"T−1/2
TXt=2
(ut − ut−1)− T (αS − 1)T−3/2TXt=2
ut−1
−T (αS − 1)[T−3/2TXt=2
t(ut − ut−1)− T (αS − 1)T−5/2TXt=2
tut−1]
#/£σ2(1− T (αS − 1) + T 2(αS − 1)2/3)
¤1/2+ op(1)
We have T−1/2PT
t=2(ut−ut−1) = T−1/2PT
t=2 et+cT−3/2PT
t=2 ut−1 → σ[W (1)+cR 10Jc(r)dr] =
σJc(1) ∼ N(0, σ2(exp(2c)− 1)/2c). The result of part (a) follows using the fact that T (αS−1)→p 0, and that the following quantities areOp(1), T−3/2
PTt=2 ut−1, T
−3/2PTt=2 t(ut−ut−1),
and T−5/2PT
t=2 tut−1.
Proof of Theorem 3: In the proof, we consider ut as known. This allows us to use resultsderived in Giraitis and Phillips (2006). The results remain valid when α is constructed withthe OLS residuals ut, though the derivations are substantially more involved. We start withthe following Lemma.
Lemma A.1 Suppose that ut = αTut−1 + et with et as defined by (2), and αT = 1 + b/T h
for some b < 0 and some 0 < h < 1; then, a) T−h/2u[Tr] ⇒ (σ2/(−2b))1/2W (1); b)(−b)T−h−1/2PT
t=1 ut ⇒ σW (1); c) (−b)T−h−3/2PTt=1 tut ⇒ σ
R 10rdW (r); d) (−2b)−1/2T 1/2+h/2(α−
αT )⇒ N(0, 1); e) T h(α− 1) = b+ op(1).
25
Proof : For part (a), u[Tr] has a mean that is o(1) and
var(T−h2 u[Tr]) = σ2T−h(1 + α2T + ...+ α
2([Tr]−1)T ) =
σ2(1− α2[Tr]T )
T h(1− α2T )
=σ2(1− (1 + b/T h)2[Tr])
T h(1− (1 + b/T h)2)→ σ2
−2bThe result follows given the errors are i.i.d. and the weighted sums thereby converges toa Normal distribution. Part (b) is from Theorem 2 of Giraitis and Phillips (2006). Theirproof can be modified to prove part (c). Part (d) is from Theorem 1 of Giraitis and Phillips(2006). To prove part (e), note that
T h(α− 1) = T h(αT − 1) + T h(α− αT )
= b+ T h/2−1/2T 1/2+h/2(α− αT )
= b+ op(1)
in view of part (d) and the fact that 0 < h < 1.¥Consider first, the limit of the OLS estimate of β. We have, with t = T−1
PTt=1 t,
T 3/2−h(βols − β) =
T−3/2−hPT
t=1(t− t)ut
T−3PT
t=1(t− t)2
=T−3/2−h
PTt=1 tut − (T−2
PTt=1 t)(T
−1/2−hPTt=1 ut)
T−3PT
t=1 t2 − (T−2PT
t=1 t)2
⇒µ−σ
b
¶W (1)− R 1
0rdW (r)
(1/12)
= N(0, σ212/b2)
sinceR 10rdW (r) − (1/2)W (1) is Normally distributed with mean zero and variance 1/12.
Consider now the Feasible GLS estimate of β, defined by
βF − β =
q11r2 − q12r1q11q22 − q212
where the terms are defined at the beginning of the appendix. The following Lemma statesthe limit of each term when αT = 1+ b/T h for h < 1/2 and h > 1/2 (the case with h = 1/2is ommitted).
Lemma A.2 Suppose that ut = αTut−1 + et with et as defined by (2), and αT = 1 + b/T h
for some b < 0 and some 0 < h < 1; then, a) T−1+2hq11 → b2 if h < 1/2 and q11 → 1if h > 1/2; b) T−2+2hq12 → b2/2 for h ≶ 1/2; c) T−3+2hq22 → b2/3 for h ≶ 1/2; d)for h < 1/2, T−1/2+hr1 ⇒ −bσW (1), and for h > 1/2, r1 ⇒ u1; e) for 0 < h < 1,T−3/2+hr2 ⇒ −bσ
R 10rdW (r).
26
The proof follows for simple caluculations using the results of Lemma A.1. Using LemmaA.2, we obtain for 0 < h < 1/2,
T 3/2−h(βF − β) =
T−1+2hq11T−3/2+hr2 − T−2+2hq12T−1/2+hr1T−1+2hq11T−3+2hq22 − T−4+4hq212
⇒ (b3/2)σW (1)− b3σR 10rdW (r)
b4/3− b4/4
= (12σ/b)
∙(1/2)W (1)−
Z 1
0
rdW (r)
¸= N(0, 12σ2/b2).
Hence, in this case the limit distribution is the same as that of the OLS estimate. For thet-statistic, we have
tFβ =T−1+2hq11T−3/2+hr2 − T−2+2hq12T−1/2+hr1£
σ2T−1+2hq11 (T−1+2hq11T−3+2hq22 − T−4+4hq212)¤1/2
⇒ (b3/2)W (1)− b3R 10rdW (r)
[b2(b4/3− b4/4)]1/2= N(0, 1)
in view of the fact that σ2p→ σ2. For 1/2 < h < 1,
T 3/2−h(βF − β) =
q11T−3/2+hr2 − T 1/2−hT−2+2hq12r1
q11T−3+2hq22 − T 1−2hT−4+4hq212
=T−3/2+hr2T−3+2hq22
+ op(1)
⇒ (−b_σ R 10rdW (r)
(b2/3)
= N(0, 3σ2/b2)
sinceR 10rdW (r) is Normally distributed with mean 0 and variance 1/3. For the t-statistic,
we have,
tFβ =T−3/2+hr2
(σ2T−3+2hq22)1/2⇒ (−b) R 1
0rdW (r)
(b2/3)1/2= N(0, 1).
Finally, using part (a) of Lemma A.1 and applying a CLT, it is easily seen that the first-
differnce estimator βFD= (T−1)−1PT
t=2∆yt is such that T 1−h/2(βFD−β)⇒ N(0, σ2/(−2b))
and tFDβ ⇒ 0.
27
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29
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