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,

11

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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