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Working Paper Series



Estimating Deterministic Trends in the
Presence of Serially Correlated Errors

Eugene Canjels and Mark W. Watson

Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
September (WP-94-19)

E s tim a tin g D ete rm in istic T re n d s in the
Presence o f S erially C orrela ted E rro rs
E u g e n e Canjels
D e p a r t m e n t of Economics, Northwestern University
M a r k W . Watson*
D e p a r t m e n t of Economics, Northwestern University
a n d Federal Reserve B a n k of Chicago
A u g u s t 1994

Abstract
This paper studies the problems of estimation and inference in the linear
trend model: y t = a + fit + u t , where u t is follows an autoregressive process
with largest root p, and /? is the parameter of interest. We contrast asymp­
totic results for the cases \p\ < 1 and p = 1 and argue that the most useful
asymptotic approximations obtain from modeling p as local-to-unity. Asymp­
totic distributions are derived for the OLS, first-difference, infeasible GLS and
three feasible GLS estimators. These distributions depend on the local-to-unity
parameter and a parameter that governs the variance of the initial error term,
ac. The feasible Cochrane-Orcutt estimator has poor properties, and the fea­
sible Prais-Winsten estimator is the prefered estimator unless the researcher
has sharp a p r i o r i knowledge about p and k . The paper develops methods for
constructing confidence intervals for (3 that account for uncertainty in p and k .
We use these results to estimate growth rates for real per capita G DP in 128
countries.
JEL: C22,040
•This is a substantially revised and expanded version on WA Note on Estimating Deterministic
Trends in the Presence ofSerially Correlated Errors” by Watson. We thank Sergio Rebelo forposing
a question that motivated this research. We have benefited from discussions and comments by two
referees, Edwin Denson, Michael Horvath, Robert King, James MacKinnon, Gauri Prakash and
James Stock. Financial support was provided by the National Science Foundation through grants
SES-91-22463 and SBR-94-09629.




1

1

Introduction

Many economic time series display clear trends well represented by deterministic
linear or exponential functions of time. The slope of the trend function represents
the average growth in the series (or rate of growth, if the series is in logarithms) and
is often a parameter of primary interest. Serial correlation in the data complicates
efficient estimation and statistical inference about the trend function, and this paper
studies trend estimation and inference when this problem is severe.
To be specific, assume that a series can be represented as.
yt = cx + 0 t + u t

(1 ~

p L)u t

=

vt

(1)

(2)

where y t is the level or log-level of the series, and u t denotes the deviations of the series
from trend. These deviations are serially correlated, with a largest autoregressive root
of p . The error term v t is an 1(0) process. If the u'ts are jointly normally distributed,
and the precise pattern of serial correlation is known, then efficient estimators of a
and 0 can be constructed by GLS, and statistical inference can be conducted using
standard regression procedures. In practise, the distribution of the errors and the
pattern of serial correlation is unknown, so that GLS estimation and exact inference
are infeasible.
Applied researchers typically use one of three feasible estimators, motivated by
the asymptotic equivalence of these estimators to the infeasible GLS estimator. If
\p\ < 1,so that u t is 1 ( 0 ), then the feasible GLS estimator is asymptotically equivalent
to the infeasible GLS estimator, under general conditions. Moreover, the classic
result of Grenander and Rosenblatt (1957) implies that the OLS estimators of a and
0 are asymptotically equivalent to the GLS estimators. Thus, if u t is 7(0), OLS
or feasible GLS applied to the level of y t is asymptotically efficient. On the other
hand, when p = 1, so that ut is 7(1), a can no longer be consistently estimated by
any method, and the OLS estimator of 0 is no longer asymptotically efficient. In
this case, the data should be differenced and the Grenander and Rosenblatt result
implies that the sample mean of A y t (the OLS estimator of 0 in the differenced
regression) is asymptotically equivalent to the efficient, but infeasible, GLS estimator
of 0 . In summary, if u t is 7(0) then OLS from the levels regression produces the
asymptotically efficient estimator, while if tt* is 7(1) then the sample mean of A y t is
the asymptotically efficient estimator.
Inference is just as dependent on the 1 ( 0 ) / 1 ( 1 ) dichotomy. Ideally, in either sit­
uation, inference should be carried out using the t-statistic from the infeasible GLS
regression. When u{ is 7(0), this t-statistic can approximated using the OLS estima­
tor together with a serial correlation robust standard error estimated from the OLS
residuals. Alternatively, when p = 1 and the data are 7(1), this t-statistic can be
approximated using the sample mean of Ay* together with a serial correlation robust




2

variance estimated from the first differences of the data. Of course, since most re­
searchers can’t know a p r i o r i whether their data are 7(0) or 7(1), these results are of
limited value. In this paper we study inference problems and the behavior of OLS,
first-difference and feasible GLS estimators when the data are either 7(0) or 7(1) and
p is unknown.
Our analysis builds on two literatures. The first is the literature on the linear
regression model with AR(1) errors exemplified by Cochrane and Orcutt (1949) and
Prais and Winsten (1954). The second is the literature on inference in regressions with
7(1) variables exemplified by Dickey and Fuller (1979), Durlauf and Phillips (1988)
and Elliott, Rothenberg and Stock (1992). Much of the former literature focuses on
efficient estimation of regression parameters when the errors follow a stationary AR(1)
process, and is directly relevant for our analysis when \p\ < 1 and v t is i i d .1 There are
few exact analytic results in this literature because of the dependence of results on the
regressors and the nonlinearity introduced by feasible GLS estimation.2 Moreover,
the asymptotic results summarized above rely on |p| < 1 and are not refined enough
to discriminate between OLS and feasible GLS estimators. Thus, the majority of
work in this area has relied on Monte Carlo simulations. Equations (1) and (2) have
also been extensible studied in the unit root literature, primarily with a focus on tests
for the hypothesis that p = 1. In most of this literature, the regression coefficients a
and 0 are nuisance parameters and p is the parameter interest.3 One of the purposes
of this paper is to highlight what this analysis says about the feasible estimators of
0 and statistical inference.
W e begin our analysis in Section 2 by presenting results on the asymptotic dis­
tributions of estimators of 0 . These include the OLS, first-difference, infeasible GLS
and three different, but commonly used, feasible GLS estimators. W e avoid the sharp
|/?| < 1 and p = 1 dichotomy in the asymptotic distributions by using local-to-unity
asymptotics, with the hope that these provide better finite sample approximations.
The asymptotic results for |p| < 1 and p = 1 are not new: they are reported here for
completeness and because, particularly w’hen p = 1, the results may not be widely
appreciated by applied researchers. In any event, the local-to-unity results are the
most relevant, since in most econometric applications the errors are highly serially
correlated, although perhaps not characterized by an exact unit root. These results
1There islarge literatureon this topic, including Beach and MacKinnon (1978), Chipman (1979),
Kadiyala (1968), Maeshiro (1976 and 1979), Magee (1987), Park and Mitchell (1980), Rao and
Griliches (1969), Spitzer (1979), and Thornton (1987).
2Two exceptions directly relevant for our analysis are Prais and Winsten (1954) and Chipman
(1979). The first paper studies equations (1) and (2) when a = 0 and t>» is iid , and calculates the
relative efficiency of the OLS and first-difference estimators as a function ofp and the sample size,
T. Chipman (1979) relaxes the assumption on a and calculates the greatest lower bound of the
efficiency of the OLS estimator for all T and p < 1. W’e discuss the Chipman (1979) analysis in
more detail in Section 2.2.1
3A notable exception is Durlauf and Phillips (1988), which isdiscussed in more detail in Section
2.2.1.




3

show sharp differences in the relative efficiencies of the estimators and four conclu­
sions emerge from the analysis. First, the Cochrane-Orcutt estimator performs very
poorly when p is large. Second, the OLS estimator is more robust to variations in p
than the first-difference estimator. Third, the variance of the initial error term has an
important effect on the relative efficiencies of the estimators. Finally, the asymptotic
results suggest that the feasible Prais-Winsten estimator isthe best estimator in most
applied situations. Section 2 concludes with a small finite-sample experiment that
indicates that the asymptotics provide reasonable approximations to the finite-sample
relative efficiencies.
Section 3 studies the problem of statistical inference about /?. Existing Monte
Carlo evidence suggests that methods relying on 7(0) asymptotic approximations
greatly understate the uncertainty in /? when \p\ < 1 but large. This leads to confi­
dence intervals that are much too small and hypothesis tests with sizes that are too
large. Asymptotic approximations that rely on p = 1 have analogous problems. This
section uses the local-to-unity asymptotic approximations from Section 2 to construct
bounds tests and conservative confidence intervals building on methods developed in
Dufour (1990) and Cavanagh, Elliott and Stock (1993).
In Section 4 we apply the methods to estimate and construct confidence intervals
for real per-capita G D P growth rates for one hundred and twenty-eight countries using
post-war data. Consistent with the analysis in Section 2, we find large differences
between the Cochrane-Orcutt and other estimators for many of the countries. There
are smaller, but economically important differences in the other estimators, and this
highlights the importance of estimator choice. Finally, for most countries, the high
degree of serial correlation and short sample leads to wide confidence intervals for /?.
Finally, we offer a summary and some conclusions in Section 5, and the appendix
contains proofs and other detailed calculations.

2

E stim ators

2.1

T h e M od el

The statistical model for the observations {yt}JC_x is conveniently summarized in the
following assumptions:
1. The data y t are generated by
yt = a + 0t + ut

2 . The error term

is generated by (1 —

3.

ui

4.

v t = d ( L ) e t,with d ( L )




=

ut

for t = 1,... ,T.
p rL )u t

=

vu

for t = 2,..., T

Pi’Ui-*-.
= £ £ 0 d,Z,', and £ £ 0i I
4

l< °°-

(3)
.

5. The error term t t isa martingale difference sequence with E ( t ] |et- i , 2,...) =
1 and with sup, E t 4 < 0 0 .
Assumption (1) says that the data are generated as a linear trend plus noise; the
parameter /? is the average trend growth in the series and is the parameter of interest.
Assumptions (2) and (3) are written to include both 7(0) and 7(1) processes. When
p T = p, with |p| < 1, then u, is 7(0); while when p j = 1, then u t is 7(1). More
generally, when />r = (1 + j ) , then u t follows a “local-to-unity” 7(1) process, with
c = 0 corresponding to an exact unit root and values of c 0 generating data that
are less (c < 0) or more (c > 0) persistent then the exact unit root process.4
Assumption (3) incorporates a range of assumptions about the initial condition
ui, depending on the value of k and p j . For example, when k = 0,then u\ = uj ,so
that the initial value is assumed to be an O p(l) random variable. When k > 0, then
ui is O p ( T */2) when u t is 7(1), but is O p(l) when tt, is 7(0). When p r = p , with
\p\ < 1 and k T —► 00,then u i is drawn from the unconditional distribution of u t, and
the process is covariance stationary.5
Assumption (5) implies that the functional central limit applies to the partial
sums of Ct, i.e., T ~ *
tt
W(s), where W(s) is a standard Wiener process.6
Assumption (4) insures that the functional central limit theorem also applies the
partial sums of v t , specifically T ~ 2
vt =
*>• d(l)W(s).

2.2

A sy m p to tic P roperties o f E stim ators

2 .2.1

OLS, First-Difference and G L S Estimators

Let P o l s denote the OLS estimator of ft in (1), let P f q = ( T — l)-1 Y lJ =2 & y t denote
the first-difference estimator, and let P g l s denote the infeasible GLS estimator that
corrects for non-zero p j - Specifically, P g l s is the OLS estimator in the transformed
regression
yt - p T V t - i

= (1 ~

P

t

1)] + ut -

)q + P [ t - p r{ t -

p r u t - 1,

< = 2,3,...,r.

(4)

together with
a ~ Xy\

=

<r~xot

4- <r~l P

+ (T~l u i ,

(5)

where a 2 = (1 — P j ^ +1^)/(1 — p \ ) f°r />t / 1 and c 7 = [/cT] + 1 for p r = 1 For
simplicity, the GLS estimator ignores the 7(0) serial correlation associated with d ( L ) .
4These “local-to-unity” processes have been used extensively to study local power properties of
unit root tests, construct confidence intervals for autoregressive parameters for highly persistent
processes, and more generally, to study the behavior ofstatistics whose distribution depends on the
persistence properties of the data. Some notable examples are Bobkoski (1983), Cavanagh (1985),
Cavanagh, Elliot and Stock (1993), Chan and Wei (1987), Chan (1988), Phillips (1987), and Stock
(1991).
5See Elliott (1993) for related discussion ofthe initialerror in the 1(1) model.
6A range of alternative assumptions will also suffice; see Phillips and Solo (1992) for discussion.




5

This allows us to focus on the major source of serial correlation, p r ^ 0, and leads
to no loss of asymptotic efficiency for the models considered here (Grenander and
Rosenblatt (1957)).
'
'
A
A
A
In large samples, the behavior of P o l S i P f d and P g l s is summarized in Theorems
1 and 2:
Theorem 1

( B e h a v io r o f

P

o l s

>P

f d

U n d e r a s s u m p t io n s ( l ) - ( 5 ) w ith p r

Tl(

(a )

(h) T ( P
ance
T (P
(c)

Po

-

ls

—

fd

+ uar(ui),

~ P)

Vj = 12(1 - />)-*<£(l)2.

w h e re

/,• = ^ ‘_0 p ^ ~ ^ d j.

T h e l i m i t i n g d is t r ib u t io n o f

d e p e n d s o n th e d is t r ib u t io n o f th e e 's , a n d s o in g e n e r a l is n o n - n o r m a l,

T*( Pq

Proof.

w ith 1 ( 0 ) E r r o r s ) :

|/>| < 1:

c o n v e r g e s in d is t r ib u t io n to a r a n d o m v a r ia b le w ith z e r o m e a n , v a r i ­

P )

Vi =

fd

g ls

p, a n d

N ( 0,Vi), w h e re

P )

P

and

=

ls

— P ) — + N(0,Vi), w h ere V i is s p e c ifie d in ( a ) .

P a r t (a ) a n d (c ) f o llo w f r o m

l i m i t th e o re m .

a s t r a ig h t f o r w a r d a p p lic a t io n o f the c e n t r a l

T o s h o w p a r t (b ), n o te th a t T ( P

fd

— P) =

u j

—

fro m

w h ic h the

re s u lt f o llo w s im m e d ia t e ly .□

Theorem 2 ( B e h a v i o r o f P o l s > P f d
L e t S c ( t ) = (— 2c)-1(l — e2rc). T h e n
(a)

T 3

(P

o ls

-

P)

and

P

q ls

w ith

7(1)

E rro rs):

u n d e r a s s u m p t io n s ( l ) - ( 5 ) , w ith p r

N ( 0,7?0, w h e re

= d(l)V5[18(c - 2 ) V C + 72c(c - 2)ec + 12c3 + 54c2 + 72c - 72]

R i

cec+ c — 2(ec — 1) |2
+d(l)2144Sc(«)[
2c2
^

(b)THPF D ~ P )

N ( 0 , R 2), w h e re

7?2 = d(l)2[5c(l) + (l-ec)25c(/c)].
( c) T H P

gls

- P ) - ^

Rz

Proof.

N ( 0 , R 3), w h e re

5c(/c)c2 + 1
= d(l)2[
(Sc(/c)c2 + 1)(1 - c + §c2) - 5e(/c)(ic2 - c)2

S e e A p p e n d ix .

Corollary 3

( B e h a v io r o f

P

o ls

>P

f d

U n d e r a s s u m p t io n s ( l ) - ( 5 ) , w ith p r —
(<•) T l 0 o i . s - 0 )

and

1.'

N ( 0 ,id ( in

(b) T i ( P F D - l ) ) ± N ( 0 , 4 i n
(c) Tl( f a s - 0 ) -i-N(0,J(in




= (1 + y);

6

P

o ls

w hen p — l ) :

W e highlight five features of these results. First, P o l s > P f d and P g l s converge
to ft faster in the 7(0) model than the 7(1) model. This results obtains because the
variance of the errors is bounded in the 7(0) model and increases linearly with t in
the 7(1) model. Sampson (1991) discusses the implication of this result for long-run
forecast confidence intervals.
Second, the averaging in P o l s in the 7(0) and 7(1) cases and in P f d in the 7(1)
case leads to asymptotically normal estimators. In contrast, since T ( P f d — P ) =
[ T / ( T — l)](ur — ui), no such averaging occurs for P f d in the 7(0) case, so that P f d
is not asymptotically normally distributed in general. (See Quah and Wooldridge
(1988) and Schmidt (1993) for related discussion.)
1
Third, P g l s is the asymptotically efficient estimator regardless of the value of p
and it corresponds to the B L U E estimator when d ( L ) = d , a constant. The efficiency
of the FD and the OLS estimator relative to the GLS estimator differs dramatically
in the 7(0) and 7(1) c a s e s . When the errors are 7(0), then P f d converges to P
more slowly than does P g l s > and thus has an asymptotic relative efficiency of 0. In
this case, P o l s is asymptotically efficient, the familiar result from Grenander and
Rosenblatt (1957). When the errors are 7(1), P o l s i P f d and P g l s converge at the
same rate and the relative efficiency depends on the parameters c and k . Figure
1 plots the asymptotic relative efficiencies (defined as the ratio of the asymptotic
variances of P o l s and P f d to the asymptotic variance of P g l s ) in the 7(1) model
for a range of values of c and k . When c = 0, both P o l s and P f d are invariant
to u i and so their variances and the relative efficiency do not depend on k . In this
case P f d is asymptotically efficient and P o l s has an efficiency of 5/6. This result is
derived in Durlauf and Phillips (1988), who study the properties of trend estimators
in the model with p = 1 (equivalently, c = 0). When c is sufficiently negative, P o l s
dominates P f d f°r all values of «. The intersection point of the P o l s and P f d relative
efficiency curves depends on k . For example, when k = 0, P f d is efficient relative to
P o l s for values of — 18.6 < c < 1.2,and P o l s dominates P f d for c outside this range.
When k = 1.0, the range narrows to — 7.6 < c < 0.9.
Fourth, when k = 0, so that u\ is O p ( 1), the relative efficiency of both P o l s
and P f d increases monotonically with c. The relatively poor performance of these
estimators when uj is O p(l) has been noted elsewhere, notably by Elliott, Rothenberg
and Stock (1992) in the context of unit root tests. On the other hand, when k > 0,so
that
is O p ( T * ) , the relative efficiency of P o l s is U-shaped, with a minimum that
depends on the specific value of k . For example, when k = 1, the minimum relative
efficiency of P o l s occurs at c = — 3.006 where it takes on the value of 0.7535. As
k — ► oo, the minimum relative efficiency of P o l s i
s .7538 and occurs at c = — 3.076,
a result that was also derived by Chipman (1979) using methods different from those
employed here.7
7Chipman (1979) also shows that, when d (L ) = d, this asymptotic relative efficiency value isthe
greatest lower bound for the relative efficiency of P o l s for all n > 2. Because of a slight numerical
error in Chipman’s paper, his reported numerical results are different from those reported here.




7

Finally, when the errors are 7(1), the variances of 0 o l s i P f d , and 0 g l s depend
on c and k in important ways. For example, Figure 2 plots the variance of 0 g i s as a
function of cand k . A s c increases, the persistence of the errors increases and so does
the associated variance of P g l s - Similarly, as k increases, the variance of uj increases,
leading to an increased variance in P g l s 2 .2.2

Feasible G L S Estimators

The efficient GLS estimator relies on two parameters, p and /c, whose values are typi­
cally unknown. In this section we analyze feasible analogues of P g l s - The parameter
p is easily estimated from the data, and as we show below, replacing p with an es­
timate has little effect on 0 g l S ‘ On the other hand, it is impossible to construct
accurate estimates of k , since this parameter only affects the data through the vari­
ance of the single observation uj. W e therefore analyze three feasible GLS estimators
that differ in their treatment of the initial observation. W e find large differences in
the relative performance of these estimators across different values of k .
To focus attention on the parameter k , we begin by analyzing the estimators
assuming that p is known; a simple modification of these results yields the results
for unknown p . As above, the GLS estimators ignore the serial correlation associated
with the 7(0) dynamics in d ( L ) , since the Grenander-Rosenblatt (1957) results imply
that OLS or GLS treatment of d ( L ) has no asymptotic effect on the estimators of
/?that we consider. Let f l c o denote the Cochrane-Orcutt (1949) GLS estimator that
ignores the levels information in the first observation; that is, (3co denotes the OLS
estimator of f i in equation (4). Let f i c c denote the GLS estimator constructed under
the assumption that uo = 0. This assumption is often made in the unit root literature
(see, e.g., Elliott, Rothenberg and Stock (1992)) and is referred to as the “conditional
case.” Thus, j 3 c c is the OLS estimator of (3 from (4) together with:
j/i = a + 13 + ux.

(6)

Finally, let f j p w denote the Prais-Winsten (1954) estimator; that is, the OLS estima­
tor of (3 from (4) together with:
(1 -

p r ) xl2y i

= (1 -

P2
t Y /2o

+ (1 -

P t ) 1,2P

+ (1 “

Pt

)1/2u»•

(7)

The Prais-Winsten estimator is defined for p j < 1, and we limit our discussion to
this situation. In the notation introduced in the last section, f r e e corresponds to the
GLS estimator constructed using k = 0, and /3pw is the limiting value of the GLS
estimator as k —► oo.
When p r = p , with |/>| < 1 (i.e., u t is 7(0)), each of the GLS estimators is
asymptotically efficient and the large sample distribution is given in Theorem 1.
(Specifically, the value of c that we report (c = —3.07558) is a more accurate estimate of the root
to his polynomial (3.3) than the value reported in his paper (c= —3.09485).)




8

Thus, we need only consider the behavior of the estimators in the 7(1) model, and
this is done in the following lemma:
Lemma 4

( B e h a v io r o f G L S E s t i m a t o r s w ith

/>r = (1 +

U n d e r a s s u m p t io n s ( l ) - ( 5 ) , w ith
(a ) T ' l \ 0 c o -

W(0,Gi),

0)-^

„
G\

7(1)

E rro rs):

f):

w h ere

12d(l)2

.

= -- -— ,f o r c / 0, a n d
c1

G\ —
(b)

T W tfc c - 0 ) ~ ^

N ( 0 , G 2), w h ere

J(1)2
fl I Sfrl (c “ 2°2)2 1
- 1 _ C + I C2[1 +
c— & ]-

C

(c )

d(l)2,f o r c = 0 .

T l t* (0 p w - 0 ) - ^

N ( 0 , G 3 ), w h ere

G3 —

<i(1)I

■[cI5,(«) + 1 + i c !].

Proof.
See Appendix.
Part (a) of the lemma implies that G \ , the limiting variance of T * { 0 c o — 0 ) , is
discontinuous at c = 0. This occurs because the regression constant term, a , becomes
unidentified as c —+ 0. For values of c close to zero, a is very poorly estimated, and
the collinearity between the two regressors (1,<) in equation (4) means that f i c o is
also a poor estimate of 0 . When c = 0, a disappears from equation (4) and so this
source of variance disappears from 0 c o - Figure 3 shows the efficiency of each of
the estimators relative to 0 g l s • The Cochrane-Orcutt estimator, 0 c o i performs very
poorly for small values of c regardless of the value of k . This result is consistent with
a large literature on the poor performance of the Cochrane-Orcutt estimator with
trending regressors and p close to unity.8
The relative performance of the other two estimators depends on the values of k
and c. When k — 0, fa c e is the asymptotically efficient estimator; while f ip w is the
efficient estimator as k — ► 00. From Figure 3, 0 p w is approximately efficient even
when k is very small. For example, for k = .01 the relative efficiency of
is larger
than 0.73 for all values of c; for k = .05 the relative efficiency is larger than .92; and
for all values of k > .10, 0 p w is essentially efficient. While f i c c is efficient when
k = 0 ,this efficiency gain disappears quickly for moderate values of c as k increases.
8See Prais and Winsten (1954), Maeshiro (1976,1978), Beach and MacKinnon (1978), Park and
Mitchell (1980), Thornton (1987) and Davidson and MacKinnon (1993, Section 10.6).




9

W e are now ready to discuss the feasible GLS estimators with p r unknown. These
estimators are calculated like their infeasible counterparts, using an estimator of p r
in equations (4) and (7). These estimators will be denoted as 0 f c o ,0 f c c , and 0 f p w Analysis of these estimators is complicated by the fact that they implicitly depend on
the estimator for pr, and a variety of estimators of p j have been suggested. For 0 f c o
the non-linear least squares estimator is often employed, and this estimator is studied
by Nagaraj and Fuller (1991) for the model with general regressors. Their analysis
can be simplified here because of the special structure of the regressors: equation (4)
together with assumption (2) can be combined as:
y t = a + bt + p r y t - i

+ ft, for t

=

2,3,..., T,

(8)

where a = a(l — p i ) 4-0 p r and 6 = /?(1 — pj).^Thus, 0 f c o can be formed from the
OLS estimators from equation (8) as 0 f c o = 6/(1 — p r ) for p T ^ 0 and 0 f c o = 2
for p r = 1, where 2, 6,and p j are the OLS estimators of the coefficients in equation
(8). Equivalently, 0 f c o can be constructed as the OLS estimator of 0 in (4) using
p x in place of pr* Since the asymptotic distribution T(l— p r ) is readily deduced
when p T = (1 + f ) , (see Stock (1991), for example), the asymptotic distribution of
o — 0 ) can also be readily deduced.
The problem is more complicated when analyzing 0 f c c and 0 f p w , since these
estimators are generally based on iterative schemes for estimating pr,or, and 0 . Iter­
ative schemes are often used to construct 0 f c o as well. Since the limiting distribution
of p r depends in important ways on the precise way the data are “detrended” (for
example, see Schmidt and Phillips (1992) and Elliott, Rothenberg and Stock (1992)),
the limiting distribution of 0 f c c > and 0 f p w will depend on the precise specification
of the iterations. Rather than present results for specific versions of these estima­
tors, we present limiting representations of 0 f c C i and 0 f p w written as functions of
c = plimT(l — pr). Different estimators of pr will lead to different limiting random
variables c and different asymptotic distributions for the estimator of 0 . A specific
example is contained in Durlauf and Phillips (1988, Theorem 4.1), who derive the lim­
iting distribution of 0 f c o when c = 0 and c is constructed from the Durbin-Watson
statistic calculated from the levels OLS regression.
Before presenting the limiting distributions for the feasible GLS estimators, it
is useful to introduce some additional notation. The error term in the feasible GLS
version of (4) is v t = V t — p r u t - u and the limiting values of the feasible GLS estimators
can be written in terms of initial condition Ui and partial sums of v t . In the appendix
we show that T ~ * u i =» IV^/c) ~ A^O, 5c(k )), where 5C(«) is defined in Theorem 2;
we also show that T ~ j
=» W ( s ) where IV’(s) is a functional of IF(s) and
W e{,c).
With this notation established, we now present the limiting distribution of the
feasible GLS estimators:

T

i

(/3f c




10

Theorem 5

( B e h a v io r o f F e a s ib le G L S E s t i m a t o r s ) :

S u p p o s e th a t a s s u m p t io n s ( l ) - ( 5 ) a re s a tis fie d , p p

= (1+ f),

and p U m ( p T - l)

= 8^0.

Then:
(a ) T L
> 0 FCO ~ 0 ) = >

c-42Jftl -

s)d W (s),

=> [1 - 3 + i c V ^ - j c 2) ^ * ) - f 0l (c s - l ) d W ( s ) ) ,
-/?)=► [1 - § 3 + i c 2] - 1[3W'e(« ) - f o i l + \ c - t s ) d W ( s ) ] .

( b) T H f o c c ~ 0 )
(c )

T \ 0 fpw

Proof.

__
and

S e e A p p e n d ix .

This theorem allows us to offer practical advice about choice of estimators. First,
notice that 8 appears in the denominator of the limiting representation of
(0 p c o ~
0 ) . For most commonly used estimators of p , p F can take on values arbitrarily close
to 1 with positive probability, so that 8 can be very close to zero. This means that
0 f c o can be very badly behaved, since realizations of 8 close to zero will often lead to
extreme realizations of 0 F c o • On the other hand, 0 p c c and 0 F p w are better behaved,
since [1— 8 + 58s] > 0 and [1 — ^8 + j^c*] > 0 for all values of 8. This can be seen in
Figure 4 which plots the limiting probability densities of T * { 0 p c o — /?),T’a[ 0 F c c — 0 )
and T i { 0 p p w — 0 ) , for the case with c = 0, k = 1,and d(l) = l.9 Also plotted is the
probability density of the exact (infeasible) GLS estimator (which in this case is the
standard normal). The estimators 0 F c c > and 0 F p w have probability distributions
very close to the infeasible efficient estimator. On the other hand, the distribution
of 0 p c o is much more disperse, with thicker tails than the other distributions. For
example, the limiting probability that \ T * ( 0 p c o — 0)\ exceeds 2 isapproximately 20%;
while the corresponding values for 0 F c c and 0 F p w are approximately 5%. Figure 4
suggests that littleislost using in using either 0 F c c and 0 F p w in place of the infeasible
efficient estimator, at least for this value of cand k , and that 0 F c o performs poorly.
Additional calculations (not shown) indicate that the relative efficiencies of 0 p c c and
0 p p w are close to their infeasible analogues for a wide range of values of c and k .
Table 1 summarizes many of the results in this section by presenting the average
mean squared error for the different feasible estimators and different values of k ,av­
eraged over different ranges of c.10 As a benchmark, the first row of the table shows
results for the efficient, but infeasible, GLS estimator. The next two rows are the
OLS and first-difference estimators, followed by two of the feasible GLS estimators.
(Since the asymptotic mean squared error of 0 F c o does not exist, this estimator is
not included in the table.) The last row of the table shows results for a “pre-test”
9The densities for the feasible GLS estimators are estimates based on 5000 draws from approxi­
mations to the asymptotic distributions (constructed usingT=500). The estimators0 f c o and 0 p c c
were constructed using p r constructed as the OLS estimator of (8). The Prais-Winsten estimator
used min(l,/>r).
l0These MSE’s were estimated using the simulations described in footnote 9.




11

estimator ( P p t ) constructed from the OLS and F D estimator. Figure 1 provides the
motivation for this estimator. Since the OLS estimator dominates the first-difference
estimator for large negative values of c and is dominated by the first-difference esti­
mator for small values of c, the pre-test estimator corresponds to the OLS estimator
when 8 is large and negative and corresponds to the FD estimators when 8 is close to
zero. Specifically, P p t = P o l s when 8 < c and P p t = P f d when 8 > c, where c is
pre-specified threshold. The results shown in the table are for c = — 15, a value that
produced good results over the range of values of k and c that we considered.
Table 1 and the figures shown above suggest five conclusions:
(i) The infeasible GLS estimator P c o performs very poorly for values of c close to
0. This poor performance is inherited by the feasible GLS estimator. For all values
of c ^ 0 and for all values of /e, this estimator is dominated by P o l s • Thus, this
estimator should not be used and is ignored in the remaining discussion.
(ii) For very small values of c (say, — 2 < c < 0), f i p p isthe preferred estimator with
a mean squared error approximately 5% lower than P f c c and P f p w • For this range
of values of c, the OLS estimator, P o l s i has a relative efficiency of approximately
.75. The pre-test estimator performs well, and is l%-2.5 % less efficient than P f D i
depending on the value of k .
(iii) For values of c in the range — 10 < c < — 2 , the relative performance of the
estimators depends critically on the value of the initial error, parameterized by k .
When k = 0,P f c c dominates the other estimators; P f p w is the preferred estimator
when k > .10. When k = .05 the feasible GLS estimators and P f d are comparable.
(iv) For values of — 30 < c < —10 and when k = 0,P f c c isthe preferred estimator.
When k > 0.05, the variance of P f c c is more than twice as large as the variance of
the best estimator, P f p w • The first difference estimator also performs poorly relative
to P f p w when k > .05.
(v) Items (ii)-(iv) show clearly that the best estimator depends on the values of c
and k . Neither of these parameters can be consistently estimated from the data, and
so a good choice must depend on either prior knowledge or robustness considerations.
Our reading of the results suggests that P f p w is the most robust estimator, with a
M S E close to the optimum for all values of the parameters considered. The pretest
estimator is a reasonable alternative to P f p w iit has slightly better performance w’hen
c close to 0 but somewhat worse performance for large negative c.

2.3

Sm all Sam ple P rop erties o f E stim ators

The asymptotic results summarized in Theorems 1, 2 and 5 are potentially useful
for two reasons. First, the asymptotic relative efficiencies can provide a criterion
for choosing among the estimators even in finite samples. Second, the asymptotic
distributions provide a basis for constructing confidence intervals and carrying out
hypothesis tests. In this section we evaluate the first of these uses, and ask whether
the 7(0) and 7(1) asymptotic variances provide a useful guide for choosing among the




12

estimators in small samples. In the following section, we discuss confidence intervals
and statistical inference.
Table 2 shows the exact relative efficiencies of 0 o l s i 0FD i 0FCC, 0 f p w , and 0 P t
for the model with d(L) = d, et ~ NIID( 0,1), for various values of T , p, and for
k = 0 (panel A) and k = 1.0 (panel B).n Also shown in the table are the relative
efficiencies implied by the 7(1) asymptotics, calculated using c = T(p — 1). The 7(0)
asymptotic relative efficiencies are not shown because they do not vary with T, £ or
«; from Theorem 1 they are 1.00 for P o l s , 0 f c c > 0 f p w , and 0 p j and 0.00 for 0 f d In all cases, the 7(0) asymptotic relative efficiency suggests indifference between the
four estimators 0 o l s > 0 f c c > 0 f p w and 0 p t , and suggests that these estimators are
preferred to 0 f d W h e n p = 0.5, the finite sample results in Table 2 suggest that 0 olsi 0FCCi and
0 f p w are essentially efficient for all of the sample sizes considered. These estimators
are significantly better than 0 f d • The pre-test estimator has a relative efficiency
intermediate between 0 o l s and 0pp when T = 30, and very close to 0 ol s for larger
values of T. Thus the 7(0) relative efficiency predictions are quite accurate when
p = 0.5. The predictions based on the 7(1) asymptotic relative efficiencies are off
the mark. The 7(1) asymptotics suggests that 0 f c c strongly dominates the other
estimators when k = 0 and is strongly dominated by both 0ol s an<l 0 f p w when
k =
1. O n the other hand, the estimator with the largest 7(1) asymptotic relative
efficiency coincides with the largest finite sample relative efficiency, even when p = 0.5.
For all of the other values of p that are considered (0.8, 0.9, 0.95, 1.0), the rank­
ings implied by the 7(1) asymptotic relative efficiencies are more accurate the 7(0)
rankings. Indeed in all cases studied in the tables, the estimator with the largest 7(1)
asymptotic relative efficiency has the largest finite sample relative efficiency as well.
Thus, this experiment suggests that the 7(1) asymptotic relative efficiencies provide
a useful criterion for ranking estimators in typical econometric settings.

3
3.1

Confidence intervals
C onstruction o f confidence intervals.

In this section we discuss methods for constructing confidence intervals for 0. W h e n
p < 1 (so that the errors are 7(0)) confidence intervals can be constructed in the usual
way by inverting the “t-statistic” constructed from any of the asymptotically equiv­
alent estimators 0 o l s » 0 f c o , 0 f c c , 0 f p w , or 0 p t • These t-statistics can be formed
using an estimator for the variance Vi in Theorem 1, constructed by replacing p and
d( 1) with consistent estimators. While these confidence intervals are asymptoticallyl

llThe mean squared errors for 0pcc> 0FPW< and 0 p r , were estimated using 10,000 Monte Carlo
draws, using p = £lT=2 2iUi-i/£^=2'2?, where u, are the OLS residuals from the regression of
y, onto (1,t). This estimator ofp issuggested by the simulation results in Park and Mitchell (1980).




13

valid, they can greatly understate the uncertainty about 0 when p is large and the
sample size is small. (See Park and Mitchell (19S0) for simulation evidence.) Thus,
in most situations of practical interest, confidence intervals based on 7(0) approxima­
tions are not satisfactory.
An alternative method pursued here is to construct confidence intervals using
approximations based on 7(1) asymptotics. As we show below, this method yields
confidence intervals with coverage rates closer to the nominal size than the 7(0) ap­
proximations. Unfortunately, the method isalso more complicated. The complication
arises because in the 7(1) model, the asymptotic distribution pf the various estimators
of 0 depends on the nuisance parameters c and k , and these parameters cannot be
consistently estimated from the data. Thus, the variances of the estimators cannot
be consistently estimated, so that t-statistics will not have the appropriate limiting
standard normal distribution. While this problem cannot be circumvented entirely, it
is possible to construct asymptotically conservative confidence intervals following the
procedures developed by Dufour (1990) and Cavanagh, Elliott and Stock (1993).12
Specifically, let B K(c) denote a 100(1 — c*i)% confidence interval for 0 constructed
conditional on a specific value of c and k . Similarly, let C K denote a 100(1 —
o i l )% confidence interval for c conditional on k . Assume that 0< k < 7c, where 7c is
pre-specified constant. Then the Bonferoni confidence interval, Uo<*<*Uc€C. B K( c ) ,
is a conservative 100(1 — ct\ — 0 2 )% confidence interval for 0 .
This confidence interval requires the conditional confidence interval for 0 , B K( c ) ,
and the marginal confidence interval for c, denoted C K. Since B K(c) conditions on
the nuisance parameters c and «, an asymptotically valid approximation can be con­
structed using any of the estimators 0 o i s > 0 F D , 0 c c , or 0 p w , and their asymptotic
variances given in Theorem 1 and Lemma 4. (These variances require d ( 1), which
can be consistently estimated using standard spectral estimators.) The marginal
confidence intervals for c, C K, can be constructed using the methods developed in
Stock(1991).13
12Dufour (1990) considers the problem ofstatistical inference in the regression model with Gaus­
sian AR(1) disturbances. He develops “bounds” tests and associated confidence intervals based on
exact distributions. Cavanagh, Elliott and Stock (1993) consider testing for Granger-Causality in
a regression with a highly serially correlated regressor modeled as a local-to-unity process. They
develop bounds tests and associated confidence intervals based on asymptotic distributions.
13Stock (1991) considers the case with k = 0 and, using our notation, develops methods for con­
structing confidence sets Co- However, it iseasy to modify his analysis for k > 0. Specifically, fol­
lowing Stock, we construct confidence intervals by inverting the Dickey-Fuller t-statistic, rT. Under
the assumption that k = 0, Stock shows fT => ( f * VFer(s)Jds)J[c+ f g W J ( s ) d l V ( s ) / ( f g WT (*)*</*)],
where W J (s ) isthe “detrended" diffusion: W J (s ) = W e(s) — a i( r ) W e( r)d r — s fg aj(r)lVe(r)dr,
where the diffusion W e(s) is defined in the appendix, ai = 4 — 6r, and aj = —6 + 12r. These
results rely on the fact that
=> d(l)lVe(s) when k = 0. As shown in the appendix, when
k ^ 0,T~£ u[
,t ] => d(l)[We(s) + e,e W e(K)], where W e(/c) ~ JV(0,Se(«)) and is independent of
W e($). Using this, itisstraightforward to show that allof Stock’s analysis continues to hold, with
lVc(s) + e, c \Vc(K) replacing W e(s) in the above limiting representation for rT.




14

In general, this procedure is quite demanding. For each 0 < k < 7c, C K must be
formed, then B K(c) must be constructed for all c € C«, and the union taken over all of
these confidence sets. There are three special features of the linear trend model that
simplify this procedure. First, from Theorem 2,the asymptotic variances of P o l s and
P f d are monotonically increasing in c. Thus, when B K(c) are formed using t-statistics
constructed from P o l s or
then Uc€C„ B * i c ) = #«(c), where c = supc{c € C K ) .
While this simplification does not necessarily hold for the GLS estimators P e c and
P p w , experiments that we have performed suggest that Ucgc* B * ( c ) v B K(c) appears
to be a good approximation for confidence sets constructed from these estimators as
well. The second simplifying feature is that the distributions of the statistics used
to form C K change little as k changes, so that C o as C K for all k . u Finally, for all
of the estimators, the asymptotic variance is increasing in k and the limit exists as
k — ► oo, so that B k ( c ) C B oo( c ) for all k . Putting these three results together implies
that U o<*<k Ucec* B K(c) as .^(c), where c = supc{c € Co}. Thus approximate
100(1 — g-i — 0:2)% confidence intervals can be formed by (i) choosing the largest value
of c in the 100(1 — a^)% confidence interval constructed using the procedure from
Stock (1992), and (ii) constructing a^l00(l —
confidence interval for P using
this value of c together with P o l s i P f d , P c c , or P p w and an associated variance
from Theorem 2 or Lemma 4 evaluated at k = oo.
W e make two final points before evaluating the small sample properties of this
procedure. First, since the variance of all of the estimators is increasing in c, smaller
confidence intervals for P can be obtained by constructing 1-sided confidence intervals
for c. Second, when the B K(c) confidence intervals are constructed by inverting the tstatistics for the estimators, the widths of the intervals will be non-random conditional
on c and «. This implies that the narrowest of the confidence intervals (across all
estimators) will also have coverage rate exceeding 100(1 — Qi — aj)%. Thus, for
example, since P o l s is efficient relative to P f d when c < — 7.6 and k is large, the
confidence interval can be constructed using P o l s when c < — 7.6 and using P f d
when c > — 7.6.

3.2

Sm all sam ple perform ance o f confidence intervals

Table 3 shows estimated coverage rates for confidence intervals for different values
of T and c, calculated as described above. In panel A, the confidence intervals are
calculated as the narrowest of the OLS and FD confidence intervals. Panel B shows
results for confidence intervals constructed from the Prais-Winsten estimator. The
design was much the same as in Section 2.3, i.e., d ( L ) = d and et ~ iV(0,1). Results
u YVhen c = 0 the distribution of ?T isinvariant to k . This isnot strictly true for other values of
but the distribution changes very little. For example, when c = —1.0 the 97.5 percentiles for r*1
are -3.72, -3.70, -3.70 and -3.70 when k = 0.0, 0.5, 1.0, and 10.0, respectively. The corresponding
percentiles are -3.89, -3.84, -3.84, -3.84 for c = -5.0; -4.20, 4.20, 4.20, -4.20 for c = —10.0; and 4.52,
-4.54, -4.54, -4.54 for c = —20.0. These percentiles are based on 5,000 simulations with T = 500.
c,




15

are reported for conservative 90%, 95% and 99% confidence intervals constructed with
Qi = aj. Results for non-symmetric ou and or? are similar and are not reported. The
confidence interval for p was constructed from the ?T statistic constructed from the
regression of y t onto A y t_i and (l,f) using the sample t = 2
,
The sample
residual variance from this regression was used as the estimator of <f(l)2 in the con­
struction of the confidence intervals for 0 . Finally, since the Prais-Winsten estimator
is defined for \p\ < 1 we restricted the upper limit of the confidence interval to p = 1.
For comparability, this restriction was also used in the 0 o l s and 0 f d confidence
intervals.
The coverage rates are close to their nominal level for c = 0. When c < 0, the
confidence intervals are conservative, with coverage rates exceeding the nominal level.
This occurs because of the sharp increase in the variance of estimators for small c.
So for example, when the true value of c = — 5, then c = 0 is often in the confidence
set Co, the variance of the estimators is much larger when c = 0 than when c = — 5
(see Figure 2) and this leads to a wide confidence interval for 0 .

4

E conom ic G row th R ates for th e Postw ar P e ­
riod

Table 4 shows estimated annual growth rates of real G D P per capita for 128 coun­
tries over the postwar period. The data are annual observations from the Penn World
Table (version 5.5) described in Summers and Heston (1991) (series RGDPCH). The
data set contains 150 countries, and we limited our analysis to those 128 countries
with 20 or more annual observations. The first column of the table shows the country
identification number from the Penn World tables, and the next column shows the
country name. Columns 3-6 present four estimates of average trend growth (0 o l s ,
0 f d »0 f c o »and 0 F P W i respectively); column 5 shows the estimate of c used to con­
struct the feasible GLS estimates (c); column 6 shows the Dickey-Fuller unit-root
test statistic (?T) used to construct a confidence interval for c, and columns 7 and 8
present lower and upper limits of the approximated 95% confidence interval for 0 con­
structed from the 0 p w (0min and 0 max, respectively). The estimate c was calculated
as explained in footnote 11. The fT statistic was calculated from the regression of A y t
onto y t -\,Ayi_i and (1,t) using data from t = 3,.. .T, and the point estimates from
this regression were used to estimate d(l). W e highlight five features of the results.
First, for the majority of the countries, the different estimators give similar results.
For example, for the Congo (country 12) the estimates range from 2.8% ( 0 f d ) to 3.4%
(0 f c o )• Second, while the 0 f c o estimates are usually similar to the other estimates,
they occasionally deviate substantially. For example, the estimates for Suriname
(country 81) constructed from 0 o i S i 0FD > and 0 f p w range from 0.4% to 1.4%, while
the estimate constructed from 0 f c o is — 212%. Indeed for 31 of the 128 countries,
0 f c o differs from 0 o i s by more than 5 percentage points. Third, while the differences




16

in the other three estimators are much smaller, these differences can be quantitatively
important. For example, 0o l s > 0FD differ by more than 1% in 5 cases and by more
than |% in 35 cases.
Fourth, the confidence intervals are often wide and include negative values for /?.
This results from three factors: a small sample size, a large error variance and a high
degree of persistence in the annual growth rates. For example, the approximate 95%
confidence interval for Algeria (country 1) is — 1.18 < /? < 4.10. For Algeria, the
Dickey-Fuller t-statistic is — 1.46 which implies that c = 0 (i.e., p = 1) is contained in
the 97.5% confidence interval for c. Thus, for this value of c, f ip w corresponds to the
first-difference estimator. The mean growth rate for Algeria over the sample period is
1.45% (= 0 f d ) and this is the center of the confidence interval. The standard devia­
tion of the annual growth rates is 7.3%; thus, ifthe annual growth rates were serially
uncorrelated, the standard deviation of the sample mean (= 0 f d — P p w ) would be
1.33% (= 7.3%/>/30). For Algeria, the growth rates are slightly negatively corre­
lated and the estimated standard deviation of f ip w used to construct the confidence
interval was 1.18%.
Finally, a few of the confidence intervals are quite narrower. For example, the
estimated confidence interval for the U K (series 140) is 2.07 < 0 < 2.44. This series is
less persistent than most of the others, and the Dickey-Fuller t-statistic is -4.51. This
leads to a confidence interval for c with an upper limit of c = — 14.1 (corresponding
to p = 0.66). From Figure 2,estimates of /? are much more precise when c = — 14.1
than when c = 0 . Indeed the ratio of the asymptotic standard deviation for f ip w for
c = — 14 and c = 0 is 0.2, which approximately corresponds to the difference between
the widths of the confidence intervals for 0 for the U K and the US (country 71).

5

C oncluding Rem arks

In this paper we study the problems of estimation and inference in the deterministic
model. While the structure of the model is very simple, serial correlation in the errors
can make efficient estimation and inference difficult. Asymptotic results are presented
for 7(0) and local-to-unity 7(1) error processes, with the latter being the most rele­
vant for econometric applications. The asymptotic distribution of the estimators is
shown to depend on two important parameters: (i) the local-to-unity parameter that
measures the persistence in the errors and (ii) a parameter that governs the variance
of the initial error term.
Three conclusions emerge from our analysis. First, the Cochrane-Orcutt estimator
is dominated by the other feasible estimators and should not be used. When the data
are highly serially correlated (i.e., the local-to-unity parameter is close to zero), the
distribution of the Cochrane-Orcutt estimator has very thick tails, and large outliers
are common. Second, the feasible Prais-Winsten estimator is the most robust across
the parameters governing persistence and initial variance. This is the preferred es­
timator unless the researcher has sharp a p r i o r i knowledge about these parameters.




17

Finally, inference that ignores uncertainty about p or the variance in the initial er­
ror term can be seriously flawed and lead to large biases in confidence intervals for
trend growth rates. It is not clear how to optimally account for uncertainty in these
parameters, but conservative confidence intervals and tests are easily constructed.




18

A
A.l

Appendix: Theorem Proofs
Preliminaries:

From assumption (5), T ~ i
*t =► W(s); in addition, this result, together with as­
sumption (4) implies T ~ »
Vt =► d(l)VF(s), where W(s) is a standard Weiner pro­
cess. Analogously, accumulating the errors backwards from time 0, T ~ a
et =*►
W(s) and T -1
d(l)W(s), where W(s) is a standard Weiner process,
independent of W(s).
Let it, = E ‘=o P T v t - i
p j - (1+f). Then
=*• d(l)Wc(a), where W c(s)
denotes the diffusion process generated by d W e( s ) = c W e(s)<fs + d l V ( s ) . Similarly,
T ~ 2U\ — T~%
p'jV\-i_= > < f ( l w h e r e W c(k ) denotes the diffusion process
generated by d W c(.s) = cVVr(.(s)ds -I-dW(s). Note that W c(k ) ~ N ( Q , S c ( k )), where
S c ( k ) = (— 2c)-1(l — e 2cK). Finally, write u t = u t 4-p {
j l u i , so that
=»
d(l)[Wc(s) + e“ W c(/c)].

A.2

P r o o f o f T h e o r e m 2:

A.2.1

Proof of (a):

By direct calculation:
t

_1

—( T - ‘ EL, j- ) ( T - i EL,

T -i Z L ,

H P o l s -0)

T -'

E Z U t )1 ~ (T-‘T

.L ,( W

Thus,
T

Ti0OLS - 0) =

=* <f(l)12j
where i?i =

A i

+

\ s

1 2

T-i’
E u . d - i) + 0 ,(1 )
t= 1
1
^

- ^)[Wc(s) +

e “ W e( K ))d s

~

N ( 0, R x),

A i ,with

A\

=

var

(<1(1)12 f

j ,

- i)ny»)<M

and
A 2

= rar{Wc(/c)<f(l)12 J

(s — ^)e*cds}.

To calculate Aj, note that:




jf (* -

= j \ s -

19

j) J

‘ ^

- TU \ V ( r ) i s

= j \ j \ * - \ y is ) '- " d W

( r )

with 6 (r) = { / (s — i)d“ ik}e“eT.

= /
Jo

2

Jt

Thus,
A, = 144d(l)2

[ l b ( s ) 2d s ,

Jo

and
A 2 = 144d(l)25c(/c)[jf (s - |)e'cds]2.
The first term in i?i is Aj after simplification, and the second term is Aj.

A .2.2

P ro o f of (b):
t

H P fd - P ) = T - > ut - T - i u , = T - i u T - T - i u ^ l -

=>

<f(l)[^(l) - (1 - e‘)VTt(«)) ~ JV(0,i(I)*[S«(l) + (1 - e‘)!S,M].

A .2.3

P ro o f of (c):

This GLS estimator is constructed by OLS applied to an equation of the form y t =
x \8 + e t, where 8 = (a /?)', X\ = (a-1 a ~*)', x t = [(1 — pr) t — p j { t — 1)]' for
= 2,... , T . Let Q = J 2 x tx ti an<^ r =
with elements q ij and r; for i , j = 1,2.
Then 0 g l s — ft) = (?n922 — 922)-1(9n r2 — 9i2ri)- The various parts of the theorem
will be proved by evaluating the relevant expressions for
and r,-.
Specifically,
t

T

9n =

a Zi

+ (T — 1)(1 — pr)2; 912 —

922 =

<*Zi + ( T —

1)pj + 2pr(l —

r
ri = ^ ’ “ 1 + (1 ~ P t ) XZ
t=2
W e consider the cases with « = 0 and
k

— 0 :

<?Zi + { T —

k

l)pr(l —

Pt )

^Z * + U —
<=2
r
r2 =
+ 5Z
<=2
> 0 in turn.
p t

)

.

By direct calculation:




T-l(9n 922 - 922) - * (1 -

T~ *qn r 2

=

JZ^C1 <=2
20

c

+ -c2)

c ^r) + ° p ( 1)'

1

+ (1 — Pr)2^Z
<=2

P t )2 5Z

(=2
“

P t ) + P t ]-

T~>ql2ri

0

.

So that,
T* (PcLS ~ 0) -

+ op(l)

(l-o+W)

(9)
The result follows by noting that (1 —c + jc2 ) - 1 =
k

R3

evaluated at

k

= 0.

> 0:

By direct calculation:
(?n<?22 ~

qu)

( S c ( k )~1

+ c2)(l -

c +

^c2) - (^c2 - c)2

T i q n r 2 = ( S c{ k )~1 + c2)(T"J £ v t{l - c ~ )) + op(l);
t=2
1
T i q i 2r l = (^c2

- c)(5e(K)-1 T-*ti, - cT ’ ]>%<)
<=2

Thus,
T h

-(5

/9

c 2

- c)&(«)

1

tPG/,S P j

d(l)[-(lc2 -

-irr
r - U 1 + r-^Er=2^[(i-cjt)(5e(Ac) - 1 + c2) + (^c3 - c 2)]
/O
(•^(/c) " 1 J./-2V1
+ c2)(l -- n c ++ k 2) _
- U
( rk*2 --A c*) 2

+ c2)(l ~

c ) W e( K ) S e( K ) - ' +

cs) +

(1c2 -

c ) c]

< W ( j )]

N ( Q , R 3 ),

(c2 + 5c(/c)~1)(l - c + ic
2) - ^(ic
2 - c)2
3 ^
2 '
where
i?3 = d(l)2[

A.3

c2 +
(5 c(k )c2 +

1 ) ( 1

- c+

3

1

C2) - 5 c(k )(|c2

-

c)>] ‘

P r o o f o f L e m m a 4:

As in the proof the part (c) of Theorem 2, each of the estimators can be written
as the OLS estimator from an equation y t = x't 6 + et, where 6 = ( a /?)', and
the estimators differ in their definition of xj and e 2. As above, let Q = £ x<x«i
and r = £ x te{, with elements ^ and r,- for i ,j = 1,2. Then, for each estimator
( 0 - / 3 ) = (q u q 22 — q i 2)~ l { q u r 2 — q\2r i ) and the for the proof we evaluate these
expressions for each estimator.




21

A.3.1
When

Proof of (a):
c

= 0, T

i 0 Co

—

0)

=

T~ j

v t,

and the result follows directly. For c ^ 0,
r

9n =

(T -

1)(1 -

9i2 =

P t )2\

(T -

l)/>r(l -

Pt )

+ (1 -

P t )2 JE2 ^

f= 2

922 = ( T — l)/>r + 2px(l - p t )£ 1 + (1 -

P t )2 £

i= 2

T
r\

= (1 -

Pt )

t= 2

T
r2 = 51 v*[^(l -

Pt )

+ Pr]-

Thus,
9 i W22 - 9 n

c2( ! ~ c + ^ 2) “ ( | c2 ~ c ) 2 = ^ c 4;

T*qn r2

= -c2T~J

r 29i2n = c2(l -

Y i v *(c f

~ !) + OpU):

\c)T ~ i Y , vt

+

0, ( 1 ).

So that,
T h h c o - 0 ) = -(^)[r-i Z m 4

= (~)[r "’ !C ^

- 1) + (1 - ;c)T-i £

+ °?(1 )

^C-f)d(i)Jo c - - S)dvns).

The result follows by noting that
(T W ) jf(5 " ' W * ) ~ ^(».G >).
where




„

,12,,.,,,, /',1

12d(l)3

G\ = (—rd(l)
c
Jq/ (0
z - *) * = —3
c*— ■

22

»,] +

0

,(1 )

(1 0 )

(1 1 )

A.3.2

Proof of (b):

For f r e e .
T

9u

=

1)(1 - />t )2;

1 + {T -

912

= 1+

-

(T —

Pt )

+ (1 -

Pt )2 $3
E

t=2

=

922

- / > t ) 5 Z< +

1 + { T - \)p \ + 2pr{l

(1

~/> t )2 $3<2;

(= 2

T
ri

= ux + (l -

T

r2 = ui + ]Tt'([*(l-/?r) + />r]-

P t )'5 2 ,v *
1 = 2

1 = 2

Thus,

*(911922 - 9n) -♦ (1 - C+ ^c2);

T

T~±qn r2

f
- , - 1
- 1
= T-Jtix
- rn
T-*
] [ > ( 4 - 1) + op(1);
(= 2

T

39l2n = (1 - C + ^C2)T JU! + Op(l).

So that,
t i ,a

m

T»(fec

c(l - 5c)r ‘*ui - r ^ ^ ( c f - 1) | - m

---------- (i - c + jc»)-------- + M 1 )

=*•
1

— C+

5

C*

- =c)ive(ic) - / (cs - l)<iU,(s)].

L

J0

The result follows by noting that
<1(1)(1 - c + I c V M i -

\ c )W M

-

£ (c s

- i)<nr(.)] ~ a '(o ,g ,),

where




^

- (1 _

+ j

y

-

[i + 5 (K)i£__ifi2 L]
l - C + i C2 ll + ^ ( 1 — C + r c 2 *

23

(1 2 )

(13)

A .3.3

P r o o f o f (c):

For $ p w i
qn = (l-p2
T) + (T-l)(l-pT?\
t= 2

922 = (1 - Pt ) + (T - 1)Pt + 2pr(l ~ pT)52t + (l - p T f ^ t 2]
1 = 2
t= 2
T

r i = (1 — />r)«i + (1 “

PT)

T

S

r 2 = (1 - P r)« i +

1 = 2

“

t l

Pt )

+

P t ]-

1 = 2

Thus,
911922

-

9?2

-» (c2 -

2

= (c2 -

2

r * 9 nr 2 = -(c2 -

2

c)(l - c + ^c2) - (^c2 - c) 2

c)(l - | c + ^ c 2);

c)T“i X >
t= 2

(

4

-

1

)+

T ^ r , = - i ( c J - 2 c ) ( 2 c T - i u , + c T ' l £ > , ) + o„(l).
1 = 2

So that,
rr\,o
a\
T*(ppw — p)

cT-tlH-T-iZvticji-ic-l)
J..— ^----- + Op(l)
1 jC-t- 12C

--------

=*• 1 — :C + TZC* !<*%(■») - J0
A «
2
1
12

- b
i - W

4

The result follows by noting that
rf(l)(l - jC + i c 3)[cWc(«) - j \ c s - i e - 1)<W M] ~ N ( 0 , G 3),
where




*

= (14 + W

-

C^ M + /°’(cs “ 5C- 1)V,>

J+

^(l) 2

n _
(1 ~

2

C + U C2)2'

24

(14)

(15)

A.4

P r o o f o f T h e o r e m 5:

It is straightforward to verify that the analogues of (6), (8 ), and (10) continue to hold
for the feasible GLS estimators, with 8 replacing c and v t = u t — P T u t - u replacing v t.
The theorem then follows from (7), (9) and (11) using T ~ »
v t => W ( c ) . To see
this, and to derive an expression for VF(c), write
Vt = Ut- PTVt-i = v t- (pT - Pt )vt—\
t-2
= Vt - (CT - c)r_,( 2 / 4 w«-i-i + />r’1“i]i
j=0

where 87 = T (1 —

p j).

Thus
[sT]

H E ? , *
t= 1
T - i

E
«=1

t- 2

(jT]

[»T]
«• -

(w -

c)T -'

[,T]

E r-i(E
t=l

-

(Sr -

c ) ( T - iu ,) T - '

j =0

E

P t ''
1

(=1

^ d ( l) W ( s ) ,

where
iy(s) = W'(a) - (8 - c) f
Jo
and the last line follows from c t

W c( T ) d r

c,

P?Tv [TT\-j

—cJC
—c
□
1




- (8 - c)We(/c)—

25

—
—C

=* Wc(^)? and T ""1

Pr"1

References
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[2] Bobkosky, M.J. (1983), H y p o t h e s is T e s t in g in N o n s t a t i o n a r y
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T im e S e r i e s ,Ph.D.

[3] Cavanagh, C.L. (1985), “Roots Local to Unity,” manuscript, Department of
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[7] Chipman, John S. (1979), “Efficiency of Least-Squares Estimation of Linear
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[8] Cochrane, D. and G.H. Orcutt (1949), “Applications of Least Squares Regres­
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A m e r i c a n S t a t is t ic a l A s s o c i a t io n , 44, pp. 32-61.
[9] Davidson, R. and J.G. MacKinnon (1993), E s t i m a t i o n
m e t r ic s , Oxford University Press: New York.

a n d I n fe r e n c e in E c o n o ­

[10] Dickey, D.A. and W.A. Fuller (1979), "Distribution of the Estimators for Au­
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A s s o c i a t io n , 74, no. 366, pp. 427-31.
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1
[13] Elliott, Graham (1993),“Efficient Tests for a Unit Root when the Initial Ob­
servation is Drawn from its Unconditional Distribution,” manuscript, Harvard
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26

[14] Elliott, Graham, Thomas J. Rothenberg and James H. Stock (1992), “Efficient
Tests of an Autoregressive Unit Root,” N B E R Technical Working Paper 130.
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S e r ie s ,John Wiley and Sons: New York.

A n a l y s i s o f S t a t io n a r y T im e

[16] Kadiyala, K.R. (1968), “A Transformation Used to Circumvent the Problem of
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o f E c o n o m e t­

[20] Nagaraj, N.K. and W.A. Fuller (1991), “Estimation of the Parameters of Linear
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[22] Phillips, P.C.B. (1987), “Toward a Unified Asymptotic Theory for Autoregres­
sion,” B io m e t r i k a , 74, pp. 535-47.
[23] Phillips P.C.B. and V. Solo (1992), “Asymptotics for Linear Processes,”
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A n n a ls

[24] Prais, S.J. and C.B. Winsten (1954), “Trend Estimators and Serial Correlation,”
Cowles Foundation Discussion Paper 383.
[25] Quah, D. and J. Wooldridge, (1988), “A Common Error in the Treatment of
Trending Time Series,” Manuscript, Department of Economics, M.I.T..
[26] Rao, P. and Z. Griliches (1969), "Small-Sample Properties of Several Two-Stage
Regression Methods in the Context of Auto-Correlated Errors,” J o u r n a l o f the
A m e r i c a n S t a t is t ic a l A s s o c i a t io n , 64, pp. 253-72.7
2
[27] Sampson M. (1991), “The Effect of Parameter Uncertainty on Forecast Vari­
ances and Confidence Intervals for Unit Root and Trend Stationary Time-series
Models,” J o u r n a l o f A p p li e d E c o n o m e t r ic s , 6,no 1,pp. 67-76.




27

[2S] Schmidt, Peter (1993), "Some Results on Testing for Stationarity Using Data
Detrended in Differences,” Economics Letters, 41. pp. 1-6.
[29] Schmidt, P. and P.C.B. Phillips (1992), “L M Tests for a Unit Root in the Pres­
ence of Deterministic Trends,” O x f o r d B u ll e t i n o f E c o n o m i c s a n d S t a t is t ic s , 54,
pp. 257-87.
[30] Spitzer, J.J. (1979), “Small Sample Properties of Nonlinear Least Squares and
Maximum Likelihood Estimators in the Context of Autocorrelated Errors,” J o u r ­
n a l o f th e A m e r i c a n S t a t is t ic a l A s s o c i a t io n , 74, pp. 41-47.
[31] Stock, James H. (1991), “Confidence Intervals of the Largest Autoregressive Root
in U.S. Macroeconomic Time Series,” J o u r n a l o f M o n e t a r y E c o n o m i c s , 28, pp.
435-60.
[32] Summers, Robert and Alan Heston (1991), “The Penn World Table (Mark 5):
An Expanded Set of International Comparisons, 1950-1988,” Q u a r t e r ly J o u r n a l
o f E c o n o m i c s , 106, pp. 327-368.
[33] Thornton, D.L. (1987), “A Note the Effect of the Cochrane-Orcutt Estimator in
the AR(1) Regression Model,” J o u r n a l o f E c o n o m e t r ic s , 36, pp. 369-76.




28




k

— 0.00

**

*= 0.01

*-

0 . 1 0

Figure 1
A

Asymptotic Relative Efficiencies of Pq ^

A

and

1.00

0 .2

0 .3

0.4

o




Figure 2
Asymptotic Variance of $ q l s




k

— 0.00

0.01

**■

0 . 1 0

Figure 3
A

A

A

Asymptotic Relative Efficiencies of Pq q , Pqq anc* Ppy

1.00

0 .4
0 .3
0 .2
0.1
0 .0



Figure 4
Densities of Feasible GLS Estimators

Table 1
Average Mean Square Error of Estimators
A. Average MSE for -30<c<0
Estimator
A
£g l s
£o l s
^FD
'
a FCC
£f p w
^PT

0 . 0 0 0

0 , 0 1 0

0.057
0.108
0.077
0.065
0.077
0.082

0.067
0 . 1 1 0

0.085
0.083
0.081
0.085

0.050
0.081
0.116

0 . 1 2 0

0.250
0.097
0.126

0 . 1 0 2

0 . 1 1 1

0 . 1 2 1

0.113
0.088
0.094

0.128
0.094

0.142

0 . 1 0 2

0 . 1 1 0

0 . 1 0 0

0.089

0 . 1 0 1

1 . 0 0 0

0.105
0.133
0.129
0.152
0.108
0.118

£g l s
£o l s
£ fd
£fcc
£f p w
^PT

B . Average MSE for -2<c<0
K ------0 . 0 0 0
0.050
0 . 1 0 0
0 - 0 1 0
0.493
0.497
0.512
0.529
0.678
0.682
0.695
0.710
0.498
0.502
0.516
0.532
0.598
0.641
0.595
0.615
0.544
0.529
0.531
0.559
0.509
0.512
0.526
0.541

Estimator
A
£g l s
£o l s
^FD
£fcc
£fpw
^PT

C. Average MSE for -10<c<i-2
tc ------.
0 . 0 0 0
0 . 0 1 0
0.050
0 , 1 0 0
0.069
0.083
0.115
0.136
0.168
0.172
0.186
0.197
0.134
0.154
0.097
0.107
0.075
0.093
0.135
0.168
0.114
0.106
0.131
0.146
0 . 1 1 0
0.141
0.118
0.160

0.179
0.206
0.165
0.182

0.173
0.192

D. Average MSE for -■30<c<-10
K ------0 . 0 1 0
0 . 0 0 0
0.050
0 . 1 0 0
0.008
0.018
0.025
0.026
0.028
0.028
0.030
0.030
0.049
0.027
0.035
0.052
0.054
0.008
0.028
0.060
0.026
0 . 0 2 0
0 . 0 2 2
0.025
0.028
0.029
0.035
0.033

0,250
0.026
0.030
0.053
0.063
0.027
0.035

0.026
0.030
0.053
0.063
0.027
0.035

Estimator

Estimator
£g l s
£o l s
£ fd
£fcc
£f p w
^PT

0-250
0.566
0.743
0.567
0.676
0.590
0.574

0-250
0.158
0 . 2 1 1

1 . 0 0 0

0.634
0.808
0.635
0.766
0.664
0.649

l.ooo
0.168
0.219
0.190
0 . 2 2 0

1 . 0 0 0

Notes: The entries in the table are the mean squared error averaged over the
indicated range of c.




A. k -

0.00

(i) p-0.50

1 . 0 0 0

0.301
0.258

0.273
0.259
0.997
0.355
0.260

0.488
0.859
0.917
0.786
0.815

(iii) p-0.90
0.511
0.386
0.803
0.698
0.944
0.940
0.632
0.772
0.778
0.631

0.498
0.616
0.925
0.695
0.560

0.305
0.451
0.962
0.442
0.343

0.678
0.978
0.889
0.924
0.978

0.635
0.971
0.797
0.874
0.905

(iv) p-0.95
0.572
0.529
0.923
0.902
0.899
0.875
0.840
0.830
0.906
0.860

0.452
0.760
0.939
0.697
0.720

0.386
0.698
0.942
0.619
0.622

0.860

0.833

0.842

0.833

1.000

1.000

1.000

1.000

0.907
0.944
0.969

0.830
0.965
0.986

0.892
0.977

0.821
0.974
0.996

0.358
0.632
0.950
0.560
0.529

0.579
0.912
0.887
0.809
0.882

A

&>LS
^FD
a FCC
^FPW
^PT

0.941

1 . 0 0 0

0.350
0.272

m )
0.258
0.113

0.714
0.466
0.947
0.839
0.675

0.607
0.810
0.922
0.838
0.805

A

£o l s
^FD
^FCC
£fpw
^PT

1 . 0 0 0

•
•

hi

1 . 0 0 0

... TExact
0.958
0.185
0.974
0.975
0.953

(ii) p-0.80
0.305
0.451
0.968
0.454
0.363

A

£o l s
^FD
^FCC
^FPW

--- T-50 --Exact
■1 (1 )
0.924
0.268
0.335
0.213

1

£o l s
£ fd
£fcc
£f p w
^PT

--- T-30 --Exact
.I,( 1 1
0.891
0.283
0.492
0.330
0.990
0.958
0.390
0.957
0.705
0.276

o
o
r-t




Table 2
Relative Efficiencies of Estimators
Exact and 1(1) Approximation

0.631
0.670
0.937
0.824
0.664

(v)

^OLS
^FD
^FCC
^FPW
^PT

p-1.00
0.833
1.000
1.000
0.859
0.903
0.964
0.966
0.993
0.992

0.850

1.000

Table 2 (Continued)
Relative Efficiencies of Estimators
Exact and 1(1) Approximation
B. * -

£o l s
/?FD
£fcc
£fpw
^PT

... t -30 --Exact
i m
0.856
0.950
0.550
0.463
0.470
0.981
1 . 0 0 0
0.979
0.698
0.735

1 . 0 0

(i) p-0.50
--- T-50 --i m
Exact
0.966
0.902
0.308
0.381
0.960
0.328
0.979
0.974
0.932
0.859
(ii) P-0.80
0.867
0.817
0.667
0.698
0.870
0.591
0.974
0.958
0.699
0.721

£o l s
£ fd
a FCC
£fpw
^PT

0.839
0.841
0.932
0.975
0.834

0.774
0.859
0.756
0.990

£o l s
£ fd
£f c c
£fpw
^PT

0.801
0.970
0.916
0.950
0.942

0.753
0.971
0.863
0.989
0.971

(Hi) p-0.90
0.800
0.764
0.895
0.902
0.870
0.786
0.942
0.960
0.858
0.869

0.764
0.997
0.861
0.959
0.975

(iv) p-0.95
0.781
0.755
0.983
0.983
0.876
0.829
0.948
0.957
0.970
0.962

£o l s
^FD
^FCC
^FPW
^PT

^OLS
^FD
^FCC
^FPW

hi

0.803
0.997
0 . 8 8 6

0.951
0.994

0 . 8 6 6

... T-100 ...
Exact
i m
0.982
0.946
0.166
0.213
1 . 0 0 0
0.174
1 . 0 0 0
0.994
1 . 0 0 0
0.949

0.915
0.420
0.844

0.883
0.451
0.393

1 . 0 0 0

1 . 0 0 0

0.836

0.809

0.842
0.683
0.805
0.980
0.720

0.817
0.698
0.621
0.994
0.746

0.782
0.898
0.863
0.983
0 . 8 8 8

0.764
0.902
0.786
0.979
0.898

0.842

0.833

0.860

0.833

(v) p-1.00
0.850
0.833

1 . 0 0 0

1 . 0 0 0

1 . 0 0 0

1 . 0 0 0

1 . 0 0 0

1 . 0 0 0

0.936
0.989

0.895
0.991

0.927

1 . 0 0 0

1 . 0 0 0

1 . 0 0 0

0.879
0.965
0.996

0.863
0.958
0.980

0.838
0.959
0.981

1 . 0 0 0

Notes: The relative efficiency is the ratio of the variance of the infeasible
GLS estimator to the variance of the estimator given in column 1. The columns
labeled 1(1) are the asymptotic relative efficiencies using c-T(p-l). The
corresponding 1 (0 ) relative efficiencies are 1 , 0 , 1 , 1 , 1 , 1 , respectively
for the estimators in column 1 and for all T and |p|<1.







Table 3
Confidence Interval Coverage Rates (%)
A

A

A. Smallest of

anc*

(i)
Level
90.0
90.0
90.0
95.0
95.0
95.0
99.0
99.0
99.0

K
0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

95.0
95.0
95.0
99.0
99.0
99.0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

95.0
95.0
95.0
99.0
99.0
99.0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

.1

z!0 _

z2Q_

96.0
95.6
94.0

97.8
97.4
97.1

97.3
96.6
96.8

94.5
94.5
94.3

93.0
92.9
93.6

97.9
97.9
96.7

98.8
98.7
98.5

98.6
98.3
98.5

96.9
97.1
96.9

97.9
98.0
97.9

99.4
99.5
99.2

99.7
99.6
99.6

99.5
99.6
99.7

99.0
99.1
99.2

T - 50

90.7
90.8
90.5

97.3
96.3
95.2

98.3
97.9
97.9

98.2
97.9
98.1

96.
96.
96.

94.9
94.8
94.9

98.7
98.3
97.7

99.2
99.0
99.0

99.3
99.1
99.1

98.
98.
98.

98.6
98.7
98.8

99.7
99.6
99.6

99.8
99.8
99.8

99.9
99.8
99.9

99.
99.
99.

r - 100
97.8
97.5
96.4

98.8
98.4
98.3

98.6
98.8
98.8

98.5
98.3
98.3

(ill)
90.0
90.0
90.0

T - 30

88.7
88.7
89.2

0

( W

90.0
90.0
90.0

Confidence Intervals

91.4
92.3
91.9
95.6
95.8
95.9

99.0
99.0
98.5

99.5
99.3
99.3

99.5
99.5
99.6

99.4
99.4
99.4

99.1
98.8
99.0

99.8
99.9
99.8

99.9
99.9
99.9

100.0
99.9
100.0

99.9
99.9
99.9

Table 3 (Continued)
Confidence Interval Coverage Rates (%)
B. Confidence Intervals Constructed from

1 . 0

89.0
93.0
92.9
93.4

97.8
97.8
96.6

98.7
98.5
98.5

98.5 | 96.5
98.2
96.5
98.4
96.5

97.8
97.9
97.9

99.3
99.5
99.2

99.7
99.6
99.7

99.5
99.5
99.6

98.9
99.0
99.0

(ii) T - 50
90.5
97.1
96.2
90.7
90.3
95.1

98.4
97.8
97.8

98.4
97.9
98.0

96.5
96.4
96.4

0 . 1

90.0
90.0
90.0
95.0
95.0
95.0
99.0
99.0
99.0

l
ro
O

88.5
8 8 . 6

99.0
99.0
99.0

99.0
99.0
99.0

97.2
96.5
96.5

0 . 0
0 . 1

0 . 1

95.0
95.0
95.0

-5
97.7
97.3
97.1

-l
96.0
95.4
93.7

95.0
95.0
95.0

90.0
90.0
90.0

0

K

1

Level
90.0
90.0
90.0

K.

(i)

0 . 0

1 . 0

0 . 0

1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0 . 0
0 . 1
1 . 0

0

- 1 0

93.9
93.6
93.6

94.8
94.7
94.9

98.6
98.2
97.6

99.1
99.0
99.0

99.3
99.1
99.1

98.5
98.2
98.3

98.6
98.7
98.8

99.7
99.5
99.6

99.8
99.8
99.7

99.9
99.8
99.9

99.7
99.6
99.7

98.9
98.4
98.3

98.9
98.8
98.8

98.6
98.2
98.2

99.6
99.5
99.6

99.4
99.3
99.3

(Hi) T - 100
91.3
97.7
92.1
97.5
91.8
96.3
95.6
95.8
95.8

99.0
98.9
98.5

99.5
99.3
99.3

99.1
98.8
99.0

99.8
99.8
99.8

99.9
99.9
99.9

1 0 0 . 0

99.9
99.9

99.9
99.8
99.9

Notes: The table shows the exact coverage rates (in percent) for conservative
confidence intervals constructed with an asymptotic level given in the first
column. The confidence intervals in panel A were constructed as the narrowest
of the intervals constructed from the OLS and first-difference estimators.
The confidence intervals from panel B were constructed from the Prais-Winsten
estimator.



Table 4
Annual Real Per-Capita Growth Rates
- 1 2

C ou ntry

Sn irl P e r i o d

c

t

l

ALGERIA

1960

1990

^OLS2.736

^FD—
1.459

^FCO2.788

^FFW2.098

-6 .0 7 0

2

ANGOLA

1960

1989

-2 .0 3 8

-1 .0 0 4

-2 .7 5 9

-1 .3 4 5

-3 .7 5 3

-2 .2 9 8

3

B EN IN

1959

1989

-0 .4 0 3

-0 .3 7 0

-0 .5 8 1

-0 .3 9 7

-1 5 .1 8 0

-3 .2 3 0

4

BOTSWANA

1960

1989

5.839

6.079

6 .1 7 3

5 .9 0 6

-1 0 .9 5 5

-3 .2 6 2

5

BURKINA FASO

1959

1990

0.859

0.006 -2 0 .3 7 8

0.006

0 .2 9 0

-3 .4 9 8

6

BURUNDI

1960

1990

0.567

-0 .4 2 5

2.043

0 .0 4 1

-5 .5 8 6

-4 .2 4 7

7

CAMEROON

1960

1990

2.698

1.888

1.455

2.052

-2 .4 3 9

-1 .8 1 6

8

CAFE VERDE I S .

1960

1989

3.880

3 .3 3 8

4.693

3.545

-4 .3 6 9

-2 .5 4 8

9

CENTRAL A F R . R .

1960

1990

-0 .4 8 6

-0 .5 8 9

-0 .8 7 9

-0 .5 4 2

-5 .3 3 1

j- l . 164

10

CHAD

1960

1990

-2 .5 8 4

-2 .0 1 0

-2 .5 8 0

-2 .4 4 2

-1 2 .2 0 0

-2 .6 2 8

-1 .4 8 4

1 1

COMOROS

1960

1987

-0 .0 4 4

0.520

-0 .7 8 0

0.219

-6 .4 9 0

-3 .0 9 6

12

CONGO

1960

1990

3.314

2.788

3 .3 6 0

3.082

-7 .0 9 1

-2 .4 5 5

14

EG YP T

1950

1990

3.002

2.385

3.696

2.481

-2 .0 1 8

-3 .7 2 5

15

ETHIOPIA

1951

1986

0.831

0.669

0.688

0.761

-7 .4 5 2

-1 .6 1 4

16

GABON

1960

1990

2.298

2 .6 2 0 - 3 7 . 3 4 2

2.616

-0 .3 5 9

-1 .4 1 3

17

GAMBIA

1960

1990

1.1 5 0

0 .9 3 5

-0 .2 2 2

1.015

-4 .2 4 3

-1 .1 1 2

18

GHANA

1955

1989

-0 .2 6 6

-0 .0 8 9

-0 .4 2 5

-0 .2 0 5

-9 .5 0 1

-2 .4 1 8

19

G U INEA

1959

1989

-0 .3 0 4

-0 .2 1 5

-0 .5 9 0

-0 .2 6 0

-6 .1 3 0

-2 .1 0 8

1.036

-5 .6 5 4

-2 .0 8 9

20

G U IN E A -B IS S

1960

1990

0 .3 7 7

-0 .2 1 2

0.724

21

IV O R Y COAST

1960

1990

1 .0 7 3

0.633

5.807

0 .6 3 3

2.305

0.611

22

KENYA

1950

1990

1 .1 7 9

1 .1 6 6

1.191

1.177

-1 7 .9 1 9

-2 .8 7 8

23

L ESOTHO

1960

1990

4.402

4 .0 5 3

2.0 9 9

4.129

-2 .5 8 5

-1 .6 8 1

1986

0.682

0 . 3 1 0 449 .3 8 3

0.310

0 .0 2 3

0.194

24

LIB E R IA

1960

25

MADAGASCAR

1960

1990

-1 .9 6 2

-1 .8 1 8

-2 .4 4 1

-1 .8 8 0

-5 .0 8 2

-2 .0 6 4

26

MALAWI

1954

1990

1 .1 7 1

1 .2 2 6

0.684

1 .197

-6 .7 7 4

-2 .0 1 6

27

MAL I

1960

1990

0.877

0 .1 5 0

3.057

0 .2 7 4

-2 .1 1 7

-2 .7 5 8

28

M AUR IT ANIA

1960

1990

-0 .1 6 4

-0 .2 0 7

-0 .6 4 1

-0 .1 8 2

-7 .6 5 2

-1 .6 6 9

29

M AUR IT IUS

1950

1990

1.340

1.399

3.893

1.385

-2 .7 2 9

-1 .6 7 7

30

MOROCCO

1950

1990

2.814

2.355

2 .8 0 4

2.600

-7 .0 1 7

-2 .6 8 9

31

MOZAMBIQUE

1960

1990

-2 .3 0 9

-1 .4 2 6

-7 .8 5 6

-1 .4 9 3

-1 .2 0 7

-2 .2 4 0

32

N AM IB IA

1960

1989

0 .3 8 4

0.509

-5 .5 9 1

0 .4 9 3

-1 .7 3 4

-1 .4 0 9

33

N IG E R

1960

1989

-0 .4 1 5

-0 .2 5 6

-4 .6 1 2

-0 .2 8 8

-2 .3 8 7

-1 .9 2 4

34

NIG ER IA

1950

1990

1.989

1 .3 3 7

-0 .2 5 5

1.475

-2 .5 7 8

-1 .8 0 8

35

REUNION

1960

1988

3.764

3.799

2.695

3.784

-4 .6 4 1

-2 .2 5 2

36

RWANDA

1960

1990

1 .974

0.791

1 .9 1 6

1.366

-5 .8 4 2

-2 .5 1 6

37

SENEGAL

1960

1990

0.138

0.204

0.101

0.139

-2 8 .5 8 2

-3 .9 6 2

38

SE YC HE LLE S

1960

1989

3.896

3.449

4.072

3 .7 8 3

-1 1 .8 6 7

-2 .2 8 9

39

S IE R R A LEONE

1961

1990

0.049

0.593

-4 .6 2 7

0.519

-1 .7 8 9

-2 .2 3 9

40

SOMALIA

1960

1989

-0 .4 4 8

-0 .5 5 1

-0 .3 8 7

-0 .4 6 0

-1 8 .5 2 9

-2 .4 8 6

41

SOUTH A F R IC A

1950

1990

1.792

1.3 4 3 - 1 6 . 3 0 4

1.3 44

-0 .2 0 8

-0 .1 8 6

42

SUDAN

1971

1990

-0 .2 8 6

-0 .8 8 6

-0 .3 7 3

-0 .5 0 9

-7 .4 1 6

-2 .5 1 8

43

SWAZILAND

1960

1989

1.626

1.985

-4 .7 0 4

1 .9 3 2

-1 .9 1 0

-1 .2 6 1

44

T A N Z A N IA

1960

1988

1.5 3 2

1.686

0 .7 1 0

1.626

-4 .4 3 1

-1 .9 1 9

45

TOGO

1960

1990

1. 777

1.7 9 7

-1 .4 8 5

1.7 9 3

-2 .1 3 0

-1 .4 0 8

46

TU N ISIA

1960

1990

3.761

3.222

2.980

3.298

-1 .8 5 5

-1 .1 2 3

47

UGANDA

1950

1989

-0 .1 8 8

0.946

-0 .0 3 1

0 .1 7 7

-1 0 .5 5 8

-2 .5 7 5

48

ZAIRE

1950

1989

0.339

0.648

19.1 57

0.648

0.527

-1 .2 5 0

49

ZAMBIA

1955

1990

-0 .6 1 3

-0 .5 9 7

-2 .6 3 3

-0 .6 0 2

-3 .1 9 5

-1 .1 2 0

50

ZIMBABWE

1954

1990

0.904

1.018

0.795

0 .9 5 0

-8 .2 9 2

-2 .7 6 8




Table 4 (Continued)
Annual Real Per-Capita Growth Rates
ID

Country

Sm nl P e r i o d

c

r

52

BARBADOS

1960

1989

^OLS3.4 7 1

2 .2 5 7

^FP W 3 .5 8 5

-2 .7 2 2

- 1 . 572

54

CANADA

1950

1990

2 .7 6 2

2.503

2.849

2.626

-5 .9 7 4

- 2 . 255

55

COSTA R I C A

1950

1990

2.310

2.363

0.049

2 .3 5 5

-1 .8 6 7

- 2 . 151

57

DOMINICAN REP.

1950

1990

2.425

1.9 7 9

2 .0 5 0

2.2 5 7

-9 .0 6 8

- 1 . 127

58

EL SALVADOR

1950

1990

1.0 63

1.0 0 9

-4 .3 2 0

1.0 1 1

-0 .8 1 3

- 1 . 965

60

GUATEMALA

1950

1990

1 .239

0.790

-6 .3 9 2

0. 794

-0 .3 1 8

- 0 . 985

61

H A ITI

1960

1989

0.1 4 7

-0 .3 3 1

0.083

-0 .0 8 1

-6 .4 0 5

- 1 . 177
- 1 . 664

^FD—
3.619

& TCCT

62

HONDURAS

1950

1990

1 .101

0.788

0 .6 6 0

0 .8 8 6

-3 .6 9 2

63

JAMAICA

1953

1989

1.4 20

2 .0 2 0

13 .2 76

2.020

0.732

- 1 . 810

64

MEXICO

1950

1990

2 .5 3 6

2.259

1 .738

2 .3 3 5

-3 .2 4 8

- 1 . 718

65

NICARAGUA

1950

1987

1 .0 2 1

0.943 -1 1 .8 2 8

0.946

-0 .8 6 9

- 1 . 186

66

PANAMA

1950

1990

2 .8 2 1

2.181

1 .626

2.331

-2 .8 1 3

-1 .

202

67

PUERTO R IC O

1955

1989

3 .6 4 9

3 .9 3 0

0 .1 9 3

3.911

-1 .1 2 3

- 1 . 902

70

T R I N I D AD&TOB AG

1950

1990

2.870

2 .5 9 6

-4 .6 1 3

2 .6 1 5

-1 .1 5 4

- 0 . 750

71

U .S .A .

1950

1990

1.9 40

1.8 94

1 .959

1 .926

-1 1 .4 8 8

- 2 . 776

72

ARG ENTIN A

1950

1990

0 .9 2 2

0.366

-1 .1 2 3

0 .453

-2 .0 3 4

- 0 . 307

73

B OLIVIA

1950

1990

1.3 17

0 .6 3 2

0 .4 9 8

0 .7 1 1

-1 .6 2 3

- 1 . 319

74

B R A Z IL

1950

1990

3.469

2 .8 5 8

-1 .3 2 2

2.881

-0 .7 7 9

-o.613

75

C HILE

1950

1990

0.925

1.234

0.801

1.0 38

-9 .3 9 1

- 3 . 067

76

COLOMBIA

1950

1990

2.146

1.92 7

2 .2 0 2

2.017

-4 .9 6 3

- 1 . 607

77

ECUADOR

1950

1990

2 .7 5 1

2 .1 6 5

0.848

2 .2 1 7

-1 .3 6 4

- 1 . 174

78

GUYANA

1950

1990

-0 .2 1 8

-0 .9 9 8

-0 .7 6 3

-0 .7 0 2

-4 .5 2 1

- 1 . 592

79

PARAGUAY

1950

1990

2.068

1.4 07

2.636

1 .596

-3 .3 7 6

- 2 . 082

80

PERU

1950

1990

1 .4 0 6

0.886

6.271

0 .8 8 6

1. 957

- 0 . 197

81

SURINAME

1960

1989

1.398

0.418 -2 1 .1 9 5

0 .4 3 7

-0 .5 4 0

- 0 . 232

82

URUGUAY

1950

1990

0 .3 7 2

0 .5 7 9

0.251

0 .4 3 7

-1 0 .9 1 0

- 2 . 907

83

VENEZUELA

1950

1990

0 .4 3 9

0 .5 4 9

6.766

0 .5 4 9

1. 237

- 1 . 338

85

BANGLADESH

1959

1990

1 .2 0 8

1.392

1 .183

1.2 61

-1 0 .8 9 5

- 2 . 813

87

C HI NA

1968

1990

5 .7 5 2

5 .984

5 .5 5 6

5.8 84

-4 .7 2 8

- 2 . 310
- 3 . 896

88

HONG KONG

1960

1990

6 .264

6.250

6 .0 5 1

6.261

-1 1 .7 0 5

89

IN D IA

1950

1990

1.43 7

1.79 4

1.6 5 5

1.6 0 8

-6 .7 9 4

- 1 . 473

90

IN D O N E S IA

1960

1990

4.471

3 .7 7 9 - 4 8 . 7 2 3

3 .7 7 9

0.0 9 7

- 3 . 505

91

IRAN

1955

1989

1.91 3

1 .528 - 5 5 . 1 9 4

1 .529

-0 .2 3 3

- 1 . 186

92

IRAQ

1953

1987

1.7 67

0 .4 7 6

-2 .7 6 7

0.8 0 1

-2 .9 5 6

- 0 . 791

93

ISRAEL

1953

1990

3 .6 2 0

3.6 3 7

10 .6 01

3 .6 3 7

0 .9 2 6

- 1 . 004

94

JAPAN

1950

1990

5.781

5.742

14 .0 77

5.742

0 .7 2 9

- 0 . 743

95

JORDAN

1954

1990

3.589

3.110

1 .6 9 8

3 .2 7 8

-4 .1 0 3

- 1 . 845

96

KOREA, REP.

1953

1989

5 .944

5 .6 9 2 -3 6 7 . 4 7 0

99

M ALA YSIA

1955

1990

4.2 64

3.871

101

MYANMAR

1950

1989

2.411

102

NEPAL

1960

1986

1.92 4

104

PAKIS TAN

1950

1990

2.355

105

P H IL IP PIN E S

1950

1990

2.001

108

SINGAPORE

1960

1990

6 .724

109

S R I LANKA

1950

1989

1. 854

110

SY R IA

1960

1990

3 .7 0 2




5.69

0.017

- 1 . 906

4.5 4 7

4.0 54

-5 .6 8 5

- 2 . 818

2.563

2 .3 1 8

2 .4 4 8

-1 3 .4 6 8

1.5 47

2 .1 2 7

1.8 00

-9 .2 3 3

2 .1 2 6

2 .6 1 4

2 .2 3 4

-5 .8 9 9

- 2 . 323

2 .0 7 3

-7 .9 9 6

2 .0 7 2

-0 .4 5 9

- 2 . 569

6.190

6 .3 4 7

6 .2 8 7

-2 .2 2 4

- 1 . 638

1. 838

2.454

1.844

-4 .1 0 2

- 1 . 325

3 .2 2 2

2 .9 5 8

3.489

-7 .0 4 8

- 0 . 608

-2 .

110

- 2 . 342

Table A (Continued)
Annual Real Per-Capita Growth Rates
ID

C ountry

S to o l

P a rlo d

-r
^FD —
5 .6 0 3

^FCO6.659

^F£W5 .6 1 3

c

_

r

111

TAIWAN

1951

1990

A dls5 .6 5 3

-2 .3 7 1

-2 .5 0 2

112

TH AILAN D

1950

1990

3.922

3 .5 7 0

3 7 .9 8 1

3.571

-0 .1 8 7

-3 .6 1 4

1.9 8 9

5.151

114

YEMEN

1969

1989

4.727

5.676

3.739

5 .3 2 9

-3 .9 2 5

-2 .0 8 3

2.950

8.402

115

AUSTRIA

1950

1990

3 .6 4 0

3.664

25.762

3 .6 6 4

0.176

-0 .9 6 7

2.578

4.750

115

BELGIUM

1950

1990

2.908

2 .7 6 7

2.788

2.803

-3 .0 4 3

-1 .3 6 6

1 .7 9 5

3.739

113

CYPRUS

1950

1990

3.970

4 .0 9 8

3.994

3.994

-1 6.3 47

-3 .7 3 3

2.363

5.788

119

CZECHOSL OVAKIA

1960

1990

3.315

3 .0 4 1

-0 .3 9 4

3.061

-1 .1 7 9

-0 .6 8 5

1 .1 2 1

4 .9 6 1

4.288

6 .9 1 8

120

DENMARK

1950

1990

2.644

2.412

2.378

2.487

-3 .8 4 3

-1 .2 8 5

1 .3 2 2

3 .5 0 1

121

F IN LA N D

1950

1990

3 .4 3 4

3.452

3.249

3.441

-9 .0 6 3

-2 .6 6 2

2.075

4.828

122

FRANCE

1950

1990

3.080

3.008

11.8 23

3.008

0.351

-0 .5 5 5

2.014

4.002

123

GERM ANY, WEST

1950

1990

3.199

3.576

5.273

3.576

2.718

-2 .8 8 1

2.174

4.978

124

GREECE

1950

1990

4.328

3.887

1 9.217

3 .8 8 7

0.318

0.130

2.369

5.4 0 4

125

HUNGARY

1970

1990

2 .2 3 4

2.322

8.202

2.322

1 .4 0 2

-0 .8 3 6

0.4 7 5

4.169

126

IC ELA N D

1950

1990

3.422

2.969

3.322

3 .2 8 0

-1 0 .9 1 3

-3 .1 4 3

1 .0 0 4

4.933

127

IR ELA N D

1950

1990

3.207

3.102

3.445

3 .1 5 1

-5 .8 0 1

-2 .7 0 2

1.716

4 .4 8 9

128

ITA LY

1950

1990

3.752

3 .7 4 9

14.7 7 9

3.749

0.359

-0 .8 4 6

2.7 0 5

4.793

129

LUXEMBOURG

1950

1990

2.185

2.246

2.297

2.199

-1 4 .2 7 8

-3 .1 1 0

0.843

3.649

130

M ALTA

1954

1989

5 .4 9 6

5 .024

6.840

5.1 0 4

-2 .1 4 8

-1 .8 4 8

3.072

6.976

131

NETHERLANDS

1950

1990

2.763

2.588

1.6 04

2.611

-1 .7 9 5

-1 .4 6 7

1.222

3 .9 5 4

132

NORWAY

1950

1990

3.346

3.051

3.125

3.138

-3 .4 9 0

-2 .0 7 0

2.114

3 .9 8 8

133

POLAND

1970

1990

0 .6 9 4

1.242 -1 8 .3 5 4

1 .224

-0 .7 2 6

-2 .7 8 5

-1 1 .7 9 5

1 4.280

134

PORTUGAL

1950

1990

4 .3 2 0

4.213

2.890

4.228

-1 .8 8 5

-1 .2 5 0

2.450

5.975

136

S P A IN

1950

1990

3.786

3.998

-9 .2 5 2

3.995

-0 .3 8 6

-0 .9 8 8

2.297

5.699

137

SWEDEN

1950

1990

2.375

2.312

0.048

2 .3 1 4

-0 .8 1 7

-1 .1 7 0

1 .4 7 8

3.146

138

SW ITZER LAND

1950

1990

2.083

2.219

1 .174

2.1 8 9

-2 .6 7 0

-1 .6 9 0

0.918

3.519

139

TURKEY

1950

1990

2.7 4 6

3 .1 4 4

2 .4 6 0

2.875

-1 0 .5 8 9

-2 .5 3 4

1.511

4.778

140

U .K .

1950

1990

2.241

2.306

2.249

2.253

-1 6 .9 1 5

-4 .5 0 5

2.072

2.438

141

U .S .S .R .

1970

1989

3.272

3.377

-9 .9 2 7

3.376

-0 .2 1 5

-1 .9 8 1

2.679

4.074

142

YU G O S LA VIA

1960

1990

3.630

2.812

13.4 77

2.812

1.129

0.919

0.236

5 .3 8 8

143

A U S TR A LIA

1950

1990

2 .1 8 4

1.8 7 0

2.158

2.086

-1 0 .9 7 3

-2 .0 6 5

0.752

2.987

144

F IJ I

1960

1990

2.043

2.0 0 6

1 .853

2.021

-4 .6 2 9

-1 .5 1 6

-0 .3 9 2

4.405

145

NEW ZEALAND

1950

1990

1 .674

1 .3 8 8

1 .5 9 9

1.559

-8 .3 5 3

-2 .0 2 1

0.035

2 .7 4 2

146

PAPUA N .G U IN E A

1960

1990

0.215

0.643

3.196

0 .6 4 3

3 .4 2 0

-2 .6 8 9

-1 .2 2 9

2 .5 1 5

Notes: The column labeled ID shows the country ID from the Penn World Tables.
The estimators ^q l s * ^FD’ ^FCO’ ^PW are described in the text; c *s an
estimate of the local-to-unity paramater, constructed as T(p-l); rr is the
augmented Dickey-Fuller t-statistic;
and /J
are the endpoints of the 95%
confidence interval for /} constructed using the Prais-Winsten estimator, as
described in the text.




Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.

REGIONAL ECONOMIC ISSUES
Estimating Monthly Regional Value Added by Combining Regional Input
With National Production Data

WP-92-8

P h ilip R . Isra ile v ic h a n d K en n eth N . K u ttn er

Local Impact of Foreign Trade Zone

WP-92-9

D a v id D . W eiss

Trends and Prospects for Rural Manufacturing

WP-92-12

W illiam A . T esta

State and Local Government Spending-The Balance
Between Investment and Consumption

WP-92-14

R ic h a rd H . M a tto o n

Forecasting with Regional Input-Output Tables

WP-92-20

P J i. Isra ile v ic h , R . M a h id h a ra , a n d G J D . H e w in g s

A Primer on Global Auto Markets

WP-93-1

P a u l D . B a lle w a n d R o b e rt H . S ch n orbu s

Industry Approaches to Environmental Policy
in the Great Lakes Region

WP-93-8

D a v id R . A lla rd ic e , R ic h a rd H . M a tto o n a n d W illiam A. T esta

The Midwest Stock Price Index-Leading Indicator
of Regional Economic Activity

WP-93-9

W illiam A . S tra u ss

Lean Manufacturing and the Decision to Vertically Integrate
Some Empirical Evidence From the U.S. Automobile Industry

WP-94-1

T hom as H . K lie r

Domestic Consumption Patterns and the Midwest Economy

W P-94-4

R o b e rt S ch n orbu s a n d P a u l B a lle w




1

Working paperseriescontinued

To Trade or Not to Trade: Who Participates in RECLAIM?

WP-94-11

Thom as H . K lie r a n d R ic h a r d M attoon

Restructuring & Worker Displacement in the Midwest

WP-94-18

P a u l D . B a lle w a n d R o b e rt H . S chn o rb u s

ISSUES IN FINANCIAL REGULATION
Incentive Conflict in Deposit-Institution Regulation: Evidence from Australia

WP-92-5

E d w a rd J . K a n e a n d G eo rg e G . K aufm an

Capital Adequacy and the Growth of U.S. Banks

WP-92-11

H e rb e rt B a e r an d Jo h n M c E lra v e y

Bank Contagion: Theory and Evidence

WP-92-13

G e o rg e G . Kaufm an

Trading Activity, Progarm Trading and the Volatility of Stock Returns

WP-92-16

Ja m e s T . M o s e r

Preferred Sources of Market Discipline: Depositors vs.
Subordinated Debt Holders

WP-92-21

D o u g la s D . E v a n o ff

An Investigation of Returns Conditional
on Trading Performance

WP-92-24

Ja m e s T . M o se r a n d J a c k y C . So

The Effect of Capital on Portfolio Risk at Life Insurance Companies

WP-92-29

E lija h B r e w e r I I I , Thom as H . M o n d sch ea n , an d P h ilip E . Strahan

A Framework for Estimating the Value and
Interest Rate Risk of Retail Bank Deposits

WP-92-30

D a v id E . H u tch iso n , G eo rg e G . P en n a cch i

Capital Shocks and Bank Growth-1973 to 1991

WP-92-31

H e rb e rt L . B a e r a n d Jo h n N . M c E lra v e y

The Impact of S&L Failures and Regulatory Changes
on the CD Market 1987-1991

WP-92-33

E lija h B r e w e r a n d Thom as H . M on dschean




2

Working paperseriescontinued

Junk Bond Holdings, Premium Tax Offsets, and Risk
Exposure at Life Insurance Companies

WP-93-3

E lija h B r e w e r III a n d T h om as H . M on dsch ean

Stock Margins and the Conditional Probability of Price Reversals

WP-93-5

P a u l K ofm an a n d J a m e s T. M o se r

Is There Lif(f)e After DTB?
Competitive Aspects of Cross Listed Futures
Contracts on Synchronous Markets
P a u l K o fm a n , T ony B ou w m an a n d J a m e s T. M o se r

W P-93-11

Opportunity Cost and Prudentiality: A RepresentativeAgent Model of Futures Clearinghouse Behavior
H e rb e rt L . B a e r , V irgin ia G. F ra n c e a n d J a m e s T. M o se r

WP-93-18

The Ownership Structure of Japanese Financial Institutions

WP-93-19

H esn a G en a y

Origins of the Modem Exchange Clearinghouse: A History of Early
Clearing and Settlement Methods at Futures Exchanges

WP-94-3

J a m e s T. M o se r

The Effect of Bank-Held Derivatives on Credit Accessibility
A. M in ton a n d J a m e s T. M o se r

WP-94-5

E lija h B r e w e r III , B e rn a d ette

Small Business Investment Companies:
Financial Characteristics and Investments

WP-94-10

E lija h B r e w e r III a n d H e sn a G en a y

MACROECONOMIC ISSUES
An Examination of Change in Energy Dependence and Efficiency
in the Six Largest Energy Using Countries-1970-1988

WP-92-2

J a c k L . H e rv e y

Does the Federal Reserve Affect Asset Prices?

WP-92-3

Vefa T arhan

Investment and Market Imperfections in the U.S. Manufacturing Sector

WP-92-4

P a u la R. W orthin gton




3

Working paperseriescontinued

Business Cycle Durations and Postwar Stabilization of the U.S. Economy

W P-92-6

M a r k W. W atson

A Procedure for Predicting Recessions with Leading Indicators: Econometric Issues
and Recent Performance

WP-92-7

Ja n ie s H . S to c k a n d M a r k W. W atson

Production and Inventory Control at the General Motors Coxporation
During the 1920s and 1930s

WP-92-10

A n il K . K a sh ya p an d D a v id W . W ilco x

Liquidity Effects, Monetary Policy and the Business Cycle

WP-92-15

L a w re n c e J . C h ristia n o an d M a rtin Eich en bau m

Monetary Policy and External Finance: Interpreting the
Behavior of Financial Flows and Interest Rate Spreads

WP-92-17

K en n eth N . K u ttn e r

Testing Long Run Neutrality

WP-92-18

R o b e rt G . K in g a n d M a rk W. W atson
A Policymaker's Guide to Indicators of Economic Activity
C h a rle s E v a n s , Steven Stro n g in , and F ra n c e s c a E u g e n i

WP-92-19

Barriers to Trade and Union Wage Dynamics

WP-92-22

E lle n R . R issm a n

Wage Growth and Sectoral Shifts: Phillips Curve Redux

WP-92-23

E lle n R . R issm a n

Excess Volatility and The Smoothing of Interest Rates:
An Application Using Money Announcements

WP-92-25

Steven S tro n g in

Market Structure, Technology and the Cyclicality of Output

WP-92-26

B r u c e P e te rse n a n d Steven Stron gin

The Identification of Monetary Policy Disturbances:
Explaining the Liquidity Puzzle

WP-92-27

Steven S trongin




4

Working paperseriescontinued

Earnings Losses and Displaced Workers

WP-92-28

L o u is S . J a c o b s o n , R o b e r t J. L a L o n d e, a n d D a n ie l G . Sullivan

Some Empirical Evidence of the Effects on Monetary Policy
Shocks on Exchange Rates

WP-92-32

M a rtin E ich en bau m a n d C h a rle s E v a n s

An Unobserved-Components Model of
Constant-Inflation Potential Output

WP-93-2

K en n eth N . K u ttn er

Investment, Cash Flow, and Sunk Costs

W P-93-4

P a u la R . W orthin gton

Lessons from the Japanese Main Bank System
for Financial System Reform in Poland

WP-93-6

T akeo H o sh i, A n il K a sh y a p , a n d G a ry L ovem an

Credit Conditions and the Cyclical Behavior of Inventories
C. S tein

WP-93-7

A n il K . K a s h y a p , O w en A . L a m o n t a n d J erem y

Labor Productivity During the Great Depression

WP-93-10

M ic h a e l D . B o rd o a n d C h a rle s L . E va n s

Monetary Policy Shocks and Productivity Measures
in the G-7 Countries

WP-93-12

C h a rle s L . E v a n s a n d F ern a n d o S a n to s

Consumer Confidence and Economic Fluctuations

WP-93-13

Joh n G . M a tsu sa k a a n d A rg ia M . S b o rd o n e

Vector Autoregressions and Cointegration

WP-93-14

M a rk W. W atson

Testing for Cointegration When Some of the
Cointegrating Vectors Are Known

WP-93-15

M ic h a e l T. K . H o rv a th a n d M a rk W. W atson

Technical Change, Diffusion, and Productivity

WP-93-16

J effrey R . C a m p b e ll




5

Working paperseriescontinued

Economic Activity and the Short-Term Credit Markets:
An Analysis of Prices and Quantities
B en ja m in M , F ried m a n a n d K en n eth N . K u ttn er

WP-93-17

Cyclical Productivity in a Model of Labor Hoarding

WP-93-20

A r g ia M . S b o rd o n e

The Effects of Monetary Policy Shocks: Evidence from the Flow of Funds

WP-94-2

L a w re n c e J. C h ristia n o , M a rtin E ich en b a u m a n d C h a rle s E v a n s

Algorithms for Solving Dynamic Models with Occasionally Binding Constraints
/. C h ristia n o a n d J o n a s D M , F ish e r

WP-94-6

Identification and the Effects of Monetary Policy Shocks

WP-94-7

L a w re n c e

L a w re n c e J . C h ristia n o , M a rtin E ich en b a u m a n d C h a rle s L . E v a n s

Small Sample Bias in GMM Estimation of Covariance Structures

WP-94-8

J o se p h G . A lto n ji a n d L e w is M . S e g a l

Interpreting the Procyclical Productivity of Manufacturing Sectors:
External Effects of Labor Hoarding?

WP-94-9

A r g ia M . S b o rd o n e

Evidence on Structural Instability in Macroeconomic Time Series Relations

WP-94-13

J a m e s H . S to ck a n d M a rk W. W atson

The Post-War U.S. Phillips Curve: A Revisionist Econometric History

WP-94-14

R o b e r t G . K in g a n d M a rk W. W atson

The Post-War U.S. Phillips Curve: A Comment

WP-94-15

C h a rle s L . E v a n s

Identification of Inflation-Unemployment

WP-94-16

B e n n e tt T. M c C a llu m

The Post-War U.S. Phillips Curve: A Revisionist Econometric History
Response to Evans and McCallum
R o b e r t G . K in g a n d M a rk W. W atson




W P-94-17

6

Working paperseriescontinued

Estimating Deterministic Trends in the
Presence of Serially Correlated Errors

WP-94-19

E u g en e C a n je ls a n d M a rk W. W atson




7