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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Confidence Intervals for Half-Life Deviations From
Purchasing Power Parity
Barbara Rossi
To cite this article: Barbara Rossi (2005) Confidence Intervals for Half-Life Deviations From
Purchasing Power Parity, Journal of Business & Economic Statistics, 23:4, 432-442, DOI:
10.1198/073500105000000027
To link to this article: http://dx.doi.org/10.1198/073500105000000027

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Published online: 01 Jan 2012.

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Date: 12 January 2016, At: 23:56

Confidence Intervals for Half-Life Deviations
From Purchasing Power Parity
Barbara R OSSI
Department of Economics, Duke University, Durham, NC 27708 (brossi@econ.duke.edu )
Existing point estimates of half-life deviations from purchasing power parity (PPP), around 3–5 years,
suggest that the speed of convergence is extremely slow. This article assesses the degree of uncertainty
around these point estimates by using local-to-unity asymptotic theory to construct confidence intervals
that are robust to high persistence in small samples. The empirical evidence suggests that the lower bound
of the confidence interval is between four and eight quarters for most currencies, which is not inconsistent
with traditional price-stickiness explanations. However, the upper bounds are infinity for all currencies,
so we cannot provide conclusive evidence in favor of PPP either.

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KEY WORDS: Half-life; Persistence; Purchasing power parity; Roots close to unity.

1. INTRODUCTION
What determines nominal exchange rates in the long run?
According to Purchasing Power Parity (PPP), because the
(bilateral) nominal exchange rate (Et ) is the relative price of
two currencies (the bilateral nominal exchange rate is defined
here as the price of the foreign country’s currency in terms of
the home country’s currency), in equilibrium it should reflect
their relative purchasing powers. So if Pt is the price level in
the home country and P∗t is the price level in the foreign country, then PPP requires
Et =

Pt
.
P∗t

(1)

Thus the ∗logarithm of the real exchange rate, defined as
EP
yt = ln( Pt t t ), should be constant if PPP holds at every point
in time. A weaker version of the PPP, which is followed in this
article and in most of the literature, requires only that (1) holds
in the long run.
The empirical evidence on PPP is mixed. Although casual
evidence suggests that the two series, Et and Pt /P∗t , tend to revert toward each other over time, there are protracted periods in
which the nominal exchange rate deviates from its PPP level.
How persistent are these deviations? A measure of persistence
is the half-life of PPP deviations. To motivate this measure, suppose that the deviations of the logarithm of the real exchange
rate yt from its long-run value y0 , which is constant under PPP,
follow an autoregressive process of order 1,
yt − y0 = ρ( yt−1 − y0 ) + ǫt ,

(2)

where ǫt is a white noise. Then, at horizon h, the percentage deviation from equilibrium is ρ h . The half-life deviation

from PPP is defined as the smallest value of h such that
E( yt+h − y0 |yt−s − y0 , s ≤ 0) ≤ 21 ( yt − y0 ), where E is the expectation operator, that is,
ρh =

1
2

⇒

h=

ln(1/2)
.
ln(ρ)

(3)

Using data under floating exchange rate regimes, estimates of
h range between 2 and 5 years for most countries, with an average of 3.7 years (see table 7.2 in Mark 2001). [Note that although these point estimates are introduced here to motivate the
article, they no longer represent the current status of our knowledge; for example, Murray and Papell (2002) noted that after

accounting for serial correlation and small-sample bias, these
point estimates become very difficult to believe.]
The existing point estimates of half-life deviations from PPP
are difficult to reconcile with conventional explanations for the
failure of short-run PPP based on price stickiness. According to Rogoff (1996, p. 654), deviations from PPP can be attributed to transitory disturbances, like financial and monetary
shocks, that buffet the nominal exchange rate and translate into
real exchange rate variability because of nominal price stickiness. Thus, whereas conventional explanations for the failure
of PPP based on nominal prices stickiness are compatible with
the enormous short-term volatility of real exchange rates, they
also imply that deviations should be short-lived, because they
can occur only during a time frame in which nominal wages
and prices are sticky (i.e., 1–2 years). The existing point estimates imply instead that deviations are much more persistent
than that. Rogoff (1996, p. 647) called this empirical inconsistency the “PPP puzzle.”
This article makes two contributions. First, it introduces a
measure of the half-life for a general AR( p) process that allows better asymptotic approximations in the presence of a root
close to unity. Although the methods for deriving the half-life
are quite standard, there is no such result in the literature.
Abuaf and Jorion (1990) discussed half-lives in the context
of an AR(1) process only. Mark (2001) discussed measures

of half-lives for general AR( p) processes, but for stationary
processes only. Andrews (1993) proposed a measure of halflife for an AR(1) process that is robust to the presence of high
persistence. Andrews and Chen (1994) generalized the method
to obtain an approximate median-unbiased estimate of AR( p)
coefficients in the presence of high persistence. They showed
how to construct an approximately median-unbiased estimator
of the impulse-response function (IRF), but did not provide an
analytic measure of the half-life for AR( p) processes.
Second, the article uses this measure to provide a simple
method for constructing confidence intervals for the half-life.
The issue is complicated by both the high persistence in real exchange rates and the small samples usually available. For these
reasons, this article considers an alternative asymptotic theory

432

© 2005 American Statistical Association
Journal of Business & Economic Statistics
October 2005, Vol. 23, No. 4
DOI 10.1198/073500105000000027

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Rossi: Confidence Intervals for Half-Lives

433

based on local-to-unity asymptotics and a half-life that grows
to infinity at the rate of the sample size, as was done by Stock
(1996) and Phillips (1998). How good this approximation is relative to the conventional (normal sampling) asymptotic theory
is explored in a Monte Carlo experiment.
This is not the first article on inference about half-life deviations from PPP. Authors who have recently addressed this issue include Cheung and Lai (2000), Murray and Papell (2002),
Lopez, Murray, and Papell (2003), Gospodinov (2002), and
Kilian and Zha (2002). All of these authors calculated confidence intervals by estimating impulse responses with various
methods: Cheung and Lai (2000) relied on stationary, normal
sampling distributions; Murray and Papell relied on work of
Andrews and Chen (1994); Gospodinov (2002) relied on work
of Hansen (1999); and Kilian and Zha (2002) used Bayesian
methods. However, estimating the whole IRF may quickly become computationally intensive.
It is also important to note that there are different methods for evaluating the persistence of a univariate AR process.
One could rely on the cumulative IRF (which is directly related

to the sum of the AR coefficients), as was done by Andrews
and Chen (1994), or alternatively, one could focus on measuring the half-life, as in the present article. In the context of the
PPP debate, it is appropriate to focus on the half-life, because
that is one of the most important statistical measures used in
the literature (see, e.g., the influential survey in Rogoff 1996).
However, methods analogous to those used in this article could
be adapted to the analysis of cumulative IRFs. It is important
to note, however, that even if our method relies on a local-tounity approximation for the largest root, it does not disregard
short-run dynamics. In fact, our method approximates the whole
IRF to approximate the half-life. The short-run dynamics of the
process is thus taken into account, although its sampling variability is not, being of smaller order by the nature of the approximation.
Overall, the results of this article are not inconsistent with traditional explanations for the short-run failure of PPP, although
they do not rule out infinite half-lives either. The existing point
estimates, although too high to be reconciled with the PPP, also
have huge variability. As a result, confidence intervals with 95%
coverage for most currencies include four to eight quarters as
their lower bound, a time interval in which deviations from PPP
are compatible with nominal price and wage stickiness. However, because we cannot rule out the possibility of an infinite
half-life, we interpret the evidence as being simply not informative enough.
The article is organized as follows. The next section introduces the data-generating process (DGP) considered and

derives the measure of half-life used. Section 3 describes
the methods used to construct the confidence intervals for h.
Section 4 discusses a small Monte Carlo experiment that compares the coverage of the various confidence intervals discussed
in Section 3. Section 5 discusses the empirical results, and Section 6 concludes.

where dt = µ0 is a deterministic component, vt is a mean-zero,
stationary, and ergodic

process with finite autocovariances
γ (k) = Evt vt−k ; ω2 = +∞
k=−∞ γ (k) is finite and non-zero; and
vt = b(L)−1 ǫt , where ǫt is a martingale difference sequence
with finite fourth moments and constant variance σǫ2 and b(L) is
finite order and has p < ∞ (stable) roots. We do not allow the
presence of a deterministic time trend in the theoretical DGP
or in the empirical estimation. The reason for this is that if
a deterministic time trend is present, then PPP in levels will
not hold. (Note that if a deterministic trend is present, so that
dt = µ0 + µ1 t, then the calculations that follow continue to
hold, provided that we define a time-varying long-run equilibrium, that is, such that the long-run equilibrium at time τ is
defined as yτ = µ0 + µ1 τ . This is the equilibrium path that

would have prevailed in the absence of the shock. The empirical results for detrended real exchange rates are similar to those
reported in this article and are available on request.)
To provide better asymptotic approximations to the statistics
of interest in small samples when variables are highly persistent, we use local-to-unity asymptotic theory (see Stock 1991,
among others),
c
(5)
ρ = ec/T ≃ 1 + ,
T
where c is a constant (negative, if the process is highly persistent but mean-reverting) and T is the sample size. To provide
better small-sample approximations in situations where the true
half-life, h, can be “big” relative to the sample size, we derive
the asymptotic distributions by letting h increase as the sample
size T increases in such a way that their ratio remains a fixed
number δ, that is,
h
→ δ.
T T→∞

We refer to δ as the “half-life as a fraction of the sample size.”

The persistence of the process in small samples, measured
by c, is relevant for our purposes, because we are trying to estimate at which horizon the deviations from PPP are back to onehalf after a shock. (We provide detailed empirical evidence on
the degree of persistence in the bilateral exchange rates considered in this article in the empirical section.) As we show later,
the speed at which the effect of a shock dies away depends on
a function of the largest root of the process, ρ h , and, under assumption (6),
ρ h → ecδ .
T→∞

Let the DGP for the log of the real exchange rate, yt , be
yt = d t + u t ,
ut = ρut−1 + vt ,

t = 1, 2, . . . , T,

(4)

(7)

To derive an expression for the half-life in this general AR( p)
process, we need to derive an expression for the effect of the
shock ǫt on yt after h periods. We derive it in terms of the eigenvalues of the process. [We follow Hamilton 1994 in referring to
the inverse of the roots of the polynomial b(L)(1 − ρL) as the
eigenvalues (or the roots) of the DGP.] We factorize (4) as
(1 − λ1 L)(1 − λ2 L) · · · (1 − λp L)( yt − dt ) = ǫt ,

2. MEASURING THE HALF–LIFE

(6)

(8)

where, for convenience, λ1 = ρ is the root close to unity and
λ2 , λ3 , . . . , λp are the (stable) roots, the inverse of the roots
of the polynomial b(L). We also define λ to be a ( p × 1)
vector containing all of the eigenvalues of the DGP, λ =
[λ1 , λ2 , . . . , λp ]′ . We assume that the eigenvalues are distinct.

434

Journal of Business & Economic Statistics, October 2005

Suppose that we start at time (t − 1) in the long-run equilibrium µ0 , and at time t there is a shock ǫt . No other shocks hit the
economy subsequently. The shock ǫt measures the initial deviation from equilibrium, which we denote by 
y t ≡ y t − µ 0 = ǫt .
It follows that the deviation from equilibrium after h periods
will be 
yt+h = c′ λh ǫt , where c is a ( p × 1) vector with generic
element,
p−1

ci =  p

λi

k=1,k =i (λi

− λk )

(9)

,

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and λh is the ( p × 1) vector containing all of the eigenvalues
to the h power (see Hamilton 1994, p. 12). After h periods, the
percentage deviation from equilibrium relative to the initial percentage deviation from equilibrium is
yt+h
( yt+h − µ0 ) ∂
=
= c′ λh .
( yt − µ0 )
∂ǫt

(10)

(Recall that in this article, yt is the logarithm of the real exchange rate, so yt+h − µ0 measures a percentage deviation.)
We call ∂
yt+h /∂ǫt (which is the usual definition of an impulse
response) “the effect of a shock ǫt after h periods.” By combining (9) and (10) and isolating the largest root λ1 (= ρ), we have
h+p−1

λ1
∂
yt+h
=
∂ǫt
(λ1 − λ2 )(λ1 − λ3 ) · · · (λ1 − λp )
+

p

i=2

h+p−1

p

λi

k=1,k =i (λi

− λk )

.

(11)

Because all eigenvalues except the first one are in moduh+p−1
lus