are real interest rates really nonstationary? new evidence from tests with good size and power

Journal of Macroeconomics 26 (2004) 409–430

www.elsevier.com/locate/econbase

Are real interest rates really nonstationary?New evidence from tests with

good size and power q

David E. Rapach a,*, Christian E. Weber b

a Department of Economics, Saint Louis University, 3674 Lindell Boulevard,

Saint Louis, MO 63108-3397, USAb Department of Economics and Finance, Albers School of Business and Economics,

Seattle University, 900 Broadway, Seattle, WA 98122-4340

Received 16 October 2002; accepted 20 March 2003

Abstract

In this paper, we re-examine the stationarity of international real interest rates, an issue first

investigated by Rose [Journal of Finance 43(5) (1988) 1095], using a new set of unit root tests

developed by Ng and Perron [Econometrica 69(6) (2001) 1519] with good size and power.

Using conventional unit root tests, Rose finds that the nominal interest rate is I(1), while

the inflation rate is I(0), for each of the many countries he considers, indicating a nonstation-

ary real interest rate for each country. Using an extended sample period and the Ng and Per-

ron unit root tests, we find that the nominal interest rate is I(1) and the inflation rate is I(0) for

only three of the 16 countries we examine. For a number of countries, the Ng and Perron tests

indicate that the nominal interest rate and inflation rate are both I(1), so that we need to test

for cointegration in order to decipher the integration properties of the real interest rate. Using

0164-0704/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmacro.2003.03.001

q This is a significantly revised version of a paper presented at the July 2001 meetings of the Western

Economic Association, and the authors are grateful to session participants, especially William Crowder

and Mark Wohar, for helpful comments. We also thank Alan Isaac, John Galbraith, and an anonymous

referee for insightful comments on earlier drafts. The usual disclaimer applies. The results reported in this

paper were generated usin GAUSS 3.6GAUSS 3.6. The GAUSSGAUSS programs are available on request from the authors.* Corresponding author. Address: Department of Economics, Saint Louis University, 3674 Lindell

Boulevard, Saint Louis, MO 63108-3397, USA. Tel.: +1 314 977 3601; fax: +1 314 977 1478.

E-mail addresses: [email protected], [email protected] (D.E. Rapach).

mailto:[email protected]

410 D.E. Rapach, C.E. Weber / Journal of Macroeconomics 26 (2004) 409–430

either the Ng and Perron unit root tests in conjunction with a pre-specified cointegrating vec-

tor or the Perron and Rodriguez [Residual based tests for cointegration with GLS detrended

data (2001)] cointegration tests for an unspecified cointegrating vector, there is little robust

evidence of cointegration. While our results are mixed, they usually provide support for the

Rose finding that international real interest rates are nonstationary, albeit often for different

reasons than Rose.

� 2004 Elsevier Inc. All rights reserved.

JEL classification: C22; E43; G12

Keywords: Nominal interest rate; Inflation rate; Real interest rate; Unit root; Cointegration

1. Introduction

In a provocative paper, Rose (1988) investigates the integration properties of the

real interest rate. Using postwar data from 18 OECD countries and conventionalDickey and Fuller (1979) and Phillips and Perron (1988) unit root tests, he examines

the integration properties of the nominal interest rate and inflation rate for each

country. Conventional unit root tests cannot reject the nonstationary null hypothesis

for the nominal interest rate levels, while the nonstationary null is easily rejected for

the first differences. Based on these results, Rose (1988) concludes that the nominal

interest rate (Rt) contains a single unit root (Rt � I(1)) in each country. In contrast,

conventional unit root tests reject the nonstationary null hypothesis for the inflation

rate levels, indicating that the inflation rate (pt) is stationary (pt � I(0)) in each coun-try. Under the relatively weak assumption that errors in inflationary expectations are

stationary, Rose (1988) points out that the real interest rate in each country must be

nonstationary, as any linear combination of an I(1) process and an I(0) process is

unambiguously I(1).

Rose (1988) observes that the finding of a nonstationary real interest rate is prob-

lematic for consumption-based asset-pricing models. Following Lucas (1978) and

Hansen and Singleton (1982), among others, consider the problem of choosing the

consumption sequence fctg1t¼0 to maximizeP1

t¼0btc1�a

t =ð1� aÞ, where b is a discountfactor and a is the coefficient of relative risk aversion, subject to the usual asset-accu-

mulation condition. The first-order condition for this problem is the intertemporal

Euler equation: Et[b(1 + rt)(ct+1/ct)�a] = 1, where 1 + rt = (1 + Rt)/(1 + pt) is the

gross one-period real interest rate. If consumption growth is stationary—as it almost

certainly is—it is clear that the first-order condition cannot be satisfied if the real

interest rate is nonstationary. 1 Furthermore, both of the commonly employed strat-

egies for estimating the Euler equation—and testing the validity of the assumptions

which underlie it—require assumptions on the stationarity properties of the realinterest rate and consumption growth processes. The generalized method of mo-

ments (GMM) procedure pioneered by Hansen (1982) requires both the real interest

1 See Rose (1988) for a detailed argument.

D.E. Rapach, C.E. Weber / Journal of Macroeconomics 26 (2004) 409–430 411

rate and consumption growth processes to be stationary, so that its use is not appro-

priate if the real interest rate is nonstationary. Extensions of the Lucas–Hansen–Sin-

gleton model to account for possible time nonseparability (for example, Dunn and

Singleton, 1986; Eichenbaum et al., 1988; Gallant and Tauchen, 1989) or for non-

expected utility maximizing behavior (Epstein and Zin, 1991) still require the realinterest rate to be stationary.

The alternative to GMM (actually, a special case of GMM) is to estimate the log-

linear version of the Euler equation, typically using linear instrumental variables (IV)

regression. While this approach does not necessarily require the real interest rate to

be I(0), it does require the real interest rate and consumption growth processes to

have the same order of integration. Thus, if the real interest rate is I(1), linear IV esti-

mation of log-linearized Euler equations requires consumption growth to be I(1) as

well. Since it is almost certainly not the case that consumption growth is I(1), thepresence of a unit root in the real interest rate would clearly require rethinking exist-

ing theoretical and empirical models of consumption.

Evidence of nonstationary real interest rate behavior is also problematic from the

standpoint of the canonical neoclassical growth model with explicitly optimizing,

infinitely lived agents. 2 The canonical growth model predicts that, absent changes

in households� rate of time preference, the steady-state real interest rate will be con-

stant. Following Barro (1981), consider a permanent tax-financed increase in govern-

ment purchases. Households experience a permanent reduction in the present valueof their lifetime wealth equal to the present value of the permanent increase in gov-

ernment purchases. Households thus decrease their consumption in each period by

an amount equal to the increase in government purchases each period, leaving the

steady-state capital stock and real interest rate unchanged. While a temporary

change in government purchases can affect the real interest rate in the canonical

growth model, the effect will only be temporary, so that the steady-state real interest

rate is still unchanged. 3 In light of these theoretical considerations, Mishkin and

Simon (1995, p. 223) opine that ‘‘any reasonable model of the macro economy wouldsurely suggest that real interest rates have mean-reverting tendencies which make

them stationary,’’ and, indeed, the stationarity of the real interest rate is often as-

sumed in empirical work despite pre-test evidence to the contrary; see, for example,

Shapiro and Watson (1988) and Galı (1992). Galı (1992, p. 717) contends that ‘‘the

assumption of a unit root in the real (interest) rate seems rather implausible on a pri-

ori grounds, given its inconsistency with standard equilibrium growth models.’’

Given the important theoretical implications and controversial nature of the Rose

(1988) findings, we re-examine the integration properties of international real interest

2 This model is often associated with Ramsey (1928), Cass (1965), and Koopmans (1965). See Romer

(1996, Chapter 2) for a textbook exposition of the canonical neoclassical growth model with explicitly

optimizing, infinitely lived agents.3 Government deficits have neither temporary nor permanent effects in the canonical growth model, as

households are Ricardian.


rates in the present paper. 4 We update the Rose (1988) study in two key ways.

Firstly, we use an extended sample of quarterly nominal interest rate and inflation

rate data covering the period 1957–2000 for a large number of OECD countries. Sec-

ondly, we employ a set of unit root tests recently developed by Ng and Perron (2001).

These tests modify conventional unit root tests in a number of ways in order to de-rive tests with better size and power. By employing the Ng and Perron (2001) tests,

we can thus be more confident that rejections of the null hypothesis of nonstation-

arity in unit root tests are not due to size distortions, while nonrejections do not re-

sult from a low probability of rejecting a false null hypothesis. We find that the Ng

and Perron (2001) unit root tests can have important effects on inferences regarding

the integration properties of international nominal interest rates and inflation rates.

When we use our extended sample and the Ng and Perron (2001) unit root tests, we

obtain the Rose (1988) finding that Rt � I(1) and pt � I(0) for three of the 16 coun-tries we consider. For the other 13 countries, our unit root test results differ from

Rose (1988) concerning the integration properties of the nominal interest rate and/

or inflation rate. For one country, we find that Rt � I(0) and pt � I(1) (which also

results in a nonstationary real interest rate). For two countries, we find that both

the nominal interest rate and inflation rate are stationary, resulting in a stationary

real interest rate. For the ten remaining countries, the Ng and Perron (2001) unit

root tests indicate that Rt,pt � I(1), so that both the nominal interest rate and infla-

tion rate are nonstationary. In these cases, we cannot conclude on the basis of theindividual unit root test results alone whether the real interest rate is stationary or

nonstationary. If the nominal interest rate and the inflation rate are each I(1), the

real interest rate can still be stationary if the linear combination Rt–pt is stationary;that is, the nominal interest rate and inflation rate may be cointegrated

(Rt,pt � CI(1,1)) with cointegrating vector (1,�1) 0. We thus test for cointegration

between Rt and pt with a pre-specified cointegrating vector of (1,�1) 0 in each country

where we find evidence that Rt,pt � I(1). We do this using the same univariate unit

root tests that we use for the individual series. Following a number of extant studies,we also test for cointegration with an unspecified cointegrating vector in order to al-

low for a nonzero tax rate on nominal interest income. In addition to the conven-

tional cointegration tests of Engle and Granger (1987) and Phillips and Ouliaris

(1990), we use the recently developed cointegration tests of Perron and Rodriguez

(2001). Analogous to the Ng and Perron (2001) unit root tests, the Perron and

Rodriguez (2001) tests modify the conventional Engle and Granger (1987) and Phil-

lips and Ouliaris (1990) tests to arrive at cointegration tests with better size and

power. Overall, the Ng and Perron (2001) and Perron and Rodriguez (2001) test re-sults provide little robust evidence of a stationary real interest rate for most

countries.

In the end, although our results are somewhat mixed, they often support the con-

troversial conclusion of Rose (1988) that the real interest rate is nonstationary in

4 As the main purpose of this paper is to re-examine the findings of Rose (1988), we concentrate on

testing whether real interest rates are I(0) or I(1) and do not consider fractionally integrated alternatives.


many OECD countries. For a few countries, in agreement with Rose (1988), we con-

clude that the real interest rate is nonstationary because the nominal interest rate and

inflation rate are I(1) and I(0), respectively. However, for a number of other coun-

tries, we find that the real interest rate is nonstationary for quite different reasons

than Rose (1988). For many countries, we conclude that the real interest rate is non-stationary because, while the nominal interest rate and inflation rate are both I(1),

there is not strong evidence that they are cointegrated.

The rest of this paper is organized as follows: Section 2 presents the testing strat-

egy and describes the unit root and cointegration tests; Section 3 reports unit root

and cointegration test results for nominal interest rates and inflation rates for 16

OECD countries; Section 4 concludes.

2. Preliminaries

2.1. Long-run testing strategy

The nominal interest rate is typically decomposed into the ex ante real interest

rate and the ex ante expected inflation rate,

Rt ¼ ret þ pet ; ð1Þ

where Rt is the nominal interest rate, ret is the ex ante real interest rate, and pet is the

ex ante expected inflation rate, all at period t. Following Rose (1988), we make the

relatively weak assumption that the ex ante expected inflation rate and the actual

inflation rate differ by a stationary, zero-mean forecasting error term, �t,5

pet ¼ pt þ �t: ð2Þ

We can then write the nominal interest rate as

Rt ¼ ret þ pt þ �t; ð3Þand the real interest rate as

ret ¼ Rt � pt � �t: ð4ÞEq. (4) forms the basis for a long-run testing strategy. Given that �t is stationary

by assumption, the integration properties of ret are determined by the integration

properties of Rt and pt. Assuming that Rt and pt have at most one unit root each, 6

there are three possible outcomes:

(i) Rt,pt � I(0), so that ret � Ið0Þ. That is, if both the nominal interest rate and the

inflation rate are stationary, then the real interest rate is stationary, as any linear

combination of two I(0) variables is I(0).

5 See Pelaez (1995) for evidence that the difference between actual and expected inflation is stationary in

the United States.6 This is clearly evident in nominal interest rate and inflation rate data for developed countries.


(ii) Rt � I(1), pt � I(0) or pt � I(1), Rt � I(0), so that ret � Ið1Þ. That is, if the nom-

inal interest rate is nonstationary and the inflation rate is stationary, or vice-

versa, then the real interest rate is nonstationary, as any linear combination

of an I(1) variable and an I(0) variable is I(1).

(iii) Rt,pt � I(1). In this case, the real interest rate is stationary if Rt,pt � CI(1,1)with cointegrating vector (1,�1) 0. That is, the real interest rate is stationary if

the linear combination Rt–pt is stationary. (We can allow for a nonzero nominal

interest income tax rate by not restricting the cointegrating vector to be

(1,�1) 0.) Otherwise, the real interest rate is nonstationary.

Using quarterly postwar nominal interest rate and price index data for a large

number of industrialized countries and conventional unit root tests, Rose (1988)

finds that Rt � I(1) and pt � I(0) for each country, so that the first part of outcome(ii) obtains. Rose (1988) thus concludes that the real interest rate is nonstationary for

every country he examines. As discussed above, this presents difficulties for some

canonical theoretical models. In Section 3 below, we re-examine the integration

properties of postwar nominal interest rates and inflation rates for 16 OECD coun-

tries. We are especially interested in how inferences concerning the integration prop-

erties of these variables are affected when we use the recently developed unit root

tests of Ng and Perron (2001) with good size and power.

2.2. Unit root test descriptions

We first test for a unit root in international nominal interest rates and inflation

rates using the familiar augmented Dickey–Fuller (ADF) test of Dickey and Fuller

(1979) and Said and Dickey (1984) and the Za test of Phillips (1987) and Phillips and

Perron (1988). The ADF test is based on the OLS t-statistic corresponding to b0 inthe regression model

Dyt ¼ b0 þ b0yt�1 þXkj¼1

bjDyt�j þ etk; ð5Þ

where t = 1, . . .,T. The null hypothesis that b0 = 0 (corresponding to a unit root in yt)

is tested against the one-sided (lower-tail) alternative hypothesis that b0 < 0 (corre-

sponding to a stationary yt).7 The inclusion of the lagged Dyt terms in Eq. (5) is a

parametric adjustment for serial correlation. We follow Campbell and Perron

(1991) and Ng and Perron (1995) and select the lag order k in Eq. (5) via top–down

testing (considering a maximum lag order of eight, as we use quarterly data). 8 The

Za test is based on the regression model

7 Given that we allow for a constant term in Eq. (5), yt is stationary about a possibly nonzero mean

under the alternative hypothesis.8 That is, we first estimate Eq. (5) with k = 8. If the t-statistic corresponding to b8 is greater than or

equal to 1.645 in absolute value, we select k = 8. If the t-statistic is less then 1.645 in absolute value, we set

k = 7. We continue in this fashion until we obtain a significant t-statistic. If no t-statistic is significant, we

select k = 0.


yt ¼ a0 þ ayt�1 þ ut: ð6ÞIn order to test the null hypothesis that a = 1 (corresponding to a unit root in yt)against the one-sided (lower-tail) alternative hypothesis that a < 1 (corresponding

to a stationary yt), the Za test uses a statistic that combines T ða� 1Þ with a semipar-

ametric adjustment for serial correlation,

Za ¼ T ða� 1Þ � 0:5ðT 2r2a=s

2Þðk� c0Þ; ð7Þwhere a is the OLS estimate of a in Eq. (6) and ra is its standard error;

s2 ¼ ðT � 2Þ�1PTt¼1u

2t and ut is the OLS residual from Eq. (6); k is an estimate of

the spectral density at frequency zero of ut (sometimes called the ‘‘long-run’’ variance

of ut) that is based on the covariance estimator, cv ¼ T�1PT

t ¼ vþ1utut�v. In forming

k, we use the quadratic spectral kernel recommended by Andrews (1991) in conjunc-

tion with the Andrews (1991) automatic bandwidth selection procedure. 9 As is well-

known, the ADF and Za test statistics have nonstandard distributions, and criticalvalues are available from a number of sources; see, for example, Fuller (1976). We

use the critical values computed by Sam Ouliaris and Peter C.B. Phillips in the

GAUSS module COINT, which are similar to, but more accurate than, those re-

ported in Fuller (1976).

Especially due to Schwert (1987), Phillips and Perron (1988), Schwert (1989), and

Ng and Perron (1995), it is now well-established that the ADF and, especially, Za

tests can have severe nominal size distortions when the underlying data-generating

process contains a moving-average root near unity. For example, using Monte Carlosimulations, Schwert (1989) finds that the nonstationary null hypothesis is incor-

rectly rejected 43.4 and 99.7% of the time for a sample of 100 observations using

the ADF (for a lag order of four) and Za (for a bandwidth of four) statistics, respec-

tively, when the underlying data-generating process for yt is an ARIMA(0,1,1)

model with a moving-average coefficient of �0.8. This is a relevant concern for us

in the present paper, as international inflation rates have been found to have a siz-

able negative moving-average component. 10 Ng and Perron (1995) find that using

the top–down testing procedure to select the lag length goes some way in controllingfor possible size distortions for the ADF test. However, this tends to overparameter-

ize Eq. (5) in the absence of a sizable moving-average component, thus leading to

inefficient tests and decreased power.

Ng and Perron (2001) have recently developed a set of state-of-the-art unit root

tests that have good size and power properties. They modify the standard ADF test

in two key ways in constructing what they label the ADFGLS test. Firstly, following

Elliott et al. (1996), Ng and Perron (2001) use local-to-unity GLS detrending (quasi-

differencing) instead of OLS when estimating the deterministic component (b0) inEq. (5). Define the series fy�at g

Tt¼0 as ðy�a0; y�at Þ ¼ ðy0; ð1� �aLÞytÞ for t = 1, . . .,T and

�a ¼ 1þ �c=T . Also define fz�at gTt¼0 as ðz�a0; z�at Þ ¼ ð1; ð1� �aLÞ1Þ. The GLS detrended ser-

ies for yt is given by ~yt ¼ yt � d, where d ¼ argmind

PTt¼0ðy�at � dz�at Þ

2. Following the

9 Our results are not sensitive to the kernel function.10 See, for example, Perron and Ng (1996).


recommendation of Elliott et al. (1996), Ng and Perron (2001) set �c ¼ �7:0. By using

GLS detrending, Ng and Perron (2001) take advantage of the considerable power

gains associated with GLS detrending demonstrated by Elliott et al. (1996). The next

step is to estimate an ADF-type regression model for the detrended data,

D~yt ¼ b0~yt�1 þXkj¼1

bjD~yt�j þ etk; ð8Þ

where the lag length in Eq. (8) is selected using a modified AIC (MAIC), which is

given by

MAICðkÞ ¼ lnðr2kÞ þ

2ðsT ðkÞ þ kÞT � kmax

; ð9Þ

where sT ðkÞ ¼ ðr2kÞ

�1PTt¼kmaxþ1~y

2t�1, r

2k ¼ ðT � kmaxÞ�1PT

t¼kmaxþ1e2tk, kmax is the maxi-

mum value of k considered (kmax = 8 in our applications), and etk ¼ ~yt � b0~yt�1 �Pkj¼1bjD~yt�j, with b0 and bj obtained via OLS estimation of Eq. (8). The use of

the MAIC is the second key modification that Ng and Perron (1997) make to the

conventional ADF test. The MAIC is designed to select a relatively long lag length

in the presence of a moving-average root near unity (to avoid size distortions) and a

shorter lag length in the absence of such a root (so that power does not suffer). The

ADFGLS test uses the OLS t-statistic corresponding to b0 in Eq. (8).Ng and Perron (2001) also develop a modified Za test, what they call the MZ

GLS

a

test. This test modifies the conventional Za test in a number of ways. As with the

ADFGLS test, the deterministic component (a0) in Eq. (6) is estimated using GLS

detrending, again taking advantage of the increased power offered by GLS detrend-

ing. Building on Perron and Ng (1996), they use a variant of the Za test based on a

class of M-tests originally proposed by Stock (1990):

MZGLS

a ¼ ðT�1~y2T � kARÞ 2T�2XTt¼1

~y2t�1

!�1

: ð10Þ

Here, kAR ¼ r2k=ð1� bð1ÞÞ2 is an autoregressive spectral density estimator, where

bð1Þ ¼Pk

i¼1bi, r2k ¼ ðT � kÞ�1PT

t¼1e2tk, and etk ¼ ~yt � b0~yt�1 �

Pkj¼1bjD~yt�j, with b0

and bj obtained via OLS estimation of Eq. (8). As with the ADFGLS test, k is selected

in Eq. (8) using the MAIC. In Monte Carlo simulations, Perron and Ng (1996) find

that theM-form of the Za test combined with the autoregressive spectral density esti-

mator has much better size properties than the Za test. The use of the MAIC in

selecting the lag length for the autoregressive spectral density estimator is designedto further help avoid size distortions while preserving power. In extensive Monte

Carlo simulations, Ng and Perron (2001) find that the ADFGLS andMZGLS

a tests have

significantly better size and power properties than conventional unit root tests. The

ADFGLS test appears slightly more powerful than the MZGLS

a , but it is also subject to

slightly larger size distortions. Critical values for the ADFGLS and MZGLS

a statistics

are reported in Ng and Perron (2001, Table 1).


2.3. Cointegration test descriptions

As discussed above, if both the nominal interest rate and inflation rate are I(1), the

real interest rate can still be stationary if Rt and pt are cointegrated with cointegrat-

ing vector (1,�1) 0. We can straightforwardly test for such a cointegrating relation-ship by applying the unit root tests described above to the series Rt–pt. In

addition, we can allow for a nonzero nominal interest income tax rate by testing

for cointegration with an unspecified cointegrating vector. 11 Two of the most com-

mon and straightforward cointegration tests are the OLS-residual-based tests of

Engle and Granger (1987) and Phillips and Ouliaris (1990). Consider two variables,

wt and xt, both of which are I(1). The first step in each test involves estimating the

potential cointegrating relationship between wt and xt using the regression model 12

wt ¼ f0 þ f1xt þ gt: ð11ÞThe second step of each test is to examine the stationarity properties of the OLS res-

iduals from Eq. (11). Denote the OLS residual by gt, so that gt ¼ wt � f0 � f1xt,where f0 and f1 are the OLS estimates of f0 and f1, respectively, in Eq. (11). If the

residuals from the cointegrating regression are stationary, this indicates that wt

and xt are cointegrated with cointegrated vector (1,�f1) 0. Engle and Granger

(1987) test the stationarity properties of the cointegrating residuals using the ADFtest, while Phillips and Ouliaris (1990) use the Za test. To apply the ADF cointegra-

tion test, we use the t-statistic corresponding to b0 in the regression model

Dgt ¼ b0gt�1 þXkj¼1

bjDgt�j þ etk: ð12Þ

We again use the top–down procedure to select the lag length for the ADF cointe-

gration test.

For the Za cointegration test, we estimate the regression model

gt ¼ agt�1 þ ut; ð13Þand form a statistic analogous to Eq. (7). We again use the quadratic spectral kernel

and the Andrews (1991) automatic bandwidth selection procedure in calculating the

semiparametric adjustment. Nonstationarity serves as the null hypothesis for both

the ADF and Za cointegration tests, so that wt and xt are not cointegrated under

the null hypothesis. Since f0 and f1 need to be estimated in the first step of the tests,

critical values from the conventional unit root tests are inappropriate. We use the

11 As described in Section 3.1 below, we use long-term nominal bond rates in our empirical analysis.

Following Shiller and Siegel (1977), Engsted (1995) observes that we should also allow the cointegrating

vector between Rt and pt to differ from (1,�1) if we use a long-term nominal interest rate in order to allow

for coupon payments on long-term bonds. By allowing for an unspecified cointegrating vector, we can

allow for both coupon payments and a nonzero nominal interest income tax rate.12 Including a constant term in the first-step regression model allows for a constant term in the

cointegrating relationship, which is tantamount to allowing for a nonzero steady-state real interest rate in

our applications.


critical values computed by Sam Ouliaris and Peter C.B. Phillips in the GAUSS

module COINT. In general, results for both the ADF and Za cointegration tests de-

pend on which variable serves as the dependent variable in Eq. (11). To check the

robustness of our results, we report results with each variable appearing in turn as

the regressand. 13

Perron and Rodriguez (2001) modify the ADF and Za cointegration tests in an

analogous manner to Ng and Perron (2001). Consider first the ADFGLS cointegra-

tion test. Begin by applying GLS detrending to the wt and xt series, generating the

detrended series ~wt and ~xt. We next estimate the OLS regression model

~wt ¼ f1~xt þ gt: ð14ÞLet gGLS

t denote the OLS residual from Eq. (14), so that gGLSt ¼ ~wt � f

GLS

1 ~xt, wherefGLS

1 is the OLS estimate of f1 in Eq. (14). The ADFGLS cointegration test usesthe t-statistic corresponding to b0 in

DgGLSt ¼ b0g

GLSt�1 þ

Xkj¼1

bjDgGLSt�j þ etk: ð15Þ

We select the lag length k in Eq. (15) using the MAIC. 14

Ng and Perron (1997) also modify the Za cointegration test in order to generate

the MZGLS

a cointegration test. It uses the statistic

MZGLS

a ¼ ½T�1ðgGLST Þ2 � kAR� 2T�2

XTt¼1

ðgGLSt Þ2

" #�1

; ð16Þ

where kAR ¼ r2k=ð1� bð1ÞÞ2, bð1Þ ¼

Pkj¼1bj, r

2k ¼ ðT � kÞ�1PT

t¼1e2tk, and etk ¼ gGLS

t �b0g

GLSt�1 �

Pkj¼1bjDg

GLSt�j , with b0 and bj obtained via OLS estimation of Eq. (15). We

again use the MAIC to select k. Critical values for the ADFGLS and MZGLS

a cointe-

gration tests are reported in Perron and Rodriguez (2001, Table 2). In Monte Carlo

simulations, Perron and Rodriguez (2001) find that the use of GLS detrending gen-

erates significant power gains relative to conventional residual-based cointegration

tests.

3. Empirical results

3.1. Data

Our quarterly nominal interest rate and inflation rate data are from the June 2000

International Monetary Fund International Financial Statistics (IFS) CD-ROM. We

use the yield on long-term government bonds as the nominal interest rate measure

13 When we allow for a nonzero nominal interest income tax rate, we expect f1 > 1 ðf1 < 1Þ when the

nominal interest rate (inflation rate) serves as the dependent variable in Eq. (11).14 Perron and Rodriguez (2001) use the BIC in their Monte Carlo simulations, as their primary focus in

on the use of GLS detrending. We use the MAIC in an effort to obtain the best size and power possible.


(IFS series 61. . .ZF. . .). We use the bond rate instead of the treasury bill rate, as

bond rate data going back at least to 1970 are available for a substantially larger

number of countries than treasury bill rate data. The 16 countries for which we have

nominal interest rate data are Australia, Austria, Belgium, Canada, Denmark,

France, Germany, Ireland, Italy, Japan, the Netherlands, New Zealand, Norway,Switzerland, the United Kingdom, and the United States. Our inflation data for this

same set of countries are based on consumer price indexes (IFS series 64. . .ZF. . .).With a few exceptions, the start and end dates are 1957:1 and 2000:1, respectively,

for our nominal interest rate and consumer price index data. 15

Note that, unlike Rose (1988), we use long-term nominal government bond rates

instead of nominal treasury bill rates. As mentioned above, we do this because nom-

inal treasury bill rate data going back to at least 1970 are only available for a rela-

tively small number of countries on the IFS CD-ROM. Wallace and Warner (1993)use an expectations model of the term structure of interest rates to show that the

integration properties of short-term bonds will pass-through to long-term bonds. In-

deed, Wallace and Warner (1993) find strong evidence that treasury bill and bond

rates are cointegrated with cointegrating vector (1,�1) 0 in the United States. 16

Given these theoretical and empirical results, Wallace and Warner (1993) argue that

studies examining the long-run relationship between Rt and pt should not be limited

to short-term nominal interest rate data (as is often the case). In light of these

considerations, and because it is widely believed that long-term rates are the mostrelevant for saving and investment decisions, we concentrate on the results for the

long-term nominal government bond rate data. 17

3.2. Unit root tests

Conventional ADF and Za unit root test results for our nominal interest rate and

inflation rate data are reported in Table 1. Rose (1988) finds that the nominal interest

rate is nonstationary for all of the countries he considers. From columns (2) and (3)of Table 1, we see that the nonstationary null hypothesis cannot be rejected for the

nominal interest rate using either the ADF or Za test at conventional significance lev-

els in all of the countries we consider, with two exceptions. The exceptions are Ger-

many, where both the ADF and Za statistics reject the null hypothesis at the 5%

level, and Switzerland, where the null is rejected at the 5% level using the ADF sta-

tistic and the 10% level using the Za statistic. Further test results (not reported to con-

serve space) overwhelmingly indicate that the nominal interest rate first differences

15 The exceptions to the 1957:1 start date are Austria (1970:1) and Japan (1966:4). The exceptions to

the 2000:1 end date are Ireland (1998:4) and Japan (1999:3). All samples are based on data availability.16 For the countries for which treasury bill rate data going back to at least to 1970 are available on the

IFS CD-ROM, we also find that the difference between the treasury bill and treasury bond rates is

stationary. All unreported results are available upon request from the authors.17 The inflation rates in Rose (1988) are based on GDP price deflators, while ours are based on

consumer price indexes. We use consumer price index data, as again data are available for more countries

on the IFS CD-ROM.

Table 1

Nominal interest rate and inflation rate unit root test results

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Nominal interest rate Inflation rate

Country ADFa Zab ADFGLSc MZ

GLS

ad ADFa Za

b ADFGLSc MZGLS

ad

Australia �1.57 �3.03 �1.09 �2.49 �2.03 �36.30** �1.88� �6.84�

Austria �2.20 �5.33 �1.75� �6.87� �1.31 �92.35** �1.37 �2.34

Belgium �1.97 �8.16 �1.40 �4.29 �2.47 �34.48** �1.47 �3.81

Canada �1.72 �4.83 �0.81 �1.44 �2.15 �27.62** �1.62� �4.77

Denmark �1.27 �4.22 �0.97 �2.10 �2.60� �97.86** �0.94 �1.78

France �1.50 �4.89 �1.26 �3.37 �1.61 �55.43** �1.50 �1.70

Germany �3.14* �14.52* �2.89** �17.36** �2.68� �111.90** �2.72** �8.31*

Ireland �1.02 �3.77 �0.83 �1.46 �1.95 �58.73** �1.63� �4.09

Italy �2.04 �5.06 �1.43 �4.16 �1.69 �21.70** �0.89 �1.71

Japan �1.22 �1.23 �1.16 �5.09 �1.82 �52.35** �1.36 �3.67

Netherlands �2.26 �7.37 �1.29 �3.66 �1.79 �130.00** �1.61 �1.49

New Zealand �1.27 �4.69 �0.89 �1.64 �2.64� �37.45** �1.56 �4.75

Norway �1.44 �2.92 �0.91 �1.98 �2.08 �83.74** �2.14* �6.86�

Switzerland �3.27* �13.70� �1.80� �7.24� �3.29* �74.33** �2.50* �9.58*

United Kingdom �1.42 �4.34 �1.04 �2.21 �2.32 �60.93** �1.62� �4.52

United States �1.88 �6.08 �0.89 �1.66 �2.70� �23.70** �2.11* �8.11*

Notes: �, *, ** indicates significance at the 10%, 5%, and 1% levels, respectively.a One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �3.46, �2.91, and �2.59, respectively.b One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �19.95, �13.99, and �11.16, respectively.c One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �2.58, �1.98, and �1.62, respectively.d One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �13.8, �8.1, and �5.7, respectively.


are stationary in each country. Overall, conventional unit root tests indicate that the

nominal interest rate is I(1) for every country we consider, with the exceptions of

Germany and Switzerland. Apart from these two countries, the results for our inter-

est rate data conform to those in Rose (1988).

The results for our inflation rate data are in agreement with Rose (1988) when we

use the Za test, but they begin to diverge when we use the ADF test. From column

(7) of Table 1, we see that the nonstationary null is rejected at the 1% level for every

country when we use the Za statistic. This matches the Rose (1988) finding that theinflation rate is stationary for every country he considers. However, from column (6)

of Table 1, we see that the nonstationary null is only rejected for one country (Swit-

zerland) at the 5% level and four additional countries (Denmark, Germany, New

Zealand, and the United States) at the 10% level when we use the ADF statistic. 18

Overall, if we focus on the Za test results reported in Table 1, our results accord

well with Rose (1988) in that the nominal interest rate appears I(1), while the infla-

18 Unreported results overwhelmingly indicate that the inflation rate first differences are stationary for

each country.


tion rate appears I(0), for a large number of OECD countries. According to the

first part of outcome (ii) above, this indicates a nonstationary real interest rate for

these countries. However, the ADF test results indicate that both the nominal inter-

est rate and inflation rate are nonstationary for a number of countries, so that

outcome (iii) above applies. We next turn to the Ng and Perron (2001) unit root testswith improved size and power properties in an effort to gain a better understanding

of the stationarity properties of international nominal interest rates and inflation

rates.

Ng and Perron (2001) unit root test results are also reported in Table 1. 19 From

columns (4) and (5) of Table 1, we see that the inferences delivered by the ADFGLS

and MZGLS

a tests for the nominal interest rates match those of the conventional tests

for every country, with the exception of Austria, where the two modified tests reject

the unit root null hypothesis. This is likely due to the increased power of the modified

tests. When both of the modified tests reject the unit root null hypothesis, wetake this as evidence that a variable is stationary. According to the Ng and Perron

(1997) unit root test results, the nominal interest rate is stationary for Austria, Ger-

many, and Switzerland and nonstationary for the remaining 13 countries.

The Ng and Perron (1997) unit root test results for the inflation rates are reported

in columns (8) and (9) of Table 1. The most striking feature of the results is the

change in inferences when we use the MZGLS

a test as opposed to the Za test. While

we reject the nonstationary null hypothesis for every country�s inflation rate at the

1% level using the Za test, we only reject the null hypothesis at the 10% or 5% levelfor five countries (Australia, Germany, Norway, Switzerland, United States) using

the MZGLS

a test. It thus appears that many of the rejections of the nonstationary null

hypothesis using the Za test in column (7) of Table 1––and in Rose (1988)––involve a

Type I error. The nonstationary null is rejected for eight countries using the ADFGLS

test. Both Ng and Perron (1997) unit root tests indicate a stationary real interest rate

for Australia, Germany, Norway, Switzerland, and the United States, and we take

this as strong evidence that the inflation rate is stationary for these five countries.

Taking the Ng and Perron (1997) unit root test results for the nominal interestrate and inflation rate together, we reach the following conclusions regarding the real

interest rate. For Australia, Norway, and the United States, we reach the same con-

clusion as Rose (1988): the real interest rate is nonstationary, as Rt � I(1) and

pt � I(0) (the first part of outcome (ii) above). For all of the remaining countries,

our inferences concerning the integration properties of the nominal interest rate

and/or inflation rate differ from Rose (1988). For Austria, like Rose (1988), we con-

clude that the real interest rate is nonstationary—not because Rt � I(1) and pt � I(0),

as in Rose (1988)—but because Rt � I(0) and pt � I(1) (the second part of outcome(ii) above). For Germany and Switzerland, unlike Rose (1988), we conclude that the

real interest rate is stationary, as Rt,pt � I(0). For the remaining ten countries, we

cannot reach a conclusion on the integration properties of the real interest rate based

19 We implemented the Ng and Perron (2001) unit root tests using the GAUSSGAUSS program available from

Serena Ng�s web page at http://www.econ.lsa.umich.edu/~ngse/.

http://www.econ.lsa.umich.edu/~ngse/

Table 2

Nominal interest rate and inflation rate cointegration test results with a pre-specified cointegrating vector

of (1,�1) 0

(1) (2) (3) (4) (5)

Country ADFa Zab ADFGLSc MZ

GLS

ad

Belgium �2.22 �46.00** �1.99* �6.34�

Canada �2.12 �44.46** �1.78� �5.49

Denmark �2.93* �113.98** �1.33 �3.04

France �2.08 �55.43** �2.23* �2.59

Ireland �2.35 �114.19** �1.10 �1.82

Italy �2.42 �41.61** �1.44 �4.60

Japan �2.49 �75.88** �2.06* �6.72�

Netherlands �1.44 �133.85** �0.99 �0.74

New Zealand �1.35 �48.21** �1.11 �2.57

Unted Kingdom �2.98* �119.96** �2.64** �9.31*

Notes: �, *, ** indicates significance at the 10%, 5%, and 1% levels, respectively.a One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �3.46, �2.91, and �2.59, respectively.b One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �19.95, �13.99, and �11.16, respectively.c One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �2.58, �1.98, and �1.62, respectively.d One-sided (lower-tail) test of the null hypothesis that the variable is nonstationary; 1%, 5%, and 10%

critical values equal �13.8, �8.1, and �5.7, respectively.


on the individual unit root test results alone, as Rt,pt � I(1) (outcome (iii) above).

For these ten countries, we need to test for possible cointegration between the nom-

inal interest rate and inflation rate. 20

3.3. Cointegration tests

We first test for cointegration with a pre-specified cointegrating vector of (1,�1) 0

by applying the unit root tests used for the individual series to the linear combinationRt–pt in each of the ten countries. 21 The results are reported in Table 2. From col-

20 The inflation rates for eight countries (Austria, Denmark, Germany, Ireland, Japan, the Netherlands,

Norway, United Kingdom) exhibit significant seasonal variation. When we seasonally adjust the inflation

rates for these countries using seasonal dummy variables, the results reported in columns (8) and (9) of

Table 1 are qualitatively unchanged for Austria, Denmark, Germany, the Netherlands, Norway, and the

United Kingdom. For Ireland, the MZGLS

a statistic becomes significant at the 10% level (the ADFGLS

statistic remains significant). For Japan, the ADFGLS and MZGLS

a statistics both become significant at the

10% level. Using seasonally adjusted inflation rates, we thus obtain the Rose (1988) result that Rt � I(1)

and pt � I(0) for Ireland and Japan, so that the real interest rate is nonstationary for these two countries.21 In our cointegration tests, we follow the convention in the literature by lining up the nominal interest

rate in a given quarter with the inflation rate figure for the immediately following quarter. For example, we

line up the nominal interest rate for the first quarter in a given year with the annualized growth rate of the

consumer price index from the first to the second quarter in the same year.


umn (2) of Table 2, we see that the nonstationary null hypothesis is only rejected for

Denmark and the United Kingdom (at the 5% level) when we use the conventional

ADF test. When we use the conventional Za test in column (3), we reject the nonsta-

tionary null hypothesis for all ten countries. We again apply the Ng and Perron

(2001) unit root tests, so that we have tests with good size and power, and the resultsare reported in columns (4) and (5) of Table 2. We have five rejections of the null

hypothesis at the 10% level or lower when we use the ADFGLS test (Belgium, Can-

ada, France, Japan, United Kingdom) and three rejections using theMZGLS

a test (Bel-

gium, Japan, United Kingdom). As with the results in Table 1, it appears that many

of the rejections in column (3) using the Za test involve a Type I error, as many of the

rejections are overturned when we use the MZGLS

a test. For Belgium, Japan, and the

United Kingdom, both Ng and Perron (2001) tests indicate that Rt–pt � I(0), and

this can be interpreted as strong evidence that the real interest rate is stationary inthese countries. For the other seven countries in Table 2, we do not find robust evi-

dence of a stationary real interest rate. 22

We next test for cointegration between Rt and pt with an unspecified cointegrating

vector, in order to allow for a nonzero nominal interest income tax rate. 23 Engle and

Granger (1987) and Phillips and Ouliaris (1990) OLS residual-based cointegration

test results are reported in columns (3) and (4) of Table 3, where the nominal interest

rate serves as the regressand in Eq. (11), and columns (3) and (4) of Table 4, where

the inflation rate serves as the regressand in Eq. (11). From column (3) of Table 3, wesee that we cannot reject the null hypothesis of no cointegration for any country

using the ADF cointegration test when the nominal interest rate serves as the

dependent variable; from column (3) of Table 4, we only reject the null hypothesis

for Denmark using the ADF test when the inflation rate serves as the dependent var-

iable. There is thus little evidence of cointegration according to the conventional

Engle and Granger (1987) ADF test. We reject the null hypothesis for four countries

(Belgium, Denmark, Italy, United Kingdom) using the Za test with the nominal

interest rate serving as the regressand in Eq. (11) (see column (4) of Table 3); we re-ject the null hypothesis for every country using the Za test when the inflation rate

serves as the regressand (see column (4) of Table 4).

Again in an effort to use tests with good size and power, we also report results for

the Perron and Rodriguez (2001) ADFGLS and MZGLS

a cointegration tests in Tables 3

and 4. There is little evidence of cointegration according to the Perron and Rodriguez

(2001) tests. For the ADFGLS test, the null hypothesis of no cointegration is only re-

jected for Italy when the nominal interest rate serves as the regressand (see column (6)

22 When we seasonally adjust the inflation rates for Denmark, the Netherlands, and the United

Kingdom using seasonal dummy variables (see footnote 20 above), the results in columns (4) and (5) of

Table 2 are qualitatively unchanged. (Recall from footnote 20 above that we conclude on the basis of the

individual unit root tests for Rt and pt alone that the real interest rate is nonstationary for Ireland and

Japan when we use seasonally adjusted inflation rates.)23 Summers (1983) argues that the increase in the nominal interest rate needs to be 1.3–1.5 times as

large as any increase in inflation in industrialized countries in order to maintain a constant real interest

rate.

Table 3

Nominal interest rate and inflation rate cointegration test results (dependent variable: nominal interest

rate)

(1) (2) (3) (4) (5) (6) (7)

Country f1 ADFa Zab f

GLS

1 ADFGLSc MZGLS

ad

Belgium 0.34 �1.92 �16.55� 0.50 �1.91 �7.90

Canada 0.42 �2.12 �15.08 0.69 �1.34 �3.31

Denmark 0.41 �1.27 �21.73* 0.44 �1.35 �3.62

France 0.31 �1.78 �14.14 0.33 �1.72 �4.51

Ireland 0.37 �1.47 �35.68 0.18 �0.61 �0.80

Italy 0.43 �2.24 �17.25� 0.40 �2.61� �13.29�

Japan 0.21 �1.05 �11.07 0.21 �1.09 �4.34

Netherlands 0.11 �2.07 �10.25 0.05 �0.97 �2.08

New Zealand 0.28 �1.16 �11.91 0.33 �0.73 �1.25

United Kingdom 0.33 �1.54 �43.38** 0.48 �1.34 �2.97

Notes: �, *, ** indicates significance at the 10%, 5%, and 1% levels, respectively.a One-sided (lower-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �4.02, �3.40, and �3.09, respectively.b One-sided (lower-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �25.77, �19.19, and �16.19, respectively.c One-sided (upper-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �3.33, �2.76, and �2.47, respectively.d One-sided (upper-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �22.84, �15.84, and �12.80, respectively.

Table 4

Nominal interest rate and inflation rate cointegration test results (dependent variable: inflation rate)

(1) (2) (3) (4) (5) (6) (7)

Country f1 ADFa Zab f

GLS

1 ADFGLSc MZGLS

ad

Belgium 0.62 �2.28 �49.18** 0.80 �1.96 �6.05

Canada 0.68 �2.22 �47.44** 0.57 �2.12 �7.47

Denmark 0.58 �3.47* �135.73** 1.23 �1.46 �3.69

France 0.72 �1.94 �58.57** 1.41 �2.36 �3.31

Ireland 1.14 �2.39 �115.18** 0.36 �1.52 �3.25

Italy 0.88 �2.36 �41.62** 1.39 �1.53 �5.07

Japan 1.18 �2.55 �76.76** 1.29 �2.17 �7.54

Netherlands 0.57 �1.53 �140.89** 0.11 �1.57 �1.38

New Zealand 0.68 �1.44 �50.36** 0.53 �1.45 �4.01

United Kingdom 1.32 �3.00 �125.37** 1.08 �2.63� �9.10

Notes: �, *, ** indicates significance at the 10%, 5%, and 1% levels, respectively.a One-sided (lower-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �4.02, �3.40, and �3.09, respectively.b One-sided (lower-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �25.77, �19.19, and �16.19, respectively.c One-sided (upper-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �3.33, �2.76, and �2.47, respectively.d One-sided (upper-tail) test of the null hypothesis that the variables are not cointegrated; 1%, 5%, and

10% critical values equal �22.84, �15.84, and �12.80, respectively.



of Table 3) and for the United Kingdom when the inflation rate serves as the regres-

sand (see column (6) of Table 4). Using the MZGLS

a test, there is only one rejection

(Italy) when the nominal interest rate serves as the regressand in Eq. (11) (see column

(7) of Table 3) and no rejections when the inflation rate serves as the regressand (see

column (7) of Table 4). Following the pattern in Table 2, the contrast between the re-sults in columns (4) and (7) of Table 4 suggests that the rejections in column (4) using

the Za cointegration test entail a Type I error. We only have one case where both the

ADFGLS andMZGLS

a tests indicate cointegration: Italy when the nominal interest rate

serves as the regressand in Eq. (14). Even in this case, the point estimate of f1 in Eq.

(14) is 0.40, and, if anything, we expect this coefficient to be greater than unity with a

nonzero nominal interest income tax rate. 24 When we use the Perron and Rodriguez

(2001) tests, there is no robust evidence of a plausible cointegrating relationship be-

tween the nominal interest rate and inflation rate for any country. 25

3.4. Discussion

Up to this point, we have focused on how our results compare to those in Rose

(1988). Next, we briefly compare our results to those reported in some other well-

known studies. The earliest test of a cointegrating relationship between nominal

interest rates and inflation rates appears to be MacDonald and Murphy (1989).

Using quarterly consumer price index and three-month treasury bill rate data forBelgium, Canada, the United Kingdom, and the United States covering 1955:1–

1986:4, MacDonald and Murphy (1989) test for cointegration between the nominal

interest rate and inflation rate using the Engle and Granger (1987) procedure. With

either the nominal interest rate or the inflation rate serving as the dependent variable

in the first-step OLS regression, the null hypothesis of no cointegration cannot be

rejected for any country at conventional significance levels. 26 These results match

24 Note that results in Ng and Perron (1997) indicate that test results with the inflation rate serving as

the regressand in Eq. (11) or (14) are likely to be more reliable than those with the nominal interest rate

serving as the regressand. When we seasonally adjust the inflation rates for Denmark and the Netherlands

using seasonal dummy variables (see footnote 20 above), the results in columns (6) and (7) of Tables 3 and

4 are qualitatively unchanged. For the United Kingdom, the results in columns (6) and (7) of Table 3 and

column (6) of Table 4 are qualitatively unchanged, while the MZGLS

a statistic becomes significant at the

10% level when the inflation rate serves as the dependent variable in Eq. (14). (Recall from footnote 20

above that we conclude on the basis of the individual unit root tests for Rt and pt alone that the real

interest rate is nonstationary for Ireland and Japan when we use seasonally adjusted inflation rates.)25 We also tested for cointegration using the system-based tests of Stock and Watson (1988) and

Johansen (1991). There is no evidence of cointegration for any country using the Stock and Watson (1988)

test. This is somewhat surprising, given the Monte Carlo simulation results in Haug (1996), which indicate

that the Stock and Watson (1988) test is among the most powerful of the cointegration tests he considers.

The null hypothesis of no cointegration is rejected for Canada, France, Ireland, and the United Kingdom

using the Johansen (1991) test. It is not clear how these tests perform in the presence of a significant

moving-average component in the inflation rate.26 There is some evidence of cointegration for Canada, the United Kingdom, and the United States

when a 1955–1972/1973 subsample is used; there is no evidence of cointegration for any country when a

1972/1973–1986 subsample is used.


ours in Table 3 for Belgium, Canada, and the United Kingdom. For the United

States, the Ng and Perron (2001) unit root test results indicate that the inflation rate

is stationary for our data, so it is inappropriate to test for cointegration between the

nominal interest rate and inflation rate. Our results using the Perron and Rodriguez

(2001) cointegration tests show that the results in MacDonald and Murphy (1989)for Belgium, Canada, and the United Kingdom are robust to cointegration tests with

good size and power.

Mishkin (1992) tests for a cointegrating relationship between the nominal interest

rate and inflation rate in the United States using monthly consumer price index and

treasury bill rate data from 1953:1 to 1990:12. With the inflation rate serving as the

dependent variable in the first-step OLS regression, the null hypothesis of no cointe-

gration is rejected using either the Engle and Granger (1987) ADF or Phillips and

Ouliaris (1990) Za tests.27 Using quarterly consumer price index and 3-month treas-

ury bill or 10-year government bond rate data, Wallace and Warner (1993) find evi-

dence of cointegration between the nominal interest rate and inflation rate for the

United States using the Johansen (1991) test for samples covering 1948:1–1990:4

and 1953:1–1990:4. Normalizing on the nominal interest rate, their f1 estimates ap-

pear reasonable, ranging from 1.20 to 1.69. Using implicit price deflator for total

consumption expenditures and 3-month treasury bill rate data from 1953:1 to

1991:4 and the Johansen (1991) test, Crowder and Hoffman (1996) also find evidence

of cointegration between the nominal interest rate and inflation rate in the UnitedStates. Normalizing on the nominal interest rate, their f1 estimate is 1.22. Crowder

and Wohar (1999) estimate a cointegrating relationship between either the treasury

bill rate or the municipal bond yield and the inflation rate for the United States.

They obtain estimates of f1 that are consistent with a long-run Fisher effect using

either nominal interest rate or the inflation rate as the dependent variable in Eq.

(11). These four studies thus indicate a stationary real interest rate for the United

States. However, as with MacDonald and Murphy (1989), all four of these studies

require a nonstationary United States inflation rate, while the Ng and Perron(2001) unit root test results in Table 1 indicate that the United States inflation rate

is stationary over our sample. With the United States nominal interest rate and infla-

tion rate apparently being I(1) and I(0), respectively, according to unit root tests with

good size and power, we obtain the Rose (1988) finding that the real interest rate is

nonstationary for the United States. 28

Using quarterly data from 1962:1 to 1993:1, Engsted (1995) investigates the

spread between the long-term bond rate and one-period inflation rate as a predictor

27 The evidence for cointegration is mixed for various subsamples.28 Evans and Lewis (1995) find that estimates of f1 in Eq. (11), with the nominal interest rate serving as

the dependent variable, are likely to be biased downward in small samples—and thus fail to signal a long-

run Fisher effect—when changes in inflation are generated by a two-state Markov-switching process.

Using data from the United States, they are unable to reject the hypothesis that nominal interest rates and

expected inflation move together one-for-one in the long run, supporting a stationary real interest rate for

the United States. Again, they model inflation as an I(1) process, while the Ng and Perron (2001) unit root

test results in Table 1 indicate that the United States inflation rate is stationary over our sample.


of future inflation for 13 OECD countries (Canada, United States, Japan, Australia,

Belgium, Denmark, France, Germany, Ireland, Italy, Sweden, Switzerland, United

Kingdom). As part of his analysis, Engsted (1995) tests for unit roots in nominal

long-term bond rates and inflation rates using the ADF test. He finds that Rt � I(1)

for all of the countries he considers, with the exceptions of Germany and Switzer-land, where Rt � I(0). This matches our results reported in columns (4) and (5) of

Table 1 using the Ng and Perron (2001) tests for all of the countries common to both

studies. Engsted (1995) also finds that pt � I(1) for Canada, the United States, Aus-

tralia, Belgium, France, Germany, Ireland, and Italy, while he finds that pt � I(0) for

Japan, Denmark, Sweden, Switzerland, and the United Kingdom. Comparing these

results to those in columns (8) and (9) of Table 1 using the Ng and Perron (2001)

tests, the results match for Belgium, Canada, France, Ireland, Italy and Switzerland.

However, for Japan, Denmark, and the United Kingdom, the Ng and Perron (2001)test results in Table 1 suggest that pt � I(1) instead of I(0), while for Australia, Ger-

many, and the United States, the Ng and Perron (2001) tests suggest that pt � I(0)

instead of I(1). Overall, the Ng and Perron (2001) tests affect inferences concerning

the inflation rate for a number of countries when we compare the results in the pre-

sent paper to those in Engsted (1995). 29

Finally, some researchers have analyzed real interest rates under regime shifts.

Using quarterly data for the United States spanning 1961–1986, Garcia and Perron

(1996) find evidence of two structural breaks in the mean of Rt–pt in 1972 and 1980.They also estimate a three-state Markov-switching model, and their results suggest

that the ex post real interest rate for the United States experiences mean shifts

around 1973 and 1980, close to the dates indicted by their tests for structural breaks.

Using the methodology of Bai and Perron (1998), Bai and Perron (2003) find evi-

dence of three structural breaks in the mean of the United States ex post real interest

rate using the Garcia and Perron (1996) data, while Caporale and Grier (2000) find

evidence of four structural breaks in the mean of the United States ex post real inter-

est rate when they extend the Garcia and Perron (1996) data through 1992. Alsousing the Bai and Perron (1998) methodology, Rapach and Wohar (2003) find evi-

dence of multiple structural breaks in ex post real interest rates for a large number

of OECD countries over the postwar period. These studies suggest that real interest

rates can be viewed as stationary processes around a shifting mean. While the present

paper is primarily concerned with testing for I(1) vs. I(0) behavior in international

real interest rates, in the spirit of Rose (1988), the ‘‘mean shifts’’ studies cited above

also detect long-lived changes in the level of real interest rates. Whether we model

shocks to the level of the real interest rates in an I(1) framework or by allowingfor occasional shifts in the mean of an otherwise stationary process, long-lived

29 Engsted (1995) also tests for cointegration between the nominal interest rate and the inflation rate for

each country with a pre-specified cointegrating vector of (1,�0.97)0 in order to account for coupon

payments on long-term bonds (see footnote 11 above). The ADF tests in Engsted (1995) indicate that Rt–

0.97pt � I(0) for Canada, Japan, France, Germany, Sweden, Switzerland, and the United Kingdom,

although this is inconsistent with Rt and pt being integrated of different orders according to the individual

unit root tests in Engsted (1995) for Japan, Germany, and the United Kingdom.


changes in the level of the real interest rate pose the same types of problems discussed

in Section 1 above for a number of prominent theoretical models. For example, con-

sumption growth rates do not exhibit the structural breaks evident in real interest

rates, so that the Euler equation at the heart of the canonical consumption-based as-

set-pricing model cannot hold in the presence of mean shifts in the real interestrate. 30

4. Conclusion

In this paper, we re-examine the controversial finding of Rose (1988) that interna-

tional real interest rates are nonstationary by updating the Rose (1988) study in two

important ways: (i) we use an extended sample, due to the passage of over a decadesince the original Rose (1988) study; (ii) we use a set of state-of-the-art unit root tests

due to Ng and Perron (2001) (obviously not available to Rose, 1988) with better size

and power than conventional tests. For three of the 16 countries we consider—Aus-

tralia, Norway, and the United States—our unit root test results match those of

Rose (1988) in that the nominal interest rate is I(1), while the inflation rate is I(0).

For the other 13 countries, our unit root test results for the nominal interest rate

and/or inflation rate differ from those in Rose (1988). For two countries—Germany

and Switzerland—we find evidence that the nominal interest rate and inflation rateare both I(0), indicating a stationary real interest rate in these two countries. For

one country—Austria—the Ng and Perron (2001) unit root tests indicate that the

nominal interest rate is I(0), while the inflation rate is I(1). This points to a nonsta-

tionary real interest rate for Austria, but for different reasons that in Rose (1988).

For the other ten countries we consider, the Ng and Perron (2001) unit root test re-

sults indicate that the nominal interest rate and inflation rate are both I(1). For these

ten countries, the real interest rate could be stationary if the nominal interest rate

and inflation rate are cointegrated. We find robust evidence of cointegration betweenthe nominal interest rate and inflation rate for Belgium, Japan, and the United King-

dom when we impose a cointegrating vector of (1,�1) 0. When we leave the cointe-

grating vector unspecified, we do not obtain robust evidence of cointegration for

any of the ten relevant countries using the Perron and Rodriguez (2001) cointegra-

tion tests with good size and power. Overall, our results concerning the integration

properties of international real interest rates are more mixed than Rose (1988). Nev-

ertheless, our results frequently support the Rose (1988) finding that the real interest

rate is nonstationary in many countries, albeit often for different reasons than Rose(1988). As discussed in Rose (1988) and Section 1 of the present paper, nonstationary

30 Using the Johansen (1991) procedure, Crowder (1997) finds evidence of a cointegrating relationship

between Rt and pt for Canada over the 1960:1–1991:4 period consistent with a nonzero nominal interest

income tax rate. However, in order for the cointegrating relationship to be stable, two dummy variables

accounting for regime shifts in 1971:2 and 1982:2 need to be included in the vector error-correction

specification. These dummy variables are tantamount to allowing for mean shifts in the Canadian real

interest rate in 1971:2 and 1982:2.


real interest rate behavior poses important problems for some prominent theoretical

models.

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are real interest rates really nonstationary? new evidence from tests with good size and power

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