Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 953, October 2008  Screen Reader
Version*
NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.
Abstract:
Nominal interest rates are unlikely to be generated by unitroot processes. Using data on short and long interest rates from eight developed and six emerging economies, we test the expectations hypothesis using cointegration methods under the assumption that interest rates are near integrated. If the null hypothesis of no cointegration is rejected, we then test whether the estimated cointegrating vector is consistent with that suggested by the expectations hypothesis. The results show support for cointegration in ten of the fourteen countries we consider, and the cointegrating vector is similar across countries. However, the parameters differ from those suggested by theory. We relate our findings to existing literature on the failure of the expectations hypothesis and to the role of term premia.
Keywords: Bonferroni tests, cointegration, expectations hypothesis, near integration, term premium
JEL classification: C22, G12
Empirical tests of the expectations hypothesis of the term structure often fail to find support for the theory. The logic underlying the theory, that expectations of future short interest rates shape the term structure of longer interest rates, is intuitive, appealing, and a common assumption in macroeconomic modelling. However, the predictability of excess returns shown by Fama and Bliss (1987), Campbell and Shiller (1991) and more recently by Cochrane and Piazzesi (2005) undermines the premise that long interest rates are rational expectations of future short rates up to a constant term premium. Rather, such evidence points strongly toward timevarying risk premia. Indeed, Dai and Singleton (2002) demonstrate that interest rates adjusted for timevarying risk premia estimated from dynamic term structure models meet the predictions of the expectations hypothesis in traditional excessreturn regressions.
One strand of the empirical literature on interest rates has sought to test the expectations hypothesis using the techniques of cointegration. As pointed out by Engle and Granger (1987) in their seminal paper on cointegration, if nominal interest rates are generated by a unitroot process, cointegration between yields of different maturities is a necessary condition for the validity of the expectations hypothesis. Intuitively, if interest rates are integrated of order one, the expectations hypothesis implies that the spread between any pair of yields is stationary. Following Engle and Granger's early work, several studies have taken a similar path and have found only mixed evidence for the expectations hypothesis; see, for example, Campbell and Shiller (1987), Boothe (1991), Hall et al. (1992), Zhang (1993) and Lardic and Mignon (2004).
It is an empirical fact that nominal interest rates are highly persistent and the poor power of traditional univariate DickyFuller type tests against the null of a unit root (Stock, 1994) has led many researchers to conclude that interest rates are integrated of order one. Moreover, the convenience of working with established results for integrated processes has made it attractive to assume the presence of a unit root for empirical purposes. As such, nominal interest rates have been treated as integrated of order one in numerous empirical papers, including Karfakis and Moschos (1990), Bremnes et al. (2001), Chong et al. (2006), Kleimeier and Sander (2006), De Graeve et al. (2007) and Liu et al. (2008). For theoretical purposes, the unitroot assumption can also be a useful modelling device to capture stylistically the highly persistent nature of interest rates in finite samples, exemplified by Cogley and Sargent's (2001) approach to modelling the dynamics of a system of macroeconomic variables.
However, the exact unitroot assumption for nominal interest rates can be questioned on both empirical and theoretical grounds. Empirically, standard tests for unit roots have difficulty discriminating true integration from highlypersistent dynamics but panel unitroot tests  which tend to be more powerful than univariate tests  tend to find support for mean reversion in nominal interest rates (Wu and Chen, 2001). Theoretically, it might be unsatisfactory to model interest rates as unbounded in the limit.^{1} More importantly though, most economic models predict that real interest rates possess a longrun equilibrium value, determined by the longrun rate of potential output growth and population growth, and agents' rate of time preference and risk aversion. Similarly, consumption growth is typically viewed as stationary  albeit slowly meanreverting (Bansal and Yaron, 2004). The standard consumption Euler equation thus implies stationary real interest rates. Nominal interest rates have varied substantially in recent decades, partly reflecting the undulations of inflation expectations, but over the longrun, have wandered within reasonable bounds. Indeed, from an historical perspective, shortterm nominal interest rates were in the range of four to eight percent during the years of the Roman Empire, and western European commercial and mortgage borrowing rates moved in the fourtoeight percent range from the 13th to 17th century (Homer and Sylla, 1996). Private nominal interest rates are in a similar range today, an outcome that would be virtually impossible if interest rates possessed a unit root but consistent with a highlypersistent data generating process.
Unfortunately, standard cointegrationbased inference designed for unitroot data is typically not robust to even small deviations from the unitroot assumption. Results will generally be biased when the autoregressive roots in the data are close, but not identical, to unity; this is true both for actual tests of the cointegrating rank (Hjalmarsson and Österholm, 2007a,b) as well as for inference on the cointegrating vector (Elliott, 1998). Conclusions of empirical tests of the expectations hypothesis that rely on traditional cointegration are therefore called into question. As such, empirical analysis within a framework that acknowledges the high persistence in nominal interest rates but that does not impose the strict assumption of a unit root is desirable but very few examples exist in the literature. To our knowledge, exceptions include one study of the Fisher effect (Lanne, 2001) and one on the term structure of interest rates (Lanne, 2000).
In this paper, we revisit the question of cointegration between yields of different maturities using methods that allow for valid inference when data are near integrated. That is, we assume that interest rates are highly persistent with autoregressive roots that are close to unity. The empirical framework nests the standard unitroot assumption used in the traditional cointegration studies mentioned above, but also permits interest rates to be (slowly) mean reverting. We test for cointegration using the recently developed methods of Hjalmarsson and Österholm (2007a) that are robust to deviations from the pure unitroot assumption and apply the tests to monthly data of the term structure of interest rates in several developed and emerging economies, namely Australia, Canada, Hungary, India, Japan, Mexico, New Zealand, Poland, Singapore, South Africa, Sweden, Switzerland, the United Kingdom and the United States. The results provide strong support for cointegration between long and short interest rates in ten of these countries. Among the developed countries, results show support for cointegration in Australia, Canada, New Zealand, Sweden, Switzerland and the United States. Among the emerging economies, a similar result is returned for Hungary, Mexico, Poland and Singapore.
Cointegration is only one of two necessary conditions for the validity of the expectations hypothesis; the theory also contains strong predictions about the parameters of the cointegrating vector. However, earlier work has largely overlooked the interpretation of the parameters of the cointegrating vector. We test whether the parameters are consistent with the theoreticallysuggested values using an extension of fullymodified estimation that is again robust to deviations from the pure unitroot assumption. Fully modified estimation of cointegrated systems was initially developed by Phillips and Hansen (1990) and discussed by Hjalmarsson (2007) in the context of predictive regressions with near unitroot variables; in the current paper, we extend those ideas further to accommodate inference in a general bivariate cointegrating relationship with nearly integrated variables.^{2} For the ten countries in which cointegration is detected, the cointegrating vector does not coincide with that suggested by theory. In each case, long rates move by less than short rates, with the similarity of the estimated vectors across countries hinting at a common explanation. Indeed, the estimated cointegrating vectors suggest that another nearintegrated variable that covaries inversely with the short rate also affect the term structure of longer interest rates. This is consistent with timevarying bond risk premia put forward by Campbell and Shiller (1991) and Dai and Singleton (2002) to explain the puzzling patterns of coefficients from yieldspread regressions, and with the countercyclical pattern of excess returns noted by Fama and Bliss (1987) and Cochrane and Piazzesi (2005). We conjecture that the phenomenon underlying the rejection of the expectations hypothesis in excessreturn regressions may also be responsible for the failure of the expectations hypothesis in cointegration methods.
The remainder of the paper is organized as follows. Section 2 presents the theoretical framework regarding the term structure of interest rates and the econometric methodology. In Section 3, the empirical analysis is conducted and the results discussed and Section 4 concludes. Some sensitivity analysis is presented in the Appendix.
This section presents the theoretical motivation for our empirical tests. We briefly lay out the expectations hypothesis of the term structure then move on to the econometric methodology of testing for cointegration between nearintegrated processes.
We begin with a statement of the expectations hypothesis of the term structure similar to that found in Campbell and Shiller (1991):
(1) 
Simply put, the expectations hypothesis posits that the interest rate on a longerterm period bond, , is equal to the average of the expected path of future oneperiod interest rates over the life of the bond, , plus a constant term premium, , that may differ across maturities, . Expectations of future short interest rates are assumed to be rational.
Subtracting from both sides of equation (1) and rearranging terms, the expectations hypothesis states that the yield spread between an period bond and the current oneperiod rate can be viewed as the weighted average of expected changes in short interest rates over the life of the bond:
(2) 
Paired with the assumption that oneperiod nominal interest rates are integrated or near integrated, the sum of first differences on the right hand side of equation (2) must be stationary. This in turn implies that any period yield should be cointegrated with the oneperiod interest rate and that the cointegrating vector will be (1, 1). Note that the term premium in equation (2) has been generalised to include a time subscript, . This has been done deliberately, acknowledging that testing for cointegration between the lefthandside variables cannot discriminate between theories of the term structure in which term premia are constant versus theories in which term premia are timevarying but stationary, as noted by Miron (1991) and Lanne (2000). As such, evidence for cointegration does not speak to the strict form of the expectations hypothesis shown in equation (1) but does substantially narrow the class of models that fit the data. For example, it rules out market segmentation and preferred habitat models of the term structure.
Cointegration alone is insufficient to conclude that the expectations hypothesis fits the data. If the cointegrating vector differs from the theoretically suggested value of (1, 1), the requirements of the theory have not been met.^{3} However, this has quite different implications from an outright rejection of cointegration in the data. A vector that differs from (1, 1) is compatible, for example, with the presence of additional nearintegrated processes driving the term structure, which systematically covary with the spread or the short interest rate.
The main starting point of our empirical analysis is the assumption that both the long and short interest rates follow near unitroot processes with localtounity roots and , respectively, where is the sample size; there is no requirement that . Thus, if , it is assumed that
(3) 
where is a matrix with , and . The innovations are assumed to satisfy a general linear, or infinite moving average, process. Although not necessary for the formal econometric procedures that are used, we will maintain the assumption throughout the paper that , which rules out explosive processes. Since we are analysing interest rates, this is clearly a weak assumption.
This specification of the datagenerating process for the interest rates at different horizons is a generalisation of a pure unitroot model where interest rates at all maturities contain a unit root. Compared to a pure unitroot process, equation (3) offers considerably more flexibility. First, the model allows for deviations from the strict unitroot assumption while still preserving, also asymptotically, the empirically observed feature that interest rates are highly persistent. Second, the model above also allows for different degrees of persistence in interest rates with different maturities. Although it is natural to believe that interest rates at different horizons share the same persistence if the expectations hypothesis is true, this does not necessarily hold under an alternative hypothesis.
As shown in the empirical analysis below, the near unitroot model suggested here finds more support in the data than the pure unitroot model. For several series, the null of a unit root can be rejected and unbiased estimates of and are typically negative, suggesting some mean reversion in the data.
While the concept of cointegration readily carries over to variables with near unit roots, as discussed in Phillips (1998), traditional cointegration tests based on a unitroot assumption are biased (Hjalmarsson and Österholm, 2007a,b). Furthermore, if the presence of cointegration between near unitroot variables is established, subsequent tests on the cointegrating vector will be biased if performed under the assumption of unit roots in the data (Elliott, 1998). Moreover, even if the null of a unit root in the variables cannot be rejected, there is no guarantee that the data contain an exact unit root, as unitroot tests have low power against alternatives close to a unit root. Thus, unless there are very strong a priori reasons to believe that there is an exact unit root in the data, based on some theoretical argument for instance, any cointegration test based on the unitroot assumption is subject to a potential bias.
In the next two sections, we outline methods for dealing with an unknown local deviation from the pure unitroot assumption, both for actual tests of cointegration as well as for inference on the cointegrating vector.
We focus on residualbased tests of cointegration and rely on the methods proposed by Hjalmarsson and Österholm (2007a) to deal with the issues raised by near unitroot variables. In effect, a residualbased cointegration test evaluates whether the residuals from the empirical regression contain a unit root. However, if the original data are in fact near integrated, with a root less than unity, the test will overreject as the residuals will not contain a unit root even if there is no cointegration.
The idea behind Hjalmarsson and Österholm's test is therefore to replace the critical values of the test under the unitroot assumption with critical values based on a conservative estimate of the localtounity root in the original data. Intuitively, if one views a residualbased test of cointegration as a test of whether there is less persistence in the residuals than in the original data, this test is only valid if the persistence of the original data is not overstated.
In particular, we will use a modified version of the traditional EngleGranger (EG) test for cointegration. Consider the cointegrating regression
(4) 
where the fitted residuals are tested for a unit root according to
(5) 
The EG test is defined as the statistic on from equation (5).
As shown by Hjalmarsson and Österholm (2007a), under the near unitroot assumption, the limiting distribution of the EG test statistic depends on the unknown matrix of localtounity parameters , and the critical values of the EG test are thus unknown in the near unitroot case. In order to obtain a practically feasible procedure, Hjalmarsson and Österholm first show that the relevant critical values of this limiting distribution are primarily a function of , the persistence of the `` dependent'' variable, and are almost invariant to the value of , the persistence of the regressor variable. They therefore suggest using critical values based on , rather than , which greatly simplifies the feasible implementation.
Furthermore, the critical values for the EG testare increasing in . Therefore, in order to form a correctly sized test, all that is needed is a `` sufficiently'' conservative estimate of . As shown by Stock (1991), a confidence interval for can be obtained by inverting a unitroot teststatistic for ; since only the lower bound matters here, a onesided confidence interval is appropriate. According to Bonferroni's inequality, if the confidence level of this onesided confidence interval is 95 percent and the nominal size of the EG test, evaluated using the critical values based on the lower bound for , is five percent, then the actual size of the cointegration test is less than or equal to ten percent.
As shown by Hjalmarsson and Österholm, however, Bonferroni's inequality tends to be strict and the actual size of the test procedure just described is, in fact, very close to zero. Instead, they find that if the EG test is evaluated at the five percent level and the critical values are based on the lower bound of a onesided confidence interval for with confidence level of percent, the overall test of cointegration will have an actual size of approximately five percent. Thus, the median unbiased estimate of can be used to form the critical values.
In summary, the following procedure to test the null of no cointegration for nearintegrated variables will thus be used in the analysis here:
We will refer to the test of cointegration constructed in the above manner as the Bonferroni EG test.
If one does establish that variables are cointegrated, it is often of interest to perform inference on the cointegrating vector. In the current application, we are interested in whether the cointegrating vector is (1, 1), since this is an additional condition for the expectations hypothesis to hold once cointegration between the short and long interest rates has been established.
In the case with unitroot regressors, inference on the cointegrating vector is typically performed using standard tests based on the estimates from some efficient estimation procedure of the cointegrating vector, such as the dynamic OLS of Saikkonen (1991) and Stock and Watson (1993) or the fully modified OLS (FMOLS) of Phillips and Hansen (1990). Using these efficient estimation methods, which are asymptotically equivalent, the resulting test statistics have standard distributions. However, as shown by Elliott (1998), this is no longer true when the data are nearly integrated, and tests based on the unitroot assumption can be highly misleading.
In this paper, we therefore apply an extension of the FMOLS procedure for nearly integrated regressors. The idea was developed by Hjalmarsson (2007) for predictive regressions with nearly integrated regressors. Given the predictive nature of that model, the FMOLS estimator needs to be slightly modified to fit the standard (contemporaneous) cointegration regression studied here. The following results are derived under the assumption of cointegration, such that equation (4) represents a true relationship, with a stationary error term .
To fix notation, let be the innovations to the short interest rate variable; that is,
(6) 
where . The model is now given by equations (4) and (6). Denote the joint innovations and suppose that satisfy a functional law such that , where denotes a twodimensional Brownian motion with variancecovariance matrix ; that is, is the longrun twosided variancecovariance matrix for . Further, let and be the onesided longrun covariance and variance, respectively.
In general, the OLS estimator of in equation (4) is not efficient, and the resulting test statistics have nonstandard asymptotic distributions whenever there is a nonzero correlation between and . In the pure unitroot case, Phillips and Hansen (1990) therefore suggest that the OLS estimator be `` fully modified''. As shown in Hjalmarsson (2007), in the near unitroot case, a similar method can also be considered. However, the modification makes use of the innovations , which can only be obtained by knowledge of , which is unknown.^{4} We therefore first derive the estimator under the assumption that is known, and then discuss feasible methods to deal with an unknown . Define
(7) 
and let and where and are consistent estimates of the respective parameters and and denote the demeaned data. The fully modified OLS estimator is given by
(8) 
Define and as shown in Hjalmarsson (2007), as ,
(9) 
where , and denotes a mixed normal distribution. The mixed normality implies that the asymptotic distribution is normal with a random variance. For practical purposes, the main implication is that corresponding tests will have asymptotically standard distributions; in particular, the corresponding statistic is normally distributed and confidence intervals for can be constructed in a standard manner. The parameters and can be consistently estimated from the residuals of first stage OLS regressions, using standard longrun covariance estimation methods such as those of Newey and West (1987).^{5} In fact, for a given , once the innovations are obtained, the FMOLS procedure for near unitroot variables is identical to the standard one used in the unitroot case.
For a given value of , a confidence interval for at the confidence level is given by , where
(10) 
(11) 
and denotes the quantile of the standard normal distribution. Here is explicitly written as a function of to facilitate the discussion below when is unknown. The estimate of is simply given by .
In order to get around the issue of an unknown parameter, we rely on similar methods to those used for the cointegration test. That is, given a confidence interval for , we can calculate for all values in this confidence interval. It is easy to show that will be a monotone function of and it is sufficient to calculate the values at the endpoints of the interval. In fact, if the longrun correlation between and is negative (positive), then will be decreasing (increasing) in . Suppose the confidence level of the lower bound of is such that and that of the upper bound such that , with . If , a robust confidence interval for , with a confidence level of at least is then given by
(12) 
and if by
(13) 
Thus, if and , a percent confidence interval for is obtained. As in the case of the cointegration test in the previous section, the actual confidence level may be higher, and for a given , and can be chosen to achieve a desired actual confidence level. Campbell and Yogo (2006) discuss in detail the methods for adjusting and . In fact, as shown by Hjalmarsson (2007), the methods in Campbell and Yogo (2006) can be interpreted as a special case of the FMOLS framework adapted to processes, with the same asymptotic properties under this more restrictive assumption. We can therefore rely on their values for and as given in Table 2 of Campbell and Yogo (2006).^{6} Setting , and using these values of and result in a confidence interval for with confidence level of percent. For completeness, in the empirical section we also show the plain nonsizeadjusted 90 percent confidence interval that is obtained by setting and .
In terms of practical implementation, we obtain a confidence interval for by inverting the ADFGLS unitroot test statistic, and use the NeweyWest estimator to calculate all longrun variances and covariances.
We use monthly data on short and long nominal interest rates in Australia, Canada, Hungary, India, Japan, Mexico, New Zealand, Poland, Singapore, South Africa, Sweden, Switzerland, the United Kingdom and the United States. For all countries, the long interest rate is the yield on a benchmark government bond and the short interest rate is a representative threemonth rate. Details of the data for each country are given in the Appendix. Sample starting dates vary from 1955 to 1997, with the sample for the emerging economies generally shorter owing to poorer market functioning and data keeping. Countryspecific samples are shown in Table 1.
We begin by documenting the univariate properties of the data series, applying the ADFGLS test to all variables. Results are shown in Table 1. The table also shows median unbiased estimates of and , and their 90 percent confidence intervals which are computed by inverting the ADFGLS test statistic following the methodology suggested by Stock (1991).^{7} The estimate of is used in the Bonferroni EG test if it is smaller than zero.
Two important features stand out in Table 1; all of the interest rate series are highly persistent, but the pure unitroot assumption is likely to be too restrictive. Unbiased estimates of are in most cases negative, suggesting mean reversion in the data. Moreover, the null hypothesis of a unit root is rejected at the five percent level for the threemonth interest rate in Canada, New Zealand, Singapore and Switzerland. The benefits of the Bonferroni EG test are evident in these cases. Most researchers would  rightly so  be reluctant to use traditional cointegration analysis after having detected evidence of stationarity. The Bonferroni EG test, on the other hand, provides a tool for valid inference also in the nearintegrated case.
Having investigated the univariate timeseries properties of the data, we now turn to the issue of cointegration between long and short interest rates. As described in Section 2.2.1, cointegration is tested by calculating the EG test statistic for the residuals of the following regression,
(14) 
where and are the long and short nominal interest rates, respectively, for each country . The five percent critical value for the traditional EG test, based on a pure unitroot assumption, is 3.341. For the Bonferroni EG test, however, the critical value depends on and is given for each country in Table 2.
Among the developed countries, the null of no cointegration is rejected for Australia, Canada, New Zealand, Sweden, Switzerland and the United States at the five percent level. The null can similarly be rejected for Hungary, Mexico, Poland and Singapore. These countries thus satisfy the first of the two necessary conditions for the expectations hypothesis to be a valid description of their term structures. The null of no cointegration cannot be rejected for Japan and the United Kingdom, nor India and South Africa. For Japan, this finding is not surprising given the complex zeronominalbound problem that disrupted the Japanese economy for much of the sample.^{8} It is more surprising that cointegration does not hold for the United Kingdom. Nonetheless, the finding is in line with evidence from previous studies, such as Cuthbertson (1996). The failure to detect a cointegrating relationship between long and short rates in the United Kingdom may owe to the substantial and longlived decline in long rates midway through the sample. Following the shift to inflation targeting in the mid 1990s, longrun inflation expectations declined and the risk premia embedded in longhorizon bond yields likely compressed, perhaps mimicking a structural break in the cointegrating relationship between short rates and long rates. However, this is only one of several factors that may have been at work; other countries in our panel also shifted to inflation targeting without detracting from the cointegration result.
Given the relative immaturity of financial markets in emerging economies, it is noteworthy that the null hypothesis of no cointegration is rejected in four of our six emerging economies. Only India and South Africa fail the test, with structural change the likely reason in both cases. In 1998, the Reserve Bank of India adopted a multipleindicator approach, a departure from earlier monetary policy that appeared to foster a permanent reduction in long and short interest rates. The relatively long sample available for South Africa clearly makes structural breaks a potential problem. Following the end of apartheid in the early 1990s, international interest in South Africa resumed, accompanied by strong inflows of foreign financial capital, and like several other countries, South Africa adopted inflation targeting in early 2000, with a concomitant reduction in long interest rates.
An important feature of Table 2 is that the critical values of the Bonferroni EG test are to the left of the standard unitroot EG critical value, once uncertainty regarding the exact value of the largest autoregressive root in the data has been taken into account. This adjustment of critical values is crucial, as it is key to valid inference when variables do not have exact unit roots. Failing to make this adjustment tends to make traditional tests of cointegration oversized. As seen here, the standard EG cointegration test, based on the unitroot assumption, and the robust Bonferroni test lead to similar conclusions. However, the Bonferroni test, which is correctly sized under more general conditions than the standard test, gives us greater confidence in rejections of the null hypothesis of no cointegration. That is, by performing inference with the Bonferroni test, one no longer needs to condition on the auxiliary assumption that the data were generated by a pure unitroot process when interpreting the cointegration results. The evidence in favour of cointegration is valid also if the data were generated by a weakly meanreverting process.
Having found evidence for cointegration in several countries, we turn to testing whether the cointegrating vector matches the theoreticallysuggested value (1, 1). As mentioned in the introduction, researchers have typically overlooked interpretation of the estimated vector, either because of a lack of suitable econometric tools in the case of earlier research, or because it has not been the focus. With the methods developed in this paper, we are able to conduct valid inference about the parameter values. Our focus is the 90 percent confidence interval for ; if this does not cover unity, the null hypothesis of is rejected, which is equivalent to rejecting the cointegrating vector (1, 1). Point estimates of are shown in Table 3, both based on OLS estimation and based on standard unitroot FMOLS. Confidence intervals are also shown. As discussed in Section 2.2.2, two Bonferroni confidence intervals are presented: (i) the sizeadjusted interval and (ii) the nonsizeadjusted interval. Both methods lead to qualitatively similar results, and it is evident that the size adjustment is not crucial in this particular application.
For all ten countries in which cointegration was detected using the Bonferroni EG test, the cointegrating vector differs significantly from the values predicted by theory. That is, despite the presence of cointegration, the expectations hypothesis is rejected as a description of the term structure. Interestingly, the values are similar across countries, with significantly smaller than one and clustered between and . The confidence intervals also overlap substantially. The finding of a slope coefficient less than one is not unique to this paper; for example, Engle and Granger (1987), Boothe (1991) and MacDonald and Speight (1991) report similar coefficients using data from different countries and sample periods. Unlike previous studies, however, we have the tools to reliably conclude that the deviations from the theoreticallyimplied relationship are statistically significant.
The joint finding of cointegration with a slope coefficient less than one is intriguing, as it suggests that the term structures of many countries are driven not just by expectations of future short interest rates and stationary term premia but by an additional nearintegrated component that covaries systematically with the short interest rate. Rather than an outright rejection of the expectations hypothesis, it suggests that something more is at work. One candidate for this additional component is a timevarying but persistent risk premium linked to the state of the term structure. Campbell and Shiller (1991) propose exactly this to explain why coefficients in their yieldspread regressions deviate from theory. Specifically, they suggest timevarying risk premia as a rationale for why ``long rates underreact to shortterm interest rates'' (p. 513). Dai and Singleton (2002) go further, showing that yieldspread regressions using riskpremium adjusted yields from a broad class of affine term structure models recover expectationshypothesis predicted coefficients. To match the empirical findings of Fama and Bliss (1987) and Campbell and Shiller (1991), Dai and Singleton's modelestimated risk premia are negatively correlated with the short interest rate.
To see how this explanation fits into the present application, consider a generalisation of equation (2) that retains the basic structure of the expectations hypothesis but generalises the term premium:
(15) 
The new term premium , is the sum of two components, the stationary component posited earlier, , plus a new component, , that is nearintegrated and covaries with the short interest rate. Specifically, assume that such that if is near integrated, so too is and they will in fact be cointegrated. Moving this term to the left hand side,
(16) 
our empirical finding of a slope coefficient of less than unity leads us to infer that is also negative. That is, the nearintegrated element of appears to covary inversely with the short interest rate, consistent with the existing literature on the term structure for the United States described above. While the empirical analysis in this paper has by no means identified such risk premia, it is a plausible explanation for the results in Table 3, as it is likely to be a common characteristic of the behaviour of investors across countries and markets. Naturally, other explanations for the failure of the expectations hypothesis abound, such as departures from rational expectations or sluggish adjustment of expectations about persistent changes in short rates. For some countries in our sample, gradual adjustment of longrun inflation expectations during the shift to a lowerinflation environment may have delayed adjustment of long interest rates, as suggested by Kozicki and Tinsley (2005) for the United States during the 1980s. However, given the similar results we find across countries and time periods, an explanation based on the failure of expectations to adjust quickly or rationally seems less likely to be the common explanation.
Nominal interest rates are likely to be stationary, albeit slowly mean reverting. The theoretical determinants of interest rates, such as agents' risk aversion and longrun growth rates of the macroeconomy, imply that real interest rates possess a longrun equilibrium value. Historically, nominal interest rates have fluctuated for many centuries within a fairly narrow band that encompasses values common today. However, their typically slow rate of mean reversion poses a problem for traditional unitroot tests, and confronted with series that behave as integrated in finite samples, many researchers have proceeded to apply cointegration methods based on the unitroot assumption. Unfortunately, empirical analysis based on this assumption can be misleading. Even small deviations from the unitroot assumption can lead to large size distortions and thus mistaken inference.
This paper illustrates the application of nearintegration methods of cointegration by testing the expectations hypothesis on termstructure data from numerous countries. The econometric framework is robust to deviations from the unitroot hypothesis by nesting the highpersistence and unitroot cases. The empirical results strongly support cointegration between long and short interest rates in several developed and emerging economies, a necessary condition for the expectations hypothesis to hold. However, for all ten countries in which cointegration is present, the theoreticallypredicted vector (1, 1) is rejected. Moreover, the estimated vectors are similar across countries, with long rates moving by less than short rates, on average. While the causes for this are far from obvious, our empirical findings are consistent with the presence of timevarying bond term premia that are themselves highly persistent and covary inversely with the short rate. Such reasoning is in a similar spirit to explanations put forward for the failure of the expectations hypothesis in the excessreturns literature, and with the stylised fact that risk premia are countercyclical. Our empirical analysis is not sufficiently structural to test this hypothesis, and looking deeper requires reliable estimates of risk premia, a complex issue in itself. We nevertheless believe it to be an interesting direction for future research.
This appendix details the data types and sources used in the analysis. For Australia, Canada, India, Japan, New Zealand, Sweden, Switzerland, the United Kingdom and the United States, the long interest rate is the yield on the benchmark tenyear government bond, for Hungary, Poland and Singapore, the yield on the fiveyear government bond, for Mexico, the yield on the threeyear government bond and, finally, for South Africa, the yield on the twentyyear government bond. The long interest rates are measured as yields on coupon securities. Turning to short interest rates, these are the three month bank bill rate for Australia and New Zealand, the three month corporate paper rate for Canada, the three month certificate of deposit rate for India and Japan, the three month interbank rate for Switzerland and the United Kingdom, and the three month treasury bill rate for Hungary, Mexico, Poland, Singapore, South Africa, Sweden and the United States. The data were sourced from the Board of Governors of the Federal Reserve System, the International Monetary Fund, and Global Financial Data.
Following Campbell and Shiller (1987), the long interest rates in our sample are the yields on specific coupon securities, as synthetic, constantmaturity, zerocoupon yields are not readily available for all countries. As pointed out by Shea (1992), conducting our analysis using yields from coupon securities relies on approximations that could distort inference. For the United Kingdom and the United States, we are able to check whether our choice materially affects our conclusions by performing our analysis on synthetic constantmaturity yields provided by the respective central banks.
The results of this exercise (reported below) indicate that our findings are not sensitive to the choice of how the long interest rate is measured. Denote the tenyear zerocoupon yield as and for the United Kingdom and United States respectively  as the long interest rate. The sample period for the United Kingdom is the same as for the main data. For the United States, the sample period is considerably shorter than originally, namely January 1973 to October 2006; in order to be to able make direct comparisons for the U.S. data, we therefore also redo the analysis using the couponbearing bond rate over this shorter sample.
Table A1 shows the results from unitroot tests and provides estimates of the localtounity parameters for the different series. In Table A2, results from the cointegration tests are presented. As can be seen from the tables, the results are qualitatively the same as when we use the original data  cointegration is supported in the United States but not in the United Kingdom. As we found no evidence of cointegration for the United Kingdom, inference regarding the cointegrating vector can only be carried out for the United States. Results from this exercise are shown in Table A3. While there are some minor quantitative differences, the two datasets still communicate a message very similar to that seen previously. The cointegrating vector (1, 1) is clearly rejected and estimates suggest a value of around (1, 0.7).
Table 1: Results From UnitRoot Tests
% confidence interval for  Sample  

1.841 
6.231 
[13.211, 0.426] 
1970M01  2006M10 

1.043 
1.663 
[6.341, 2.826] 

2.300 
9.968 
[18.345, 1.982] 
1956M01  2006M10 

0.952 
1.281 
[5.748, 3.001] 

0.576 
0.178 
[3.378, 3.426] 
1997M03  2004M05 

0.582 
0.190 
[3.762, 3.419] 

0.403 
0.171 
[3.058, 3.567] 
1995M01  2006M10 

0.279 
0.974 
[1.293, 3.942] 

0.377 
1.055 
[1.113, 3.992] 
1989M01  2006M10 

0.296 
0.342 
[2.708, 3.630] 

0.506 
0.030 
[3.479, 3.505] 
1995M02  2004M05 

0.938 
1.231 
[5.671, 3.015] 

2.145 
8.612 
[16.479, 1.105] 
1974M01  2006M10 

1.447 
3.662 
[9.402, 1.884] 

0.759 
1.329 
[0.888, 4.046] 
1994M03  2004M0 

0.516 
1.163 
[0.556, 4.138] 

2.009 
7.517 
[14.998, 0.364] 
1988M01  2004M05 

0.227 
0.438 
[2.479, 3.672] 

1.552 
4.313 
[10.365, 1.525] 
1976M01  2004M05 

1.536 
4.213 
[10.212, 1.592] 

0.877 
1.018 
[5.263, 3.082] 
1987M01  2006M10 

0.041 
0.744 
[1.793, 3.819] 

2.059 
7.901 
[15.518, 0.634] 
1975M10  2006M10 

0.480 
0.023 
[3.366, 3.521] 

0.880 
1.030 
[5.289, 3.079] 
1978M01  2006M10 

0.097 
0.799 
[1.670, 3.848] 

1.087 
1.852 
[6.640, 2.740] 
1955M01  2006M10 

0.841 
0.895 
[5.069, 3.122] 
Notes:^{*}indicates significance at the five percent significance level.
Table 2: Results From Cointegration Tests
Sample  

Australia  4.231  3.387  1970M01  2006M10 
Canada  4.789  3.375  1956M01  2006M10 
Hungary  4.991  3.346  1997M03  2004M05 
India  2.493  3.341  1995M01  2006M10 
Japan  2.220  3.341  1989M01  2006M10 
Mexico  5.757  3.374  1995M02  2004M05 
New Zealand  6.278  3.485  1974M01  2006M10 
Poland  5.275  3.341  1994M03  2004M05 
Singapore  4.358  3.341  1988M01  2004M05 
South Africa  2.374  3.516  1976M01  2004M05 
Sweden  4.243  3.341  1987M01  2006M10 
Switzerland  4.186  3.341  1975M10  2006M10 
United Kingdom  2.913  3.341  1978M01  2006M10 
United States  4.034  3.364  1955M01  2006M10 
Notes: Critical values are for a test with a nominal size of five percent. The five percent critical value of the standard EG test under a unitroot assumption is equal to 3.341.
Table 3: Estimates of the Cointegrating Vector
% confidence interval for (i)  % confidence interval for (ii)  Sample  

Australia  0.680 
0.651 
1970M01  2006M10 

Canada  0.715 
0.696 
1956M01  2006M10 

Hungary  0.798 
0.799 
1997M03  2004M05 

India   
 
 
 
1995M01  2006M10 
Japan   
 
 
 
1989M01  2006M10 
Mexico  0.788 
0.802 
1995M02  2004M05 

New Zealand  0.647 
0.624 
1974M01  2006M10 

Poland  0.794 
0.786 
1994M03  2004M0 

Singapore  0.677 
0.636 
1988M01  2004M05 

South Africa   
 
 
 
1976M01  2004M05 
Sweden  0.727 
0.704 
1987M01  2006M10 

Switzerland  0.385 
0.378 
1975M10  2006M10 

United Kingdom   
 
 
 
1978M01  2006M10 
United States  0.859 
0.836 
1955M01  2006M10 
Notes: The 90 percent coinfidence interval (i) has been size adjusted, whereas (ii) has not.
Table A1: Results from UnitRoot Tests
% confidence interval for  Sample  

0.880 
1.030 
[5.289, 3.079] 
1978M01  2006M10 

0.349 
1.034 
[1.162, 3.978] 

0.880 
1.030 
[5.289, 3.079] 
1978M01  2006M10 

0.097 
0.799 
[1.670, 3.848] 

1.176 
2.225 
[7.269, 2.541] 
1973M01  2006M10 

1.215 
2.412 
[7.562, 2.461] 

1.176 
2.225 
[7.269, 2.541] 
1973M01  2006M10 

1.121 
2.387 
[7.520, 2.472] 
Notes: ^{*} indicates significance at the five percent significance level.
Table A2: Results From Cointegration Tests
EG  Sample  

United Kingdom, zero coupon yield  2.832  3.341  1978M01  2006M10 
United Kingdom  2.913  3.341  1978M01  2006M10 
United States, zero coupon yield  3.601  3.419  1973M01  2006M10 
United States  3.553  3.418  1973M01  2006M10 
Notes: Critical values are for a test with a nominal size of five percent. The five percent cirtical value of the standard EG test under a unitroot assumption is equal to 3.341.
Table A3: Estimates of the Cointegrating Vector
% confidence interval for (i)  % confidence interval for (ii)  Sample  

United Kingdom, zero coupon yield   
 
 
 
1978M01  2006M10 
United Kingdom   
 
 
 
1978M01  2006M10 
United States, zero coupon yield  0.697 
0.677 
1973M01  2006M10 

United States  0.773 
0.755 
1973M01  2006M10 
Notes: The 90 percent coinfidence interval (i) has been size adjusted, whereas (ii) has not.
Bansal, R., Yaron, A., 2004. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59, 14811509.
Boothe, P, 1991. Interest parity, cointegration, and the term structure in Canada and the United States. Canadian Journal of Economics 24, 595603.
Bremnes, H., Gjerde, Ø., Sættem, F, 2001. Linkages among interest rates in the United States, Germany and Norway. Scandinavian Journal of Economics 103, 27145.
Campbell, J. Y., Shiller, R. J., 1987. Cointegration and tests of present value models. Journal of Political Economy 95, 10621088.
Campbell, J. Y., Shiller, R. J., 1991. Yield spreads and interest rate movements: A bird's eye view. Review of Economic Studies 58, 495514.
Campbell, J. Y., Yogo, M., 2006. Efficient tests of stock return predictability. Journal of Financial Economics 81, 2760.
Chong, B. S., Liu, M.H., Shrestha, K., 2006. Monetary transmission via the administered interest rate channels, Journal of Banking and Finance 30, 14671484.
Cochrane, J. H., Piazzesi, M., 2005. Bond risk premia. American Economic Review 95, 138160.
Cogley, T., Sargent, T. J., 2001. Evolving postWorld war II U.S. inflation dynamics, NBER Macroeconomics Annual 16, 331373.
Cuthbertson, K., 1996. The expectations hypothesis of the term structure: The UK interbank market. Economic Journal 106, 578592.
Dai, Q., Singleton, K., 2002. Expectations puzzles, timevarying risk premia, and affine models of the term structure. Journal of Financial Economics 63, 415441.
De Graeve, F., De Jonghe, O. and Vander Vennet, R., 2007. Competition, transmission and bank pricing policies: Evidence from Belgian loan and deposit markets. Journal of Banking and Finance 31, 259278.
Elliott, G., 1998. On the robustness of cointegration methods when regressors almost have unit roots. Econometrica 66, 149158.
Elliott, G., Rothenberg, T. J., Stock, J. H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64, 813836.
Engle, R., Granger C. W. J., 1987. Cointegration and error correction: Representation, estimation, and testing. Econometrica 55, 251276.
Fama, E. F., Bliss, R. R., 1987. The information in longmaturity forward rates. American Economic Review 77, 680692.
Hall, A. D., Anderson, H. D., Granger, C. W. J., 1992. A cointegration analysis of treasury bill yields. Review of Economics and Statistics 74, 116126.
Hjalmarsson, E., 2007. Fully modified estimation with nearly integrated regressors. Finance Research Letters 4, 9294.
Hjalmarsson, E., Österholm, P., 2007a. A residualbased cointegration test for near unit root variables. International Finance Discussion Papers 907, Board of Governors of the Federal Reserve System.
Hjalmarsson, E., Österholm, P., 2007b. Testing for cointegration using the Johansen methodology when variables are nearintegrated. IMF Working Paper 07/141, International Monetary Fund.
Homer, S., Sylla, R., 1996. A history of interest rates. Rutgers University Press: New Brunswick.
Karfakis, C. J., Moschos, D. M., 1990. Interest rate linkages within the European monetary system: A time series analysis. Journal of Money, Credit and Banking 22, 388394.
Kleimeier, S., Sander, H., 2006. Expected versus unexpected monetary policy impulses and interest rate passthrough in eurozone retail banking, Journal of Banking and Finance 30, 18391870.
Kozicki, S., Tinsley, P., 2005. What do you expect? Imperfect policy credibility and tests of the expectations hypothesis. Journal of Monetary Economics 52, 421447.
Lanne, M., 2000. Near unit roots, cointegration, and the term structure of interest rates. Journal of Applied Econometrics 15, 513529.
Lanne, M., 2001. Near unit root and the relationship between inflation and interest Rates: A reexamination of the Fisher effect. Empirical Economics 26, 357366.
Lardic, S., Mignon, V., 2004. Fractional cointegration and the term structure. Empirical Economics 29, 723736.
Liu, M.H., Margaritis, D., TouraniRad, A., 2008. Monetary policy transparency and passthrough of retail interest rates. Journal of Banking and Finance 32, 501511.
MacDonald, R., Speight, A. E. H., 1991. The term structure of interest rates under rational expectations: Some international evidence. Applied Financial Economics 1, 211221.
Miron, J. A., 1991. Pitfalls and opportunities: What macroeconomists should know about unit roots: Comment. NBER Macroeconomics Annual 6, 211218.
Newey, W. K., West, K. D., 1987. A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703708.
Nicolau, J., 2002. Stationary processes that look like random walks  The bounded random walk process in discrete and continuos time. Econometric Theory 18, 99118.
Phillips, P. C. B., 1988. Regression theory for nearintegrated time series. Econometrica 56, 10211043.
Phillips, P. C. B., Hansen, B., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99125.
RugeMurcia, F. J., 2006. The expectations hypothesis of the term structure when interest rates are close to zero. Journal of Monetary Economics 53, 14091424.
Saikkonen, P., 1991. Asymptotically efficient estimation of cointegrating regressions. Econometric Theory 7, 121.
Shea, G. S., 1992. Benchmarking the expectations hypothesis of the interestrate term structure: An analysis of cointegration vectors. Journal of Business and Economic Statistics 10, 347366.
Stock, J. H., 1991. Confidence intervals for the largest autoregressive root in U.S. economic timeseries. Journal of Monetary Economics 28, 435460.
Stock, J. H., 1994. Unit roots and trend breaks. In Engle, R., McFadden, D. (Eds.), Handbook of Econometrics, Vol. 4. North Holland: Amsterdam.
Stock, J. H., Watson, M. W., 1993. A simple estimator of the cointegrating vectors in higher order integrated systems. Econometrica 61, 783820.
Svensson, L. E. O., 2003. Escaping from a liquidity trap and deflation: The foolproof way and others. Journal of Economic Perspectives 17, 145166.
Ueda, K., 2005. The Bank of Japan's struggle with the zero lower bound on nominal interest rates: Exercises in expectations management. International Finance 8, 329350.
Wallis, K., 1987. Time series analysis of bounded economic variables. Journal of Time Series Analysis 8, 115123.
Wu, J.L., Chen, S.L., 2001. Mean reversion of interest rates in the eurocurrency market. Oxford Bulletin of Economics and Statistics 63, 459474.
Zhang, H.,1993. Treasury yield curves and cointegration. Applied Economics 25, 361367.
^{a} Divison of Monetary Affairs, Board of Governors of the Federal Reserve System, 20th and C Streets, Washington, DC 20551, USA Return to text
^{b} Divison of International Finance, Board of Governors of the Federal Reserve System, 20th and C Streets, Washington, DC 20551, USA Return to text
^{c} Department of Economics, Uppsala University, Box 513, 751 20 Uppsala, Sweden Return to text
1. Because nominal interest rates are bounded downward, they cannot strictly be a linear unitroot process with an additive error term fulfilling standard assumptions (Nicolau, 2002). However, the approximation error from making such an assumption is likely to be negligible, and other bounded variables, such as unemployment rates, are often treated as possessing a unit root for this reason. Moreover, the problem of boundedness can be overcome by transforming the series, for example, by taking the natural logarithm of nominal interest rates. A discussion of such transformations can be found in Wallis (1987). Return to text
2. As mentioned above, Lanne (2000) also analyses the term structure of interest rates within a framework of nearintegrated processes but takes quite a different approach to that in this paper. He applies a joint test of cointegration and the value of the cointegrating vector(s) to U.S. term structure data, whereas we sequentially test for the presence of cointegration and for specific values of the cointegrating vector in a larger international data set. The different approaches have different benefits. Our approach enables us to detect departures of the cointegrating vector from theory in the presence of cointegration. This could provide insight into how and why the expecations hypothesis fails to describe the dynamic behaviour of the term structure. Return to text
3. As pointed out by Miron (1991), even if cointegration with the vector is supported, this is still not sufficient for the validity of the expectations hypothesis, but merely another necessary condition. For example, a weighting scheme other than would be consistent with the same vector. However, in practice, very few reasonable hypotheses would remain to compete with the expectations hypothesis. Return to text
4. Note that it is possible to estimate consistently, but not precisely enough to consistently identify ; median unbiased estimates can also be obtained, as presented in the empirical section, but these are again not consistent estimates. Using an estimate of to obtain estimates of the innovations in the FMOLS procedure discussed here will lead to biased tests. Return to text
5. Since is consistently estimable, it follows that , , and are consistently estimable, even though is not. Return to text
6. The relevant `` endogeneity'' parameter that determines the choice of and ( in Table 2 of Campbell and Yogo, 2006) is now given by the longrun correlation . Return to text
7. Lag length in the ADFGLS and EG tests was determined using the Schwarz information criterion. Using the Akaike information criterion instead in both tests does affect the lag length chosen and the estimate of in quite a few cases. However, it typically does not affect our qualitative conclusion regarding the presence of cointegration. In fact, only for Poland and the United States is the qualitative conclusion different, as we find no support for cointegration when using the Akaike information criterion. Results are not reported but are available upon request. Return to text
8. Since the mid1990's, nominal interest rates in Japan have been extremely low. This problematic fact has received substantial attention in the monetary policy literature; see Svensson (2003) and Ueda (2005). For a discussion of some aspects of the expectations hypothesis and the zerobound problem, see RugeMurcia (2006). Return to text
This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text